text: Astron. Nachr. / AN (2006) 795 agram. We can see the marginal position of our target object eclipse stops to occur for eccentricities higher than about in the CM diagram. This indicates that this star is possibly a 0.38. The best ﬁt, however, follows the data rather poorly ﬁeld star only, but according to the other outliers, which are due to the proximity effects predicted by the model but not visible in the left side of the diagram, it can be also a cluster visible in the light curve. The resulting parameters for the member. ﬁxed eccentricity of e = 0.4 are listed in Table 4. The sec- The conﬁrmation process of the planet candidate, which ond alternative is not very probable: the secondary compo- is currently planned, would require direct imaging, multi- nent would have to be very cold for the secondary minima passband photometry and high-dispersion spectroscopy (see to be absent, and at the same time large (Ω < Ω ). The 2 1 Subsection 5.1). third alternative is the most probable but requires more ob- 4.2 Eclipsing binary stars servations to cover the missing phases. J221227.5+501309 (P = 0.564323 days; Fig. A1 - mid- According to the General Catalogue of Variable Stars14 dle top right) and J221536+495726 (P = 0.508895 days; (Samus et al. 2012; hereafter GCVS) there are two Fig. A3 - top left) are two short-period systems and at the known Algol-type variables in our studied region: TZ Lac same time detached binary stars. This means that their com- (J221412.3+495618) and XX Lac (J221710.2+500111). ponents must be low-mass stars of late spectral type. In Apart from that, in the studied region there is one spite of that, no signiﬁcant light-curve asymmetry caused more Algol-type eclipsing binary presented in the Inter- by surface activity (usual in such systems) is visible in our national Variable Star Index15 catalogue (Watson, Hen- data. Their further analysis would, however, require multi- den & Price 2014; hereafter VSX): NSVS6038647 passband photometry and high-dispersion spectroscopy. (J221230.9+500233), which is also described in the North- 4.2.2 Contact binary stars ern Sky Variability Survey16. Our observations led to dis- covery of 37 new eclipsing binary stars in the studied re- For contact binary stars mass ratios close to unity are gion. The R passband magnitude range of the stars is from not observed (see e.g., Rucinski 1993). For part of the sys- 17.8 mag to 11.7 mag. The light-curve analysis was per- tems χ2 was found to signiﬁcantly depend on the mass ratio formed using the Roche code as described in Section . In the (mostly totally eclipsing). For systems, where χ2 was not following two Paragraphs 4.2.1 and 4.2.2 we describe the varying signiﬁcantly with the mass ratio, the solution listed light-curve analysis of selected binaries only. Light-curve in Table 5 corresponds to q = 0.50. solutions for all systems are shown in Appendix A and B. J221609.1+500925 (Fig. B2 - middle bottom left) shows 4.2.1 Detached eclipsing binaries an O’Connell effect. The maxima difference is about 0.025 mag in the R passband. The mass ratio was ﬁxed to the value As discussed in Section , for detached eclipsing bi- of 0.4 giving the lowest χ2. The resulting parameters are nary stars the mass ratio does not affect the light-curve listed in Table 5. Fixing the spot to the equator and assuming shape signiﬁcantly and only fractional radii of the compo- ◦ spot temperature factor of k = 0.7, its radius is r = 20 . nents can be determined (in favourable cases). When both spot In the case of 8 more objects (marked in Table 4 and 5) we minima are visible, the luminosities of the componets and also assumed one spot. The light curve of these objects is their masses must be similar. Hence, q = 0.90 was adopted less asymmetric than that of J221609.1+500925. for most systems. Only in the case of J221200.9+494442, J221332.2+493046 is a contact eclipsing binary with J221428.8+495509 and J221514.4+500059, χ2 was found 0.43 and 0.40 mag deep minima in the R passband. It shows to signiﬁcantly depend on the mass ratio. Resulting photo- total eclipses (Fig. B1 - bottom right). In this case the mass metric elements for all detached systems are given in Ta- ratio was ﬁxed to several values between 0.1 and 0.4. The ble 4. minimum of χ2 was found close to q = 0.2. In the next J221515.2+495017 is a close but detached eclipsing bi- step we modelled the light curve with mass ratio as a free nary with 0.33 mag deep minima in the R passband. For parameter. The resulting photometric elements are listed in the best orbital period, about P = 2.6597 days, the sec- orb Table 5. Total eclipses are observed in 5 more cases (marked ondary minimum is not visible (Fig. A2 - bottom left). We in Table 4 and 5). can explain this fact in several ways. (1) The orbit is eccen- For J221249.5+493616 the χ2 was found to decrease tric and the secondary eclipses do not occur. (2) The orbit is with increasing ﬁxed mass ratio. The modelling showed that circular and the secondary component has a very low tem- the most probable is the W-type conﬁguration for the sys- perature, and therefore secondary eclipses are too shallow. tem with mass ratio close to 2. Thus the deeper minimum is (3) The true orbital period is two times larger and the min- the eclipse of the hotter but less massive secondary compo- ima have the same depth. Unfortunately, the phase interval − nent. Similar possibility is probable for more systems (es- < 0.4 0.6 > for the double period was not covered with pecially for short-period contact binaries) but the quality of observations. The R passband light curve was modelled as- data is insufﬁcient. Conclusive analysis would require spec- suming the ﬁrst alternative. It was found that the secondary troscopic follow-up (see e.g., the DDO radial-velocity pro- 14 http://www.sai.msu.su/gcvs/gcvs/ gram, Pribulla et al. 2008). 15 http://www.aavso.org/vsx/index.php 4.3 Pulsating variable stars 16 http://skydot.lanl.gov/nsvs/nsvs.php www.an-journal.org (cid:13)c 2006WILEY-VCHVerlagGmbH&Co.KGaA,Weinheim ##LTLine## ##LTLine##

text: Astron.

label: paragraph

text: Nachr.

text: AN

text: agram.

text: We

text: can

text: see

text: the

text: marginal

text: position

text: of

text: our

text: target

text: object

text: eclipse

text: stops

text: to

text: occur

text: for

text: eccentricities

text: higher

text: than

text: about

text: in

text: the

text: CM

text: diagram.

text: This

text: indicates

text: that

text: this

text: star

text: is

text: possibly

text: a

text: The

text: best

text: ﬁt,

text: however,

text: follows

text: the

text: data

text: rather

text: poorly

text: ﬁeld

text: star

text: only,

text: but

text: according

text: to

text: the

text: other

text: outliers,

text: which

text: are

text: due

text: to

text: the

text: proximity

text: effects

text: predicted

text: by

text: the

text: model

text: but

text: not

text: visible

text: in

text: the

text: left

text: side

text: of

text: the

text: diagram,

text: it

text: can

text: be

text: also

text: a

text: cluster

text: visible

text: in

text: the

text: light

text: curve.

text: The

text: resulting

text: parameters

text: for

text: the

text: member.

text: ﬁxed

text: eccentricity

text: of

text: e

text: are

text: listed

text: in

text: Table

text: The

text: sec-

text: The

text: conﬁrmation

text: process

text: of

text: the

text: planet

text: candidate,

text: which

text: ond

text: alternative

text: is

text: not

text: very

text: probable:

text: the

text: secondary

text: compo-

text: is

text: currently

text: planned,

text: would

text: require

text: direct

text: imaging,

text: multi-

text: nent

text: would

text: have

text: to

text: be

text: very

text: cold

text: for

text: the

text: secondary

text: minima

text: passband

text: photometry

text: and

text: high-dispersion

text: spectroscopy

text: (see

text: to

text: be

text: absent,

text: and

text: at

text: the

text: same

text: time

text: large

text: (Ω

text: Ω

text: The

text: Subsection

text: third

text: alternative

text: is

text: the

text: most

text: probable

text: but

text: requires

text: more

text: ob-

text: Eclipsing

text: binary

text: stars

text: servations

text: to

text: cover

text: the

text: missing

text: phases.

text: (P

text: days;

text: Fig.

text: mid-

text: According

text: to

text: the

text: General

text: Catalogue

text: of

text: Variable

text: dle

text: top

text: right)

text: and

text: (P

text: days;

text: (Samus

text: et

text: al.

text: hereafter

text: GCVS)

text: there

text: are

text: two

text: Fig.

text: top

text: left)

text: are

text: two

text: short-period

text: systems

text: and

text: at

text: the

text: known

text: Algol-type

text: variables

text: in

text: our

text: studied

text: region:

text: TZ

text: Lac

text: same

text: time

text: detached

text: binary

text: stars.

text: This

text: means

text: that

text: their

text: com-

text: and

text: XX

text: Lac

text: ponents

text: must

text: be

text: low-mass

text: stars

text: of

text: late

text: spectral

text: type.

text: In

text: Apart

text: from

text: that,

text: in

text: the

text: studied

text: region

text: there

text: is

text: one

text: spite

text: of

text: that,

text: no

text: signiﬁcant

text: light-curve

text: asymmetry

text: caused

text: more

text: Algol-type

text: eclipsing

text: binary

text: presented

text: in

text: the

text: Inter-

text: by

text: surface

text: activity

text: (usual

text: in

text: such

text: systems)

text: is

text: visible

text: in

text: our

text: national

text: Variable

text: Star

text: catalogue

text: (Watson,

text: Hen-

text: data.

text: Their

text: further

text: analysis

text: would,

text: however,

text: require

text: multi-

text: den

text: Price

text: hereafter

text: VSX):

text: passband

text: photometry

text: and

text: high-dispersion

text: spectroscopy.

text: which

text: is

text: also

text: described

text: in

text: the

text: North-

text: Contact

text: binary

text: stars

text: ern

text: Sky

text: Variability

text: Our

text: observations

text: led

text: to

text: dis-

text: covery

text: of

text: new

text: eclipsing

text: binary

text: stars

text: in

text: the

text: studied

text: re-

text: For

text: contact

text: binary

text: stars

text: mass

text: ratios

text: close

text: to

text: unity

text: are

text: gion.

text: The

text: R

text: passband

text: magnitude

text: range

text: of

text: the

text: stars

text: is

text: from

text: not

text: observed

text: (see

text: e.g.,

text: Rucinski

text: For

text: part

text: of

text: the

text: sys-

text: mag

text: to

text: mag.

text: The

text: light-curve

text: analysis

text: was

text: per-

text: tems

text: χ2

text: was

text: found

text: to

text: signiﬁcantly

text: depend

text: on

text: the

text: mass

text: ratio

text: formed

text: using

text: the

text: Roche

text: code

text: as

text: described

text: in

text: Section

text: In

text: the

text: (mostly

text: totally

text: eclipsing).

text: For

text: systems,

text: where

text: χ2

text: was

text: not

text: following

text: two

text: Paragraphs

text: and

text: we

text: describe

text: the

text: varying

text: signiﬁcantly

text: with

text: the

text: mass

text: ratio,

text: the

text: solution

text: listed

text: light-curve

text: analysis

text: of

text: selected

text: binaries

text: only.

text: Light-curve

text: in

text: Table

text: corresponds

text: to

text: q

text: solutions

text: for

text: all

text: systems

text: are

text: shown

text: in

text: Appendix

text: A

text: and

text: B.

text: (Fig.

text: middle

text: bottom

text: left)

text: shows

text: Detached

text: eclipsing

text: binaries

text: an

text: O’Connell

text: effect.

text: The

text: maxima

text: difference

text: is

text: about

text: mag

text: in

text: the

text: R

text: passband.

text: The

text: mass

text: ratio

text: was

text: ﬁxed

text: to

text: the

text: value

text: As

text: discussed

text: in

text: Section

text: for

text: detached

text: eclipsing

text: bi-

text: of

text: giving

text: the

text: lowest

text: The

text: resulting

text: parameters

text: are

text: nary

text: stars

text: the

text: mass

text: ratio

text: does

text: not

text: affect

text: the

text: light-curve

text: listed

text: in

text: Table

text: Fixing

text: the

text: spot

text: to

text: the

text: equator

text: and

text: assuming

text: shape

text: signiﬁcantly

text: and

text: only

text: fractional

text: radii

text: of

text: the

text: compo-

text: spot

text: temperature

text: factor

text: of

text: k

text: its

text: radius

text: is

text: r

text: nents

text: can

text: be

text: determined

text: (in

text: favourable

text: cases).

text: When

text: both

text: spot

text: In

text: the

text: case

text: of

text: more

text: objects

text: (marked

text: in

text: Table

text: and

text: we

text: minima

text: are

text: visible,

text: the

text: luminosities

text: of

text: the

text: componets

text: and

text: also

text: assumed

text: one

text: spot.

