We discuss some history of motivations for and issues arising in transferring algebra structures on complexes along homotopy equivalences. Then we discuss how, under suitably nice homotopy equivalences, a dg-algebra structure descends. Lastly, we discuss a concrete application: we use the homotopy on the tautological Koszul complex given by the weighted de Rham map to build a concrete permutation invariant dg-algebra structure on a well-known resolution. We mention briefly a homological tool called the perturbation lemma used in this application to obtain the needed homotopy equivalencies from the de Rham homotopy. This is joint work with Hamid Rahmati.
We show that complete local normal domains satisfy the uniform symbolic topology property if local hypersurface rings have uniform symbolic topologies at regular primes. We also discuss two types of bootstrapping results related to the uniform symbolic topology property. This is joint work with Craig Huneke.
Let R be a commutative 1 dimensional domain whose integral closure is a DVR. For the first part of this talk, we assume further that R is a complete k-algebra where k is any field. Under this hypothesis, there is a long standing conjecture of R.W. Berger which states that R is regular if and only if the (universally finite) differential module Ω_{R/k} is torsionfree. We introduce an invariant of Ω_{R/k} and discuss some lower bounds on it which would give some new cases of this conjecture. In particular, we can generalize the quasihomogeneous case proved by Scheja. Finally, we explore some relations between this invariant and the colength of the conductor.
We show that the Koszul homology algebra of a quotient by the edge ideal of a forest is generated by the lowest linear strand. This provides an answer, for such rings, to a question of Avramov about the Koszul homology algebra of Koszul algebras. We also recover a result of Roth and Van Tuyl on the graded Betti numbers of quotients of edge ideals of trees.
It has been an open problem since the 1960s to construct closed-form, canonical, combinatorial minimal free resolutions of arbitrary monomial ideals in polynomial rings. This talk explains how to solve the problem, in characteristic 0 and almost all positive characteristics, using sums over lattice paths of combinatorial data from simplicial complexes, one simplicial complex for each lattice point. Any minimal free resolution of any monomial ideal must — either implicitly or explicitly — produce homomorphisms between various homology groups of these simplicial complexes. Therefore an important aspect of the solution is an explicit way to write down canonical homomorphisms between these homology groups without choosing bases. This is joint work with John Eagon and Ezra Miller.
In many ways the bounded complexes of finitely generated projective modules control the regular behaviour of a local ring R. These are sometimes called the "small objects" of D(R). The Auslander-Buchbaum-Serre theorem implies that R is regular iff every object of D^{b}(R) is small. I will talk about the class of "proxy-small objects" in D(R), and how it's becoming apparent that these objects control the complete intersection behaviour of R. By results results of Dwyer, Greenlees, and Iyengar, and Pollitz, R is a complete intersection iff every object of D^{b}(R) is proxy-small. I will explain how to generalise this to the relative situation: a local homomorphism f: R → S is complete intersection iff the proxy-small property "ascends along f". As an application, we can deduce an important result of Avramov about factorisations of complete intersection homorphisms. This is joint work with with Srikanth Iyengar, Janina Letz, and Josh Pollitz.
Let S be a complete intersection presented as R/J for R a regular ring and J a parameter ideal in R. Let I be an ideal containing J. It is well known that the set of associated primes of H^{i}_{I}(S) can be infinite, but far less is known about the set of minimal primes. In 2017, Hochster and Núñez-Betancourt showed that if R has prime characteristic p > 0, then the finiteness of Ass H^{i}_{I}(J) implies the finiteness of Min H_{I}^{i-1}(S)(S), raising the following question: is Ass H^{i}_{I}(J) always finite? We give a positive answer when i = 2 but provide a counterexample when i = 3. The counterexample crucially requires Ass H^{2}_{I}(S) to be infinite. The following question, to the best of our knowledge, is open: (under suitable hypotheses on R) does the finiteness of Ass H^{i-1}_{I}(S) imply the finiteness of Ass H^{i}_{I}(J)? When S is a domain, we give a positive answer II when i = 3. When S is locally factorial, we extend this to i = 4. Finally, if R has prime characteristic p > 0 and S is regular, we give a complete answer by showing that Ass H^{i}_{I}(J) is finite for all i ≥ 0.
Ideals for studying algebraic-statistical models often have a rich algebraic structure, reflected, for example, in a large symmetry group. It is natural to expect that the properties of the ideals remain somewhat invariant when one preserves the structure of the model, but changes one of its parameters. We discuss an algebraic framework in order to capture such phenomena. Related questions arise in representation theory and topology. A starting point are ascending chains of symmetric ideals in more and more variables. Finiteness results in a suitable category lead to a uniform description of properties of all but finitely many ideals in such a chain. This is mainly based on joint work with Tim Römer.
A commutative noetherian local ring R is homologically persistent if every finitely generated R-module M with Tor^{R}_{I >> 0}(M,M) = 0 has finite projective dimension. In this talk, we discuss some of the recent progress on tracking the homologically persistence property along deformations with some of its applications. This talk is based on an in-progress joint work with L. Avramov, S. Iyengar, and S. Sather-Wagstaff.
Avramov, Iyengar, Nasseh, and Sather-Wagstaff show how the existence of a DG-algebra structure on a minimal resolution can be used to prove a strong Tor-rigidity result. We give an explicit construction of DG algebra resolutions for certain products of ideals. Moreover, we give sufficient conditions for this construction to be minimal and, in other cases, show how the construction can be reduced to a minimal DG algebra resolution.
The relationship between tight closure and its test ideal can be described in terms of a duality between a closure operation and a corresponding interior operation. We discuss a duality between closure operations, interior operations, and test ideals over complete local rings, and give applications of this technique to trace ideals and the core of an ideal with respect to certain closure operations. This is joint work with Neil Epstein and Janet Vassilev.
We say that a Cohen-Macaulay local ring has finite CM_{+}-representation type if there exist only finitely many isomorphism classes of indecomposable maximal Cohen-Macaulay modules that are not locally free on the punctured spectrum. In this talk, we consider finite CM_{+}-representation type from various points of view, relating it with several conjectures on finite/countable Cohen-Macaulay representation type. We prove in dimension one that the Gorenstein local rings of finite CM_{+}-representation type are exactly the local hypersurfaces of countable Cohen-Macaulay representation type; under mild hypotheses these are exactly the hypersurfaces of type (A_{∞}) and (D_{∞}). We also consider partial results and obstacles in higher dimension. This is joint work with Toshinori Kobayashi and Ryo Takahashi.
We study the resolution of R/(x^{N}, y^{N}, z^{N}, w^{N}) over the hypersurface ring R = k[x,y,z,w]/(x^{n} + y^{n} + z^{n} + w^{n}), and the related resolution of P/(x^{N}, y^{N}, z^{N}, w^{N}) : (x^{n} + y^{n} + z^{n} + w^{n}), where P = k[x,y,z,w] and k is a field of characteristic zero. This is joint work with Andrew Kustin and Rebecca R.G..
Let R be a graded commutative ring generated over a field by finitely many homogeneous elements of positive degree. Work of Avramov, Buchweitz and Sally shows that, under certain assumptions, the Laurent series expansions around 1 of the Hilbert series of two graded R-modules and those of some of their Ext modules are related. When R is a complete intersection, we show that such relations are connected to the asymptotic behavior of the Ext modules. This is joint work with David A. Jorgensen and Peder Thompson.