text: 20 ARTHUR-CÉSAR LE BRAS • Lemma 5.9. — Let K be a strict complex of Q -Fréchet spaces, let W be a Q -Banach p p space. Then one has for all k •⊗ • ⊗ Hk(K ˆ W) = Hk(K )ˆ W. Qp Qp Proposition 5.10. — Assume that C is the completion of an algebraic closure of W(k)[1/p]. Let i ≥ 0. The B+ -module Hi(Z , B+ )† is ﬁnite free. dR C dR Proof. — It is a direct consequence of Proposition 2.6, the last lemma and [43, Cor 5.21]. Remark 5.11. — If we don’t make any assumption of C, it is still true that there exists ◦ → a smooth aﬃnoid rigid space S = Spa(A, A ) over W(k)[1/p], with a map A C, such C that the motive Q (Z ) descends to a motive over S. If is a rigid motive over S, Vezzani p C ∗ C † C deﬁned in [43] the overconvergent relative de Rham cohomology H ( /S) of over S. If C dR we knew that these groups were A-modules of ﬁnite type when is quasi-compact (i.e. if we had a relative version of the ﬁniteness theorem for overconvergent de Rham cohomology), we could prove as before that one has an isomorphism : C B † (cid:39) C † ⊗ Hi( , + ) Hi ( /S) B+ C dR dR A dR C for any quasi-compact motive over S. As relative de Rham cohomology groups, the C † groups Hk ( /S) are endowed with an integrable connection, the Gauss-Manin connection dR (as deﬁned in [25]). But any coherent sheaf with an integrable connection on a space over a characteristic 0 ﬁeld is automatically locally free, so this would give the result in this case too. Remark 5.12. — Lemma 2.8 and the motivic deﬁnition of the overconvergent de Rham cohomology have another interesting application, unrelated to the rest of this paper. Let G be a locally proﬁnite group acting continuously (in the sense of [36, Def. 2.1, Lem. 2.2]) on a smooth Stein variety Z, of dimension d. Then the action of G on Hi (Z) is smooth, for ≥ dR,c any i 0. Indeed let (U ) be a Stein cover (that is, an increasing covering by aﬃnoid U j O → O j such that the transition maps (U ) (U ) are compact with dense image for all j) of ∈ j+1 j Z. Let v Hi (Z). As dR,c † Hi (Z) = −li→m Hi (U ) , dR,c dR,c j j ∈ † there exists j such that v Hi (U ) . One can ﬁnd a compact open subgroup H of G dR,c j † stabilizing U . It is thus enough to show that the action of H on Hi (U ) is smooth. As j dR,c j this space is ﬁnite dimensional, this amounts to say that the action of H factors through a ﬁnite quotient and this can be checked on the dual, which, by Poincaré duality ([24]), − † ∈ identiﬁes with H2d i(U ) . But if g H is close enough to the identity, the morphism it dR j induces on U is homotopic to the identity by Lemma 2.8, thus equal to the identity on j − † the associated motive. As H2d i(U ) depends only on the motive attached to U , g acts dR j j trivially on it. (cid:54) This argument also applies to (cid:96)-adic cohomology with compact supports ((cid:96) = p), where the result was already proved by Berkovich [5, §7]. The smoothness of the action on de Rham cohomology with compact supports was already checked by hand in [16], in the particular case where G = GL (Q ) and Z is a Drinfeld covering of the non-archimedean half plane. 2 p Corollary 5.13. — Assume that C is the completion of an algebraic closure of W(k)[1/p]. Let i ≥ 0. The B+ -module Hi(Z , B)† ⊗ B+ is ﬁnite free. dR C B dR Proof. — Combine Propositions 5.6 and 5.10. Proposition 5.14. — Assume that C is the completion of the algebraic closure of ≥ \{∞} W(k)[1/p]. Let i 0. The B -module H0(X , HFi F(Z)) is a ﬁnite free B -module. e C(cid:91) e ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ARTHUR-CÉSAR

text: LE

text: BRAS

text: Lemma

text: Let

text: K

text: be

text: a

text: strict

text: complex

text: of

text: Q

text: -Fréchet

text: spaces,

text: let

text: W

text: be

text: a

text: Q

text: -Banach

text: p

text: p

text: space.

text: Then

text: one

text: has

text: for

text: all

text: k

text: Hk(K

text: ˆ

text: W)

text: Hk(K

text: )ˆ

text: W.

text: Qp

text: Qp

text: Proposition

text: Assume

text: that

text: C

text: is

text: the

text: completion

text: of

text: an

text: algebraic

text: closure

text: of

text: Let

text: i

text: The

text: B+

text: -module

text: Hi(Z

text: B+

text: is

text: ﬁnite

text: free.

text: dR

text: C

text: dR

text: Proof.

text: It

text: is

text: a

text: direct

text: consequence

text: of

text: Proposition

text: the

text: last

text: lemma

text: and

text: Cor

text: Remark

text: If

text: we

text: don’t

text: make

text: any

text: assumption

text: of

text: C,

text: it

text: is

text: still

text: true

text: that

text: there

text: exists

text: a

text: smooth

text: aﬃnoid

text: rigid

text: space

text: S

text: Spa(A,

text: A

text: over

text: with

text: a

text: map

text: A

text: C,

text: such

text: C

text: that

text: the

text: motive

text: Q

text: (Z

text: descends

text: to

text: a

text: motive

text: over

text: S.

text: If

text: is

text: a

text: rigid

text: motive

text: over

text: S,

text: Vezzani

text: p

text: C

text: C

text: C

text: deﬁned

text: in

text: the

text: overconvergent

text: relative

text: de

text: Rham

text: cohomology

text: H

text: /S)

text: of

text: over

text: S.

text: If

text: C

text: dR

text: we

text: knew

text: that

text: these

text: groups

text: were

text: A-modules

text: of

text: ﬁnite

text: type

text: when

text: is

text: quasi-compact

text: (i.e.

text: if

text: we

text: had

text: a

text: relative

text: version

text: of

text: the

text: ﬁniteness

text: theorem

text: for

text: overconvergent

text: de

text: Rham

text: cohomology),

text: we

text: could

text: prove

text: as

text: before

text: that

text: one

text: has

text: an

text: isomorphism

text: C

text: B

text: C

text: Hi(

text: Hi

text: /S)

text: B+

text: C

text: dR

text: dR

text: A

text: dR

text: C

text: for

text: any

text: quasi-compact

text: motive

text: over

text: S.

text: As

text: relative

text: de

text: Rham

text: cohomology

text: groups,

text: the

text: C

text: groups

text: Hk

text: /S)

text: are

text: endowed

text: with

text: an

text: integrable

text: connection,

text: the

text: Gauss-Manin

text: connection

text: dR

text: (as

text: deﬁned

text: in

text: But

text: any

text: coherent

text: sheaf

text: with

text: an

text: integrable

text: connection

text: on

text: a

text: space

text: over

text: a

text: characteristic

text: ﬁeld

text: is

text: automatically

text: locally

text: free,

text: so

text: this

text: would

text: give

text: the

text: result

text: in

text: this

text: case

text: too.

text: Remark

text: Lemma

text: and

text: the

text: motivic

text: deﬁnition

text: of

text: the

text: overconvergent

text: de

text: Rham

text: cohomology

text: have

text: another

text: interesting

text: application,

text: unrelated

text: to

text: the

text: rest

text: of

text: this

text: paper.

text: Let

text: G

text: be

text: a

text: locally

text: proﬁnite

text: group

text: acting

text: continuously

text: (in

text: the

text: sense

text: of

text: Def.

text: Lem.

text: on

text: a

text: smooth

text: Stein

text: variety

text: Z,

text: of

text: dimension

text: d.

text: Then

text: the

text: action

text: of

text: G

text: on

text: Hi

text: (Z)

text: is

text: smooth,

text: for

text: dR,c

text: any

text: i

text: Indeed

text: let

text: (U

text: be

text: a

text: Stein

text: cover

text: (that

text: is,

text: an

text: increasing

text: covering

text: by

text: aﬃnoid

text: U

text: j

text: O

text: O

text: j

text: such

text: that

text: the

text: transition

text: maps

text: (U

text: (U

text: are

text: compact

text: with

text: dense

text: image

text: for

text: all

text: j)

text: of

text: j

text: Z.

text: Let

text: v

text: Hi

text: (Z).

text: As

text: dR,c

text: Hi

text: (Z)

text: −li→m

text: Hi

text: (U

text: dR,c

text: dR,c

text: j

text: j

text: there

text: exists

text: j

text: such

text: that

text: v

text: Hi

text: (U

text: One

text: can

text: ﬁnd

text: a

text: compact

text: open

text: subgroup

text: H

text: of

text: G

text: dR,c

text: j

text: stabilizing

text: U

text: It

text: is

text: thus

text: enough

text: to

text: show

text: that

text: the

text: action

text: of

text: H

text: on

text: Hi

text: (U

text: is

text: smooth.

text: As

text: j

text: dR,c

text: j

text: this

text: space

text: is

text: ﬁnite

text: dimensional,

text: this

text: amounts

text: to

text: say

text: that

text: the

text: action

text: of

text: H

text: factors

text: through

text: a

text: ﬁnite

text: quotient

text: and

text: this

text: can

text: be

text: checked

text: on

text: the

text: dual,

text: which,

text: by

text: Poincaré

text: duality

text: identiﬁes

text: with

text: i(U

text: But

text: if

text: g

text: H

text: is

text: close

text: enough

text: to

text: the

text: identity,

text: the

text: morphism

text: it

text: dR

text: j

text: induces

text: on

text: U

text: is

text: homotopic

text: to

text: the

text: identity

text: by

text: Lemma

text: thus

text: equal

text: to

text: the

text: identity

text: on

text: j

text: the

text: associated

text: motive.

text: As

text: i(U

text: depends

text: only

text: on

text: the

text: motive

text: attached

text: to

text: U

text: g

text: acts

text: dR

text: j

text: j

text: trivially

text: on

text: it.

text: This

text: argument

text: also

text: applies

text: to

text: cohomology

text: with

text: compact

text: supports

text: p),

text: where

text: the

text: result

text: was

text: already

text: proved

text: by

text: Berkovich

text: The

text: smoothness

text: of

text: the

text: action

text: on

text: de

text: Rham

text: cohomology

text: with

text: compact

text: supports

text: was

text: already

text: checked

text: by

text: hand

text: in

text: in

text: the

text: particular

text: case

text: where

text: G

text: GL

text: (Q

text: and

text: Z

text: is

text: a

text: Drinfeld

text: covering

text: of

text: the

text: non-archimedean

text: half

text: plane.

text: p

text: Corollary

text: Assume

text: that

text: C

text: is

text: the

text: completion

text: of

text: an

text: algebraic

text: closure

text: of

text: Let

text: i

text: The

text: B+

text: -module

text: Hi(Z

text: B)†

text: B+

text: is

text: ﬁnite

text: free.

text: dR

text: C

text: B

text: dR

text: Proof.

text: Combine

text: Propositions

text: and

text: Proposition

text: Assume

text: that

text: C

text: is

text: the

text: completion

text: of

text: the

text: algebraic

text: closure

text: of

text: Let

text: i

text: The

text: B

text: -module

text: HFi

text: F(Z))

text: is

text: a

text: ﬁnite

text: free

text: B

text: -module.