text: The

text: light

text: curve

text: of

text: these

text: objects

text: is

text: their

text: masses

text: must

text: be

text: similar.

text: Hence,

text: q

text: was

text: adopted

text: less

text: asymmetric

text: than

text: that

text: of

text: for

text: most

text: systems.

text: Only

text: in

text: the

text: case

text: of

text: is

text: a

text: contact

text: eclipsing

text: binary

text: with

text: and

text: χ2

text: was

text: found

text: and

text: mag

text: deep

text: minima

text: in

text: the

text: R

text: passband.

text: It

text: shows

text: to

text: signiﬁcantly

text: depend

text: on

text: the

text: mass

text: ratio.

text: Resulting

text: photo-

text: total

text: eclipses

text: (Fig.

text: bottom

text: right).

text: In

text: this

text: case

text: the

text: mass

text: metric

text: elements

text: for

text: all

text: detached

text: systems

text: are

text: given

text: in

text: Ta-

text: ratio

text: was

text: ﬁxed

text: to

text: several

text: values

text: between

text: and

text: The

text: ble

text: minimum

text: of

text: χ2

text: was

text: found

text: close

text: to

text: q

text: In

text: the

text: next

text: is

text: a

text: close

text: but

text: detached

text: eclipsing

text: bi-

text: step

text: we

text: modelled

text: the

text: light

text: curve

text: with

text: mass

text: ratio

text: as

text: a

text: free

text: nary

text: with

text: mag

text: deep

text: minima

text: in

text: the

text: R

text: passband.

text: For

text: parameter.

text: The

text: resulting

text: photometric

text: elements

text: are

text: listed

text: in

text: the

text: best

text: orbital

text: period,

text: about

text: P

text: days,

text: the

text: sec-

text: orb

text: Table

text: Total

text: eclipses

text: are

text: observed

text: in

text: more

text: cases

text: (marked

text: ondary

text: minimum

text: is

text: not

text: visible

text: (Fig.

text: bottom

text: left).

text: We

text: in

text: Table

text: and

text: can

text: explain

text: this

text: fact

text: in

text: several

text: ways.

text: The

text: orbit

text: is

text: eccen-

text: For

text: the

text: χ2

text: was

text: found

text: to

text: decrease

text: tric

text: and

text: the

text: secondary

text: eclipses

text: do

text: not

text: occur.

text: The

text: orbit

text: is

text: with

text: increasing

text: ﬁxed

text: mass

text: ratio.

text: The

text: modelling

text: showed

text: that

text: circular

text: and

text: the

text: secondary

text: component

text: has

text: a

text: very

text: low

text: tem-

text: the

text: most

text: probable

text: is

text: the

text: W-type

text: conﬁguration

text: for

text: the

text: sys-

text: perature,

text: and

text: therefore

text: secondary

text: eclipses

text: are

text: too

text: shallow.

text: tem

text: with

text: mass

text: ratio

text: close

text: to

text: Thus

text: the

text: deeper

text: minimum

text: is

text: The

text: true

text: orbital

text: period

text: is

text: two

text: times

text: larger

text: and

text: the

text: min-

text: the

text: eclipse

text: of

text: the

text: hotter

text: but

text: less

text: massive

text: secondary

text: compo-

text: ima

text: have

text: the

text: same

text: depth.

text: Unfortunately,

text: the

text: phase

text: interval

text: nent.

text: Similar

text: possibility

text: is

text: probable

text: for

text: more

text: systems

text: (es-

text: for

text: the

text: double

text: period

text: was

text: not

text: covered

text: with

text: pecially

text: for

text: short-period

text: contact

text: binaries)

text: but

text: the

text: quality

text: of

text: observations.

text: The

text: R

text: passband

text: light

text: curve

text: was

text: modelled

text: as-

text: data

text: is

text: insufﬁcient.

text: Conclusive

text: analysis

text: would

text: require

text: spec-

text: suming

text: the

text: ﬁrst

text: alternative.

text: It

text: was

text: found

text: that

text: the

text: secondary

text: troscopic

text: follow-up

text: (see

text: e.g.,

text: the

text: DDO

text: radial-velocity

text: pro-

text: gram,

text: Pribulla

text: et

text: al.

text: Pulsating

text: variable

text: stars

text: ##LTLine##

text: ##LTLine##


text: ON SUPPORT VARIETIES FOR LIE SUPERALGEBRAS AND FINITE SUPERGROUP SCHEMES CHRISTOPHER M. DRUPIESKI AND JONATHAN R. KUJAWA Abstract. We study the spectrum of the cohomology rings of cocommutative Hopf superalgebras, restricted and non-restricted Lie superalgebras, and ﬁnite supergroup schemes. We also investigate support varieties in these settings and demonstrate that they have the desirable properties of such a theory. We completely characterize support varieties for ﬁnite supergroup schemes over algebraically closed ﬁelds of characteristic zero, while for non-restricted Lie superalgebras we obtain results in positive characteristic that are strikingly similar to results of Duﬂo and Serganova in characteristic zero. Our computations for restricted Lie superalgebras and inﬁnitesimal supergroup schemes provide natural generalizations of foundational results of Friedlander and Parshall and of Bendel, Friedlander, and Suslin in the classical setting. 1. Introduction For more than three decades geometric techniques have played a fundamental role in the study of non-semisimple representation theory. Inspired by their use in the study of ﬁnite groups [17,45], Friedlander and Parshall [26, 27] introduced support varieties for restricted Lie algebras. Their results were later generalized by Bendel, Friedlander, and Suslin [46, 47] to inﬁnitesimal group schemes and then by Friedlander and Pevtsova [28] to arbitrary ﬁnite group schemes. In a diﬀerent direction, Ginzburg and Kumar [30] calculated the cohomological spectrum for Lusztig’s small quantum group. Since then support varieties have been studied for many other interesting classes of ﬁnite-dimensional algebras; cf. [1,8,11,21,24,40,41] and the references therein. In each context support varieties have provided important new insights. For example, they play an important role in the study of representation type of self-injective algebras [22, 24], in the classiﬁcation of thick tensor ideals in triangulated categories [9,10], in Premet’s proof of the Kac–Weisfeiler conjecture on the p-divisibility of the dimensions of modules for Lie algebras [43], and in the derived equivalences of Arkhipov, Bezrukavnikov, and Ginzburg [2] that relate representations of quantum groups at a root of unity to the geometry of the nilpotent cone. In contrast to ordinary Lie algebras, the category of ﬁnite-dimensional modules of a simple Lie superalgebra over the complex numbers is almost always a non-semisimple category. Supergroups, Z Z Lie superalgebras, and related -graded structures (including -graded Hopf algebras as deﬁned by 2 Milnor and Moore [39]) thus provide another natural setting for geometric methods. With Boe and Nakano, the second author initiated a study of support varieties for complex Lie superalgebras and showed that they capture information about the representation theory of these algebras, including atypicality, complexity, and the thick tensor ideals of the category [12–15]. In independent work, Duﬂo and Serganova [20] also deﬁned associated varieties for Lie superalgebras in characteristic zero and showed they too provide representation theoretic information. 2010 Mathematics Subject Classiﬁcation. Primary 17B56. Secondary 20G10. The ﬁrst author was supported in part by a Faculty Summer Research Grant from the DePaul University College of Science and Health. The second author was supported in part by NSF grant DMS-1160763. 1 ##LTLine##

text: ON

text: SUPPORT

text: VARIETIES

text: FOR

text: LIE

text: SUPERALGEBRAS

text: AND

text: FINITE

text: SUPERGROUP

text: SCHEMES

text: CHRISTOPHER

text: M.

text: DRUPIESKI

text: AND

text: JONATHAN

text: R.

text: KUJAWA

text: Abstract.

text: We

label: abstract

text: study

text: the

text: spectrum

text: of

text: the

text: cohomology

text: rings

text: of

text: cocommutative

text: Hopf

text: superalgebras,

text: restricted

text: and

text: non-restricted

text: Lie

text: superalgebras,

text: and

text: ﬁnite

text: supergroup

text: schemes.

text: We

text: also

text: investigate

text: support

text: varieties

text: in

text: these

text: settings

text: and

text: demonstrate

text: that

text: they

text: have

text: the

text: desirable

text: properties

text: of

text: such

text: a

text: theory.

text: We

text: completely

text: characterize

text: support

text: varieties

text: for

text: ﬁnite

text: supergroup

text: schemes

text: over

text: algebraically

text: closed

text: ﬁelds

text: of

text: characteristic

text: zero,

text: while

text: for

text: non-restricted

text: Lie

text: superalgebras

text: we

text: obtain

text: results

text: in

text: positive

text: characteristic

text: that

text: are

text: strikingly

text: similar

text: to

text: results

text: of

text: Duﬂo

text: and

text: Serganova

text: in

text: characteristic

text: zero.

text: Our

text: computations

text: for

text: restricted

text: Lie

text: superalgebras

text: and

text: inﬁnitesimal

text: supergroup

text: schemes

text: provide

text: natural

text: generalizations

text: of

text: foundational

text: results

text: of

text: Friedlander

text: and

text: Parshall

text: and

text: of

text: Bendel,

text: Friedlander,

text: and

text: Suslin

text: in

text: the

text: classical

text: setting.

text: Introduction

text: For

text: more

text: than

text: three

text: decades

text: geometric

text: techniques

text: have

text: played

text: a

text: fundamental

text: role

text: in

text: the

text: study

text: of

text: non-semisimple

text: representation

text: theory.

text: Inspired

text: by

text: their

text: use

text: in

text: the

text: study

text: of

text: ﬁnite

text: groups

text: Friedlander

text: and

text: Parshall

text: introduced

text: support

text: varieties

text: for

text: restricted

text: Lie

text: algebras.

text: Their

text: results

text: were

text: later

text: generalized

text: by

text: Bendel,

text: Friedlander,

text: and

text: Suslin

text: to

text: inﬁnitesimal

text: group

text: schemes

text: and

text: then

text: by

text: Friedlander

text: and

text: Pevtsova

text: to

text: arbitrary

text: ﬁnite

text: group

text: schemes.

text: In

text: a

text: diﬀerent

text: direction,

text: Ginzburg

text: and

text: Kumar

text: calculated

text: the

text: cohomological

text: spectrum

text: for

text: Lusztig’s

text: small

text: quantum

text: group.

text: Since

text: then

text: support

text: varieties

text: have

text: been

text: studied

text: for

text: many

text: other

text: interesting

text: classes

text: of

text: ﬁnite-dimensional

text: algebras;

text: cf.

text: and

text: the

text: references

text: therein.

text: In

text: each

text: context

text: support

text: varieties

text: have

text: provided

text: important

text: new

text: insights.

text: For

text: example,

text: they

text: play

text: an

text: important

text: role

text: in

text: the

text: study

text: of

text: representation

text: type

text: of

text: self-injective

text: algebras

text: in

text: the

text: classiﬁcation

text: of

text: thick

text: tensor

text: ideals

text: in

text: triangulated

text: categories

text: in

text: Premet’s

text: proof

text: of

text: the

text: Kac–Weisfeiler

text: conjecture

text: on

text: the

text: p-divisibility

text: of

text: the

text: dimensions

text: of

text: modules

text: for

text: Lie

text: algebras

text: and

text: in

text: the

text: derived

text: equivalences

text: of

text: Arkhipov,

text: Bezrukavnikov,

text: and

text: Ginzburg

text: that

text: relate

text: representations

text: of

text: quantum

text: groups

text: at

text: a

text: root

text: of

text: unity

text: to

text: the

text: geometry

text: of

text: the

text: nilpotent

text: cone.

text: In

text: contrast

text: to

text: ordinary

text: Lie

text: algebras,

text: the

text: category

text: of

text: ﬁnite-dimensional

text: modules

text: of

text: a

text: simple

text: Lie

text: superalgebra

text: over

text: the

text: complex

text: numbers

text: is

text: almost

text: always

text: a

text: non-semisimple

text: category.

text: Supergroups,

text: Z

text: Z

text: Lie

text: superalgebras,

text: and

text: related

text: -graded

text: structures

text: (including

text: -graded

text: Hopf

text: algebras

text: as

text: deﬁned

text: by

text: Milnor

text: and

text: Moore

text: thus

text: provide

text: another

text: natural

text: setting

text: for

text: geometric

text: methods.

