# Tor sheaves: what do they tell us about geometry

Hi!

I fear that I am up to ask a very vague question, but more than an answer I need a suggestion of references I should look up.

I need to know everything about Tor sheaves and what do they tell about geometry. For example if $X$ is a smooth variety and $Z$ and $B$ are subvarieties,where are the sheaves $\mathbf{Tor}_i(O_Z, O_B)$ supported? Does the vanishing of (some) of the higher Tor sheaves have some reflection on the mutual geometry of $Z$ and $B$ (for example can we evince the dimension of the intersection of $B$ and $Z$).

As I already explain you do not need to answer this (probably very silly) questions. I just need someone to point the right book to me, since right know I have not the slightest idea of where to find these informations.

Best

• As an aside, I found the answer to this question interesting mathoverflow.net/questions/89683/…. (it's not much, as the relation $Hom(E,F) = E^\vee \otimes F$ is only nice for E a vector bundle, but hey...) – Yosemite Sam Mar 12 '12 at 10:24

$\text{Tor}(O_Z, O_B)$ certainly tells you about the intersection of $Z$ and $B$ and is supported on $Z \cap B$ (essentially by definition).

One common interpretation of these sheaves is in intersection theory (on wikipedia). In particular, they are used to compute the intersection number. See for example Chapter 20 of W. Fulton's book Intersection Theory.

For example if $X$ is a smooth variety and $Z$ and $B$ are subvarieties,where are the sheaves $\mathbf{Tor}_i(O_Z, O_B)$ supported? Does the vanishing of (some) of the higher Tor sheaves have some reflection on the mutual geometry of $Z$ and $B$?

In fact just the support or vanishing of the higher Tor sheaves can tell you quite a bit about the geometry. When $Z,B$ intersect properly and $Tor_1=0$ (which is actually equivalent to all the higher $Tor$ vanish) then you can even say that $Z,B$ are Cohen-Macaulay!

See Serre's Local Algebra book, V.6, Theorem 4, p 110 (this great book also contains the intersection formula that Karl mentioned).

More precisely, Auslander's ICM 1962 note (available on this page, first in Section 2) describes the support of the $Tor_i(Z,B)$ for any two sheaves over a regular Noetherian scheme (see Theorem 2). He only stated it for the unramified local case, but we now know it in full generality. A summary of his result: the support of Tor only depends on the depths of $Z,B$ at the stalks.

• Thank you very much both do Hailong and Karl. It seems to me that many results holds just for subvarieties and not for subschemes. Am I correct? I have a smooth varieties of dimension 5, $X$ and two subschemes $B$ (reduced irred of dimension 3 and $Z$ (pure of dimension 3 (but it could have embedded components of lower dimension) and it has at least 2 non-embedd copts and one of them does not intersect $B$). They intersect properly. The higher tor's of their strct sheaves vanish. It seems to me that could not somewhat be true? If it where true, would it tell me that both $Z$ and $B$ are CM? – Rurik Mar 14 '12 at 8:31