text: e

text: e

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: DE RHAM COHOMOLOGY OVER THE FARGUES-FONTAINE CURVE 3 (cid:101) ∞ ∈ ∞ at the point P1 lying above 0 and (which is ﬁxed by the U(1)-action), using the R Hodge ﬁltration on Hi (Z, C). Similarly, if Z is proper and smooth over K, a complete Betti G discretely val(cid:98)ued extension of Qp with perfect residue ﬁeld, one Ecan modify the K- equivariant vector bundle (obtained by descending to X) HFi F(Z) := (D (Hi (Z , Q )), ⊗ pst ´et C Gp where C = K¯ , using the Hodge ﬁltration on D K. This modiﬁcation is the - ⊗ O pst K equivariant vector bundle Hi (Z , Q ) (2). The ﬁrst construction relies on the ´et Cp p Q∗p X ⊗ equivalence between ﬁltrations on V and C -equivariant lattices in V C((t)), if V is a C ﬁnite dimensional C-vector space ; the second on the equivalence between ﬁltrations on V G ⊗ and -equivariant lattices in V B , if V is a ﬁnite dimensional K-vector space. K K dR Over the ﬁeld of complex numbers, the vector bundle H∞i (Z) can still be deﬁned more generally, by the same formula : (cid:101) ⊗ O H∞i (Z) := Hi (Z, R) , Betti P1 R which makes sense whenever the Betti/de Rham cohomology is ﬁnite dimensional. Our goal in this text is to show that we can also extend the construction of the vector bundle HFi F(Z) outside the proper case, and give a geometric deﬁnition of this vector bundle. More precisely, let K be a complete valued ﬁeld containing Q (not assumed to be discretely valued), C the p completion of an algebraic closure of K (for example, K may be a ﬁnite extension of Q , or G p C ) with residue ﬁeld k, and = Gal(C/K) the absolute Galois group of K. We explain p K how to construct a cohomology theory (cid:55)→ FF Z (Z), on the category of smooth quasi-compact rigid spaces (3) deﬁned over K, taking values in the bounded derived category Db(Coh ) of coherent sheaves on the Fargues-Fontaine curve X , such that if C is the completXioCn(cid:91) of an algebraic closure of W(k)[1/p], Z is a smooth C(cid:91) ≥ quasi-compact rigid space of dimension n and i 0, the coherent sheaf FF HFi F(Z) := Hi( (Z)) is a vector bundle, which vanishes if i < 0 or i > 2n. Under this assumption on C, this cohomology theory satisﬁes several properties explained below and thus conﬁrms Conjecture 6.4 of [38] (see also [22, Conj. 1.13]). In particular, still under this assumption on C, one gets an isocrystal Hi (Z) attached isoc to Z, for all i (see Corollary 5.16 ; there is some caveat). A few remarks are in order. Remark 1.1. — Assume that K is a complete discretely valued extension of Q with ≥ p perfect residue ﬁeld and that Z is smooth and proper. We show that for all i 0, HFi F(Z) G is the -equivariant vector bundle on X attached as in [23, Ch. 10] to the D of the K C(cid:91) pst étale cohomology Hi (Z , Q ), using comparison theorems in p-adic Hodge theory. ´et C p Remark 1.2. — When the smooth rigid variety Z over K is such that Z has a smooth O C quasi-compact formal model Z over , we expect HFi F(Z) to be the vector bundle attached C to the isocrystal given by the rigid cohomology of the special ﬁber of Z. This implies in particular that this isocrystal is independent of the choice of the formal model. We prove this statement when Z is proper, which is easy using the results of Bhatt-Morrow-Scholze [8] (in the proper case, the independence of the isocrystal is already known : when K is a complete discretely valued extension of Q with perfect residue ﬁeld, it is a corollary of a p deep theorem : Fontaine’s C conjecture ; in general it was recently proved in [8]). crys 2. The slightly confusing point in this analogy is that it does not match the two trivial vector bundles... 3. All the rigid spaces we consider are assumed to be separated and taut. ##LTLine##

text: DE

label: paragraph

text: RHAM

text: COHOMOLOGY

text: OVER

text: THE

text: FARGUES-FONTAINE

text: CURVE

text: at

text: the

text: point

text: lying

text: above

text: and

text: (which

text: is

text: ﬁxed

text: by

text: the

text: using

text: the

text: R

text: Hodge

text: ﬁltration

text: on

text: Hi

text: (Z,

text: C).

text: Similarly,

text: if

text: Z

text: is

text: proper

text: and

text: smooth

text: over

text: K,

text: a

text: complete

text: Betti

text: G

text: discretely

text: extension

text: of

text: Qp

text: with

text: perfect

text: residue

text: ﬁeld,

text: one

text: Ecan

text: modify

text: the

text: K-

text: equivariant

text: vector

text: bundle

text: (obtained

text: by

text: descending

text: to

text: X)

text: HFi

text: F(Z)

text: (D

text: (Hi

text: (Z

text: Q

text: pst

text: ´et

text: C

text: Gp

text: where

text: C

text: K¯

text: using

text: the

text: Hodge

text: ﬁltration

text: on

text: D

text: K.

text: This

text: modiﬁcation

text: is

text: the

text: O

text: pst

text: K

text: equivariant

text: vector

text: bundle

text: Hi

text: (Z

text: Q

text: The

text: ﬁrst

text: construction

text: relies

text: on

text: the

text: ´et

text: Cp

text: p

text: Q∗p

text: X

text: equivalence

text: between

text: ﬁltrations

text: on

text: V

text: and

text: C

text: -equivariant

text: lattices

text: in

text: V

text: C((t)),

text: if

text: V

text: is

text: a

text: C

text: ﬁnite

text: dimensional

text: C-vector

text: space

text: the

text: second

text: on

text: the

text: equivalence

text: between

text: ﬁltrations

text: on

text: V

text: G

text: and

text: -equivariant

text: lattices

text: in

text: V

text: B

text: if

text: V

text: is

text: a

text: ﬁnite

text: dimensional

text: K-vector

text: space.

text: K

text: K

text: dR

text: Over

text: the

text: ﬁeld

text: of

text: complex

text: numbers,

text: the

text: vector

text: bundle

text: H∞i

text: (Z)

text: can

text: still

text: be

text: deﬁned

text: more

text: generally,

text: by

text: the

text: same

text: formula

text: O

text: H∞i

text: (Z)

text: Hi

text: (Z,

text: R)

text: Betti

text: R

text: which

text: makes

text: sense

text: whenever

text: the

text: Betti/de

text: Rham

text: cohomology

text: is

text: ﬁnite

text: dimensional.

text: Our

text: goal

text: in

text: this

text: text

text: is

text: to

text: show

text: that

text: we

text: can

text: also

text: extend

text: the

text: construction

text: of

text: the

text: vector

text: bundle

text: HFi

text: F(Z)

text: outside

text: the

text: proper

text: case,

text: and

text: give

text: a

text: geometric

text: deﬁnition

text: of

text: this

text: vector

text: bundle.

text: More

text: precisely,

text: let

text: K

text: be

text: a

text: complete

text: valued

text: ﬁeld

text: containing

text: Q

text: (not

text: assumed

text: to

text: be

text: discretely

text: valued),

text: C

text: the

text: p

text: completion

text: of

text: an

text: algebraic

text: closure

text: of

text: K

text: (for

text: example,

text: K

text: may

text: be

text: a

text: ﬁnite

text: extension

text: of

text: Q

text: or

text: G

text: p

text: C

text: with

text: residue

text: ﬁeld

text: k,

text: and

text: Gal(C/K)

text: the

text: absolute

text: Galois

text: group

text: of

text: K.

text: We

text: explain

text: p

text: K

text: how

text: to

text: construct

text: a

text: cohomology

text: theory

text: FF

text: Z

text: (Z),

text: on

text: the

text: category

text: of

text: smooth

text: quasi-compact

text: rigid

text: spaces

text: deﬁned

text: over

text: K,

text: taking

text: values

text: in

text: the

text: bounded

text: derived

text: category

text: Db(Coh

text: of

text: coherent

text: sheaves

text: on

text: the

text: Fargues-Fontaine

text: curve

text: X

text: such

text: that

text: if

text: C

text: is

text: the

text: of

text: an

text: algebraic

text: closure

text: of

text: Z

text: is

text: a

text: smooth

text: quasi-compact

text: rigid

text: space

text: of

text: dimension

text: n

text: and

text: i

text: the

text: coherent

text: sheaf

text: FF

text: HFi

text: F(Z)

text: Hi(

text: (Z))

text: is

text: a

text: vector

text: bundle,

text: which

text: vanishes

text: if

text: i

text: or

text: i

text: Under

text: this

text: assumption

text: on

text: C,

text: this

text: cohomology

text: theory

text: satisﬁes

text: several

text: properties

text: explained

text: below

text: and

text: thus

text: conﬁrms

text: Conjecture

text: of

text: (see

text: also

text: Conj.

text: In

text: particular,

text: still

text: under

text: this

text: assumption

text: on

text: C,

text: one

text: gets

text: an

text: isocrystal

text: Hi

text: (Z)

text: attached

text: isoc

text: to

text: Z,

text: for

text: all

text: i

text: (see

text: Corollary

text: there

text: is

text: some

text: caveat).

text: A

text: few

text: remarks

text: are

text: in

text: order.

text: Remark

text: Assume

text: that

text: K

text: is

text: a

text: complete

text: discretely

text: valued

text: extension

text: of

text: Q

text: with

text: p

text: perfect

text: residue

text: ﬁeld

text: and

text: that

text: Z

text: is

text: smooth

text: and

text: proper.

text: We

text: show

text: that

text: for

text: all

text: i

text: HFi

text: F(Z)

text: G

text: is

text: the

text: -equivariant

text: vector

text: bundle

text: on

text: X

text: attached

text: as

text: in

text: Ch.

text: to

text: the

text: D

text: of

text: the

text: K

text: pst

text: étale

text: cohomology

text: Hi

text: (Z

text: Q

text: using

text: comparison

text: theorems

text: in

text: p-adic

text: Hodge

text: theory.

text: ´et

text: C

text: p

text: Remark

text: When

text: the

text: smooth

text: rigid

text: variety

text: Z

text: over

text: K

text: is

text: such

text: that

text: Z

text: has

text: a

text: smooth

text: O

text: C

text: quasi-compact

text: formal

text: model

text: Z

text: over

text: we

text: expect

text: HFi

text: F(Z)

text: to

text: be

text: the

text: vector

text: bundle

text: attached

text: C

text: to

text: the

text: isocrystal

text: given

text: by

text: the

text: rigid

text: cohomology

text: of

text: the

text: special

text: ﬁber

text: of

text: Z.

text: This

text: implies

text: in

text: particular

text: that

text: this

text: isocrystal

text: is

text: independent

text: of

text: the

text: choice

text: of

text: the

text: formal

text: model.

text: We

text: prove

text: this

text: statement

text: when

text: Z

text: is

text: proper,

text: which

text: is

text: easy

text: using

text: the

text: results

text: of

text: Bhatt-Morrow-Scholze

text: (in

text: the

text: proper

text: case,

text: the

text: independence

text: of

text: the

text: isocrystal

text: is

text: already

text: known

text: when

text: K

text: is

text: a

text: complete

text: discretely

text: valued

text: extension

text: of

text: Q

text: with

text: perfect

text: residue

text: ﬁeld,

text: it

text: is

text: a

text: corollary

text: of

text: a

text: p

text: deep

text: theorem

text: Fontaine’s

text: C

text: conjecture

text: in

text: general

text: it

text: was

text: recently

text: proved

text: in

text: crys

text: The

text: slightly

text: confusing

text: point

text: in

text: this

text: analogy

text: is

text: that

text: it

text: does

text: not

text: match

text: the

text: two

text: trivial

text: vector

text: bundles...

text: All

text: the

text: rigid

text: spaces

text: we

text: consider

text: are

text: assumed

text: to

text: be

text: separated

text: and

text: taut.

text: ##LTLine##


text: 22 ARTHUR-CÉSAR LE BRAS 6. Relation with other cohomology theories In this section, we give proofs of the assertions of Remarks 1.1 and 1.2, and discuss brieﬂy the relation of the décalage functor Lη with the pro-étale cohomology of Q and t p syntomic cohomology on rigid spaces. Proof of Remark 1.1. Let K be a discretely valued extension of Q with perfect residue G p ﬁeld κ and V be a p-adic representation of , which is semi-stable. Let K ⊗ G Dst(V ) = (V Qp Bst) K. This a (ϕ, N)-module over K. To such a (ϕ, N)-module D, Fargues and Fontaine [23, Ch. G E G 10] attach a -equivariant vector bundle (D) on X corresponding to the -equivariant K C(cid:91) K B-pair ⊗ ⊗ ((D B )ϕ=1,N=0, D B+ ) K0 st K0 dR (K is the maximal unramiﬁed extension of W(κ)[1/p] contained in K). If V is only assumed 0 to be de Rham, there exists by Berger’s theorem a ﬁnite Galois extension L of K such that G V| is semi-stable : it gives rise to a -equivariant vector bundle on X with a descent daGtLum from L to K ([23, Def. 10.6.3]),Lthat is to a G -equivariant vectoCr(cid:91) bundle on X K G C(cid:91) ([23, Prop 10.6.4]). This vector bundle corresponds to the -equivariant B-pair : K ⊗ ⊗ ((D (V ) B )ϕ=1(cid:91),N=0, D (V ) B+ ), pst W(k)[1/p] st pst W(k)[1/p] dR where D (V ) = D (V| ). pst st G (cid:48) (cid:48) ∞ K [K :K]< ≥ Let Z be a proper smooth rigid space deﬁned over K, and i 0. We know by [35] that Hi (Z , Q ) is a de Rham representation. ´et C p G Proposition 6.1. — The -equivariant vector bundle on X attached to Hi (Z , Q ) K G C(cid:91) ´et C p by the above recipe is isomorphic to the -equivariant vector bundle HFi F(Z). K G Proof. — We will check that HFi F(Z) corresponds to the same -equivariant B-pair. By K Proposition 4.8, we have B † (cid:48) B RΓ(Z , ) = RΓ (Z , Lη Rν∗ ). C ´et Cp t After inverting t, this is just B ⊗ RΓ (Z , [1/t]) = RΓ (Z , Q ) B[1/t]. pro´et C ´et C p Qp Around ∞, it’s given by RΓ (Z , Lη Rν∗(cid:48) B+ ). Using 5.8 or [8, Ch. 13], ´et C t dR (cid:48) B ⊗ Hi(RΓ (Z , Lη Rν∗ + )) = Hi (Z) B+ . ´et C t dR dR K dR In other words, HFi F(Z) corresponds to the B-pair ⊗ ⊗ (Hi (Z , Q ) B , Hi (Z) B+ ). ´et C p Qp e dR K dR The comparison theorem gives an isomorphism ⊗ ⊗ Hi (Z , Q ) B = D (Hi (Z , Q )) B ´et C p Qp st pst ´et C p W(k)[1/p] st G compatible with ϕ, N and the -action on both sides. Taking ϕ = 1, N = 0 shows that G K the two B -modules with -action considered are the same. It is also true for the two e K B+ -modules, since dR ⊗ ⊗ Hi (Z) B+ = D (Hi (Z , Q )) B+ . dR K dR pst ´et C p W(k)[1/p] dR O Remark 6.2. — When Z has a proper semi-stable formal model Z over , the (ϕ, N)- K module D (Hi (Z , Q )) is isomorphic to the Hyodo-Kato (or log-rigid) cohomology st ´et C p Hi (Z ). logrig s ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ARTHUR-CÉSAR

text: LE

text: BRAS

label: section

text: Relation

text: with

text: other

text: cohomology

text: theories

text: In

text: this

text: section,

text: we

text: give

text: proofs

text: of

text: the

text: assertions

text: of

text: Remarks

text: and

text: and

text: discuss

text: brieﬂy

text: the

text: relation

text: of

text: the

text: décalage

text: functor

text: Lη

text: with

text: the

text: pro-étale

text: cohomology

text: of

text: Q

text: and

text: t

text: p

text: syntomic

text: cohomology

text: on

text: rigid

text: spaces.