text: With

text: Boe

text: and

text: Nakano,

text: the

text: second

text: author

text: initiated

text: a

text: study

text: of

text: support

text: varieties

text: for

text: complex

text: Lie

text: superalgebras

text: and

text: showed

text: that

text: they

text: capture

text: information

text: about

text: the

text: representation

text: theory

text: of

text: these

text: algebras,

text: including

text: atypicality,

text: complexity,

text: and

text: the

text: thick

text: tensor

text: ideals

text: of

text: the

text: category

text: In

text: independent

text: work,

text: Duﬂo

text: and

text: Serganova

text: also

text: deﬁned

text: associated

text: varieties

text: for

text: Lie

text: superalgebras

text: in

text: characteristic

text: zero

text: and

text: showed

text: they

text: too

text: provide

text: representation

text: theoretic

text: information.

text: Mathematics

text: Subject

text: Classiﬁcation.

text: Primary

text: Secondary

text: The

text: ﬁrst

text: author

text: was

text: supported

text: in

text: part

text: by

text: a

text: Faculty

text: Summer

text: Research

text: Grant

text: from

text: the

text: DePaul

text: University

text: College

text: of

text: Science

text: and

text: Health.

text: The

text: second

text: author

text: was

text: supported

text: in

text: part

text: by

text: NSF

text: grant

text: ##LTLine##


text: ON SUPPORT VARIETIES FOR LIE SUPERALGEBRAS AND FINITE SUPERGROUP SCHEMES 15 The group algebra kG is semisimple by the assumption that k is a ﬁeld of characteristic 0, so Ei,j = 0 for all i > 0. Then the spectral sequence collapses to the row i = 0. This implies that 2 restriction to Λ(V ) induces isomorphisms ∼ ∼ ∼ • • ⊗ ∗ • ⊗ ∗ • Ext (M, N) = H (A, N M ) = H (Λ(V ), N M )G = Ext (M, N)G, A Λ(V ) where the ﬁrst and last isomorphisms in this string are consequences of the tensor-hom adjunction in Lemma 2.3.4. In particular, restriction to Λ(V ) induces an algebra isomorphism ∼ • • H (Λ(V )#kG, k) = H (Λ(V ), k)G. • Finally, it is a classical result [44, 2.2(2)] that H (Λ(V ), k) is isomorphic to the symmetric algebra • ∗ ∗ (cid:3) S (V ) with V concentrated in cohomological degree 1. Theorem 3.2.2. Let G be a ﬁnite group and let V be a ﬁnite-dimensional purely odd kG-module. Then there exist isomorphisms of varieties ∼ ∼ | (cid:111) | ∗ V G = Max(S(V )G) = V/G (cid:3) Proof. This is an immediate application of [6, Proposition 5.4.8]. ∈ Given a kG-module V , write [v] for the G-orbit of an element v V . Theorem 3.2.3. Let G be a ﬁnite group(cid:8) and let V be a ﬁnite-dimens(cid:9)ional purely odd kG-module. (cid:111) Let M be a ﬁnite-dimensional V G-supermodule. Then ∼ | (cid:111) | ∈ | V G = [v] V/G : M (cid:104) (cid:105) is not free M v • → • (cid:8) (cid:9) Proof. Set E = Λ(V ). Let Φ : H (E, k) Ext (M, M) be the natural ring homomorphism, | | • E • and set I = ker(Φ). Then E = Max(H (E, k)/I). Identifying H (E, k) with the symmetric ∗ M § | | ∼ ∈ | algebra S(V ), it follows from the results in [1, 3] that E = v V : M (cid:104) (cid:105) is not free . More (cid:48) M v• • precisely, let I be the annihilator ideal for the left Y(cid:8)oneda product action of H(cid:9) (E, k) = Ext (k, k) • E on Ext (M, k). Then [1, Theorem 3.9] asserts that E ∼ | | • (cid:48) ∈ | E = Max(H (E, k)/I ) = v V : M (cid:104) (cid:105) is not free . (M,k) v Up to parity shift and isomorphism, the trivial module k is the unique irreducible E-supermodule, | | | | so Propositions 2.3.9 and 2.3.11 imply that E = E . (M,k) M Next, the map Φ is a G-module homomorphism. Since G is a ﬁnite group and since k is a ﬁeld − of characteristic zero, the ﬁxed point functor ( )G is exact. Then one obtains an exact sequence 0 −→ IG −→ H•(E, k)G −Φ→G im(Φ )G −→ 0, M • → • where•ΦG : H (E, k∼)G • ExtE(M(cid:0), M)G is the map in(cid:1)duced by(cid:0)Φ. In particul(cid:1)ar, IG = ker(ΦG), and H (E, k)G/IG = (H (E, k)/I)G as algebras. Now Theorem 3.2.1 implies that ∼ ∼ | (cid:111) | • • V G = Max H(cid:8)(E, k)G/ ker(ΦG) = Max (H(cid:9) (E, k)/I)G . M | (cid:111) | | | Then by [6, Proposition 5.4.8], the support variety V G identiﬁes with the quotient of E | (cid:111) | ∼ ∈ | M M(cid:3) by the action of G, i.e., V G = [v] V/G : M (cid:104) (cid:105) is not free . M v

text: ON

text: SUPPORT

text: VARIETIES

text: FOR

text: LIE

text: SUPERALGEBRAS

text: AND

text: FINITE

text: SUPERGROUP

text: SCHEMES

text: The

text: group

text: algebra

text: kG

text: is

text: semisimple

text: by

text: the

text: assumption

text: that

text: k

text: is

text: a

text: ﬁeld

text: of

text: characteristic

text: so

text: Ei,j

text: for

text: all

text: i

text: Then

text: the

text: spectral

text: sequence

text: collapses

text: to

text: the

text: row

text: i

text: This

text: implies

text: that

text: restriction

text: to

text: Λ(V

text: induces

text: isomorphisms

text: Ext

text: (M,

text: N)

text: H

text: (A,

text: N

text: M

text: H

text: (Λ(V

text: N

text: M

text: )G

text: Ext

text: (M,

text: N)G,

text: A

text: Λ(V

text: where

text: the

text: ﬁrst

text: and

text: last

text: isomorphisms

text: in

text: this

text: string

text: are

text: consequences

text: of

text: the

text: tensor-hom

text: adjunction

text: in

text: Lemma

text: In

text: particular,

text: restriction

text: to

text: Λ(V

text: induces

text: an

text: algebra

text: isomorphism

text: H

text: (Λ(V

text: )#kG,

text: k)

text: H

text: (Λ(V

text: k)G.

text: Finally,

text: it

text: is

text: a

text: classical

text: result

text: that

text: H

text: (Λ(V

text: k)

text: is

text: isomorphic

text: to

text: the

text: symmetric

text: algebra

text: S

text: (V

text: with

text: V

text: concentrated

text: in

text: cohomological

text: degree

text: Theorem

text: Let

text: G

text: be

text: a

text: ﬁnite

text: group

text: and

text: let

text: V

text: be

text: a

text: ﬁnite-dimensional

text: purely

text: odd

text: kG-module.

text: Then

text: there

text: exist

text: isomorphisms

text: of

text: varieties

text: V

text: G

text: Max(S(V

text: )G)

text: V/G

text: Proof.

text: This

text: is

text: an

text: immediate

text: application

text: of

text: Proposition

text: Given

text: a

text: kG-module

text: V

text: write

text: [v]

text: for

text: the

text: G-orbit

text: of

text: an

text: element

text: v

text: V

text: Theorem

text: Let

text: G

text: be

text: a

text: ﬁnite

text: and

text: let

text: V

text: be

text: a

text: purely

text: odd

text: kG-module.

text: Let

text: M

text: be

text: a

text: ﬁnite-dimensional

text: V

text: G-supermodule.

text: Then

text: V

text: G

text: [v]

text: V/G

text: M

text: is

text: not

text: free

text: M

text: v

text: Proof.

text: Set

text: E

text: Λ(V

text: Let

text: Φ

text: H

text: (E,

text: k)

text: Ext

text: (M,

text: M)

text: be

text: the

text: natural

text: ring

text: homomorphism,

text: E

text: and

text: set

text: I

text: ker(Φ).

text: Then

text: E

text: Max(H

text: (E,

text: k)/I).

text: Identifying

text: H

text: (E,

text: k)

text: with

text: the

text: symmetric

text: M

text: algebra

text: S(V

text: it

text: follows

text: from

text: the

text: results

text: in

text: that

text: E

text: v

text: V

text: M

text: is

text: not

text: free

text: More

text: M

text: v•

text: precisely,

text: let

text: I

text: be

text: the

text: annihilator

text: ideal

text: for

text: the

text: left

text: product

text: action

text: of

text: (E,

text: k)

text: Ext

text: (k,

text: k)

text: E

text: on

text: Ext

text: (M,

text: k).

text: Then

text: Theorem

text: asserts

text: that

text: E

text: E

text: Max(H

text: (E,

text: k)/I

text: v

text: V

text: M

text: is

text: not

text: free

text: (M,k)

text: v

text: Up

text: to

text: parity

text: shift

text: and

text: isomorphism,

text: the

text: trivial

text: module

text: k

text: is

text: the

text: unique

text: irreducible

text: E-supermodule,

text: so

text: Propositions

text: and

text: imply

text: that

text: E

text: E

text: (M,k)

text: M

text: Next,

text: the

text: map

text: Φ

text: is

text: a

text: G-module

text: homomorphism.

text: Since

text: G

text: is

text: a

text: ﬁnite

text: group

text: and

text: since

text: k

text: is

text: a

text: ﬁeld

text: of

text: characteristic

text: zero,

text: the

text: ﬁxed

text: point

text: functor

text: )G

text: is

text: exact.

text: Then

text: one

text: obtains

text: an

text: exact

text: sequence

text: IG

text: H•(E,

text: k)G

text: −Φ→G

text: im(Φ

text: )G

text: M

text: where•ΦG

text: H

text: (E,

text: k∼)G

text: M)G

text: is

text: the

text: map

text: In

text: IG

text: ker(ΦG),

text: and

text: H

text: (E,

text: k)G/IG

text: (H

text: (E,

text: k)/I)G

text: as

text: algebras.

text: Now

text: Theorem

text: implies

text: that

text: V

text: G

text: Max

text: k)G/

text: ker(ΦG)

text: Max

text: (E,

text: k)/I)G

text: M

text: Then

text: by

text: Proposition

text: the

text: support

text: variety

text: V

text: G

text: identiﬁes

text: with

text: the

text: quotient

text: of

text: E

text: M

text: by

text: the

text: action

text: of

text: G,

text: i.e.,

text: V

text: G

text: [v]

text: V/G

text: M

text: is

text: not

text: free

text: M

text: v


text: 16 CHRISTOPHER M. DRUPIESKI AND JONATHAN R. KUJAWA As a corollary of the “rank variety” description in Theorem 3.2.3 we get the following tensor product property: Corollary 3.2.4. Let G be a ﬁnite group and let V be a ﬁnite-dimensional purely odd kG-module. (cid:111) Let M and N be ﬁnite-dimensional V G-supermodules. Then | (cid:111) | | (cid:111) | ∩ | (cid:111) | V G ⊗ = V G V G . M N M N Proof. We use the description of support varieties provided by Theorem 3.2.3. Since Λ(V )#kG is a cocommutative Hopf superalgebra, one has | (cid:111) | ⊆ | (cid:111) | ∩ | (cid:111) | V G ⊗ V G V G M N M N ∈ | (cid:111) | ∩ | (cid:111) | by Proposition 2.3.9(1). Now let [v] V G V G . Viewing V as an abelian purely (cid:104) (cid:105) M N odd Lie superalgebra, the subalgebra v of Λ(V )#kG generated by v is isomorphic to a one- variable exterior algebra, Λ(v). Up to isomorphism and parity change, the only indecomposable Λ(v)-supermodules are the trivial module and its projective cover (see, e.g., [13, Proposition 5.2.1]). (cid:104) (cid:105) Consequently, a v -supermodule is not free if and only if it contains the trivial module as a direct (cid:104) (cid:105) summand. In particular, when written as a direct sum of indecomposable v -supermodules, both ⊗ ⊗ M and N must have a trivial direct summand and hence so must M N. Thus, M N is not free (cid:104) (cid:105) ∈ | (cid:111) | (cid:3) as a v -supermodule and [v] V G ⊗ . M N Theorem 3.2.5. Let G be a ﬁnite group and let V be a ﬁnite-dimensional purely odd kG-module. (cid:111) | (cid:111) | − | (cid:111) | Let M be a ﬁnite-dimensional V G-supermodule and let d = dim V G dim V G be the | (cid:111) | | (cid:111) | | M codimension of V G in V G . Then 2d dim M. If d > 0, then sdim(M) = 0. M k | (cid:111) | Proof. Set E = Λ(V ). By the proof of Theorem 3.2.3, V G identiﬁes with the quotient of | | M | (cid:111) | | (cid:111) | E by the action of G. Since G is a ﬁnite group, then the codimension of V G in V G M | | | | | | M is equal to the codimension of E in E . Next, E is deﬁned by an ideal in the polynomial • ∼ ∗ M M ring H (E, k) = S(V ), so it follows from the Noether Normalization Theorem [36, Theorem II.3.1] ∩ | | { } that there exists a d-dimensional subspace H of V such that H E = 0 . Considering the M | | { } restriction of M to the subalgebra Λ(H) of E generated by H, this implies that Λ(H) = 0 , M and hence implies by Proposition 2.3.13 that M is projective as a Λ(H)-supermodule. Since Λ(H) is indecomposable over itself, M is then isomorphic as a Λ(H)-supermodule to a direct sum of (cid:3) copies of Λ(H), each of which is of dimension 2d and (if d > 0) of superdimension 0. (cid:111) (cid:111) K Let V G-mod denote the category of all ﬁnite-dimensional V G-modules and let denote the (cid:111) K stable module category of V G-mod. Using Lemma 2.3.2, it follows that is a tensor triangulated K category. We expect that the spectrum of (in the sense of Balmer [4]) is homeomorphic to V/G K and that the thick tensor ideals of are classiﬁed by the specialization closed subsets of V/G via support varieties; see the results of Pevtsova and Witherspoon [42] in a setting with close similarities to the one considered here.

