KUMUNU 2017 Commutative Algebra
Eric Canton, University of Nebraska
Asymptotic Invariants of Ideal Sequences in Positive Characteristic via Berkovich Spaces
I will describe my recent work extending results of Jonsson and Mustata regarding log canonical thresholds of graded sequences of ideals to the positive characteristic setting. The techniques are a blend of F-singularity machinery and non-Archimedean geometry.
Justin Chen, University of California, Berkeley
Flat Maps to and from Noetherian Rings
We investigate flat maps where the source or target is a Noetherian ring, giving necessary and/or sufficient conditions on a ring for such maps to exist. Along the way, we develop some general facts about flat ring maps, and exhibit many examples, including a new class of zero-dimensional local rings with nice properties.
Alessandra Costantini, Purdue University
Cohen-Macaulay Property of Rees Algebras of Modules
We provide a class of modules having Cohen-Macaulay Rees algebra. Our result generalizes a well-known result by Johnson and Ulrich, where the Rees algebra of an ideal I was shown to be Cohen-Macaulay under suitable assumptions on the reduction number of I and on the depths of powers of I.
Rankeya Datta, University of Michigan
Uniform Approximation of Abhyankar Valuation Ideals in Functions Fields of Prime C
Using the theory of asymptotic test ideals, we prove the prime characteristic analogue of a characteristic 0 result of Ein, Lazarsfeld and Smith on uniform approximation of valuation ideals associated to real-valued Abhyankar valuations centered on regular varieties over perfect fields.
Benjamin Drabkin, University of Nebraska
Symbolic Defect of Squarefree Monomial Ideals
A central area of interest in the study of symbolic powers of ideals is the relationship between symbolic and ordinary powers. The symbolic defect of an ideal is a numerical invariant which measures the difference between the m-th ordinary and symbolic powers of an ideal. More specifically, the m-th symbolic defect of an ideal, I, is the number of minimal generators of the m-th symbolic power of I quotiented by the m-th ordinary power of I. The growth of the symbolic defect of a monomial ideal can be studied through Presburger counting functions and various combinatorial constructions.
Zachary Flores, Colorado State University
The Weak Lefschetz for a Graded Module
Over a field of characteristic zero, it is well known that complete intersections in codimension 3 have the Weak Lefschetz Property. This result follows from a beautiful blend of commutative algebra and algebraic geometry. We discuss a modest generalization of this result by extending these techniques to investigate when the cokernel of a linear map between free modules over a polynomial ring in three variables has the Weak Lefschetz Property.
Brent Holmes, University of Kansas
Generalized Nerves and Depth
We introduce and study generalized notions of the nerve complex for a collection of subsets of a topological space. For simplicial complexes, we show that these notions afford an elegant and efficient characterization of the depth of the complex. We also prove several results relating these new complexes and Serre conditions.
Hang Huang, University of Wisconsin
Equations of Kalman Varieties
Given a subspace L of a vector space V, the Kalman variety consists of all matrices of V that have a nonzero eigenvector in L. We applied Kempf Vanishing technique with some more explicit constructions to get a long exact sequence involving coordinate ring of Kalman variety, its normalization and some other related varieties in characteristic zero. We also extracted more information from the long exact sequence including the minimal defining equations for Kalman varieties.
Takumi Murayama, University of Michigan
Frobenius-Seshadri Constants: Limits in Algebraic Geometry
We introduce higher-order variants of the Frobenius-Seshadri constant due to Mustata and Schwede. These are constants defined as a limit supremum involving both ordinary and Frobenius powers of ideals defining closed points on a projective variety, and seem to be related to Hilbert-Kunz multiplicity and F-signature. As applications, we use these constants to give geometric information about ample line bundles on projective varieties, and give a characterization of projective space. Both applications were previously known only in characteristic zero.
Josh Pollitz, University of Nebraska
Koszul Varieties and an Application in the Derived Category
The support varieties of Avramov and Buchweitz were used to show certain (co)homological properties of a complete intersection hold. One example is that, over a complete intersection, one can can show the eventual vanishing of Ext modules for a pair of modules is equivalent to the eventual vanishing of Tor modules for that same pair of modules using these varieties. I will discuss how to define a generalization of these varieties when your ring is not necessarily a complete intersection. Moreover, these varieties will be the same as the classical support varieties when the underlying ring is a complete intersection. I will also give a sketch of a result that recovers the aforementioned result of Avramov and Buchweitz over complete intersections, and I will present an application in the derived category of non-complete intersections.
Tony Se, University of Mississippi
Semidualizing Modules of Ladder Determinantal Rings
A semidualizing module is a generalization of the canonical module of a Cohen-Macaulay local ring. We aim to determine the semidualizing modules of so-called ladder determinantal rings.
William Taylor, University of Arkansas
Interpolating Between Hilbert-Samuel and Hilbert-Kunz Multiplicity
We present a function that interpolates continuously between the Hilbert-Samuel and Hilbert-Kunz multiplicities of ideals in a ring of positive characteristic, and show how it can be calculated using volumes of polyhedra in Euclidean space.