text: Proof

text: of

text: Remark

text: Let

text: K

text: be

text: a

text: discretely

text: valued

text: extension

text: of

text: Q

text: with

text: perfect

text: residue

text: G

text: p

text: ﬁeld

text: κ

text: and

text: V

text: be

text: a

text: p-adic

text: representation

text: of

text: which

text: is

text: semi-stable.

text: Let

text: K

text: G

text: Dst(V

text: (V

text: Qp

text: Bst)

text: K.

text: This

text: a

text: (ϕ,

text: N)-module

text: over

text: K.

text: To

text: such

text: a

text: (ϕ,

text: N)-module

text: D,

text: Fargues

text: and

text: Fontaine

text: Ch.

text: G

text: E

text: G

text: attach

text: a

text: -equivariant

text: vector

text: bundle

text: (D)

text: on

text: X

text: corresponding

text: to

text: the

text: -equivariant

text: K

text: K

text: B-pair

text: ((D

text: B

text: D

text: B+

text: st

text: dR

text: (K

text: is

text: the

text: maximal

text: unramiﬁed

text: extension

text: of

text: contained

text: in

text: K).

text: If

text: V

text: is

text: only

text: assumed

text: to

text: be

text: de

text: Rham,

text: there

text: exists

text: by

text: Berger’s

text: theorem

text: a

text: ﬁnite

text: Galois

text: extension

text: L

text: of

text: K

text: such

text: that

text: G

text: V|

text: is

text: semi-stable

text: it

text: gives

text: rise

text: to

text: a

text: -equivariant

text: vector

text: bundle

text: on

text: X

text: with

text: a

text: descent

text: daGtLum

text: from

text: L

text: to

text: K

text: Def.

text: is

text: to

text: a

text: G

text: -equivariant

text: bundle

text: on

text: X

text: K

text: G

text: Prop

text: This

text: vector

text: bundle

text: corresponds

text: to

text: the

text: -equivariant

text: B-pair

text: K

text: ((D

text: (V

text: B

text: D

text: (V

text: B+

text: pst

text: st

text: pst

text: dR

text: where

text: D

text: (V

text: D

text: (V|

text: pst

text: st

text: G

text: K

text: [K

text: :K]<

text: Let

text: Z

text: be

text: a

text: proper

text: smooth

text: rigid

text: space

text: deﬁned

text: over

text: K,

text: and

text: i

text: We

text: know

text: by

text: that

text: Hi

text: (Z

text: Q

text: is

text: a

text: de

text: Rham

text: representation.

text: ´et

text: C

text: p

text: G

text: Proposition

text: The

text: -equivariant

text: vector

text: bundle

text: on

text: X

text: attached

text: to

text: Hi

text: (Z

text: Q

text: K

text: G

text: ´et

text: C

text: p

text: by

text: the

text: above

text: recipe

text: is

text: isomorphic

text: to

text: the

text: -equivariant

text: vector

text: bundle

text: HFi

text: F(Z).

text: K

text: G

text: Proof.

text: We

text: will

text: check

text: that

text: HFi

text: F(Z)

text: corresponds

text: to

text: the

text: same

text: -equivariant

text: B-pair.

text: By

text: K

text: Proposition

text: we

text: have

text: B

text: B

text: RΓ(Z

text: RΓ

text: (Z

text: Lη

text: Rν∗

text: C

text: ´et

text: Cp

text: t

text: After

text: inverting

text: t,

text: this

text: is

text: just

text: B

text: RΓ

text: (Z

text: RΓ

text: (Z

text: Q

text: pro´et

text: C

text: ´et

text: C

text: p

text: Qp

text: Around

text: it’s

text: given

text: by

text: RΓ

text: (Z

text: Lη

text: B+

text: Using

text: or

text: Ch.

text: ´et

text: C

text: t

text: dR

text: B

text: Hi(RΓ

text: (Z

text: Lη

text: Rν∗

text: Hi

text: (Z)

text: B+

text: ´et

text: C

text: t

text: dR

text: dR

text: K

text: dR

text: In

text: other

text: words,

text: HFi

text: F(Z)

text: corresponds

text: to

text: the

text: B-pair

text: (Hi

text: (Z

text: Q

text: B

text: Hi

text: (Z)

text: B+

text: ´et

text: C

text: p

text: Qp

text: e

text: dR

text: K

text: dR

text: The

text: comparison

text: theorem

text: gives

text: an

text: isomorphism

text: Hi

text: (Z

text: Q

text: B

text: D

text: (Hi

text: (Z

text: Q

text: B

text: ´et

text: C

text: p

text: Qp

text: st

text: pst

text: ´et

text: C

text: p

text: st

text: G

text: compatible

text: with

text: ϕ,

text: N

text: and

text: the

text: -action

text: on

text: both

text: sides.

text: Taking

text: ϕ

text: N

text: shows

text: that

text: G

text: K

text: the

text: two

text: B

text: -modules

text: with

text: -action

text: considered

text: are

text: the

text: same.

text: It

text: is

text: also

text: true

text: for

text: the

text: two

text: e

text: K

text: B+

text: -modules,

text: since

text: dR

text: Hi

text: (Z)

text: B+

text: D

text: (Hi

text: (Z

text: Q

text: B+

text: dR

text: K

text: dR

text: pst

text: ´et

text: C

text: p

text: dR

text: O

text: Remark

text: When

text: Z

text: has

text: a

text: proper

text: semi-stable

text: formal

text: model

text: Z

text: over

text: the

text: (ϕ,

text: N)-

text: K

text: module

text: D

text: (Hi

text: (Z

text: Q

text: is

text: isomorphic

text: to

text: the

text: Hyodo-Kato

text: (or

text: log-rigid)

text: cohomology

text: st

text: ´et

text: C

text: p

text: Hi

text: (Z

text: logrig

text: s

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: DE RHAM COHOMOLOGY OVER THE FARGUES-FONTAINE CURVE 25 which obviously induces an isomorphism between the truncations τ≤rLη Rν∗(cid:48) trB and τ≤rRν∗(cid:48) trB, by choice of the function ε . But we know (Proposition 3.4) thaεtr,tfor all j, r Hj (Z , trB) = Hj (Z , Q (r)) ⊗ trB, pro´et C ´et C p Qp ≤ E ⊗ O so for i r, the sheaf i(Z, r) will just be the sheaf Hi (Z , Q (r)) . Taking global sections, we get that the above map induces an iso´emt orCphismp Qp XC(cid:91) (cid:39) Hi (ZO , r)[1/p] Hi (Z , Q (r)), syn C ´et C p ≤ for every i r. That this isomorphism is the same as the one induced by the Fontaine- Messing period map can probably be done as in [15, §4.7] (the author did not check this). Relation with the pro-étale cohomology of Q . Assume that C is the completion of an p algebraic closure of W(k)[1/p]. Let Z be a Stein rigid space over K. Choose a Stein covering (Z ) of Z (see Remark 5.12). k k One has an exact sequence (15) of pro-étale sheaves on Z : C → → B → B B → 0 Q [1/t]ϕ=1 / + 0. p dR dR Assume (for simplicity) that dim Z = 1. The cohomology of the pro-étale sheaf B /B+ dR dR can be analyzed via the Poincaré lemma : this was done in [29, Rem. 2.3.21] (16). One ﬁnds that H0 (Z , B /B+ ) is an extension of (O(Z )/C)(−1) by B /B+ , that H1 (Z , B pro/eB´t+ C) =dRH1 d(RZ)⊗ˆ B /t−1B+ and thatCHi (Z , B /BdR+ ) d=R 0 for proe´t C dR dR dR K dR dR proe´t C dR dR i > 1. One gets a short exact sequence : (cid:16) (cid:17) → O − → → 0 ( (Z )/C)( 1) H1 (Z , Q ) C proe´t C p B → ⊗ − → Ker H1 (Z , [1/t]ϕ=1) H1 (Z)ˆ B /t 1B+ 0. proe´t C dR K dR dR ≥ E ⊗ O (cid:98) (cid:98) Let k 0 and = HF1 F(Z ) (1). The exact sequence, for any k, k → E → \{∞} E → E E → 0 (cid:16)H0(X , ) H0(X , ) ∞[1/t]/ ∞ 0(cid:17) C(cid:91) C(cid:91) tells us that B → †⊗ − Ker H1 (Z , [1/t]ϕ=1) H1 (Z ) ˆ B /t 1B+ proe´t C dR k K dR dR is isomorphic to ⊗ O B † H0(X , HF1 F(Z ) (1)) = (H1(Z , ) )ϕ=p. C(cid:91) k k,C Taking the inverse limit over k (17), one gets an exact sequence : (2) 0 → (O(Z )/C)(−1) → H1 (Z , Q ) → H1 (Z , Lη Rν∗(cid:48) B)ϕ=p → 0. C proe´t C p ´et C t We now turn to the relation with the results of [14]. Assume that K is a discretely valued extension of Q with perfect residue ﬁeld (18) and that Z has a semi-stable formal model Z O p over . If the semi-stable version of Conjecture 6.3 was known to be true, this could be K rewritten using log-rigid cohomology. Indeed, let (Z ) be an aﬃne covering of Z such that k k ≥ (Z ) is a Stein covering of Z. Then we would know that for all k and all i 0, HFi F(Z ) k k † k is the vector bundle attached to the (ϕ, N)-module Hi (Z ) . In particular, logrig k,s † ⊗ H0(X , HFi F(Z )(1)) = (Hi (Z ) B+)ϕ=p,N=0. C(cid:91) k logrig k,s K0 st 15. That a variant of such an exact sequence may be relevant to the study of the pro-étale cohomology of Qp was suggested to the author by Gabriel Dospinescu and was used in [29, §2.3] to compute the pro-étale cohomology of Qp on the aﬃne space and the open disk. 16. One needs to write the Tate twists, which was not done in loc. cit., but is easy to do. 17. And using that the R1 l←im− term vanishes, as it only involves H0 which can be computed. 18. This assumption is probably unnecessary. ##LTLine##

text: DE

text: RHAM

text: COHOMOLOGY

text: OVER

text: THE

text: FARGUES-FONTAINE

text: CURVE

text: which

text: obviously

text: induces

text: an

text: isomorphism

text: between

text: the

text: truncations

text: τ≤rLη

text: trB

text: and

text: trB,

text: by

text: choice

text: of

text: the

text: function

text: ε

text: But

text: we

text: know

text: (Proposition

text: thaεtr,tfor

text: all

text: j,

text: r

text: Hj

text: (Z

text: trB)

text: Hj

text: (Z

text: Q

text: (r))

text: trB,

text: pro´et

text: C

text: ´et

text: C

text: p

text: Qp

text: E

text: O

text: so

text: for

text: i

text: r,

text: the

text: sheaf

text: i(Z,

text: r)

text: will

text: just

text: be

text: the

text: sheaf

text: Hi

text: (Z

text: Q

text: (r))

text: Taking

text: global

text: sections,

text: we

text: get

text: that

text: the

text: above

text: map

text: induces

text: an

text: iso´emt

text: orCphismp

text: Qp

text: Hi

text: (ZO

text: Hi

text: (Z

text: Q

text: (r)),

text: syn

text: C

text: ´et

text: C

text: p

text: for

text: every

text: i

text: r.

