Algebraic Geometry seminar: Thursday, 26 April at 11.30 am.

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Host: Sudarshan Gurjar : Venue: Ramanujan Hall

Speaker: Nitin Nitsure : Affiliation: TIFR (retd)

Title: Derived Categories and derived functors: a first look-2

Abstract: The category of abelian groups, and that of modules over a ring is part of common knowledge. These are examples of Abelian categories.  More advanced examples include sheaves of abelian groups on a topological space, quasi-coherent sheaves on a scheme, etc. Functors such as Hom, tensor product or global sections on these categories are not exact but have higher derived functors, such as Ext, Tor, and sheaf cohomology, which give long exact sequences that are basic tools in both theory and computations. A more evolved version of abelian categories and derived functors are the so-called `derived categories' made from them, which support a corresponding version of derived functors. These are indispensable in various topics in mathematics, such as duality theory, deformation theory, D-modules, etc. At first sight, this subject appears to be forbiddingly technical. These two lectures are a `first look' at the subject. We will focus on conceptual points and simple examples so that the students can go on to read the more complete technical accounts in the literature with
greater ease.