UC Berkeley

Commutative Algebra and Algebraic Geometry Seminar

September 23, 2014

939 Evans Hall

3:45PM: On limits in commutative algebra

Hai Long Dao

The Hilbert-Samuel multiplicity can be viewed as a limit of the colength of higher powers of the maximal ideal divided by a suitable numerical function. It has been studied intensively form both algebraic and geometric sides. In this talk I will describes a host of new limits that came up recently in commutative algebra. They often involve derived functors, or the Frobenius homomorphism (sometimes both). Proving that these limits exists are often non-trivial, but they do capture very subtle information. For example, they can tell us about torsions in the local Picard groups or numerical triviality of certain cycles.

5:00PM: Tate resolution on products of projective spaces

Frank-Olaf Schreyer

The Tate resolution of a coherent sheaf F on a single projective space is a double infinite complex of finitely generated graded modules over the exterior algebra, which incodes cohomology groups of all twists. It has many applications, e.g. resultant formulas, computations of Beilinson monads, characterization of higher regularity via linear monads. In this talk, I will report on joint work with David Eisenbud and Daniel Erman, which is a generalization to products of projective space. The Tate resolution on a product of projective spaces is no longer a complex of finitely generated modules. Nevertheless we have an algorithm which computes cohomology in any bounded range of twists. Applications include computation of direct image complexes of coherent sheaves along morphism between projective varieties.

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