Commutative Algebra Seminar

Fall 2018, Friday 2:30–3:20, LCB 215

Date Speaker Title — click for abstract
August 24 Michael Brown
University of Wisconsin, Madison
Chern-Weil theory for matrix factorizations
This is a report on joint work with Mark Walker. Classical algebraic Chern-Weil theory provides a formula for the Chern character of a projective module P over a commutative ring in terms of a connection on P. In this talk, I will discuss an analogous formula for the Chern character of a matrix factorization. Along the way, I will provide background on matrix factorizations, and also on classical Chern-Weil theory.
August 31 Ben Briggs
University of Utah
Lusternik-Schnirelmann category in commutative algebra
L-S category is a numerical invariant originating in differential geometry, which has since found an important place in rational homotopy theory. In the early 80s commutative algebraists and rational homotopy theorists began to uncover deep parallels between their disciplines, greatly enriching both sides. Constructions and theorems on one side were found to have "looking-glass" analogues on the other. For example, L-S category can be defined for local rings or homomorphisms, and this invariant has been used to prove important results in local commutative algebra. Nonetheless, L-S category has not been well studied in commutative algebra in the years since. I will talk about an attempt to develop some of the properties of this invariant. In particular, I'll present a looking glass analogue of the mapping theorem, which is invaluable for computing L-S category in rational homotopy theory. I will also be very careful to explain any topology I might want to use.
September 7 Thomas Polstra
University of Utah
Prime characteristic invariants and birational maps
We explore the behavior of prime characteristic invariants, such as F-signature and Hilbert-Kunz multiplicity, under birational morphisms. We will discuss both positive and negative behavior.
September 14 Jenny Kenkel
University of Utah
Local cohomology of thickenings
Let R be a standard graded polynomial ring that is finitely generated over a field, and let I be a homogenous prime ideal. Bhatt, Blickle, Lyubeznik, Singh, and Zhang examined the local cohomology of R/I^t, as t goes to infinity, which led to the development of an asymptotic invariant by Dao and Montanõ. I will discuss their results, and give concrete examples of the calculation of this new invariant in the case of determinantal rings.
September 21 Henning Krause
University of Bielefeld
The Morita theory for derived categories revisited
A fundamental theorem of Rickard states for a pair of coherent rings that the categories of perfect complexes are triangle equivalent if and only if their bounded derived categories of finitely presented modules are triangle equivalent. The proofs of both directions are somewhat delicate and not treated properly in any text book. The aim of this talk is to propose new proofs, which should be of independent interest; they involve the notion of Cauchy completion for one direction, and the notion of a distinguished t-structure for the other direction.
September 28
October 5 Rebecca R.G.
George Mason University
October 12
October 19
October 26 Cris Negron
M.I.T., Boston
November 2 Jon Carlson
University of Georgia, Athens
November 9 Linquan Ma
Purdue University
November 16 Lars Winther Christensen
Texas Tech., Lubbock
November 30 Robert M. Walker
University of Michigan

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