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Hi Hailong,

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.

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., ${\rm Spec}\, B$ is a closed subset of ${\rm Spec}\, A$. Now both ${\rm depth}\, M$ and ${\rm supp}\, M$ 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 second 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..."
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Hi Hailong,

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 $i$ strictly between the depth and 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 out to 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 $\dim X$.
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Hi Hailong,

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 $i$ strictly between the depth and 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 out to 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 $\dim X$.