text: CHRISTOPHER

text: M.

text: DRUPIESKI

text: AND

text: JONATHAN

text: R.

text: KUJAWA

text: As

text: a

text: corollary

text: of

text: the

text: “rank

text: variety”

text: description

text: in

text: Theorem

text: we

text: get

text: the

text: following

text: tensor

text: product

text: property:

text: Corollary

text: Let

text: G

text: be

text: a

text: ﬁnite

text: group

text: and

text: let

text: V

text: be

text: a

text: ﬁnite-dimensional

text: purely

text: odd

text: kG-module.

text: Let

text: M

text: and

text: N

text: be

text: ﬁnite-dimensional

text: V

text: G-supermodules.

text: Then

text: V

text: G

text: V

text: G

text: V

text: G

text: M

text: N

text: M

text: N

text: Proof.

text: We

text: use

text: the

text: description

text: of

text: support

text: varieties

text: provided

text: by

text: Theorem

text: Since

text: Λ(V

text: )#kG

text: is

text: a

text: cocommutative

text: Hopf

text: superalgebra,

text: one

text: has

text: V

text: G

text: V

text: G

text: V

text: G

text: M

text: N

text: M

text: N

text: by

text: Proposition

text: Now

text: let

text: [v]

text: V

text: G

text: V

text: G

text: Viewing

text: V

text: as

text: an

text: abelian

text: purely

text: M

text: N

text: odd

text: Lie

text: superalgebra,

text: the

text: subalgebra

text: v

text: of

text: Λ(V

text: )#kG

text: generated

text: by

text: v

text: is

text: isomorphic

text: to

text: a

text: one-

text: variable

text: exterior

text: algebra,

text: Λ(v).

text: Up

text: to

text: isomorphism

text: and

text: parity

text: change,

text: the

text: only

text: indecomposable

text: Λ(v)-supermodules

text: are

text: the

text: trivial

text: module

text: and

text: its

text: projective

text: cover

text: (see,

text: e.g.,

text: Proposition

text: Consequently,

text: a

text: v

text: -supermodule

text: is

text: not

text: free

text: if

text: and

text: only

text: if

text: it

text: contains

text: the

text: trivial

text: module

text: as

text: a

text: direct

text: summand.

text: In

text: particular,

text: when

text: written

text: as

text: a

text: direct

text: sum

text: of

text: indecomposable

text: v

text: -supermodules,

text: both

text: M

text: and

text: N

text: must

text: have

text: a

text: trivial

text: direct

text: summand

text: and

text: hence

text: so

text: must

text: M

text: N.

text: Thus,

text: M

text: N

text: is

text: not

text: free

text: as

text: a

text: v

text: -supermodule

text: and

text: [v]

text: V

text: G

text: M

text: N

text: Theorem

text: Let

text: G

text: be

text: a

text: ﬁnite

text: group

text: and

text: let

text: V

text: be

text: a

text: ﬁnite-dimensional

text: purely

text: odd

text: kG-module.

text: Let

text: M

text: be

text: a

text: ﬁnite-dimensional

text: V

text: G-supermodule

text: and

text: let

text: d

text: dim

text: V

text: G

text: dim

text: V

text: G

text: be

text: the

text: M

text: codimension

text: of

text: V

text: G

text: in

text: V

text: G

text: Then

text: dim

text: M.

text: If

text: d

text: then

text: sdim(M)

text: M

text: k

text: Proof.

text: Set

text: E

text: Λ(V

text: By

text: the

text: proof

text: of

text: Theorem

text: V

text: G

text: identiﬁes

text: with

text: the

text: quotient

text: of

text: M

text: E

text: by

text: the

text: action

text: of

text: G.

text: Since

text: G

text: is

text: a

text: ﬁnite

text: group,

text: then

text: the

text: codimension

text: of

text: V

text: G

text: in

text: V

text: G

text: M

text: M

text: is

text: equal

text: to

text: the

text: codimension

text: of

text: E

text: in

text: E

text: Next,

text: E

text: is

text: deﬁned

text: by

text: an

text: ideal

text: in

text: the

text: polynomial

text: M

text: M

text: ring

text: H

text: (E,

text: k)

text: S(V

text: so

text: it

text: follows

text: from

text: the

text: Noether

text: Normalization

text: Theorem

text: Theorem

text: that

text: there

text: exists

text: a

text: d-dimensional

text: subspace

text: H

text: of

text: V

text: such

text: that

text: H

text: E

text: Considering

text: the

text: M

text: restriction

text: of

text: M

text: to

text: the

text: subalgebra

text: Λ(H)

text: of

text: E

text: generated

text: by

text: H,

text: this

text: implies

text: that

text: Λ(H)

text: M

text: and

text: hence

text: implies

text: by

text: Proposition

text: that

text: M

text: is

text: projective

text: as

text: a

text: Λ(H)-supermodule.

text: Since

text: Λ(H)

text: is

text: indecomposable

text: over

text: itself,

text: M

text: is

text: then

text: isomorphic

text: as

text: a

text: Λ(H)-supermodule

text: to

text: a

text: direct

text: sum

text: of

text: copies

text: of

text: Λ(H),

text: each

text: of

text: which

text: is

text: of

text: dimension

text: and

text: (if

text: d

text: of

text: superdimension

text: K

text: Let

text: V

text: G-mod

text: denote

text: the

text: category

text: of

text: all

text: ﬁnite-dimensional

text: V

text: G-modules

text: and

text: let

text: denote

text: the

text: K

text: stable

text: module

text: category

text: of

text: V

text: G-mod.

text: Using

text: Lemma

text: it

text: follows

text: that

text: is

text: a

text: tensor

text: triangulated

text: K

text: category.

text: We

text: expect

text: that

text: the

text: spectrum

text: of

text: (in

text: the

text: sense

text: of

text: Balmer

text: is

text: homeomorphic

text: to

text: V/G

text: K

text: and

text: that

text: the

text: thick

text: tensor

text: ideals

text: of

text: are

text: classiﬁed

text: by

text: the

text: specialization

text: closed

text: subsets

text: of

text: V/G

text: via

text: support

text: varieties;

text: see

text: the

text: results

text: of

text: Pevtsova

text: and

text: Witherspoon

text: in

text: a

text: setting

text: with

text: close

text: similarities

text: to

text: the

text: one

text: considered

text: here.


text: 24 CHRISTOPHER M. DRUPIESKI AND JONATHAN R. KUJAWA (cid:48)| ∈ − | | ∼ ∗ − e Ext2pr 1(km 0(r), km 0(r)) = Hom (g (r), H2pr 1(G, k)), i G G k m (cid:48)(cid:48)| ∈ − | | ∼ ∗ − e Ext2pr 1(k0 n(r), k0 n(r)) = Hom (g (r), H2pr 1(G, k)), i G G k n (5.1.2) | ∈ | | ∼ ∗ − c Extpr(k0 n(r), km 0(r)) = Hom (g (r), H2pr 1(G, k)), and r G G k +1 | ∈ | | ∼ ∗ − cΠ Extpr(km 0(r), k0 n(r)) = Hom (g−(r), H2pr 1(G, k)). r G G k 1 (cid:48) (cid:48)(cid:48) | | • Viewing (e +e ) and (c +cΠ) as linear maps into H (G, k), they extend uniquely to graded i i G r r G superalgebra homomorphisms (cid:48) (cid:48)(cid:48) | | ∗ − → • (e + e ) : S(gl(m n) [2pi 1])(r) H (G, k), and i i G 0 | | ∗ → • (c + cΠ) : S(gl(m n) [pr])(r) H (G, k). r r G 1 (cid:32) (cid:33) Taking the product of these homomorphisms, we get a graded superalgebra homomorphism (cid:79) r | ∗ − ⊗ | ∗ → • (5.1.3) φ : S(gl(m n) [2pi 1])(r) S(gl(m n) [pr])(r) H (G, k). G 0 1 i=1 § § • The main consequence of the results in [19, 5.5] and [18, 5.4] is that H (G, k) is ﬁnite over the • image of φ and hence that H (G, k) is a ﬁnitely-generated algebra. G | | Now consider GL(m n) , the ﬁrst Frobenius kernel of GL(m n). A key step in the proof that • 1 (cid:48) (cid:48)(cid:48) | | H (G, k) is ﬁnite over φ involved verifying that (e + e ) | and (c + cΠ) | admit G r r GL(m n)1 r r GL(m n)1 particular descriptions. Speciﬁcally, consider the May spectral sequence [18, Corollary 5.2.3] | ∗ ⊗ | ∗ ⇒ | (5.1.4) Ei,j = Λj(gl(m n) ) Si/2(gl(m n) )(1) Hi+j(GL(m n) , k). 0 s 0 1 Here the superscript i/2 means that Ei,j = 0 unless i is even. In the proof of [19, Theorem 5.5.1], 0 the ﬁrst author veriﬁed the following properties: (cid:48) (cid:48)(cid:48) (5.1.5) Replacing e + e by a scalar multiple if necessary, r r (cid:48) (cid:48)(cid:48) | | ∗ − → • | (e + e ) : S(gl(m n) [2pr 1])(r) H (GL(m n) , k) r r G 0 1 − | ∗ − → | ∗ is equal to the composition of the pr 1-power map S(gl(m n) [2pr 1])(r) S(gl(m n) [2])(1) • → • | 0 0 with the horizontal edge map E ,0 H (GL(m n) , k) of (5.1.4). 0 1 (5.1.6) Replacing c + cΠ by a scalar multiple if necessary, the composition of r r | | ∗ → • | (c + cΠ) | : S(gl(m n) [pr])(r) H (GL(m n) , k) r r GL(m n)1 1 1 • | → • with the vertical edge map H (GL(m n) , k) E0, of (5.1.4) is equal to the composition | ∗ →1 |0 ∗ | ∗ of the pr-power map S(gl(m n) [pr])(r) S(gl(m n) ) and the inclusion into Λ (gl(m n) ). s 1 1 § (cid:48) (cid:48)(cid:48) The veriﬁcation in [19, 5.5] of these properties treated the classes e , e , c , and cΠ separately, (cid:48) (cid:48)(cid:48) (cid:48) (cid:48)(cid:48) r r r r but since (e )Π = e , the classes e and e can be rescaled if necessary by the same scalar factor, r r r r and similarly for c and cΠ. r r | More generally, let G be a sub-supergroup scheme of GL(m n) and set g = Lie(G). Then 1 G = G and the May spectral sequence for G takes the form 1 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: CHRISTOPHER

text: M.

text: DRUPIESKI

text: AND

text: JONATHAN

text: R.

