Commutative Algebra Seminar

Frobenius thresholds are a family of invariants associated to singularities in positive characteristic. Though originally defined in terms of the splitting properties of Frobenius, Mustaţă, Takagi, and Watanabe showed in 2005 that Frobenius thresholds are closely related to the Bernstein-Sato polynomial and other invariants of singularity in characteristic zero. In this talk, we will present algorithms to compute Frobenius thresholds of a natural family of irreducible power series in two variables. These algorithms look at computing Frobenius thresholds as an integer programming problem. The integer programs discussed here build upon those recently developed by Hernández and Witt to compute the roots of the Bernstein-Sato polynomial for this family of curves.

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Let A be a Noetherian ring (not necessarily commutative). When is there a uniform upper bound on the projective dimensions of all (left) A-modules of finite projective dimension? When A is commutative, it follows from the works of Bass and Gruson-Raynaud, that this is the case if and only if A has finite Krull dimension. The question of whether such a uniform upper bound exists for Artin algebras, even when restricted to finitely generated modules only, was first publicized by Bass in the 1960s. This question, since known as the finitistic dimension conjecture, remains open even after half a century. In this talk, based on ongoing joint work with Jan Stovicek, we will present some criteria for the existence of such uniform upper bounds in terms of certain form of generation in singularity categories. One ingredient of our approach is based on a generalization of the "delooping level" of Gélinas.

Let I be a monomial ideal. The k'th squarefree power of $I$ is the ideal generated by the squarefree monomials in I^k. In this talk, we investigate the regularity and depth function of squarefree powers and consider the question of when such powers have linear resolution.

Let I be the maximal minors of generic matrix X in R=\mathbb{C}[X]. In this talk I will discuss the module structure, in particular that annihilators, of the R-modules H^i_\mathfrac{m}(R/I^t) and Ext^i(R/I^t,R).

Weakly F-regular rings (i.e. Noetherian rings in which all ideals are tightly closed) are among the mildest singularity types in prime characteristic commutative algebra. They are always normal, and under excellent hypotheses are Cohen-Macaulay. In this talk, I will explain how in the absence of excellence, weakly F-regular rings need not be Cohen-Macaulay or even catenary. As a consequence, we extend a theorem of Ogoma concerning the existence of non-catenary splinters of equal characteristic zero to prime characteristic. This is joint work with Susan Loepp.

A map in the derived category is ghost if it induces the zero map on homology. Ghost maps play an important role in understanding the homological behavior of modules. In this talk, we shift our perspective to commutative ring homomorphisms and introduce an analogous class of maps to this setting. The frobenius endomorphism will be a primary example of a ghost map. We begin with an overview of the necessary tools for handling derived functors in the category of rings, and use this framework to define ghost maps. We conclude with an analog of the ghost lemma in this setting and discuss applications to the study of singularities.

A landmark theorem of Orlov relates the derived category of coherent sheaves on a projective complete intersection to the singularity category of its affine cone. I'll discuss joint work with Michael K. Brown generalizing this result to dg-algebras.

Conjectures about the finite generation of various classes of rings in function fields have been instrumental in the development of algebraic geometry and commutative algebra. In this talk we will introduce one such conjecture in prime characteristic using a variant of F-regularity, which is a prime characteristic analog of KLT singularities. We will mention the connection of this problem to other long-standing questions in the theory of F-singularities. Our main goal is to discuss evidence in favor of the conjecture by considering the class of algebras determined by lattice points inside convex cones of finite dimensional vector spaces. This talk is based on joint work with Karl Schwede and Kevin Tucker.

Given a graph G, we define the ideal J_t(G) to be the monomial ideal generated by the t-connected subsets of the graph. In this talk, I will discuss some homological and combinatorial properties associated to monomial ideals that are of interest, namely linear resolutions and linear quotients. The two main results are (i) characterizing when J_t of a tree has linear quotients, and (ii) noting some necessary and sufficient conditions for arbitrary graphs. This is joint work with H. Ananthnarayan and Omkar Javadekar.

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A local ring is said to have finite Cohen-Macaulay type if there are only finitely many indecomposable maximal Cohen-Macaulay modules up to isomorphism. The classification of such rings is far from complete; we understand small dimensions and the case of a hypersurface ring, but that is essentially it. This talk will focus on the key tool used in proving the hypersurface case (matrix factorizations). We will introduce the basics of the theory of matrix factorizations and explain how they are used in the classification of hypersurface rings of finite Cohen-Macaulay type. Along the way, we will address some natural questions inspired by the classification regarding “d-fold” matrix factorizations. This is joint work with Graham Leuschke.

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Finite F-representation type is an important notion in characteristic-p commutative algebra, and is closely connected to the behavior of differential operators. Despite this, explicit examples of varieties with or without this property are few. We prove that a large class of homogeneous coordinate rings (essentially, those of Calabi-Yau or general-type varieties) will fail to have finite F-representation type, via an analysis of their rings of differential operators. This illustrates a connection between the commutative-algebraic property of FFRT, and the algebro-geometric properties of positivity/negativity of tangent sheaves of varieties. This also provides instructive examples of the structure of the ring of differential operators for non-F-pure rings, which to this point have largely been unexplored. We will also discuss the case of Fano varieties: Recent work has provided non-toric smooth Fano varieties that do have FFRT (Grassmannians Gr(2,n) and the quintic del Pezzo surface). However, it seems unlikely that this will be true for all Fano varieties; we will present conjectural evidence that “in general” smooth Fano varieties will often fail to have FFRT.