text: That

text: this

text: isomorphism

text: is

text: the

text: same

text: as

text: the

text: one

text: induced

text: by

text: the

text: Fontaine-

text: Messing

text: period

text: map

text: can

text: probably

text: be

text: done

text: as

text: in

text: (the

text: author

text: did

text: not

text: check

text: this).

text: Relation

text: with

text: the

text: pro-étale

text: cohomology

text: of

text: Q

text: Assume

text: that

text: C

text: is

text: the

text: completion

text: of

text: an

text: p

text: algebraic

text: closure

text: of

text: Let

text: Z

text: be

text: a

text: Stein

text: rigid

text: space

text: over

text: K.

text: Choose

text: a

text: Stein

text: covering

text: (Z

text: of

text: Z

text: (see

text: Remark

text: k

text: k

text: One

text: has

text: an

text: exact

text: sequence

text: of

text: pro-étale

text: sheaves

text: on

text: Z

text: C

text: B

text: B

text: B

text: Q

text: p

text: dR

text: dR

text: Assume

text: (for

text: simplicity)

text: that

text: dim

text: Z

text: The

text: cohomology

text: of

text: the

text: pro-étale

text: sheaf

text: B

text: /B+

text: dR

text: dR

text: can

text: be

text: analyzed

text: via

text: the

text: Poincaré

text: lemma

text: this

text: was

text: done

text: in

text: Rem.

text: One

text: ﬁnds

text: that

text: (Z

text: B

text: /B+

text: is

text: an

text: extension

text: of

text: (O(Z

text: by

text: B

text: /B+

text: that

text: (Z

text: B

text: pro/eB´t+

text: C)

text: d(RZ)⊗ˆ

text: B

text: and

text: thatCHi

text: (Z

text: B

text: /BdR+

text: d=R

text: for

text: proe´t

text: C

text: dR

text: dR

text: dR

text: K

text: dR

text: dR

text: proe´t

text: C

text: dR

text: dR

text: i

text: One

text: gets

text: a

text: short

text: exact

text: sequence

text: O

text: (Z

text: )/C)(

text: (Z

text: Q

text: C

text: proe´t

text: C

text: p

text: B

text: Ker

text: (Z

text: (Z)ˆ

text: B

text: /t

text: proe´t

text: C

text: dR

text: K

text: dR

text: dR

text: E

text: O

text: Let

text: k

text: and

text: F(Z

text: The

text: exact

text: sequence,

text: for

text: any

text: k,

text: k

text: E

text: E

text: E

text: E

text: tells

text: us

text: that

text: B

text: Ker

text: (Z

text: (Z

text: ˆ

text: B

text: /t

text: proe´t

text: C

text: dR

text: k

text: K

text: dR

text: dR

text: is

text: isomorphic

text: to

text: O

text: B

text: F(Z

text: )ϕ=p.

text: k

text: k,C

text: Taking

text: the

text: inverse

text: limit

text: over

text: k

text: one

text: gets

text: an

text: exact

text: sequence

text: (O(Z

text: (Z

text: Q

text: (Z

text: Lη

text: B)ϕ=p

text: C

text: proe´t

text: C

text: p

text: ´et

text: C

text: t

text: We

text: now

text: turn

text: to

text: the

text: relation

text: with

text: the

text: results

text: of

text: Assume

text: that

text: K

text: is

text: a

text: discretely

text: valued

text: extension

text: of

text: Q

text: with

text: perfect

text: residue

text: ﬁeld

text: and

text: that

text: Z

text: has

text: a

text: semi-stable

text: formal

text: model

text: Z

text: O

text: p

text: over

text: If

text: the

text: semi-stable

text: version

text: of

text: Conjecture

text: was

text: known

text: to

text: be

text: true,

text: this

text: could

text: be

text: K

text: rewritten

text: using

text: log-rigid

text: cohomology.

text: Indeed,

text: let

text: (Z

text: be

text: an

text: aﬃne

text: covering

text: of

text: Z

text: such

text: that

text: k

text: k

text: (Z

text: is

text: a

text: Stein

text: covering

text: of

text: Z.

text: Then

text: we

text: would

text: know

text: that

text: for

text: all

text: k

text: and

text: all

text: i

text: HFi

text: F(Z

text: k

text: k

text: k

text: is

text: the

text: vector

text: bundle

text: attached

text: to

text: the

text: (ϕ,

text: N)-module

text: Hi

text: (Z

text: In

text: particular,

text: logrig

text: k,s

text: HFi

text: F(Z

text: (Hi

text: (Z

text: k

text: logrig

text: k,s

text: st

text: That

text: a

text: variant

text: of

text: such

text: an

text: exact

text: sequence

text: may

text: be

text: relevant

text: to

text: the

text: study

text: of

text: the

text: pro-étale

text: cohomology

text: of

text: Qp

text: was

text: suggested

text: to

text: the

text: author

text: by

text: Gabriel

text: Dospinescu

text: and

text: was

text: used

text: in

text: to

text: compute

text: the

text: pro-étale

text: cohomology

text: of

text: Qp

text: on

text: the

text: aﬃne

text: space

text: and

text: the

text: open

text: disk.

text: One

text: needs

text: to

text: write

text: the

text: Tate

text: twists,

text: which

text: was

text: not

text: done

text: in

text: loc.

text: cit.,

text: but

text: is

text: easy

text: to

text: do.

text: And

text: using

text: that

text: the

text: l←im−

text: term

text: vanishes,

text: as

text: it

text: only

text: involves

text: which

text: can

text: be

text: computed.

text: This

text: assumption

text: is

text: probably

text: unnecessary.

text: ##LTLine##


text: 4 ARTHUR-CÉSAR LE BRAS One can also ask a similar question when Z is only assumed to be semi-stable, and answer it in the proper case, using the recent work of C˘esnavi˘cius-Koshikawa [10]. Note that our construction of HFi F(Z) is entirely on the generic ﬁber and does not involve any choice of formal model. FF This cohomology theory is quite simple to describe : if Z is as before, (Z) comes by the equivalence between (the derived category of) vector bundles on X and ﬁnite projective ϕ-modules over B (which are the same as ϕ-equivariant vector bundles on Y ) from an “overconvergent version” of the complex RΓ(Z , Lη Rν∗(cid:48) B ), (ν∗(cid:48) , ν(cid:48)−1) being the morphism C t Z from the pro-étale topos of Z to the étale topos of Z . There are essentially two things to C prove about these cohomology groups : a "ﬁniteness" and a "torsion-freeness" result. For both, the key point is to show that this cohomology theory factors through the category of rigid analytic motives of Ayoub ([3]). This reduces to prove ﬁniteness in the case where Z is smooth and proper, where several tools are available. This strategy was inspired by Vezzani’s paper [43], which studies overconvergent de Rham cohomology using the motivic formalism. This works for any C. If C is the completion of an algebraic closure of W(k)[1/p], we also get torsion-freeness, using a trick : any quasi-compact motive over C descends to a motive deﬁned over a discretely valued extension of Q with perfect residue ﬁeld (a statement which p is false for smooth quasi-compact rigid spaces !) ; therefore, we can use some p-adic Hodge theory. Let us end this introduction by saying that there should exist a more geometric construc- FF tion of (Z), making the analogy with Betti cohomology more transparent and explaining the title of this text. Let us start by (trivially !) reformulating the construction of H∞i (Z) in ⊗ O the complex case. It is obtained by descending to P˜1 the vector bundle Hi (Z, C) , R Betti C A1 which, in the diagram, C × Z A1 C p q (cid:124)(cid:124) (cid:35)(cid:35) Z A1 C is simply • Riq∗(Ω × ). Z A1 /A1 C C When Z is moreover projective, the twistor is obtained by descending to P˜1 the sheaf R • Riq∗((Ω × , zd)), Z A1 /A1 C C where z is the coordinate on A1 (this is Deligne’s reinterpretation of Simpson’s construc- C tion). Similarly, in the p-adic case, we can look at the diagram of diamonds : (cid:5) × (cid:5) Z Spa(Q ) p p q (cid:120)(cid:120) (cid:38)(cid:38) (cid:5) (cid:5) Z Y (cid:5) (cid:5)× (cid:5) † • as Y = Spa(Cp) Spa(Qp) . There should exist a complex of étale sheaves ”ΩZ,(cid:5)×Spa(Qp)(cid:5)/Y ” such that the pull-back of HFi F(Z) to Y is † • Riq∗(”Ω ,(cid:5)× (cid:5) ”). Z Spa(Qp) /Y In joint work in progress with Alberto Vezzani, we show how to give a meaning to this hypothetical overconvergent relative de Rham complex (4). Still, the point of the present 4. The relation between the pro-étale cohomology of B and Deligne’s λ-connections was also noticed by Liu and Zhu, cf, [30, Rem. 3.2]. ##LTLine## ##LTLine## ##LTLine##

text: ARTHUR-CÉSAR

text: LE

text: BRAS

text: One

text: can

text: also

text: ask

text: a

text: similar

text: question

text: when

text: Z

text: is

text: only

text: assumed

text: to

text: be

text: semi-stable,

text: and

text: answer

text: it

text: in

text: the

text: proper

text: case,

text: using

text: the

text: recent

text: work

text: of

text: C˘esnavi˘cius-Koshikawa

text: Note

text: that

text: our

text: construction

text: of

text: HFi

text: F(Z)

text: is

text: entirely

text: on

text: the

text: generic

text: ﬁber

text: and

text: does

text: not

text: involve

text: any

text: choice

text: of

text: formal

text: model.

text: FF

text: This

text: cohomology

text: theory

text: is

text: quite

text: simple

text: to

text: describe

text: if

text: Z

text: is

text: as

text: before,

text: (Z)

text: comes

text: by

text: the

text: equivalence

text: between

text: (the

text: derived

text: category

text: of)

text: vector

text: bundles

text: on

text: X

text: and

text: ﬁnite

text: projective

text: ϕ-modules

text: over

text: B

text: (which

text: are

text: the

text: same

text: as

text: ϕ-equivariant

text: vector

text: bundles

text: on

text: Y

text: from

text: an

text: “overconvergent

text: version”

text: of

text: the

text: complex

text: RΓ(Z

text: Lη

text: B

text: being

text: the

text: morphism

text: C

text: t

text: Z

text: from

text: the

text: pro-étale

text: topos

text: of

text: Z

text: to

text: the

text: étale

text: topos

text: of

text: Z

text: There

text: are

text: essentially

text: two

text: things

text: to

text: C

text: prove

text: about

text: these

text: cohomology

text: groups

text: a

text: "ﬁniteness"

text: and

text: a

text: "torsion-freeness"

text: result.

text: For

text: both,

text: the

text: key

text: point

text: is

text: to

text: show

text: that

text: this

text: cohomology

text: theory

text: factors

text: through

text: the

text: category

text: of

text: rigid

text: analytic

text: motives

text: of

text: Ayoub

text: This

text: reduces

text: to

text: prove

text: ﬁniteness

text: in

text: the

text: case

text: where

text: Z

text: is

text: smooth

text: and

text: proper,

text: where

text: several

text: tools

text: are

text: available.

text: This

text: strategy

text: was

text: inspired

text: by

text: Vezzani’s

text: paper

text: which

text: studies

text: overconvergent

text: de

text: Rham

text: cohomology

text: using

text: the

text: motivic

text: formalism.

text: This

text: works

text: for

text: any

text: C.

text: If

text: C

text: is

text: the

text: completion

text: of

text: an

text: algebraic

text: closure

text: of

text: we

text: also

text: get

text: torsion-freeness,

text: using

text: a

text: trick

text: any

text: quasi-compact

text: motive

text: over

text: C

text: descends

text: to

text: a

text: motive

text: deﬁned

text: over

text: a

text: discretely

text: valued

text: extension

text: of

text: Q

text: with

text: perfect

text: residue

text: ﬁeld

text: (a

text: statement

text: which

text: p

text: is

text: false

text: for

text: smooth

text: quasi-compact

text: rigid

text: spaces

text: therefore,

text: we

text: can

text: use

text: some

text: p-adic

text: Hodge

text: theory.

text: Let

text: us

text: end

text: this

text: introduction

text: by

text: saying

text: that

text: there

text: should

text: exist

text: a

text: more

text: geometric

text: construc-

text: FF

text: tion

text: of

text: (Z),

text: making

text: the

text: analogy

text: with

text: Betti

text: cohomology

text: more

text: transparent

text: and

text: explaining

text: the

text: title

text: of

text: this

text: text.

text: Let

text: us

text: start

text: by

text: (trivially

text: reformulating

text: the

text: construction

text: of

text: H∞i

text: (Z)

text: in

text: O

text: the

text: complex

text: case.