text: KUJAWA

label: equation

text: e

text: km

text: Hom

text: (g

text: (r),

text: k)),

text: i

text: G

text: G

text: k

text: m

text: e

text: n(r),

text: n(r))

text: Hom

text: (g

text: (r),

text: k)),

text: i

text: G

text: G

text: k

text: n

text: c

text: n(r),

text: km

text: Hom

text: (g

text: (r),

text: k)),

text: and

text: r

text: G

text: G

text: k

text: cΠ

text: Extpr(km

text: n(r))

text: Hom

text: (g−(r),

text: k)).

text: r

text: G

text: G

text: k

text: Viewing

text: (e

text: +e

text: and

text: (c

text: +cΠ)

text: as

text: linear

text: maps

text: into

text: H

text: (G,

text: k),

text: they

text: extend

text: uniquely

text: to

text: graded

text: i

text: i

text: G

text: r

text: r

text: G

text: superalgebra

text: homomorphisms

text: (e

text: e

text: S(gl(m

text: n)

text: H

text: (G,

text: k),

text: and

text: i

text: i

text: G

text: (c

text: cΠ)

text: S(gl(m

text: n)

text: [pr])(r)

text: H

text: (G,

text: k).

text: r

text: r

text: G

text: Taking

text: the

text: product

text: of

text: these

text: homomorphisms,

text: we

text: get

text: a

text: graded

text: superalgebra

text: homomorphism

text: r

text: φ

text: S(gl(m

text: n)

text: S(gl(m

text: n)

text: [pr])(r)

text: H

text: (G,

text: k).

text: G

text: The

text: main

text: consequence

text: of

text: the

text: results

text: in

text: and

text: is

text: that

text: H

text: (G,

text: k)

text: is

text: ﬁnite

text: over

text: the

text: image

text: of

text: φ

text: and

text: hence

text: that

text: H

text: (G,

text: k)

text: is

text: a

text: ﬁnitely-generated

text: algebra.

text: G

text: Now

text: consider

text: GL(m

text: n)

text: the

text: ﬁrst

text: Frobenius

text: kernel

text: of

text: GL(m

text: n).

text: A

text: key

text: step

text: in

text: the

text: proof

text: that

text: H

text: (G,

text: k)

text: is

text: ﬁnite

text: over

text: φ

text: involved

text: verifying

text: that

text: (e

text: e

text: and

text: (c

text: cΠ)

text: admit

text: G

text: r

text: r

text: GL(m

text: r

text: r

text: GL(m

text: particular

text: descriptions.

text: Speciﬁcally,

text: consider

text: the

text: May

text: spectral

text: sequence

text: Corollary

text: Ei,j

text: Λj(gl(m

text: n)

text: n)

text: Hi+j(GL(m

text: n)

text: k).

text: s

text: Here

text: the

text: superscript

text: means

text: that

text: Ei,j

text: unless

text: i

text: is

text: even.

text: In

text: the

text: proof

text: of

text: Theorem

text: the

text: ﬁrst

text: author

text: veriﬁed

text: the

text: following

text: properties:

text: Replacing

text: e

text: e

text: by

text: a

text: scalar

text: multiple

text: if

text: necessary,

text: r

text: r

text: (e

text: e

text: S(gl(m

text: n)

text: H

text: (GL(m

text: n)

text: k)

text: r

text: r

text: G

text: is

text: equal

text: to

text: the

text: composition

text: of

text: the

text: pr

text: map

text: S(gl(m

text: n)

text: S(gl(m

text: n)

text: with

text: the

text: horizontal

text: edge

text: map

text: E

text: H

text: (GL(m

text: n)

text: k)

text: of

text: Replacing

text: c

text: cΠ

text: by

text: a

text: scalar

text: multiple

text: if

text: necessary,

text: the

text: composition

text: of

text: r

text: r

text: (c

text: cΠ)

text: S(gl(m

text: n)

text: [pr])(r)

text: H

text: (GL(m

text: n)

text: k)

text: r

text: r

text: GL(m

text: with

text: the

text: vertical

text: edge

text: map

text: H

text: (GL(m

text: n)

text: k)

text: of

text: is

text: equal

text: to

text: the

text: composition

text: →1

text: of

text: the

text: pr-power

text: map

text: S(gl(m

text: n)

text: [pr])(r)

text: S(gl(m

text: n)

text: and

text: the

text: inclusion

text: into

text: Λ

text: (gl(m

text: n)

text: s

text: The

text: veriﬁcation

text: in

text: of

text: these

text: properties

text: treated

text: the

text: classes

text: e

text: e

text: c

text: and

text: cΠ

text: separately,

text: r

text: r

text: r

text: r

text: but

text: since

text: (e

text: )Π

text: e

text: the

text: classes

text: e

text: and

text: e

text: can

text: be

text: rescaled

text: if

text: necessary

text: by

text: the

text: same

text: scalar

text: factor,

text: r

text: r

text: r

text: r

text: and

text: similarly

text: for

text: c

text: and

text: cΠ.

text: r

text: r

text: More

text: generally,

text: let

text: G

text: be

text: a

text: sub-supergroup

text: scheme

text: of

text: GL(m

text: n)

text: and

text: set

text: g

text: Lie(G).

text: Then

text: G

text: G

text: and

text: the

text: May

text: spectral

text: sequence

text: for

text: G

text: takes

text: the

text: form

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ON SUPPORT VARIETIES FOR LIE SUPERALGEBRAS AND FINITE SUPERGROUP SCHEMES 31 ∈ N ≤ for some a , b, c with a 1 for each i. Now making the identiﬁcation of graded superspaces i i ∗ i (5.2.3), the diﬀerential d on Hom (X(g), k) satisﬁes t V (g) ∗ ∗ · · · ∗ (cid:55)→ (y )b(x )c0 (x )cn 0, 0 n (cid:104) ∗(cid:105) ∗ ∗ · · · ∗ (cid:55)→ − · ∗ ∗ · · · ∗ ∗ x (y )b(x )c0 (x )cn ( 1)bα (y )b(x )c0 (x − )cn−1(x )cn+1, 0 0 n 0 0 n 1 n (cid:104) ∗(cid:105) ∗ ∗ · · · ∗ (cid:55)→ − ∗ ∗ ∗ · · · ∗ − − ∗ ∗ · · · ∗ x (y )b(x )c0 (x )cn ( 1)b(y )b(x )c0+1(x )c1 (x )cn ( 1)b(y )b+2(x )c0 (x )cn 1 0 n 0 1 n 0 n − · ∗ ∗ · · · ∗ + ( 1)bα (y )b(x )c0 (x )cn+1, and 1 0 n (cid:104) ∗(cid:105) ∗ ∗ · · · ∗ (cid:55)→ − ∗ ∗ · · · ∗ · · · ∗ x (y )b(x )c0 (x )cn ( 1)b(y )b(x )c0 (x − )ci−1+1 (x )cn i 0 n 0 i 1 n − · ∗ ∗ · · · ∗ ∗ ≤ ≤ + ( 1)bα (y )b(x )c0 (x − )cn−1(x )cn+1 if 2 i n. i 0 n 1 n In particular, the following polynomials are coboundaries: · ∗ α x , 0 n ∗ · ∗ − ∗ x + α x (y )2, and 0 1 n (cid:80) ∗ · ∗ ≤ x + α x for 1(cid:80) i < n. i i+1 n ∗ More generally, d maps any monomial of the form (5.3.2) in which n a = j to a linear t − i=0 i combination of monomials of the form (5.3.2) in which n a = j 1. i=0 i ≥ | To begin this section, let r 1 and let GL(m n) be the r-th Frobenius kernel of the general | r linear supergroup scheme GL(m n). As discussed in Section , the extension classes (5.1.1) give rise to a homomorphism, which(cid:0)we now write in the form(cid:1) | ∗ ⊕ ⊕ | ∗ → • | (5.4.1) φ : S (gl(m n) ) r gl(m n) (r) H (GL(m n) , k). r 0 1 r • | Since H (GL(m n) , k) is ﬁnite over φ , the induced morphism between maximal ideal spectra r r | | | → | × × | (5.4.2) Φ : GL(m n) (gl(m n) ) r gl(m n) . r r 0 1 is a ﬁnite morphism of aﬃne varieties. Deﬁnition 5.4.1. Let G be an aﬃne k-supergroup scheme and let g = Lie(G) be the restricted (cid:55)→ ≥ Lie superalge(cid:110)bra of G, with the p-map on g denoted by x x[p]. Given an integer r 1, deﬁne 0 the commuting variety C (G) by r (cid:111) ∈ × × C (G) = (α , α , . . . , α − , β) (g ) r g : [α , α ] = 0, [α , β] = 0 for all i, j, r 0 1 r 1 0 1 i j i ≤ ≤ − α[p] = 0 for 0 i r 2, and α[p−] = 1[β, β] . i r 1 2 | × Recall that the underlying purely even subgroup scheme of GL(m n) is GL GL . Then the × × m n | r-th Frobenius kernel (GL GL ) of GL GL is naturally a subgroup scheme of GL(m n) , × m → n r | m n r and the inclusion GL GL (cid:44) GL(m n) induces a corresponding morphism m n | × | → | | | (GL GL ) GL(m n) m n r r between the cohomology varieties. We now get the following analogue of [46, Proposition 5.1]: ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: ON

text: SUPPORT

text: VARIETIES

text: FOR

text: LIE

text: SUPERALGEBRAS

text: AND

text: FINITE

text: SUPERGROUP

text: SCHEMES

text: N

text: for

text: some

text: a

text: b,

text: c

text: with

text: a

text: for

text: each

text: i.

text: Now

text: making

text: the

text: identiﬁcation

text: of

text: graded

text: superspaces

text: i

text: i

text: i

text: the

text: diﬀerential

text: d

text: on

text: Hom

text: (X(g),

text: k)

text: satisﬁes

text: t

text: V

text: (g)

text: (y

text: )b(x

text: (x

text: )cn

text: n

text: x

text: (y

text: )b(x

text: (x

text: )cn

text: (y

text: )b(x

text: (x

text: n

text: n

text: n

text: x

text: (y

text: )b(x

text: (x

text: )cn

text: )b(x

text: (x

text: )cn

text: (x

text: )cn

text: n

text: n

text: n

text: (y

text: )b(x

text: (x

text: and

text: n

text: x

text: (y

text: )b(x

text: (x

text: )cn

text: )b(x

text: (x

text: (x

text: )cn

text: i

text: n

text: i

text: n

text: (y

text: )b(x

text: (x

text: if

text: i

text: n.

text: i

text: n

text: n

text: In

text: particular,

text: the

text: following

text: polynomials

text: are

text: coboundaries:

text: α

text: x

text: n

text: x

text: α

text: x

text: (y

text: and

text: n

text: x

text: α

text: x

text: for

text: i

text: n.

text: i

text: n

text: More

text: generally,

text: d

text: maps

text: any

text: monomial

text: of

text: the

text: form

text: in

text: which

text: n

text: a

text: j

text: to

text: a

text: linear

text: t

text: i

text: combination

text: of

text: monomials

text: of

text: the

text: form

text: in

text: which

text: n

text: a

text: j

text: i

text: To

text: begin

text: this

text: section,

text: let

text: r

text: and

text: let

text: GL(m

text: n)

text: be

text: the

text: r-th

text: Frobenius

text: kernel

text: of

text: the

text: general

text: r

text: linear

text: supergroup

text: scheme

text: GL(m

text: n).

text: As

text: discussed

text: in

text: Section

text: the

text: extension

text: classes

text: give

text: rise

text: to

text: a

text: homomorphism,

text: now

text: write

text: in

text: the

text: φ

text: S

text: (gl(m

text: n)

text: r

text: gl(m

text: n)

text: (r)

text: H

text: (GL(m

text: n)

text: k).

text: r

text: r

text: Since

text: H

text: (GL(m

text: n)

text: k)

text: is

text: ﬁnite

text: over

text: φ

text: the

text: induced

text: morphism

text: between

text: maximal

text: ideal

text: spectra

text: r

text: r

text: Φ

text: GL(m

text: n)

text: (gl(m

text: n)

text: r

text: gl(m

text: n)

text: r

text: r

text: is

text: a

text: ﬁnite

text: morphism

text: of

text: aﬃne

text: varieties.

text: Deﬁnition

text: Let

text: G

text: be

text: an

text: aﬃne

text: k-supergroup

text: scheme

text: and

text: let

text: g

text: Lie(G)

text: be

text: the

text: restricted

text: Lie

text: of

text: G,

text: with

text: the

text: p-map

text: on

text: g

text: denoted

text: by

text: x

text: x[p].

