UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

February 03, 2015

939 Evans Hall

3:45PM: K-theory of toric varieties revisited

Joseph Gubeladze

In the talk we report on recent progresses in understanding higher K-theory of general toric varieties, accomplished in a series of works of several people. In the second half of the talk we will discuss a conjectural description of higher K-groups of these varieties, representing a far reaching - in a sense the ultimate extension of the known results. In general terms, the theory develops around controlling the failure of homotopy invariance of Quillen's theory and the conjecture is a multi-graded refinement of the previously known results. The starting point is the positive results for the Grothendieck group of vector bundles on toric varieties, known since the 1980s.

5:00PM: The vanishing of Tor over complete intersections

Roger Wiegand

Let $(R,\mathfrak m)$ be a Noetherian local ring, and let $M$ and $N$ be non-zero finitely generated $R$-modules. One says that $M$ and $N$ are {\em Tor-independent} provided $\Tor^R_i(M,N)= 0$ for every $i>0$. In this talk we will seek conditions on $M$, $N$, and $M\otimes_RN$ that force $M$ and $N$ to be Tor-independent. One reason for seeking such conditions is that there are many situation in which Tor-independence implies the {\em depth formula}: \begin{equation*} \depth M + \depth N = \depth (M\otimes_RN) + \depth R \end{equation*} Auslander proved, more than 50 years ago, that Tor-independence implies the depth formula if one of the modules has finite projective dimension. About 20 years ago, Huneke and I proved that Tor-independence implies the depth formula if $R$ is a complete intersection (a local ring of the form $S/(\underline f)$, where $(\underline f) = (f_1,\dots,f_c)$ is a regular sequence). (There are no known examples where Tor-independence does {\em not} imply the depth formula.) The talk will be guided by the following \begin{conjecture}Suppose that $R = S/(\underline f)$ as above and, in addition, that $R$ is a domain. If $M\otimes_RN$ satisfies Serre's condition $(S_{c+1})$, then $M$ and $N$ are Tor-independent (and hence the depth formula holds). \end{conjecture} The case $c=0$ (that is, $R$ is a regular local ring) was proved by Auslander and Lichtenbaum in the sixties. The case $c=1$ was proved by Huneke and me in our 1994 paper. I will discuss some new tools for attacking this problems and give some positive results. This research is joint work with Olgur Celikbas, Srikanth Iyengar, and Greg Piepmeyer.

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