KUMUNU 2018 Commutative Algebra
Benjamin Drabkin, University of Nebraska
Symbolic Defect and Cover Ideals
Let R be a commutative Noetherian ring, and let I be an ideal in R. The symbolic defect is a numerical measurement of the difference between the symbolic and ordinary powers of I. In the case that I has sufficiently well-behaved symbolic powers (i.e. its symbolic Rees algebra is finitely generated) we prove that the symbolic defect of I grows eventually quasi-polynomially in m. Furthermore, we describe more specifically the growth of the symbolic defect in certain classes of ideals arising from combinatorial structures.
Alessandra Costantini, Purdue University
On the Cohen-Macaulayness of the Rees Algebra of the Module of Differentials
Let R be a Noetherian ring, E a finite R-module having a rank e (i.e. E is free of rank e locally at every associated prime). We say that E satisfies condition Ft if the minimal number of generators of Ep is at most dim(Rp) + e − t for every prime ideal p such that Ep is not free. It is known by work of Avramov, Huneke, and Simis and Vasconcelos that if E has projective dimension one and satisfies F1, then the Rees algebra of E is Cohen-Macaulay. While the converse does not hold in general, Simis, Ulrich and Vasconcelos proved it holds in the case when E is the module of differentials of a complete intersection over a field of characteristic zero, satisfying F0. It is an open question whether the assumption that E is F0 can be removed. I will describe how to reduce the problem to a linear algebra question about the presentation matrix of E. This is a joint work in progress with Tan Dang.
Brent Holmes, University of Kansas
A Generalized Serre Condition
We generalize Serre's condition and prove generalizations of results for rings satisfying Serre's condition. We examine an equivalent functorial condition, bounds on cohomological dimension, resolution of the Alexander dual, and a generalization of Reisner's criterion for our generalization.
Toshinori Kobayashi, Nagoya University
Syzygy Modules of Cohen-Macaulay Modules over One-dimensional Cohen-Macaulay Local Rings
The main object is the syzygy modules of maximal Cohen-Macaulay modules over a local ring. I will explain charaterizations of local rings of Krull dimension one satisfying the almost Gorenstein property or having minimal multiplicity, in terms of those syzygy modules.
Whitney Liske, University of Notre Dame
Defining Equations of Rees Algebras for a Family of Gorenstein Ideals
Let R=k[x1, ..., xd] for d>3 and let I be a Gorenstein ideal submaximally generated by quadrics. This poster will describe the defining equations of the Rees ring and the special fiber ring in two ways. First, the defining equations will be given as a sub-ideal of the 2x2 minors of a symmetric matrix in a similar way to the defining equations for the ideal m . Second, thedefining equations will be given in a more classical way relating them to the maximal minors of the Jacobian dual.
Justin Lyle, University of Kansas
Cohen-Macaulay Rings with Finite Non-Locally Free Type
We examine Cohen-Macaulay local rings with only finitely many nonisomorphic indecomposable maximal Cohen-Macaulay modules which are not locally free on the punctured spectrum of R.
Josh Pollitz, University of Nebraska
The Derived Category of Locally Complete Intersections
Let R be a commutative noetherian ring. It is well known that R is regular if and only if every complex with finitely generated homology is a perfect complex. The goal of this talk is to explain how one can characterize whether R is locally a complete intersection in terms of how each complex with finitely generated homology relates to the perfect complexes. Namely, R is locally a complete intersection if and only if each nontrivial complex with finitely generated homology can build a nontrivial perfect complex in the derived category using finitely many cones and retracts. This characterization gives a completely triangulated category characterization of locally complete intersection rings. In this talk, we will introduce a theory of support varieties and discuss how they can be applied to yield this characterization.
Tony Se, University of Mississippi
Semidualizing Modules of Ladder Determinantal Rings
Let R be a Cohen-Macaulay ring. The number of isomorphism classes of the semidualizing modules of R, in some sense, measures how far R is from being Gorenstein. We will show how to determine the isomorphism classes of the semidualizing modules of an arbitrary ladder determinantal ring. This is joint work with Sean Sather-Wagstaff and Sandra Spiroff.