TITLE: What is Serre's condition (S_n) for sheaves? 
QUESTION [37 upvotes]: The Serre's condition $(S_n)$, especially $(S_2)$, has been mentioned in a few MO answers: see  here and here for example. I am pretty sure I have seen it in other questions as well, but could not remember exactly. 
I have always been confused by this condition, especially for a sheaf $F$ on a locally Noetherian scheme $X$, mainly because as far as I know, there are at least three different definitions in the literature. Let us look at them, $F$ is said to satisfy condition $(S_n)$ if:
$$ depth_x (F_x) \ge \min (\text{dim} \mathcal O_{X,x},n) \  \forall x\in X  \ (1) $$ 
$$ depth_x (F_x) \ge \min (\dim F_x,n) \ \forall x\in X  \ (2)  $$ 
$$ depth_x (F_x) \ge \min (\dim F_x,n) \ \forall x\in \text{Supp}(F) \ (3) $$ 
Definition (1) can be found  in Evans-Grifffith book "Syzygies", definition (2) is given in EGA IV (definition 5.7.2) or Bruns-Herzog book "Cohen-Macaulay modules". Definition (3) is  what VA used in his answer to the second question quoted above (and I certainly have seen it in papers or books, but can't locate one right now, so references would be greatly appreciated). 
When $F$ is the structure sheaf or a vector bundle (of constant positive rank), then they all agree. However, they can differ when $E$ is a sheaf. For example, (1) allows us to say that if $X$ is normal, then $E$ is reflexive if and only if it is $(S_2)$. But according to (2) or (3), if $X=Spec(R)$ for $(R,m)$ local, then $k=R/m$ would satisfy $(S_n)$ for all $n$. (2) and (3) are equivalent if we assume that the depth of the $0$ module is infinity, but I have seen papers which do not use that convention, adding to the confusion. 
Since a result surely depends on which definition we use (I have certainly made mistakes because of this confusion, and I think I am not alone), I would like to ask:
Question: Is there an agreement on what exactly is condition $(S_n)$ for sheaves? If not, what are the advantages and disadvantage for each of the different definition? 
Some precise references: 
Bruns-Herzog "Cohen-Macaulay modules" : after Theorem 2.1.15, version (2)
Evans-Griffith "Syzygies": part B of Chapter 0, version (1)
Kollar-Mori "Birational geometry of algebraic varieties": definition 5.2, version (1)
EGA, Chapter 4, definition 5.7.2, version (2).

REPLY [3 votes]: The following is too long for a comment, but I hope it is useful. I would appreciate any additional information anyone might have concerning these notions and their history.
As far as I can tell, definition (1) for $(S_n)$ first appeared in Definition 1.1 of the following paper of Vasconcelos:

Wolmer V. Vasconcelos, Reflexive modules over Gorenstein rings, Proc. Amer. Math. Soc. 19 (1968), 1349–1355. DOI: 10.2307/2036210. MR: 237480.

In this paper, Vasconcelos shows that reflexive modules behave particularly well on rings satisfying $(G_1)$ and $(S_2)$ (in the notation of [Hartshorne 1994, p. 291]), which Hartshorne utilized later in a geometric context. Here, we note that rings satisfying $(G_1)$ and $(S_2)$ are also known as $(G_2)$ rings (following [Ischebeck 1969, Definition 3.16]) or 2-Gorenstein rings (following Auslander; see [Reiten and Fossum 1972, p. 35]).
One reason to use definition (1) for $(S_n)$ is the following:
Theorem (Samuel; see [Malliavin 1968, Proposition 2]). Let $A$ be a noetherian local ring and let $M$ be a finitely generated $A$-module. For every integer $n \ge 0$, the following are equivalent:

$(\mathrm{i}_n)$ For every prime ideal $\mathfrak{p}$ in $A$, we have $\operatorname{depth}(M_\mathfrak{p}) \ge \min\{n,\operatorname{depth}(A_\mathfrak{p})\}$.
$(\mathrm{ii}_n)$ Every regular sequence on $A$ of length $\le n$ is a regular sequence on $M$.

Note that if $A$ satisfies $(S_n)$, then $M$ satisfies $(\mathrm{i}_n)$ if and only if $M$ satisfies $(S_n)$ in the sense of definition (1). In fact, definition (1) for $(S_n)$ is exactly how Samuel originally stated the condition $(\mathrm{i})_n$ in [Samuel 1964, Proposition 6].
In [Ischebeck 1969, Satz 4.4 and Satz 4.6], Ischebeck proved that Samuel's conditions $(\mathrm{i})_n$ and $(\mathrm{ii})_n$ are also equivalent to $M$ being an $n$-th syzygy module under the assumption that $A$ satisfies $(G_n)$ in the sense of [Ischebeck 1969, Definition 3.16].
Ischebeck denotes Samuel's conditions $(\mathrm{i}_n)$ and $(\mathrm{ii}_n)$ by $(\mathrm{b}_n)$ and $(\mathrm{a}_n)$, respectively. Because of the Theorem above, Malliavin says that a module satisfies Samuel's condition $(\mathrm{a}_n)$ if it satisfies either of the equivalent conditions $(\mathrm{i}_n)$ or $(\mathrm{ii}_n)$.