text: It

text: is

text: obtained

text: by

text: descending

text: to

text: the

text: vector

text: bundle

text: Hi

text: (Z,

text: C)

text: R

text: Betti

text: C

text: which,

text: in

text: the

text: diagram,

text: C

text: Z

text: C

text: p

text: q

text: Z

text: C

text: is

text: simply

text: Riq∗(Ω

text: Z

text: C

text: C

text: When

text: Z

text: is

text: moreover

text: projective,

text: the

text: twistor

text: is

text: obtained

text: by

text: descending

text: to

text: the

text: sheaf

text: R

text: Riq∗((Ω

text: zd)),

text: Z

text: C

text: C

text: where

text: z

text: is

text: the

text: coordinate

text: on

text: (this

text: is

text: Deligne’s

text: reinterpretation

text: of

text: Simpson’s

text: construc-

text: C

text: tion).

text: Similarly,

text: in

text: the

text: p-adic

text: case,

text: we

text: can

text: look

text: at

text: the

text: diagram

text: of

text: diamonds

text: Z

text: Spa(Q

text: p

text: p

text: q

text: Z

text: Y

text: as

text: Y

text: Spa(Cp)

text: Spa(Qp)

text: There

text: should

text: exist

text: a

text: complex

text: of

text: étale

text: sheaves

text: such

text: that

text: the

text: pull-back

text: of

text: HFi

text: F(Z)

text: to

text: Y

text: is

text: Riq∗(”Ω

text: Z

text: Spa(Qp)

text: /Y

text: In

text: joint

text: work

text: in

text: progress

text: with

text: Alberto

text: Vezzani,

text: we

text: show

text: how

text: to

text: give

text: a

text: meaning

text: to

text: this

text: hypothetical

text: overconvergent

text: relative

text: de

text: Rham

text: complex

text: Still,

text: the

text: point

text: of

text: the

text: present

text: The

text: relation

text: between

text: the

text: pro-étale

text: cohomology

text: of

text: B

text: and

text: Deligne’s

text: λ-connections

text: was

text: also

text: noticed

text: by

text: Liu

text: and

text: Zhu,

text: cf,

text: Rem.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: DE RHAM COHOMOLOGY OVER THE FARGUES-FONTAINE CURVE 5 paper is that, even if we don’t know what this complex is, Y being Stein, to construct HFi F(Z), it is enough to know what † • Rp∗(”Ω ,(cid:5)× (cid:5) ”) Z Spa(Qp) /Y is : this is precisely (the overconvergent version of) Lη Rν∗(cid:48) B . t Z Plan of the text. Sections and are devoted to some remainders and some fundamental results about the category of rigid motives, pro-étale period sheaves and the décalage F functors. The key technical statement of this text (showing that our cohomology theory FF factors through the category of motives) is proved in Section . We deﬁne and prove that its cohomology groups are vector bundles on the Fargues-Fontaine curve when C is the completion of an algebraic closure of some discretely valued ﬁeld in Section . Finally, we brieﬂy explain in Section how our constructions are related to other cohomology theories in some cases : étale, crystalline, syntomic and pro-étale cohomologies. Conventions and notations. The ﬁeld K is a complete valued ﬁeld containing Q ; C p the completion of an algebraic closure of K, with residue ﬁeld k (in particular K may very † well be C itself !) ; D is the closed unit disk over Q and D its overconvergent analogue. p∈ (cid:54) We ﬁx a compatible system of pn-th roots of unity ε C(cid:91), with ε(0) = 1, ε(1) = 1, and let ∈ p t = log([ε]) B be “Fontaine’s 2iπ”. All rigid spaces are assumed to be separated and taut. Acknowledgments. The idea of this paper originated from a conversation with Peter Scholze in Bonn in February 2017. I would like to thank him heartily for this, as well as for his numerous very helpful suggestions and comments. Some of the key cohomological computations of this text were started when redacting Chapter 2 of [29] as part of my PhD thesis and I thank Laurent Fargues for everything he taught me about period sheaves and the Fargues-Fontaine curve. Special thanks go to Alberto Vezzani for his interest and the time he took to answer my many naive questions about motives. I am indebted to Wieslawa Nizioł who pointed out a gap in a ﬁrst version of this text. Finally, I would also like to thank K¸estutis C˘esnavi˘cius and Teruhisa Koshikawa for interesting conversations and Eugen Hellmann for an invitation to talk about this work. 2. Motives of (overconvergent) rigid analytic varieties For what follows, the main reference is [43]. We take Λ = Q as a coeﬃcient ring ; it will p be implicit in the notation. In all this section, let F be a characteristic 0 complete valued ﬁeld with a non archimedean valuation of rank 1 and residue characteristic p. We recall the deﬁnition of the category of rigid motives (and overconvergent rigid motives), which can be thought of as the Verdier quotient of the derived category of étale sheaves of Q -vector p spaces on the category of smooth rigid varieties over F, obtained by requiring that all the × → projections maps (Z D )[i] Z[i] are invertible, for Z any rigid smooth space over F and ∈ F i Z, and we state two results which will be useful later on (Theorem 2.5 and Proposition 2.6 below). Let Rig (resp. Aﬀ ) be the category of rigid spaces over F (resp. aﬃnoid rigid spaces F † F † over F), and let Rig (resp. Aﬀ ) be the category of overconvergent rigid spaces over F F F (resp. overconvergent aﬃnoid rigid spaces), in the sense of Große-Klönne [24]. The natural † → (cid:55)→ functor Rig Rig , Z Zˆ is denoted by (cid:96). Let RigSm be the subcategory of smooth F F † F rigid analytic varieties over F and RigSm be the subcategory of smooth overconvergent F rigid varieties over F (an overconvergent rigid space Z is said to be smooth if the associated rigid space Zˆ is).

text: DE

text: RHAM

text: COHOMOLOGY

text: OVER

text: THE

text: FARGUES-FONTAINE

text: CURVE

text: paper

text: is

text: that,

text: even

text: if

text: we

text: don’t

text: know

text: what

text: this

text: complex

text: is,

text: Y

text: being

text: Stein,

text: to

text: construct

text: HFi

text: F(Z),

text: it

text: is

text: enough

text: to

text: know

text: what

text: Rp∗(”Ω

text: Z

text: Spa(Qp)

text: /Y

text: is

text: this

text: is

text: precisely

text: (the

text: overconvergent

text: version

text: of)

text: Lη

text: B

text: t

text: Z

text: Plan

text: of

text: the

text: text.

text: Sections

text: and

text: are

text: devoted

text: to

text: some

text: remainders

text: and

text: some

text: fundamental

text: results

text: about

text: the

text: category

text: of

text: rigid

text: motives,

text: pro-étale

text: period

text: sheaves

text: and

text: the

text: décalage

text: F

text: functors.

text: The

text: key

text: technical

text: statement

text: of

text: this

text: text

text: (showing

text: that

text: our

text: cohomology

text: theory

text: FF

text: factors

text: through

text: the

text: category

text: of

text: motives)

text: is

text: proved

text: in

text: Section

text: We

text: deﬁne

text: and

text: prove

text: that

text: its

text: cohomology

text: groups

text: are

text: vector

text: bundles

text: on

text: the

text: Fargues-Fontaine

text: curve

text: when

text: C

text: is

text: the

text: completion

text: of

text: an

text: algebraic

text: closure

text: of

text: some

text: discretely

text: valued

text: ﬁeld

text: in

text: Section

text: Finally,

text: we

text: brieﬂy

text: explain

text: in

text: Section

text: how

text: our

text: constructions

text: are

text: related

text: to

text: other

text: cohomology

text: theories

text: in

text: some

text: cases

text: étale,

text: crystalline,

text: syntomic

text: and

text: pro-étale

text: cohomologies.

text: Conventions

text: and

text: notations.

text: The

text: ﬁeld

text: K

text: is

text: a

text: complete

text: valued

text: ﬁeld

text: containing

text: Q

text: C

text: p

text: the

text: completion

text: of

text: an

text: algebraic

text: closure

text: of

text: K,

text: with

text: residue

text: ﬁeld

text: k

text: (in

text: particular

text: K

text: may

text: very

text: well

text: be

text: C

text: itself

text: D

text: is

text: the

text: closed

text: unit

text: disk

text: over

text: Q

text: and

text: D

text: its

text: overconvergent

text: analogue.

text: p∈

text: We

text: ﬁx

text: a

text: compatible

text: system

text: of

text: pn-th

text: roots

text: of

text: unity

text: ε

text: with

text: and

text: let

text: p

text: t

text: log([ε])

text: B

text: be

text: “Fontaine’s

text: All

text: rigid

text: spaces

text: are

text: assumed

text: to

text: be

text: separated

text: and

text: taut.

text: Acknowledgments.

text: The

text: idea

text: of

text: this

text: paper

text: originated

text: from

text: a

text: conversation

text: with

text: Peter

text: Scholze

text: in

text: Bonn

text: in

text: February

text: I

text: would

text: like

text: to

text: thank

text: him

text: heartily

text: for

text: this,

text: as

text: well

text: as

text: for

text: his

text: numerous

text: very

text: helpful

text: suggestions

text: and

text: comments.

text: Some

text: of

text: the

text: key

text: cohomological

text: computations

text: of

text: this

text: text

text: were

text: started

text: when

text: redacting

text: Chapter

text: of

text: as

text: part

text: of

text: my

text: PhD

text: thesis

text: and

text: I

text: thank

text: Laurent

text: Fargues

text: for

text: everything

text: he

text: taught

text: me

text: about

text: period

text: sheaves

text: and

text: the

text: Fargues-Fontaine

text: curve.

text: Special

text: thanks

text: go

text: to

text: Alberto

text: Vezzani

text: for

text: his

text: interest

text: and

text: the

text: time

text: he

text: took

text: to

text: answer

text: my

text: many

text: naive

text: questions

text: about

text: motives.

text: I

text: am

text: indebted

text: to

text: Wieslawa

text: Nizioł

text: who

text: pointed

text: out

text: a

text: gap

text: in

text: a

text: ﬁrst

text: version

text: of

text: this

text: text.

text: Finally,

text: I

text: would

text: also

text: like

text: to

text: thank

text: K¸estutis

text: C˘esnavi˘cius

text: and

text: Teruhisa

text: Koshikawa

text: for

text: interesting

text: conversations

text: and

text: Eugen

text: Hellmann

text: for

text: an

text: invitation

text: to

text: talk

text: about

text: this

text: work.

text: Motives

text: of

text: (overconvergent)

text: rigid

text: analytic

text: varieties

text: For

text: what

text: follows,

text: the

text: main

text: reference

text: is

text: We

text: take

text: Λ

text: Q

text: as

text: a

text: coeﬃcient

text: ring

text: it

text: will

text: p

text: be

text: implicit

text: in

text: the

text: notation.

text: In

text: all

text: this

text: section,

text: let

text: F

text: be

text: a

text: characteristic

text: complete

text: valued

text: ﬁeld

text: with

text: a

text: non

text: archimedean

text: valuation

text: of

text: rank

text: and

text: residue

text: characteristic

text: p.

text: We

text: recall

text: the

text: deﬁnition

text: of

text: the

text: category

text: of

text: rigid

text: motives

text: (and

text: overconvergent

text: rigid

text: motives),

text: which

text: can

text: be

text: thought

text: of

text: as

text: the

text: Verdier

text: quotient

text: of

text: the

text: derived

text: category

text: of

text: étale

text: sheaves

text: of

text: Q

text: -vector

text: p

text: spaces

text: on

text: the

text: category

text: of

text: smooth

text: rigid

text: varieties

text: over

text: F,

text: obtained

text: by

text: requiring

text: that

text: all

text: the

text: projections

text: maps

text: (Z

text: D

text: )[i]

text: Z[i]

text: are

text: invertible,

text: for

text: Z

text: any

text: rigid

text: smooth

text: space

text: over

text: F

text: and

text: F

text: i

text: Z,

text: and

text: we

text: state

text: two

text: results

text: which

text: will

text: be

text: useful

text: later

text: on

text: (Theorem

text: and

text: Proposition

text: below).

text: Let

text: Rig

text: (resp.

text: Aﬀ

text: be

text: the

text: category

text: of

text: rigid

text: spaces

text: over

text: F

text: (resp.

text: aﬃnoid

text: rigid

text: spaces

text: F

text: F

text: over

text: F),

text: and

text: let

text: Rig

text: (resp.

text: Aﬀ

text: be

text: the

text: category

text: of

text: overconvergent

text: rigid

text: spaces

text: over

text: F

text: F

text: F

text: (resp.

text: overconvergent

text: aﬃnoid

text: rigid

text: spaces),

text: in

text: the

text: sense

text: of

text: Große-Klönne

text: The

text: natural

text: functor

text: Rig

text: Rig

text: Z

text: Zˆ

text: is

text: denoted

text: by

text: Let

text: RigSm

text: be

text: the

text: subcategory

text: of

text: smooth

text: F

text: F

text: F

text: rigid

text: analytic

text: varieties

text: over

text: F

text: and

text: RigSm

text: be

text: the

text: subcategory

text: of

text: smooth

text: overconvergent

text: F

text: rigid

text: varieties

text: over

text: F

text: (an

text: overconvergent

text: rigid

text: space

text: Z

text: is

text: said

text: to

text: be

text: smooth

text: if

text: the

text: associated

text: rigid

text: space

text: Zˆ

text: is).


