KUMUNU 2018 Commutative Algebra
Raymond Heitmann, University of Texas
Extended Plus Closure in Complete Local Rings
The extended plus closure was defined as an attempt to provide a mixed characteristic analogue of tight closure. It was shown to possess a number of desirable properties. In fact, the demonstration that it possessed the colon-capturing property for excellent rings of dimension three was instrumental in proving the direct summand conjecture in that case.
More recently, we have had success in the reverse direction. Employing Andre’́s perfectoid algebra techniques, Linquan Ma and myself have shown that the extended plus closure has the colon-capturing property for mixed characteristic complete local domains of arbitrary dimension. One may paraphrase this by saying that when we embed R in its absolute integralclosure R+, obstructions to Cohen-Macaulayness become almost zero. We also obtained results of this nature for local cohomology and Tor. For example, there exists a nonzero element c ∈ R such that the map 𝑐∈ 𝐻im (𝑅) → 𝐻 (𝑅+) is zero for every positive rational number ∈ and every i less than the dimension of R.
Jack Jeffries, University of Michigan
Quantifying Singularities with Differential Operators
Rings of positive characteristic make some people squeamish, due to fears of inseparability, and the fact that the rings coming most directly from geometry have characteristic zero. However, rings of positive characteristic have the Frobenius endomorphism, which allows for many tools to study singularities. One such tool, the F-signature of a local ring of positive characteristic, is a numerical invariant that detects many interesting properties. For example, this invariant detects (non)singularity and strong F-regularity. However, it is very difficult to compute.
In this talk, we define a numerical invariant for rings of characteristic zero or p>0 that exhibits many of the useful properties of the F-signature. We also compute many examples of this invariant, including cases where the F-signature is not known. Our definition is motivated by the philosophy that differential operators often serve as a characteristic-free substitute for the Frobenius map. This is based on joint work with Holger Brenner and Luis Núñez-Betancourt.
Kyungyong Lee, University of Nebraska
How Cluster Algebras Helped Understand Indecomposable Modules Over Short Gorenstein Rings
There has been some interest in explicitly constructing indecomposable modules over rings of wild representation type. In this talk, we consider short Gorenstein rings. Motivated by a description for the generators of cluster algebras, we give an explicit expression of an indecomposable module for every possible Hilbert function. This is a joint work with Lucho Avramov and Milen Yakimov.
Jason McCullough, Iowa State University Singularities of Rees-like Algebras
Rees-like algebras were defined by I. Peeva and the speaker to define counterexamples to the Eisenbud-Goto regularity conjecture. A second process, called Step-by-step homogenization, was used to transfer properties of the non-standard graded Rees-like algebra to the standard graded setting. The resulting projective varieties are singular projective varieties whose regularity exceeds their degree. These varieties are necessarily not arithmetically Cohen-Macaulay nor normal. A natural question this is: How singular are they? In this talk I will present ongoing joint work with P. Mantero and L. Miller on (1) the size of singular locus of Rees-like algebras—before and after homogenization, (2) a better way to homogenize, (3) criteria for F-purity and seminormality, and (4) structure of the canonical module.
Ryo Takahashi, Nagoya University
When is Every Ideal Isomorphic to Some Trace Ideal?
Let R be a commutative noetherian ring. Lindo and Pande have recently posed the question asking when every ideal of R is isomorphic to some trace ideal of R. In this talk we will consider it and give some answers. In fact, some classes of Gorenstein rings, hypersurfaces and unique factorization domains will appear. This talk is based on joint work with Toshinori Kobayashi.
Parangama Sarkar, University of Missouri
Mixed Multiplicities of Filtrations
We define mixed multiplicities of (not necessarily Noetherian) filtrations of m-primary ideals in a Noetherian local ring (R,m), generalizing the classical theory for m-primary ideals. We construct a real polynomial whose coefficients give the mixed multiplicities. This polynomial exists if and only if the dimension of the nilradical of the completion of R is less than the dimension of R. We show that many of the classical theorems for mixed multiplicities of m- primary ideals hold for filtrations, including the famous Minkowski inequalities of Teissier, and Rees and Sharp. This is joint work with Steven Dale Cutkosky and Hema Srinivasan.