text: Given

text: an

text: integer

text: r

text: deﬁne

text: the

text: commuting

text: variety

text: C

text: (G)

text: by

text: r

text: C

text: (G)

text: (α

text: α

text: α

text: β)

text: (g

text: r

text: g

text: [α

text: α

text: [α

text: β]

text: for

text: all

text: i,

text: j,

text: r

text: r

text: i

text: j

text: i

text: α[p]

text: for

text: i

text: r

text: and

text: α[p−]

text: β]

text: i

text: r

text: Recall

text: that

text: the

text: underlying

text: purely

text: even

text: subgroup

text: scheme

text: of

text: GL(m

text: n)

text: is

text: GL

text: GL

text: Then

text: the

text: m

text: n

text: r-th

text: Frobenius

text: kernel

text: (GL

text: GL

text: of

text: GL

text: GL

text: is

text: naturally

text: a

text: subgroup

text: scheme

text: of

text: GL(m

text: n)

text: m

text: n

text: r

text: m

text: n

text: r

text: and

text: the

text: inclusion

text: GL

text: GL

text: GL(m

text: n)

text: induces

text: a

text: corresponding

text: morphism

text: m

text: n

text: (GL

text: GL

text: GL(m

text: n)

text: m

text: n

text: r

text: r

text: between

text: the

text: cohomology

text: varieties.

text: We

text: now

text: get

text: the

text: following

text: analogue

text: of

text: Proposition

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 8 CHRISTOPHER M. DRUPIESKI AND JONATHAN R. KUJAWA (cid:39) (cid:55)→ map on M deﬁnes an odd A-supermodule isomorphism M Π(M), m mπ. In particular, if M and N are A-supermodules, there are odd isomorphisms (cid:39) (cid:39) (2.3.1) Hom (M, N) Hom (M, Π(N)) and Hom (M, N) Hom (Π(M), N) A A A A that are natural with respect to even homomorphisms in either variable. Z Given a k-superalgebra A, one can form the smash product algebra A#k . As a superspace, Z ⊗ Z Z 2 Z A#k is simply A k , the tensor product of A with the group ring k for (considered as 2 2 2 2 Z a purely even superalgebra). Multiplication in A#k is induced by the given products in A and Z ⊗ ⊗ − ⊗ 2 k and by the relation (1 1)(a 0) = ( 1)aa 1. In the previous equation, 0 and 1 denote the 2 Z identity and non-identity elements of the group ; these elements form a vector space basis for the 2 Z Z group ring k . The category of A#k -modules naturally identiﬁes with (smod ) . Speciﬁcally, 2 2 A ev Z Z if M is an A#k -module, then M is an A-module and a k -module by restriction. Since k 2 (cid:54) 2 Z is assumed to be a ﬁeld of characteristic p = 2, M decomposes under the action of k into a 2 Z Z trivial k -submodule M and a nontrivial k -submodule M . Then M is an A-supermodule with 2 0 ⊕ 2 1 respect to the decomposition M = M M . Conversely, if M is an A-supermodule, then M lifts Z 0 1 ∈ Z (cid:55)→ − to the structure of an A#k -module by having 1 act on M by the map m ( 1)mm. 2 2 − → We say that an A-supermodule P is projective if the functor Hom (P, ) : smod svec −A e→v ev is exact, and we say that an A-supermodule Q is injective if Hom ( , Q) : smod svec is A ev ev exact. It follows from (2.3.1) that an A-supermodule is projective (resp. injective) in this sense if and only if it is projective (resp. injective) in the usual sense in the abelian category (smod ) . A ev Z Since (smod ) identiﬁes with the category of A#k -modules, it contains enough projectives and A ev 2 enough injectives. We can thus make the following deﬁnition: − §§ Deﬁnition 2.3.1. Given an A-supermodule M, deﬁne Extn (M, ) as in [18, 2.5–2.6] to be the − → A n-th right derived functor of Hom (M, ) : (smod ) svec . A A ev ev With the above deﬁnition, the extension group Extn (M, N) can have a nonzero odd subspace, A whereas ordinary extension groups in the abelian category smod , which are deﬁned in terms of the − − ev right derived functors of Hom (M, ) = Hom (M, ) , are always purely even superspaces. smodev A 0 In principle, the odd isomorphisms (2.3.1) permit us to reduce all cohomology calculations to • calculations strictly within the abelian category smod , but we prefer to work with Ext (M, N) ev A Z so that we don’t have to manually keep track of the -gradings. In terms of the smash product Z • • 2 (cid:80) algebra A#k 2, ExtA#kZ2(M, N) = ExtA(MZ, N)0. If A is a Hopf superalgebra, then A#k is naturally a Hopf algebra in the ordinary sense. 2 ∈ ⊗ Speciﬁcally, if the coproduct in A of an el(cid:88)ement a A is denoted ∆(a) = a a , then the 1 2 Z coproduct on A#k is determined by the equations 2 ⊗ ⊗ ⊗ ⊗ ∆ Z (a 0) = (a a ) (a 0), and A#k 2 1 2 2 ⊗ ⊗ ⊗ ⊗ ∆ Z (1 1) = (1 1) (1 1). A#k 2 Z Since (smod ) identiﬁes with the category of A#k -modules, the next lemma follows from the A ev 2 well-known fact that any ﬁnite-dimensional ordinary Hopf algebra is self-injective. Lemma 2.3.2. If A is a ﬁnite-dimensional Hopf superalgebra, then an A-supermodule is projective if and only if it is injective. ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: CHRISTOPHER

text: M.

text: DRUPIESKI

text: AND

text: JONATHAN

text: R.

text: KUJAWA

text: map

text: on

text: M

text: deﬁnes

text: an

text: odd

text: A-supermodule

text: isomorphism

text: M

text: Π(M),

text: m

text: mπ.

text: In

text: particular,

text: if

text: M

text: and

text: N

text: are

text: A-supermodules,

text: there

text: are

text: odd

text: isomorphisms

text: Hom

text: (M,

text: N)

text: Hom

text: (M,

text: Π(N))

text: and

text: Hom

text: (M,

text: N)

text: Hom

text: (Π(M),

text: N)

text: A

text: A

text: A

text: A

text: that

text: are

text: natural

text: with

text: respect

text: to

text: even

text: homomorphisms

text: in

text: either

text: variable.

text: Z

text: Given

text: a

text: k-superalgebra

text: A,

text: one

text: can

text: form

text: the

text: smash

text: product

text: algebra

text: A#k

text: As

text: a

text: superspace,

text: Z

text: Z

text: Z

text: Z

text: A#k

text: is

text: simply

text: A

text: k

text: the

text: tensor

text: product

text: of

text: A

text: with

text: the

text: group

text: ring

text: k

text: for

text: (considered

text: as

text: Z

text: a

text: purely

text: even

text: superalgebra).

text: Multiplication

text: in

text: A#k

text: is

text: induced

text: by

text: the

text: given

text: products

text: in

text: A

text: and

text: Z

text: k

text: and

text: by

text: the

text: relation

text: In

text: the

text: previous

text: equation,

text: and

text: denote

text: the

text: Z

text: identity

text: and

text: non-identity

text: elements

text: of

text: the

text: group

text: these

text: elements

text: form

text: a

text: vector

text: space

text: basis

text: for

text: the

text: Z

text: Z

text: group

text: ring

text: k

text: The

text: category

text: of

text: A#k

text: -modules

text: naturally

text: identiﬁes

text: with

text: (smod

text: Speciﬁcally,

text: A

text: ev

text: Z

text: Z

text: if

text: M

text: is

text: an

text: A#k

text: -module,

text: then

text: M

text: is

text: an

text: A-module

text: and

text: a

text: k

text: -module

text: by

text: restriction.

text: Since

text: k

text: Z

text: is

text: assumed

text: to

text: be

text: a

text: ﬁeld

text: of

text: characteristic

text: p

text: M

text: decomposes

text: under

text: the

text: action

text: of

text: k

text: into

text: a

text: Z

text: Z

text: trivial

text: k

text: -submodule

text: M

text: and

text: a

text: nontrivial

text: k

text: -submodule

text: M

text: Then

text: M

text: is

text: an

text: A-supermodule

text: with

text: respect

text: to

text: the

text: decomposition

text: M

text: M

text: M

text: Conversely,

text: if

text: M

text: is

text: an

text: A-supermodule,

text: then

text: M

text: lifts

text: Z

text: Z

text: to

text: the

text: structure

text: of

text: an

text: A#k

text: -module

text: by

text: having

text: act

text: on

text: M

text: by

text: the

text: map

text: m

text: We

text: say

text: that

text: an

text: A-supermodule

text: P

text: is

text: projective

text: if

text: the

text: functor

text: Hom

text: (P,

text: smod

text: svec

text: −A

text: e→v

text: ev

text: is

text: exact,

text: and

text: we

text: say

text: that

text: an

text: A-supermodule

text: Q

text: is

text: injective

text: if

text: Hom

text: Q)

text: smod

text: svec

text: is

text: A

text: ev

text: ev

text: exact.

text: It

text: follows

text: from

text: that

text: an

text: A-supermodule

text: is

text: projective

text: (resp.

text: injective)

text: in

text: this

text: sense

text: if

text: and

text: only

text: if

text: it

text: is

text: projective

text: (resp.

text: injective)

text: in

text: the

text: usual

text: sense

text: in

text: the

text: abelian

text: category

text: (smod

text: A

text: ev

text: Z

text: Since

text: (smod

text: identiﬁes

text: with

text: the

text: category

text: of

text: A#k

text: -modules,

text: it

text: contains

text: enough

text: projectives

text: and

text: A

text: ev

text: enough

text: injectives.

text: We

text: can

text: thus

text: make

text: the

text: following

text: deﬁnition:

text: Deﬁnition

text: Given

text: an

text: A-supermodule

text: M,

text: deﬁne

text: Extn

text: (M,

text: as

text: in

text: to

text: be

text: the

text: A

text: n-th

text: right

text: derived

text: functor

text: of

text: Hom

text: (M,

text: (smod

text: svec

text: A

text: A

text: ev

text: ev

text: With

text: the

text: above

text: deﬁnition,

text: the

text: extension

text: group

text: Extn

text: (M,

text: N)

text: can

text: have

text: a

text: nonzero

text: odd

text: subspace,

text: A

text: whereas

text: ordinary

text: extension

text: groups

text: in

text: the

text: abelian

text: category

text: smod

text: which

text: are

text: deﬁned

text: in

text: terms

text: of

text: the

text: ev

text: right

text: derived

text: functors

text: of

text: Hom

text: (M,

text: Hom

text: (M,

text: are

text: always

text: purely

text: even

text: superspaces.

text: smodev

text: A

text: In

text: principle,

text: the

text: odd

text: isomorphisms

text: permit

text: us

text: to

text: reduce

text: all

text: cohomology

text: calculations

text: to

text: calculations

text: strictly

text: within

text: the

text: abelian

text: category

text: smod

text: but

text: we

text: prefer

text: to

text: work

text: with

text: Ext

text: (M,

text: N)

text: ev

text: A

text: Z

text: so

text: that

text: we

text: don’t

text: have

text: to

text: manually

text: keep

text: track

text: of

text: the

text: -gradings.

text: In

text: terms

text: of

text: the

text: smash

text: product

text: Z

text: algebra

text: A#k

text: N)

text: ExtA(MZ,

text: If

text: A

text: is

text: a

text: Hopf

text: superalgebra,

text: then

text: A#k

text: is

text: naturally

text: a

text: Hopf

text: algebra

text: in

text: the

text: ordinary

text: sense.

text: Speciﬁcally,

text: if

text: the

text: coproduct

text: in

text: A

text: of

text: an

text: a

text: A

text: is

text: denoted

text: ∆(a)

text: a

text: a

text: then

text: the

text: Z

text: coproduct

text: on

text: A#k

text: is

text: determined

text: by

text: the

text: equations

text: Z

text: (a

text: (a

text: a

text: (a

text: and

text: A#k

text: Z

text: A#k

text: Z

text: Since

text: (smod

text: identiﬁes

text: with

text: the

text: category

text: of

text: A#k

text: -modules,

text: the

text: next

text: lemma

text: follows

text: from

text: the

text: A

text: ev

text: well-known

text: fact

text: that

text: any

text: ﬁnite-dimensional

text: ordinary

text: Hopf

text: algebra

text: is

text: self-injective.