text: 8 ARTHUR-CÉSAR LE BRAS (cid:98) (cid:98) (cid:98) — The sheaf A = W(O+ ). One has a morphism of sheaves θ : A → O+, which extends to θi:nfA [1/p] →Z(cid:91) O . inf Z inf Z — The sheaf B+ = l←im− A [1/p]/(ker(θ))k. It is endowed with a ﬁltration deﬁned by FiliB+ = kedrR(θ)i. k inf — Let tdbRe a g(cid:80)enerator of Fil1B+ (such an element exists locally for the pro-étale topol- ogy, is unique up to a unitdRand is not a zero divisor). Let B = B+ [1/t] and FiliB = ∈ t−jFili+jB+ . dR dR dR j Z dR ⊂ Deﬁnition 3.3. — Let Z be an adic space over C. Let I = [a, b] ]0, 1[ be a compact ∈ ∈ O | | | | subinterval, with a, b pQ. If α, β are such that α = a, β = b, one deﬁnes : C(cid:91) A A B A = ←lim− ( [[α]/p, p/[β]])/pn ; = [1/p]. I inf I I n Let : B B = l←im− . I ⊂ I ]0,1[ The pro-étale cohomology of these period sheaves can be simply described in the proper case. Proposition 3.4. — Let Z be a smooth proper rigid space over C. Let M ∈ {B B B B B B } + , , , , [1/t], [1/t] , dR dR I I and accordingly ∈ { } M B+ , B , B , B, B [1/t], B[1/t] . dR dR I I One has a quasi-isomorphism ⊗ (cid:39) M RΓ (Z, Q ) M RΓ (Z, ). ´et p Qp pro´et Moreover, the cohomology groups Hi (Z, Q ) are ﬁnite dimensional Q -vector spaces, for ≥ ´et p p all i 0. Proof. — As Z is proper, thus quasi-compact, it is enough to prove the statement when M ∈ {B+ , B , B}. Using Lemma 3.5 below, the statement for M = B is easily deduced from the statdeRmenIt for B , for all compact subintervals I ⊂]0, 1[. The sheaf B is obtained by I I p-adically completing A [[α]/p, p/[β]], for α, β as in Deﬁnition 3.3, and inverting p. So it inf ∈ O | | is enough to prove that for any x , 0 < x < 1, and any m > 0, C(cid:91) ⊗ (cid:39) A RΓ (Z, Z /pm) A /([x]) a RΓ (Z, /(pm, [x])). ´et p Zp inf pro´et inf This is a corollary of the proof of [35, Th. 8.4]. The case M = B+ is also treated in [35, dR Th. 8.4]. Finaly, the last statement is [37, Th. 3.1]. Lemma 3.5. — Let Z be a smooth proper rigid space over C. For any i > 0, one has B Ril←im− = 0, I I when I runs among compact subintervals of ]0, 1[, with extremities in pQ. Proof. — We apply [35, Lem. 3.18], by taking as a basis the set of all aﬃnoid perfectoid spaces pro-étale over Z. Condition (ii) of loc. cit. follows by acyclicity of B on aﬃnoid ∈ I perfectoid spaces. Condition (i) comes from the fact that for any U Z aﬃnoid per- fectoid, and any I ⊂ J, the morphism B (U) → B (U) is continuous morpproh´eitsm of Banach J I spaces with dense image. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ARTHUR-CÉSAR

text: LE

text: BRAS

label: list

text: The

text: sheaf

text: A

text: W(O+

text: One

text: has

text: a

text: morphism

text: of

text: sheaves

text: θ

text: A

text: O+,

text: which

text: extends

text: to

text: θi:nfA

text: O

text: inf

text: Z

text: inf

text: Z

text: The

text: sheaf

text: B+

text: l←im−

text: A

text: It

text: is

text: endowed

text: with

text: a

text: ﬁltration

text: deﬁned

text: by

text: FiliB+

text: kedrR(θ)i.

text: k

text: inf

text: Let

text: tdbRe

text: a

text: of

text: (such

text: an

text: element

text: exists

text: locally

text: for

text: the

text: pro-étale

text: topol-

text: ogy,

text: is

text: unique

text: up

text: to

text: a

text: unitdRand

text: is

text: not

text: a

text: zero

text: divisor).

text: Let

text: B

text: B+

text: and

text: FiliB

text: t−jFili+jB+

text: dR

text: dR

text: dR

text: j

text: Z

text: dR

text: Deﬁnition

text: Let

text: Z

text: be

text: an

text: adic

text: space

text: over

text: C.

text: Let

text: I

text: [a,

text: b]

text: be

text: a

text: compact

text: O

text: subinterval,

text: with

text: a,

text: b

text: pQ.

text: If

text: α,

text: β

text: are

text: such

text: that

text: α

text: a,

text: β

text: b,

text: one

text: deﬁnes

text: A

text: A

text: B

text: A

text: ←lim−

text: [[α]/p,

text: p/[β]])/pn

text: I

text: inf

text: I

text: I

text: n

text: Let

text: B

text: B

text: l←im−

text: I

text: I

text: The

text: pro-étale

text: cohomology

text: of

text: these

text: period

text: sheaves

text: can

text: be

text: simply

text: described

text: in

text: the

text: proper

text: case.

text: Proposition

text: Let

text: Z

text: be

text: a

text: smooth

text: proper

text: rigid

text: space

text: over

text: C.

text: Let

text: M

text: {B

text: B

text: B

text: B

text: B

text: B

text: dR

text: dR

text: I

text: I

text: and

text: accordingly

text: M

text: B+

text: B

text: B

text: B,

text: B

text: dR

text: dR

text: I

text: I

text: One

text: has

text: a

text: quasi-isomorphism

text: M

text: RΓ

text: (Z,

text: Q

text: M

text: RΓ

text: (Z,

text: ´et

text: p

text: Qp

text: pro´et

text: Moreover,

text: the

text: cohomology

text: groups

text: Hi

text: (Z,

text: Q

text: are

text: ﬁnite

text: dimensional

text: Q

text: -vector

text: spaces,

text: for

text: ´et

text: p

text: p

text: all

text: i

text: Proof.

text: As

text: Z

text: is

text: proper,

text: thus

text: quasi-compact,

text: it

text: is

text: enough

text: to

text: prove

text: the

text: statement

text: when

text: M

text: {B+

text: B

text: B}.

text: Using

text: Lemma

text: below,

text: the

text: statement

text: for

text: M

text: B

text: is

text: easily

text: deduced

text: from

text: the

text: statdeRmenIt

text: for

text: B

text: for

text: all

text: compact

text: subintervals

text: I

text: The

text: sheaf

text: B

text: is

text: obtained

text: by

text: I

text: I

text: p-adically

text: completing

text: A

text: [[α]/p,

text: p/[β]],

text: for

text: α,

text: β

text: as

text: in

text: Deﬁnition

text: and

text: inverting

text: p.

text: So

text: it

text: inf

text: O

text: is

text: enough

text: to

text: prove

text: that

text: for

text: any

text: x

text: x

text: and

text: any

text: m

text: A

text: RΓ

text: (Z,

text: Z

text: /pm)

text: A

text: /([x])

text: a

text: RΓ

text: (Z,

text: /(pm,

text: [x])).

text: ´et

text: p

text: Zp

text: inf

text: pro´et

text: inf

text: This

text: is

text: a

text: corollary

text: of

text: the

text: proof

text: of

text: Th.

text: The

text: case

text: M

text: B+

text: is

text: also

text: treated

text: in

text: dR

text: Th.

text: Finaly,

text: the

text: last

text: statement

text: is

text: Th.

text: Lemma

text: Let

text: Z

text: be

text: a

text: smooth

text: proper

text: rigid

text: space

text: over

text: C.

text: For

text: any

text: i

text: one

text: has

text: B

text: Ril←im−

text: I

text: I

text: when

text: I

text: runs

text: among

text: compact

text: subintervals

text: of

text: with

text: extremities

text: in

text: pQ.

text: Proof.

text: We

text: apply

text: Lem.

text: by

text: taking

text: as

text: a

text: basis

text: the

text: set

text: of

text: all

text: aﬃnoid

text: perfectoid

text: spaces

text: pro-étale

text: over

text: Z.

text: Condition

text: (ii)

text: of

text: loc.

text: cit.

text: follows

text: by

text: acyclicity

text: of

text: B

text: on

text: aﬃnoid

text: I

text: perfectoid

text: spaces.

text: Condition

text: (i)

text: comes

text: from

text: the

text: fact

text: that

text: for

text: any

text: U

text: Z

text: aﬃnoid

text: per-

text: fectoid,

text: and

text: any

text: I

text: J,

text: the

text: morphism

text: B

text: (U)

text: B

text: (U)

text: is

text: continuous

text: morpproh´eitsm

text: of

text: Banach

text: J

text: I

text: spaces

text: with

text: dense

text: image.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 10 ARTHUR-CÉSAR LE BRAS ∈ • Lemma 3.10. — Let A be a ring, f A a non zero divisor, and K a bounded complex of ≥ A-modules, such that Kj is f-torsion free for all j. Let, for i 0, δ be the function deﬁned i by − δ (j) = max(0, j i). i • • • • Note that K = η K and that K = K for any i big enough. Then one has, for all i ≥ 0, a trianδ0g,lfe in Df (A) δi,f • −→ • −→ ≥ • −→ K K τ i+1(K /f) +1 . δi,f δi+1,f O These deﬁnitions and results extend to a more general framework ([8, §6.1]) : if (T, ) ∈ O • O T is a ringed topos, f generates an invertible ideal and K is a complex of -modules T • T without f-torsion, one can deﬁne η K , for any non decreasing δ, as before and one can δ,f O O show that η extends to the derived category D( ) of -modules. δ,f T T In particular, let Z be a rigid space over C and M and M be as in Proposition 3.4. O Specializing to the case where T is the topos of étale sheaves of M-modules on Z, = M and f = t, one obtains a complex Lη Rν∗(cid:48) M in the derived category of étale sheaTves of t M-modules on Z. Proposition 3.11. — For any smooth aﬃnoid rigid space Z over C, and M ∈ {B B B B } + , , , [1/t] , dR dR I I the natural map M → (cid:48) M Lη RΓ (Z, ) RΓ (Z, Lη Rν∗ ) t pro´et ´et t is a quasi-isomorphism. Proof. — The statement is of course empty when M ∈ {B , B [1/t]}, as t in invertible and the complexes are bounded. (cid:98)We can thus assume that MdR∈ {IB+ , B }, I being such that the annulus of radius I contains at least one zero of t, in whichdcRaseIM/t is isomorphic to a ﬁnite number of copies of O. Using Lemma 3.10, applied both to Lη RΓ (Z, M) and Lη Rν∗(cid:48) M, we see that it is enough to show t(cid:98)hat for all i ≥ 0(cid:98): t pro´et t ≥ (cid:48) O (cid:39) O Hj (Z, τ iRν∗ ) Hj (Z, ) ´et pro´et ≥ (cid:98) if j i and that ≥ (cid:48) O (cid:98) Hj (Z, τ iRν∗ ) = 0 ´et if j < i (we used the fact that M/t is a ﬁnite product of copies of O). To do so, we use the (cid:98) (cid:98) spectral sequence − ≥ (cid:48) O ⇒ ≥ (cid:48) O Hj k(Z, Hk(τ iRν∗ )) = Hj (Z, τ iRν∗ ) ´et (cid:40) ´et and Scholze’s computation ([37, Prop. 3.23] : (cid:98) (cid:98) ≥ (cid:48) O 0 if k < i Hk(τ iRν∗ ) = (cid:48) O ≥ Hk(Rν∗ ) = Ωk if k i. Similarly, we have a spectral sequence (cid:98) − ⇒ O Hj k(Z, Ωk) = Hj (Z, ). ´et pro´et The diﬀerences between the abutments of these two spectral sequences could thus a priori − ≥ − come from the terms Hj k(Z, Ωk), for k < i. But then as j i we have j k > 0 and Z being aﬃnoid, these terms are zero. O Proposition 3.12. — Let Z be a semi-stable quasi-compact formal scheme over , with − C generic ﬁber Z, and (ν∗, ν 1) be the morphism from the pro-étale topos of Z to the étale ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ARTHUR-CÉSAR

text: LE

text: BRAS

text: Lemma

text: Let

text: A

text: be

text: a

text: ring,

text: f

text: A

text: a

text: non

text: zero

text: divisor,

text: and

text: K

text: a

text: bounded

text: complex

text: of

text: A-modules,

text: such

text: that

text: Kj

text: is

text: f-torsion

text: free

text: for

text: all

text: j.

text: Let,

text: for

text: i

text: δ

text: be

text: the

text: function

text: deﬁned

text: i

text: by

text: δ

text: (j)

text: j

text: i).

text: i

text: Note

text: that

text: K

text: η

text: K

text: and

text: that

text: K

text: K

text: for

text: any

text: i

text: big

text: enough.

