Tor sheaves: what do they tell us about geometry - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T01:59:48Z http://mathoverflow.net/feeds/question/90970 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/90970/tor-sheaves-what-do-they-tell-us-about-geometry Tor sheaves: what do they tell us about geometry Rurik 2012-03-12T08:25:14Z 2012-03-12T15:08:44Z <p>Hi!</p> <p>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.</p> <p>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 $Z$).</p> <p>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. </p> <p>Thank you very much for your time and your attention,</p> <p>Best</p> http://mathoverflow.net/questions/90970/tor-sheaves-what-do-they-tell-us-about-geometry/90978#90978 Answer by Karl Schwede for Tor sheaves: what do they tell us about geometry Karl Schwede 2012-03-12T12:18:52Z 2012-03-12T14:44:23Z <p>$\text{Tor}(O_Z, O_B)$ certainly tells you about the intersection of $Z$ and $B$ and is supported on $Z \cap B$ (essentially by definition).</p> <p>One common interpretation of these sheaves is in <a href="http://en.wikipedia.org/wiki/Intersection_theory" rel="nofollow">intersection theory (on wikipedia)</a>. In particular, they are used to compute the intersection number. See for example Chapter 20 of W. Fulton's book <em>Intersection Theory</em>. </p> http://mathoverflow.net/questions/90970/tor-sheaves-what-do-they-tell-us-about-geometry/90992#90992 Answer by Hailong Dao for Tor sheaves: what do they tell us about geometry Hailong Dao 2012-03-12T15:08:44Z 2012-03-12T15:08:44Z <blockquote> <p>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 $B$?</p> </blockquote> <p>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! </p> <p>See Serre's Local Algebra book, V.6, Theorem 4, p 110 (this great book also contains the intersection formula that Karl mentioned). </p> <p>More precisely, Auslander's ICM 1962 note (available on <a href="http://mathunion.org/ICM/ICM1962.1/" rel="nofollow">this page</a>, 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. </p>