# MathCS Seminar

Title: The Foxby-morphism and derived equivalences
Seminar: Algebra
Speaker: Satya Mandal of University of Kansas
Contact: David Zureick-Brown, dzb@mathcs.emory.edu
Date: 2015-02-26 at 4:00PM
Venue: W306
Abstract:
Suppose $X$ is a quasi-projective scheme over a noetherian (Cohen-Macaulay) affine scheme $Spec(A)$, with $dim X=d$. In $K$-theory and related areas (Witt theory, Grothendieck-Witt theory), bounded chain complexes $G_{\bullet}$ of Coherent sheaves or locally free sheaves play an important role. One considers the category $Ch^b(Coh(X))$ (resp. $Ch^b(V(X))$) of bounded chain complexes of coherent sheaves (resp. of locally free sheaves). One also considers, the corresponding derived categories $D^b(Coh(X)$, $D^b(V(X))$, which is obtained by inverting the quasi-isomorphisms in the chain complex categories. \vspace{4pt} Given a chain complex map $L_{\bullet}\to G_{\bullet}$, between two complexes $L_{\bullet}$, $G_{\bullet}$, with extra information on homologies, one complex can be viewed as \emph{an approximation to} the other. Given one such complex $G_{\bullet}$, constructing such a complex $L_{\bullet}$, with desired properties, and constructing a map $L_{\bullet}\to G_{\bullet}$ would be challenging. In the affine case $X=Spec(A)$, such a map was constructed by Hans-Bjorn Foxby (unpublished), several other versions of the same was given by others. In this lecture we implement the construction of Foxby to quasi-affine case and give applications. Intuitively, one can look at this implementation as a "graded" version of Foxby's construction.