text: Then

text: one

text: has,

text: for

text: all

text: i

text: a

text: in

text: Df

text: (A)

text: δi,f

text: K

text: K

text: τ

text: /f)

text: δi,f

text: O

text: These

text: deﬁnitions

text: and

text: results

text: extend

text: to

text: a

text: more

text: general

text: framework

text: if

text: (T,

text: O

text: O

text: T

text: is

text: a

text: ringed

text: topos,

text: f

text: generates

text: an

text: invertible

text: ideal

text: and

text: K

text: is

text: a

text: complex

text: of

text: -modules

text: T

text: T

text: without

text: f-torsion,

text: one

text: can

text: deﬁne

text: η

text: K

text: for

text: any

text: non

text: decreasing

text: δ,

text: as

text: before

text: and

text: one

text: can

text: δ,f

text: O

text: O

text: show

text: that

text: η

text: extends

text: to

text: the

text: derived

text: category

text: D(

text: of

text: -modules.

text: δ,f

text: T

text: T

text: In

text: particular,

text: let

text: Z

text: be

text: a

text: rigid

text: space

text: over

text: C

text: and

text: M

text: and

text: M

text: be

text: as

text: in

text: Proposition

text: O

text: Specializing

text: to

text: the

text: case

text: where

text: T

text: is

text: the

text: topos

text: of

text: étale

text: sheaves

text: of

text: M-modules

text: on

text: Z,

text: M

text: and

text: f

text: t,

text: one

text: obtains

text: a

text: complex

text: Lη

text: M

text: in

text: the

text: derived

text: category

text: of

text: étale

text: sheaTves

text: of

text: t

text: M-modules

text: on

text: Z.

text: Proposition

text: For

text: any

text: smooth

text: aﬃnoid

text: rigid

text: space

text: Z

text: over

text: C,

text: and

text: M

text: {B

text: B

text: B

text: B

text: dR

text: dR

text: I

text: I

text: the

text: natural

text: map

text: M

text: M

text: Lη

text: RΓ

text: (Z,

text: RΓ

text: (Z,

text: Lη

text: Rν∗

text: t

text: pro´et

text: ´et

text: t

text: is

text: a

text: quasi-isomorphism.

text: Proof.

text: The

text: statement

text: is

text: of

text: course

text: empty

text: when

text: M

text: {B

text: B

text: as

text: t

text: in

text: invertible

text: and

text: the

text: complexes

text: are

text: bounded.

text: can

text: thus

text: assume

text: that

text: MdR∈

text: {IB+

text: B

text: I

text: being

text: such

text: that

text: the

text: annulus

text: of

text: radius

text: I

text: contains

text: at

text: least

text: one

text: zero

text: of

text: t,

text: in

text: whichdcRaseIM/t

text: is

text: isomorphic

text: to

text: a

text: ﬁnite

text: number

text: of

text: copies

text: of

text: O.

text: Using

text: Lemma

text: applied

text: both

text: to

text: Lη

text: RΓ

text: (Z,

text: M)

text: and

text: Lη

text: M,

text: we

text: see

text: that

text: it

text: is

text: enough

text: to

text: show

text: for

text: all

text: i

text: t

text: pro´et

text: t

text: O

text: O

text: Hj

text: (Z,

text: τ

text: iRν∗

text: Hj

text: (Z,

text: ´et

text: pro´et

text: if

text: j

text: i

text: and

text: that

text: O

text: Hj

text: (Z,

text: τ

text: iRν∗

text: ´et

text: if

text: j

text: i

text: (we

text: used

text: the

text: fact

text: that

text: M/t

text: is

text: a

text: ﬁnite

text: product

text: of

text: copies

text: of

text: O).

text: To

text: do

text: so,

text: we

text: use

text: the

text: spectral

text: sequence

text: O

text: O

text: Hj

text: k(Z,

text: Hk(τ

text: iRν∗

text: Hj

text: (Z,

text: τ

text: iRν∗

text: ´et

text: ´et

text: and

text: Scholze’s

text: computation

text: Prop.

text: O

text: if

text: k

text: i

text: Hk(τ

text: iRν∗

text: O

text: Hk(Rν∗

text: Ωk

text: if

text: k

text: i.

text: Similarly,

text: we

text: have

text: a

text: spectral

text: sequence

text: O

text: Hj

text: k(Z,

text: Ωk)

text: Hj

text: (Z,

text: ´et

text: pro´et

text: The

text: diﬀerences

text: between

text: the

text: abutments

text: of

text: these

text: two

text: spectral

text: sequences

text: could

text: thus

text: a

text: priori

text: come

text: from

text: the

text: terms

text: Hj

text: k(Z,

text: Ωk),

text: for

text: k

text: i.

text: But

text: then

text: as

text: j

text: i

text: we

text: have

text: j

text: k

text: and

text: Z

text: being

text: aﬃnoid,

text: these

text: terms

text: are

text: zero.

text: O

text: Proposition

text: Let

text: Z

text: be

text: a

text: semi-stable

text: quasi-compact

text: formal

text: scheme

text: over

text: with

text: C

text: generic

text: ﬁber

text: Z,

text: and

text: (ν∗,

text: ν

text: be

text: the

text: morphism

text: from

text: the

text: pro-étale

text: topos

text: of

text: Z

text: to

text: the

text: étale

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:12) (cid:90) (cid:90) (cid:12) (cid:12) (cid:12) dH · − · = 2 (A B ) v⊥ dS 2 (A v⊥ ) B dS, (2) p h z p h z dt S S S where A is the vector potential of the potential ﬁeld. B , B and p p h B denote the tangential and normal magnetic ﬁelds, and v⊥ and v⊥ z h z are the tangential and normal components of velocity v⊥. The ﬁrst term on the right side of Equation (2) is the helicity generated from shearing motions while the second term is the helicity from emerging motions. A is p determined by the photospheric vertical magnetic ﬁeld and Coulomb gauge by equations (Berger, 1997; Berger & Ruzmaikin, 2000): ∇ × · ∇ · · A nˆ = B , A = 0, A nˆ = 0. (3) p h p p Based on the basic algebraic relations, we then obtain: · (v B) B v(cid:107) = , (4) B2 · − (v B) B v⊥ = v , (5) B2 · − (v B) B v⊥ = v h , (6) h h B2 · − (v B) B v⊥ = v z . (7) z z B2 Here, v is the photospheric plasma velocity, v(cid:107) and v⊥ denotes the velocity components that is parallel and perpendicular to the magnetic ﬁeld. Solar Dynamic Observatory (SDO) was launched in 2010. The on-board Helioseismic and Magnetic Imager (HMI) can map the full disk photospheric vector magnetic ﬁeld with high cadence and long continuity. The vector magnetograms employed in this study are from Space-weather HMI Active Region Patches (SHARPs) with a spatial resolution of 1” and a 12 mins ca- dence (Bobra et al., 2014). The photospheric plasma velocity was calculated using the Diﬀerential Aﬃne Velocity Estimator for Vector Magnetograms (DAVE4VM) algorithm (Schuck, 2008), the window size used in the calcula- tion is 19 pixels, which was determined by examining non-parametric Spear- man rank order correlation coeﬃcients, Pearson correlation coeﬃcients and · − slopes between ∆ (v B v B ) and δB /δt (Schuck, 2008). The vector h z h h z z 9 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: equation

text: dH

text: (A

text: B

text: v⊥

text: dS

text: (A

text: v⊥

text: B

text: dS,

text: p

text: h

text: z

text: p

text: h

text: z

text: dt

text: S

text: S

text: S

text: where

text: A

text: is

text: the

text: vector

text: potential

text: of

text: the

text: potential

text: ﬁeld.

text: B

text: B

text: and

text: p

text: p

text: h

text: B

text: denote

text: the

text: tangential

text: and

text: normal

text: magnetic

text: ﬁelds,

text: and

text: v⊥

text: and

text: v⊥

text: z

text: h

text: z

text: are

text: the

text: tangential

text: and

text: normal

text: components

text: of

text: velocity

text: v⊥.

text: The

text: ﬁrst

text: term

text: on

text: the

text: right

text: side

text: of

text: Equation

text: is

text: the

text: helicity

text: generated

text: from

text: shearing

text: motions

text: while

text: the

text: second

text: term

text: is

text: the

text: helicity

text: from

text: emerging

text: motions.

text: A

text: is

text: p

text: determined

text: by

text: the

text: photospheric

text: vertical

text: magnetic

text: ﬁeld

text: and

text: Coulomb

text: gauge

text: by

text: equations

text: (Berger,

text: Berger

text: Ruzmaikin,

text: A

text: nˆ

text: B

text: A

text: A

text: nˆ

text: p

text: h

text: p

text: p

text: Based

text: on

text: the

text: basic

text: algebraic

text: relations,

text: we

text: then

text: obtain:

text: (v

text: B)

text: B

text: (v

text: B)

text: B

text: v⊥

text: v

text: (v

text: B)

text: B

text: v⊥

text: v

text: h

text: h

text: h

text: (v

text: B)

text: B

text: v⊥

text: v

text: z

text: z

text: z

text: Here,

text: v

text: is

text: the

text: photospheric

text: plasma

text: velocity,

text: and

text: v⊥

text: denotes

text: the

text: velocity

text: components

text: that

text: is

text: parallel

text: and

text: perpendicular

text: to

text: the

text: magnetic

text: ﬁeld.

text: Solar

text: Dynamic

text: Observatory

text: (SDO)

text: was

text: launched

text: in

text: The

text: on-board

text: Helioseismic

text: and

text: Magnetic

text: Imager

text: (HMI)

text: can

text: map

text: the

text: full

text: disk

text: photospheric

text: vector

text: magnetic

text: ﬁeld

text: with

text: high

text: cadence

text: and

text: long

text: continuity.

text: The

text: vector

text: magnetograms

text: employed

text: in

text: this

text: study

text: are

text: from

text: Space-weather

text: HMI

text: Active

text: Region

text: Patches

text: (SHARPs)

text: with

text: a

text: spatial

text: resolution

text: of

text: 1”

text: and

text: a

text: mins

text: ca-

text: dence

text: (Bobra

text: et

text: al.,

text: The

text: photospheric

text: plasma

text: velocity

text: was

text: calculated

text: using

text: the

text: Diﬀerential

text: Aﬃne

text: Velocity

text: Estimator

text: for

text: Vector

text: Magnetograms

text: algorithm

text: (Schuck,

text: the

text: window

text: size

text: used

text: in

text: the

text: calcula-

text: tion

text: is

text: pixels,

text: which

text: was

text: determined

text: by

text: examining

text: non-parametric

text: Spear-

text: man

text: rank

text: order

text: correlation

text: coeﬃcients,

text: Pearson

text: correlation

text: coeﬃcients

text: and

text: slopes

text: between

text: (v

text: B

text: v

text: B

text: and

text: δB

text: /δt

text: (Schuck,

text: The

text: vector

text: h

text: z

text: h

text: h

text: z

text: z

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Inference on Weibull Parameters Under a Balanced Two Sample Type-II Progressive Censoring Scheme Shuvashree Mondal1, Debasis Kundu1,2 Abstract The progressive censoring scheme has received considerable amount of attention in the last ﬁfteen years. During the last few years joint progressive censoring scheme has gained some popularity. Recently, the authors Mondal and Kundu (“A new two sample Type-II progressive censoring scheme”, arXiv:1609.05805) introduced a balanced two sample Type-II progressive censoring scheme and provided the exact inference when the two populations are exponentially distributed. In this article we consider the case when the two populations follow Weibull distributions with the common shape parameter and diﬀerent scale parameters. We obtain the maximum likelihood estimators of the unknown parameters. It is observed that the maximum likelihood estimators cannot be obtained in explicit forms, hence, we propose approximate maximum likelihood estimators, which can be obtained in explicit forms. We construct the asymptotic and bootstrap conﬁdence intervals of the population parameters. Further we derive an exact joint conﬁdence region of the unknown parameters. We propose an objective function based on the expected volume of this conﬁdence set and using that we obtain the optimum progressive censoring scheme. Extensive simulations have been performed to see the performances of the proposed method, and one real data set has been analyzed for illustrative purposes. Key Words and Phrases: Type-II censoring; progressive censoring; joint progressive censor- ing; maximum likelihood estimator; approximate maximum likelihood estimator; joint conﬁdence region; optimum censoring scheme. AMS Subject Classifications: 62N01, 62N02, 62F10. 1 Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Pin 208016, India. 1, 2 Corresponding author. E-mail: kundu@iitk.ac.in, Phone no. 91-512-2597141, Fax no. 91-512- 2597500. 1

text: Inference

label: title

text: on

text: Weibull

text: Parameters

text: Under

text: a

text: Balanced

text: Two

text: Sample

text: Type-II

text: Progressive

text: Censoring

text: Scheme

text: Shuvashree

label: author

text: Debasis

text: Abstract

text: The

label: abstract

text: progressive

text: censoring

text: scheme

text: has

text: received

text: considerable

text: amount

text: of

text: attention

text: in

text: the

text: last

text: ﬁfteen

text: years.

