What is Serre's condition (S_n) for sheaves?

Question

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).

Answer 1

I would add one more observation to the other comments. Let me not worry about (2) vs (3) as the difference is only about the zero module so this is more of a philosophical question than a mathematical one.

I would just like to point out that there is a very useful characterization of depth and dimension of a module, namely Grothendieck's vanishing theorem which says that at any $x\in X$, the local cohomology of $M$ vanishes for $i$ smaller than the depth or larger than the dimension of the module and does not vanish for $i$ equal either the depth or the dimension.

In my eyes this suggest that one should use the dimension of the module in the definition, i.e., use (2).

Another argument to support the use of (2) is that we like to say that CM is equivalent to "$S_n$ for all $n$". Now if you use definition (1) then only modules supported on the entire $X$ have even a chance to be CM, but I don't see how one would gain from assuming that. More specifically, a module could never satisfy $S_n$ for any $n$ that's larger than the dimension of the module, but not larger than $\dim X$.

Kind of along the same lines, let $A\to B$ be a surjective morphism of rings (commutative with an identity) and $M$ a $B$-module. I.e., $\operatorname{Spec}B$ is a closed subset of $\operatorname{Spec}A$. Now both $\operatorname{depth}M$ and $\operatorname{supp}M$ are independent of the fact whether one views $M$ as a $B$-module or an $A$-module. It is reasonable that then whether it is $S_n$ would be also independent.

The main difference between (1) and (2) is whether one wants to compare to the support of the module (i.e., view it over ring/annihilator) or the whole ring. To me, the former seems more natural. This way a sheaf/module that is $S_n$ on a subscheme remains $S_n$ when viewed on an ambient scheme. The definition (1) seems to prefer to compare to the fixed ring. One way some people try to bridge the gap between the two definitions is to say "$M$ is $S_n$ over its support", meaning that one should mod out by annihilator first before applying (either of the) definition(s). Then the two definitions are equivalent. As for (3), some people go the distance to say "a non-zero module is $S_n$ if..."

Answer 2

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 $(\mathrm{ii}_n)$.