text: Lemma

text: If

text: A

text: is

text: a

text: ﬁnite-dimensional

text: Hopf

text: superalgebra,

text: then

text: an

text: A-supermodule

text: is

text: projective

text: if

text: and

text: only

text: if

text: it

text: is

text: injective.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 2 / Nuclear Physics A 00 (2020) 1–4 ××××Nf = 2 QCD Quenched 1st 1 s t or 2 nd h N → large f 1st ud Fig. 1. Left: The quark mass dependence of the nature of phase transition (Columbia plot) for (2 + Nf) ﬂavor QCD. Right: The slope of the eﬀective potential Veﬀ(P) for h = 0.0 - 0.4 from top to bottom. We focus on two topics. One is the ﬁnite temperature phase transition of the massless 2 ﬂavor QCD [2]. To understand the nature of the massless limit is a long standing problem. Investigating the light quark mass dependence of the boundary of the ﬁrst order region, one may ﬁnd whether the massless limit of 2 ﬂavor QCD, i.e. top left corner of Fig. 1 (left), is of ﬁrst order or second order. If the boundary runs like the dotted curve in Fig. 1 (left), the transition is of ﬁrst order in the massless limit. The other is the chemical potential dependence [3]. We will show that the critical mh increases as the chemical potential µh increases. This suggests that the critical curve can be investigated in the heavy quark region even for small Nf when µh is large. Through the study of (2 + N ) ﬂavor QCD, we discuss the QCD critical point at ﬁnite density. f 2. Histogram method We study the system where two light ﬂavors and N heavy ﬂavors exist. The hopping parameter and f the quark chemical po(cid:82)tential a(cid:16)re κl, (cid:17)µl for the light ﬂavor and κh, µh for the heavy ﬂavor, respectively. The hopping parameter is in inverse proportion to the quark mass when the mass is heavy. To investigate the nature of phase transitions, we consider the probability distribution function of the average plaquette, D − w(P; β, κl, µl, κh, µh) = Uδ P Pˆ e6βNsitePˆ (det M(κl, µl))2 (det M(κh, µh))Nf , where M is the quark ma- ≡ × trix, Nsite Ns3 Nt is the num−ber of lattice sites, and β = 6/g20 is the coupling constant. Pˆ is deﬁned from the gauge action Sg as Pˆ =− Sg/(6Nsiteβ), which is a linear combination of Wilson loops and is called the generalized plaquette. δ(P Pˆ) is the del√ta function, which constrains the operator Pˆ to P. We use the ≈ − delta function approximated by δ(x) 1/(∆ π) exp[ (x/∆)2], where the parameter ∆ must be small. For ≡ − convenience, we deﬁne the eﬀective potential as V (P) ln w(P). It is rewritten as eﬀ − Veﬀ(P; β, κl, µl, κh, µh) = V0(P; β0, κl) ln R(P; β, β0, κl, µl, κh, µh) (1) with the potential of 2 ﬂavor QCD at µl = 0, V(cid:42)(cid:32)0(P; β0, κl), an(cid:33)d t(cid:32)he reweighting(cid:33) fa(cid:43)ctor ln R(P) = 6(β − β0)NsiteP + ln ddeettMM((κκll,,µ0l)) 2 ddeettMM(κ(0h,,µ0h)) Nf (P;ﬁxed) , (2) (cid:104)· · ·(cid:105) where means the ensemble average over 2 ﬂavor conﬁgurations when the plaquette is ﬁxed (P;ﬁxed) to be P. We evaluate the quark determinant of N ﬂavors in R(P) with the leading order of the hopping f paramete(cid:34)r expansion fo(cid:35)r the standard Wilson action [4]. (cid:16) (cid:17) ln ddeettMM(κ(0h,,µ0h)) = 288Nsiteκh4Wˆ P + 12Ns3(2κh)Nt cosh(µh/T)Ωˆ R + i sinh(µh/T)Ωˆ I + · · · , (3) ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

font: VCKFCA+rtxi

text: Nuclear

text: Physics

text: A

text: ××××Nf

font: default

text: QCD

text: Quenched

text: s

text: t

text: or

text: nd

text: h

text: N

text: large

text: f

text: ud

text: Fig.

text: Left:

label: caption

text: The

text: quark

text: mass

text: dependence

text: of

text: the

text: nature

text: of

text: phase

text: transition

text: (Columbia

text: plot)

text: for

font: XKJJZT+rtxr

text: Nf)

text: ﬂavor

text: QCD.

text: Right:

text: The

text: slope

text: of

text: the

text: eﬀective

text: potential

text: Veﬀ(P)

text: for

text: h

text: from

text: top

text: to

text: bottom.

text: We

text: focus

text: on

text: two

text: topics.

text: One

text: is

text: the

text: ﬁnite

text: temperature

text: phase

text: transition

text: of

text: the

text: massless

text: ﬂavor

text: QCD

text: To

text: understand

text: the

text: nature

text: of

text: the

text: massless

text: limit

text: is

text: a

text: long

text: standing

text: problem.

text: Investigating

text: the

text: light

text: quark

text: mass

text: dependence

text: of

text: the

text: boundary

text: of

text: the

text: ﬁrst

text: order

text: region,

text: one

text: may

text: ﬁnd

text: whether

text: the

text: massless

text: limit

text: of

text: ﬂavor

text: QCD,

text: i.e.

text: top

text: left

text: corner

text: of

text: Fig.

text: (left),

text: is

text: of

text: ﬁrst

text: order

text: or

text: second

text: order.

text: If

text: the

text: boundary

text: runs

text: like

text: the

text: dotted

text: curve

text: in

text: Fig.

text: (left),

text: the

text: transition

text: is

text: of

text: ﬁrst

text: order

text: in

text: the

text: massless

text: limit.

text: The

text: other

text: is

text: the

text: chemical

text: potential

text: dependence

text: We

text: will

text: show

text: that

text: the

text: critical

text: mh

text: increases

text: as

text: the

text: chemical

text: potential

text: µh

text: increases.

text: This

text: suggests

text: that

text: the

text: critical

text: curve

text: can

text: be

text: investigated

text: in

text: the

text: heavy

text: quark

text: region

text: even

text: for

text: small

text: Nf

text: when

text: µh

text: is

text: large.

text: Through

text: the

text: study

text: of

text: N

text: ﬂavor

text: QCD,

text: we

text: discuss

text: the

text: QCD

text: critical

text: point

text: at

text: ﬁnite

text: density.

text: f

text: Histogram

text: method

text: We

text: study

text: the

text: system

text: where

text: two

text: light

text: ﬂavors

text: and

text: N

text: heavy

text: ﬂavors

text: exist.

text: The

text: hopping

text: parameter

text: and

text: f

text: the

text: quark

text: chemical

text: κl,

text: for

text: the

text: light

text: ﬂavor

text: and

text: κh,

text: µh

text: for

text: the

text: heavy

text: ﬂavor,

text: respectively.

text: The

text: hopping

text: parameter

text: is

text: in

text: inverse

text: proportion

text: to

text: the

text: quark

text: mass

text: when

text: the

text: mass

text: is

text: heavy.

text: To

text: investigate

text: the

text: nature

text: of

text: phase

text: transitions,

text: we

text: consider

text: the

text: probability

text: distribution

text: function

text: of

text: the

text: average

text: plaquette,

text: D

text: w(P;

text: β,

text: κl,

text: µl,

text: κh,

text: µh)

text: Uδ

text: P

text: Pˆ

text: (det

text: M(κl,

text: (det

text: M(κh,

text: µh))Nf

text: where

text: M

text: is

text: the

text: quark

text: ma-

text: trix,

text: Nsite

text: Nt

text: is

text: the

text: num−ber

text: of

text: lattice

text: sites,

text: and

text: β

text: is

text: the

text: coupling

text: constant.

text: Pˆ

text: is

text: deﬁned

text: from

text: the

text: gauge

text: action

text: Sg

text: as

text: Pˆ

text: which

text: is

text: a

text: linear

text: combination

text: of

text: Wilson

text: loops

text: and

text: is

text: called

text: the

text: generalized

text: plaquette.

text: δ(P

text: Pˆ)

text: is

text: the

text: del√ta

text: function,

text: which

text: constrains

text: the

text: operator

text: Pˆ

text: to

text: P.

text: We

text: use

text: the

text: delta

text: function

text: approximated

text: by

text: δ(x)

text: π)

text: exp[

text: where

text: the

text: parameter

text: must

text: be

text: small.

text: For

text: convenience,

text: we

text: deﬁne

text: the

text: eﬀective

text: potential

text: as

text: V

text: (P)

text: ln

text: w(P).

text: It

text: is

text: rewritten

text: as

text: eﬀ

text: Veﬀ(P;

text: β,

text: κl,

text: µl,

text: κh,

text: µh)

text: κl)

text: ln

text: R(P;

text: β,

text: κl,

text: µl,

text: κh,

text: µh)

text: with

text: the

text: potential

text: of

text: ﬂavor

text: QCD

text: at

text: µl

text: κl),

text: ln

text: R(P)

text: ln

text: Nf

text: (P;ﬁxed)

text: where

text: means

text: the

text: ensemble

text: average

text: over

text: ﬂavor

text: conﬁgurations

text: when

text: the

text: plaquette

text: is

text: ﬁxed

text: (P;ﬁxed)

text: to

text: be

text: P.

text: We

text: evaluate

text: the

text: quark

text: determinant

text: of

text: N

text: ﬂavors

text: in

text: R(P)

text: with

text: the

text: leading

text: order

text: of

text: the

text: hopping

text: f

text: expansion

text: the

text: standard

text: Wilson

text: action

text: ln

text: P

text: cosh(µh/T)Ωˆ

text: R

text: i

text: sinh(µh/T)Ωˆ

text: I

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 9 (a) The DAQ was in stable operation at more than 98% contribution to the total error in the eﬃciencies-corrected of the time. The trigger rate is low and the DAQ live Bulk rates B . 0 time is close to 100%. Contributions to systematic un- 4. Choice Quenching Function: Two studies were per- certainties are negligible. − formed to investigate the sensitivities to exclusion limits (b) Trigger Eﬃciency Since the analysis threshold from the choice of QF. (i) As displayed in Figure 7, the (475 eVee) is much higher than the trigger threshold (246 red line evaluated by TRIM software together with the eVee at 50%), the trigger eﬃciency of the physics events yellow band (10% systematic uncertainty) were adopted. relevant to this analysis was 100%, resulting in negligible Analysis is performed by scanning QF within 10% of their contribution to the systematic uncertainties. − nominal value. It was shown that the diﬀerence among (c) Fiducial Mass The error of the measured thick- these results are small, eg. the variation of σSI is about χN ness of the dead layer gave rise to a 1% uncertainty at 15% at m =8 GeV/c2, and the least stringent bounds χ ﬁducial mass. This corresponds to an additional 0.1% among them at a given WIMP-mass were adopted as our contribution to the total systematic uncertainty at 475 ﬁnal physics limits. (ii) The same procedure described eVee. in (i) but the QF evaluated by Lindhard (k=0.157) and CoGeNT [12] were applied. It was concluded that the dif- 2. Signal Selection: The systematic uncertainties orig- ference were small, eg. about 14% deviations in σSI at inated from the stability of BC and AC cuts, and were χN studied with the change of cut parameters around the mχ=8 GeV/c2). The results have been displayed in [11] nominal values. The BC cut contributed an additional and our formulation with TRIM provided the most con- 0.5% to contribution to the total systematic uncertainty servative limits among the alternatives. at 475 eVee, while the contribution arising from AC cut In our previous work, the 53.9 kg-days exposure is negligible. has shown that the statistical uncertainties were dom- 3. Bulk Events Selection: The evaluation of system- inant and contributed 86% relative to the total uncer- atic eﬀects follow the procedures described in our earlier tainty [11]. As the exposure expanded to 335.6 kg-days, work [11, 18]. In particular, the statistical uncertainties were secondary and system- atic uncertainties dominated 81% relative to the total (a) The leading systematic uncertainties is from the B- uncertainty. It was crucial to develop new method to event selection and (ε , λ ) calibration due to possi- BS BS evaluate the (ε , λ ) to decrease the systematic un- ble diﬀerences in locations and energy spectra between BS BS certainty further which contributes the main part of the the calibration sources and background events. The cal- systematic uncertainty. ibration sources probe the surface eﬀects due to both low energy (surface richer) and high energy (bulk richer) photons. The τ distributions for B-events are identical V. LIMITS ON WIMPS for both sources and physics background, while those for S-events showed intrinsic diﬀerence due to the diﬀerence The measured energy spectra and its evolution with in surface penetration which manifest as the diﬀerence data selection progress are depicted in Figure 17 (a). Six of slopes in the (ε , λ ) plane [11].The systematic BS BS cosmogenic nuclides can be identiﬁed clearly through the uncertainties are derived from the spread of the (ε , BS K-shell X-rays peaks, and the contributions of the corre- λ ) intersections of calibration bands, relative to the BS sponding L-shell X-rays at low energy range can be cal- combined best-ﬁt solution. This leads to a 25.0% contri- culated accurately since the ratioes of the intensities of bution to the total error in the eﬃciencies-corrected Bulk K-shell and L-shell X-rays are deﬁnite, as shown in Fig- rates B , accounting for the most signiﬁcant in the total 0 ure 18 (a). The half-lives of the dominant nuclides can systematic uncertainty. be measured by their K-shell X-rays rays. Figure 17 (b) (b) The systematic uncertainties related with diﬀerent displayed the decay of 68Ge, 65Zn and 55Fe. High energy locations are studied with the sources placed at several γ rays originated from ambient radioactivity contributed positions of the top and the side (the cylindrical sur- a ﬂat electron-recoil background. face) of the pPCGe. Among them, the 241Am γ from The nature of the interaction between WIMPs with the side are strongly attenuated due to additional thick- baryonic matter is a priori unknown. The data was ness from the cylindrical copper support structure and analysed with two benchmark χ-N cross-sections: spin- curved surface of the germanium crystal and therefore independent (SI, scalar) and spin-dependent (SD, axial- do not produce useful signals. The higher energy γ from vector) couplings: 57Co, 137Cs and 60Co at top and side, as well as those ⊗ ⊗ − from physics samples (BC AC+ and BC AC ), show dσ dσ dσ similar distributions in τ, independent of locations. The χN = ( χN ) + ( χN ) . (3) shift in (εBS, λBS) based on calibration source data at dif- dER dER SI dER SD ferent locations is less than 4%, corresponding to a 3.7% ##LTLine## ##LTLine## ##LTLine##