text: During

text: the

text: last

text: few

text: years

text: joint

text: progressive

text: censoring

text: scheme

text: has

text: gained

text: some

text: popularity.

text: Recently,

text: the

text: authors

text: Mondal

text: and

text: Kundu

text: (“A

text: new

text: two

text: sample

text: Type-II

text: progressive

text: censoring

text: scheme”,

text: introduced

text: a

text: balanced

text: two

text: sample

text: Type-II

text: progressive

text: censoring

text: scheme

text: and

text: provided

text: the

text: exact

text: inference

text: when

text: the

text: two

text: populations

text: are

text: exponentially

text: distributed.

text: In

text: this

text: article

text: we

text: consider

text: the

text: case

text: when

text: the

text: two

text: populations

text: follow

text: Weibull

text: distributions

text: with

text: the

text: common

text: shape

text: parameter

text: and

text: diﬀerent

text: scale

text: parameters.

text: We

text: obtain

text: the

text: maximum

text: likelihood

text: estimators

text: of

text: the

text: unknown

text: parameters.

text: It

text: is

text: observed

text: that

text: the

text: maximum

text: likelihood

text: estimators

text: cannot

text: be

text: obtained

text: in

text: explicit

text: forms,

text: hence,

text: we

text: propose

text: approximate

text: maximum

text: likelihood

text: estimators,

text: which

text: can

text: be

text: obtained

text: in

text: explicit

text: forms.

text: We

text: construct

text: the

text: asymptotic

text: and

text: bootstrap

text: conﬁdence

text: intervals

text: of

text: the

text: population

text: parameters.

text: Further

text: we

text: derive

text: an

text: exact

text: joint

text: conﬁdence

text: region

text: of

text: the

text: unknown

text: parameters.

text: We

text: propose

text: an

text: objective

text: function

text: based

text: on

text: the

text: expected

text: volume

text: of

text: this

text: conﬁdence

text: set

text: and

text: using

text: that

text: we

text: obtain

text: the

text: optimum

text: progressive

text: censoring

text: scheme.

text: Extensive

text: simulations

text: have

text: been

text: performed

text: to

text: see

text: the

text: performances

text: of

text: the

text: proposed

text: method,

text: and

text: one

text: real

text: data

text: set

text: has

text: been

text: analyzed

text: for

text: illustrative

text: purposes.

text: Key

text: Words

text: and

text: Phrases:

text: Type-II

text: censoring;

text: progressive

text: censoring;

text: joint

text: progressive

text: censor-

text: ing;

text: maximum

text: likelihood

text: estimator;

text: approximate

text: maximum

text: likelihood

text: estimator;

text: joint

text: conﬁdence

text: region;

text: optimum

text: censoring

text: scheme.

text: AMS

text: Subject

text: Classifications:

text: Department

text: of

text: Mathematics

text: and

text: Statistics,

text: Indian

text: Institute

text: of

text: Technology

text: Kanpur,

text: Pin

text: India.

text: Corresponding

text: author.

text: E-mail:

text: kundu@iitk.ac.in,

text: Phone

text: no.

text: Fax

text: no.


text: scheme, two sets of samples from these two products are simultaneously put on a test. Suppose the ﬁrst failure is coming from the product line A and the ﬁrst failure time is denoted by W , then at 1 − W , R units are removed randomly from the remaining m 1 surviving units of the product line 1 1 A as well as R + 1 units are chosen randomly from the remaining m surviving units of product 1 line B and they are removed from the experiment. Next, if the second failure is coming from the − − product line B at time point W , R + 1 units are withdrawn from the remaining m R 1 units 2 2 1 − − from the product line A and R units withdrawn from the remaining m R 2 units from the 2 1 product line B randomly at W . The test is continued until k failures are observed with removal 2 − of all the remaining surviving units from both the product lines at the k th failure. In this life testing experiment a new set of random variable Z ,. . . ,Z is introduced where Z = 1 or 0 if ith 1 k i failure comes from the product line A or B respectively. Under the BJPC, the data consists of (W, Z) where W = (W , . . . W ) and Z = (Z , . . . , Z ). A schematic diagram of the BJPC is 1 k 1 k provided in Figure 1 and Figure 2. A random variable X is said to follow Weibull distribution with the shape parameter α > 0 (cid:26) and the scale parameter λ > 0 if it has the following probability density function (PDF) − − αλxα 1e λxα if x > 0 f(x; α, λ) = ≤ (1) 0 if x 0, and it will be denoted by WE(α, λ). We assume the lifetimes of m units of product line A, say X , . . . , X , are independent identically distributed (i.i.d) random variables from WE(α, λ ) and 1 m 1 the lifetimes of m units of product line B, say Y , . . . , Y are i.i.d random variables from WE(α, λ ). 1 m 2 Point Estimations 3 3.1 Maximum Likelihood Estimators (MLEs) The likelihood function of the unknown parameters (α, λ , λ ) based on the observed data 1 2 (W, Z), is given by (cid:89) | k − − L(α, λ , λ w, z) = Cαkλ k1λ k2 w α 1e (λ1+λ2)A(α) (2) 1 2 1 2 i i=1 4

text: scheme,

text: two

text: sets

text: of

text: samples

text: from

text: these

text: two

text: products

text: are

text: simultaneously

text: put

text: on

text: a

text: test.

text: Suppose

text: the

text: ﬁrst

text: failure

text: is

text: coming

text: from

text: the

text: product

text: line

text: A

text: and

text: the

text: ﬁrst

text: failure

text: time

text: is

text: denoted

text: by

text: W

text: then

text: at

text: W

text: R

text: units

text: are

text: removed

text: randomly

text: from

text: the

text: remaining

text: m

text: surviving

text: units

text: of

text: the

text: product

text: line

text: A

text: as

text: well

text: as

text: R

text: units

text: are

text: chosen

text: randomly

text: from

text: the

text: remaining

text: m

text: surviving

text: units

text: of

text: product

text: line

text: B

text: and

text: they

text: are

text: removed

text: from

text: the

text: experiment.

text: Next,

text: if

text: the

text: second

text: failure

text: is

text: coming

text: from

text: the

text: product

text: line

text: B

text: at

text: time

text: point

text: W

text: R

text: units

text: are

text: withdrawn

text: from

text: the

text: remaining

text: m

text: R

text: units

text: from

text: the

text: product

text: line

text: A

text: and

text: R

text: units

text: withdrawn

text: from

text: the

text: remaining

text: m

text: R

text: units

text: from

text: the

text: product

text: line

text: B

text: randomly

text: at

text: W

text: The

text: test

text: is

text: continued

text: until

text: k

text: failures

text: are

text: observed

text: with

text: removal

text: of

text: all

text: the

text: remaining

text: surviving

text: units

text: from

text: both

text: the

text: product

text: lines

text: at

text: the

text: k

text: th

text: failure.

text: In

text: this

text: life

text: testing

text: experiment

text: a

text: new

text: set

text: of

text: random

text: variable

text: Z

text: ,Z

text: is

text: introduced

text: where

text: Z

text: or

text: if

text: ith

text: k

text: i

text: failure

text: comes

text: from

text: the

text: product

text: line

text: A

text: or

text: B

text: respectively.

text: Under

text: the

text: BJPC,

text: the

text: data

text: consists

text: of

text: (W,

text: Z)

text: where

text: W

text: (W

text: W

text: and

text: Z

text: (Z

text: Z

text: A

text: schematic

text: diagram

text: of

text: the

text: BJPC

text: is

text: k

text: k

text: provided

text: in

text: Figure

text: and

text: Figure

text: A

text: random

text: variable

text: X

text: is

text: said

text: to

text: follow

text: Weibull

text: distribution

text: with

text: the

text: shape

text: parameter

text: α

text: and

text: the

text: scale

text: parameter

text: λ

text: if

text: it

text: has

text: the

text: following

text: probability

text: density

text: function

text: (PDF)

text: αλxα

text: λxα

text: if

text: x

text: f(x;

text: α,

text: λ)

text: if

text: x

text: and

text: it

text: will

text: be

text: denoted

text: by

text: WE(α,

text: λ).

text: We

text: assume

text: the

text: lifetimes

text: of

text: m

text: units

text: of

text: product

text: line

text: A,

text: say

text: X

text: X

text: are

text: independent

text: identically

text: distributed

text: (i.i.d)

text: random

text: variables

text: from

text: WE(α,

text: λ

text: and

text: m

text: the

text: lifetimes

text: of

text: m

text: units

text: of

text: product

text: line

text: B,

text: say

text: Y

text: Y

text: are

text: i.i.d

text: random

text: variables

text: from

text: WE(α,

text: λ

text: m

text: Point

text: Estimations

text: Maximum

text: Likelihood

text: Estimators

text: (MLEs)

text: The

text: likelihood

text: function

text: of

text: the

text: unknown

text: parameters

text: (α,

text: λ

text: λ

text: based

text: on

text: the

text: observed

text: data

text: (W,

text: Z),

text: is

text: given

text: by

text: k

text: L(α,

text: λ

text: λ

text: w,

text: z)

text: Cαkλ

text: w

text: α

text: i


text: 14 (a) For = . (b) For − = 0 01. (c) For − = 0 05. p2 p1 p2 p1 . p2 p1 . Fig. 2: Location of the optimum points ( ∗ ∗). d1, d2 optimum points except the boundary points, hence: (1 − 2 ) 0 ( ∗ ) − 0 ( ) + ( − 1)(1 − 2 ) 0 ( ∗ ) = 0 for = 1 2 (21) p h d p µh d µ q h d p , i , . i b i i b i i b This non-linear system of equations dose not have any closed-form solution and it is generally solved by using numerical methods such as Newton’s method [25]. Next, the Hessian matrix = + is calculated in these roots to check whether a possible H H µH F D R point is exactly an optimum point or not. h i 2 ∂ F ∈ { } = ; 1 2 (22) H i, j , . F ∂d ∂d i j ∇ If is a positive deﬁnite matrix in a root of = [0 0], then it is a solution. Otherwise, H F R since there is not such a point, the solution is only located on the boundary points, i.e., at least one of the ∗s equals 0 or 0 5. Due to the non-convexity of the optimization problem d . i and the non-linearity of the rate and distortion expressions, we give an asymptotic analysis (cid:16) (cid:17) (cid:16) (cid:17) of the problem (14). In this regard, we consider a high resolution regime. Lemma 1: Assume K = (1 − 2p) log 1−p and K = (1 − 2p ) log 1−pi for i = 1, 2. For p i i pi → 0, x DRAFT May 30, 2020 ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine##

text: (a)

text: For

text: (b)

text: For

text: (c)

text: For

text: Fig.

text: Location

label: caption

text: of

text: the

text: optimum

text: points

text: optimum

text: points

text: except

text: the

text: boundary

text: points,

text: hence:

text: for

text: p

text: h

text: d

text: p

text: µh

text: d

text: µ

text: q

text: h

text: d

text: p

text: i

text: i

text: b

text: i

text: i

text: b

text: i

text: i

text: b

text: This

text: non-linear

text: system

text: of

text: equations

text: dose

text: not

text: have

text: any

text: closed-form

text: solution

text: and

text: it

text: is

text: generally

text: solved

text: by

text: using

text: numerical

text: methods

text: such

text: as

text: Newton’s

text: method

text: Next,

text: the

text: Hessian

text: matrix

text: is

text: calculated

text: in

text: these

text: roots

text: to

text: check

text: whether

text: a

text: possible

text: H

text: H

text: µH

text: F

text: D

text: R

text: point

text: is

text: exactly

text: an

text: optimum

text: point

text: or

text: not.

text: h

text: i

text: F

text: H

text: i,

text: j

text: F

text: ∂d

text: ∂d

text: i

text: j

text: If

text: is

text: a

text: positive

text: deﬁnite

text: matrix

text: in

text: a

text: root

text: of

text: then

text: it

text: is

text: a

text: solution.

text: Otherwise,

text: H

text: F

text: R

text: since

text: there

text: is

text: not

text: such

text: a

text: point,

text: the

text: solution

text: is

text: only

text: located

text: on

text: the

text: boundary

text: points,

text: i.e.,

text: at

text: least

text: one

text: of

text: the

text: ∗s

text: equals

text: or

text: Due

text: to

text: the

text: non-convexity

text: of

text: the

text: optimization

text: problem

text: d

text: i

text: and

text: the

text: non-linearity

text: of

text: the

text: rate

text: and

text: distortion

text: expressions,

text: we

text: give

text: an

text: asymptotic

text: analysis

text: of

text: the

text: problem

text: In

text: this

text: regard,

text: we

text: consider

text: a

text: high

text: resolution

text: regime.

text: Lemma

text: Assume

text: K

text: log

text: and

text: K

text: log

text: for

text: i

text: For

text: p

text: i

text: i

text: pi

text: x

text: DRAFT

text: May

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##