@MISC{Yekutieli08dualityin, author = {Amnon Yekutieli}, title = { Duality in Noncommutative Algebra and Geometry }, year = {2008} }

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Abstract

Duality is one of the fundamental concepts in mathematics. The most basic duality is that of linear algebra. We take a vector space V over a field K and assign to it V ∗ = HomK(V,K). If V is finite dimensional then V ∼ = V ∗∗. This can be generalized in many ways. For instance we can make V infinite, and then often we impose a topology to retain reflexivity (e.g. a Banach space). Now change the ring of coefficients: instead of the field K take a commutative ring A, and instead of a vector space V we consider an A-module M. Immediately we run into difficulties. Example 1.1. Let us look at A: = Z, the ring of integers. A Z-module is just an abelian group. There are two distinct dualities for finitely generated abelian groups. For a finite group H the dual is H ∗ = HomZ(H,Q/Z), whereas for a free group G the dual is G ∗ = HomZ(G,Z). 1 2(We want to stay within finitely generated groups, so the Pontryagin dual HomZ(G,R/Z) is ruled out.) We know that G∗ ∗ ∼ = G and H∗ ∗ ∼ = H. Is it possible to unite these two dualities into one? What about more complicated rings? 2. The Derived Category This is where the derived category enters. The idea of Grothendieck and Verdier (see [RD]) was to work with complexes of modules. A complex of A-modules consists of M = · · · →M i−1 di−1 −−−→M i di −→M i+1 → · · · where the M i are A-modules, the di are A-linear maps, and di ◦ di−1 = 0. The module M i is called the degree i component of M. The derived category D(ModA) has complexes as objects. Morphisms in D(ModA) are more complicated to explain. For any i, the i-th cohomology of a complex M is the module