text: (a)

label: list

text: The

text: DAQ

text: was

text: in

text: stable

text: operation

text: at

text: more

text: than

text: contribution

text: to

text: the

text: total

text: error

text: in

text: the

text: eﬃciencies-corrected

text: of

text: the

text: time.

text: The

text: trigger

text: rate

text: is

text: low

text: and

text: the

text: DAQ

text: live

text: Bulk

text: rates

text: B

text: time

text: is

text: close

text: to

text: Contributions

text: to

text: systematic

text: un-

text: Choice

text: Quenching

text: Function:

text: Two

text: studies

text: were

text: per-

text: certainties

text: are

text: negligible.

text: formed

text: to

text: investigate

text: the

text: sensitivities

text: to

text: exclusion

text: limits

text: (b)

text: Trigger

text: Eﬃciency

text: Since

text: the

text: analysis

text: threshold

text: from

text: the

text: choice

text: of

text: QF.

text: (i)

text: As

text: displayed

text: in

text: Figure

text: the

text: eVee)

text: is

text: much

text: higher

text: than

text: the

text: trigger

text: threshold

text: red

text: line

text: evaluated

text: by

text: TRIM

text: software

text: together

text: with

text: the

text: eVee

text: at

text: the

text: trigger

text: eﬃciency

text: of

text: the

text: physics

text: events

text: yellow

text: band

text: systematic

text: uncertainty)

text: were

text: adopted.

text: relevant

text: to

text: this

text: analysis

text: was

text: resulting

text: in

text: negligible

text: Analysis

text: is

text: performed

text: by

text: scanning

text: QF

text: within

text: of

text: their

text: contribution

text: to

text: the

text: systematic

text: uncertainties.

text: nominal

text: value.

text: It

text: was

text: shown

text: that

text: the

text: diﬀerence

text: among

text: (c)

text: Fiducial

text: Mass

text: The

text: error

text: of

text: the

text: measured

text: thick-

text: these

text: results

text: are

text: small,

text: eg.

text: the

text: variation

text: of

text: σSI

text: is

text: about

text: χN

text: ness

text: of

text: the

text: dead

text: layer

text: gave

text: rise

text: to

text: a

text: uncertainty

text: at

text: at

text: m

text: and

text: the

text: least

text: stringent

text: bounds

text: χ

text: ﬁducial

text: mass.

text: This

text: corresponds

text: to

text: an

text: additional

text: among

text: them

text: at

text: a

text: given

text: WIMP-mass

text: were

text: adopted

text: as

text: our

text: contribution

text: to

text: the

text: total

text: systematic

text: uncertainty

text: at

text: ﬁnal

text: physics

text: limits.

text: (ii)

text: The

text: same

text: procedure

text: described

text: eVee.

text: in

text: (i)

text: but

text: the

text: QF

text: evaluated

text: by

text: Lindhard

text: and

text: CoGeNT

text: were

text: applied.

text: It

text: was

text: concluded

text: that

text: the

text: dif-

text: Signal

text: Selection:

text: The

text: systematic

text: uncertainties

text: orig-

text: ference

text: were

text: small,

text: eg.

text: about

text: deviations

text: in

text: σSI

text: at

text: inated

text: from

text: the

text: stability

text: of

text: BC

text: and

text: AC

text: cuts,

text: and

text: were

text: χN

text: studied

text: with

text: the

text: change

text: of

text: cut

text: parameters

text: around

text: the

text: The

text: results

text: have

text: been

text: displayed

text: in

text: nominal

text: values.

text: The

text: BC

text: cut

text: contributed

text: an

text: additional

text: and

text: our

text: formulation

text: with

text: TRIM

text: provided

text: the

text: most

text: con-

text: to

text: contribution

text: to

text: the

text: total

text: systematic

text: uncertainty

text: servative

text: limits

text: among

text: the

text: alternatives.

text: at

text: eVee,

text: while

text: the

text: contribution

text: arising

text: from

text: AC

text: cut

text: In

text: our

text: previous

text: work,

text: the

text: kg-days

text: exposure

text: is

text: negligible.

text: has

text: shown

text: that

text: the

text: statistical

text: uncertainties

text: were

text: dom-

text: Bulk

text: Events

text: Selection:

text: The

text: evaluation

text: of

text: system-

text: inant

text: and

text: contributed

text: relative

text: to

text: the

text: total

text: uncer-

text: atic

text: eﬀects

text: follow

text: the

text: procedures

text: described

text: in

text: our

text: earlier

text: tainty

text: As

text: the

text: exposure

text: expanded

text: to

text: kg-days,

text: work

text: In

text: particular,

text: the

text: statistical

text: uncertainties

text: were

text: secondary

text: and

text: system-

text: atic

text: uncertainties

text: dominated

text: relative

text: to

text: the

text: total

text: (a)

text: The

text: leading

text: systematic

text: uncertainties

text: is

text: from

text: the

text: B-

text: uncertainty.

text: It

text: was

text: crucial

text: to

text: develop

text: new

text: method

text: to

text: event

text: selection

text: and

text: (ε

text: λ

text: calibration

text: due

text: to

text: possi-

text: BS

text: BS

text: evaluate

text: the

text: (ε

text: λ

text: to

text: decrease

text: the

text: systematic

text: un-

text: ble

text: diﬀerences

text: in

text: locations

text: and

text: energy

text: spectra

text: between

text: BS

text: BS

text: certainty

text: further

text: which

text: contributes

text: the

text: main

text: part

text: of

text: the

text: the

text: calibration

text: sources

text: and

text: background

text: events.

text: The

text: cal-

text: systematic

text: uncertainty.

text: ibration

text: sources

text: probe

text: the

text: surface

text: eﬀects

text: due

text: to

text: both

text: low

text: energy

text: (surface

text: richer)

text: and

text: high

text: energy

text: (bulk

text: richer)

text: photons.

text: The

text: τ

text: distributions

text: for

text: B-events

text: are

text: identical

text: V.

text: LIMITS

text: ON

text: WIMPS

text: for

text: both

text: sources

text: and

text: physics

text: background,

text: while

text: those

text: for

text: S-events

text: showed

text: intrinsic

text: diﬀerence

text: due

text: to

text: the

text: diﬀerence

text: The

text: measured

text: energy

text: spectra

text: and

text: its

text: evolution

text: with

text: in

text: surface

text: penetration

text: which

text: manifest

text: as

text: the

text: diﬀerence

text: data

text: selection

text: progress

text: are

text: depicted

text: in

text: Figure

text: (a).

text: Six

text: of

text: slopes

text: in

text: the

text: (ε

text: λ

text: plane

text: systematic

text: BS

text: BS

text: cosmogenic

text: nuclides

text: can

text: be

text: identiﬁed

text: clearly

text: through

text: the

text: uncertainties

text: are

text: derived

text: from

text: the

text: spread

text: of

text: the

text: (ε

text: BS

text: K-shell

text: X-rays

text: peaks,

text: and

text: the

text: contributions

text: of

text: the

text: corre-

text: λ

text: intersections

text: of

text: calibration

text: bands,

text: relative

text: to

text: the

text: BS

text: sponding

text: L-shell

text: X-rays

text: at

text: low

text: energy

text: range

text: can

text: be

text: cal-

text: combined

text: best-ﬁt

text: solution.

text: This

text: leads

text: to

text: a

text: contri-

text: culated

text: accurately

text: since

text: the

text: ratioes

text: of

text: the

text: intensities

text: of

text: bution

text: to

text: the

text: total

text: error

text: in

text: the

text: eﬃciencies-corrected

text: Bulk

text: K-shell

text: and

text: L-shell

text: X-rays

text: are

text: deﬁnite,

text: as

text: shown

text: in

text: Fig-

text: rates

text: B

text: accounting

text: for

text: the

text: most

text: signiﬁcant

text: in

text: the

text: total

text: ure

text: (a).

text: The

text: half-lives

text: of

text: the

text: dominant

text: nuclides

text: can

text: systematic

text: uncertainty.

text: be

text: measured

text: by

text: their

text: K-shell

text: X-rays

text: rays.

text: Figure

text: (b)

text: (b)

text: The

text: systematic

text: uncertainties

text: related

text: with

text: diﬀerent

text: displayed

text: the

text: decay

text: of

text: and

text: High

text: energy

text: locations

text: are

text: studied

text: with

text: the

text: sources

text: placed

text: at

text: several

text: γ

text: rays

text: originated

text: from

text: ambient

text: radioactivity

text: contributed

text: positions

text: of

text: the

text: top

text: and

text: the

text: side

text: (the

text: cylindrical

text: sur-

text: a

text: ﬂat

text: electron-recoil

text: background.

text: face)

text: of

text: the

text: pPCGe.

text: Among

text: them,

text: the

text: γ

text: from

text: The

text: nature

text: of

text: the

text: interaction

text: between

text: WIMPs

text: with

text: the

text: side

text: are

text: strongly

text: attenuated

text: due

text: to

text: additional

text: thick-

text: baryonic

text: matter

text: is

text: a

text: priori

text: unknown.

text: The

text: data

text: was

text: ness

text: from

text: the

text: cylindrical

text: copper

text: support

text: structure

text: and

text: analysed

text: with

text: two

text: benchmark

text: χ-N

text: cross-sections:

text: spin-

text: curved

text: surface

text: of

text: the

text: germanium

text: crystal

text: and

text: therefore

text: independent

text: (SI,

text: scalar)

text: and

text: spin-dependent

text: (SD,

text: axial-

text: do

text: not

text: produce

text: useful

text: signals.

text: The

text: higher

text: energy

text: γ

text: from

text: vector)

text: couplings:

text: and

text: at

text: top

text: and

text: side,

text: as

text: well

text: as

text: those

text: from

text: physics

text: samples

text: (BC

text: AC+

text: and

text: BC

text: AC

text: show

text: dσ

text: dσ

text: dσ

text: similar

text: distributions

text: in

text: τ,

text: independent

text: of

text: locations.

text: The

text: χN

text: χN

text: χN

text: shift

text: in

text: (εBS,

text: λBS)

text: based

text: on

text: calibration

text: source

text: data

text: at

text: dif-

text: dER

text: dER

text: SI

text: dER

text: SD

text: ferent

text: locations

text: is

text: less

text: than

text: corresponding

text: to

text: a

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##