Commutative Algebra Seminar

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Given a holomorphic function f, its Bernstein-Sato polynomial is a classical invariant that detects the singularities of the zero locus of f in very subtle ways; for example, its roots recover the log-canonical threshold of f and the eigenvalues of the monodromy action on the cohomology of the Milnor fibre. In this talk I will describe a characteristic-p analogue of this invariant and some recent developments that I proved in my thesis. I will also describe some open problems and future directions that I am working on at the moment.

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Over an infinite field, a generic determinantal ring is the fixed subring of an action of the general linear group on a polynomial ring; this is the natural embedding of the title. If the field has characteristic zero, the general linear group is linearly reductive, and it follows that the invariant ring is a split subring of the polynomial ring. We determine if the natural embedding is split in the case of a field of positive characteristic. Time permitting, we will address the corresponding question for Pfaffian and symmetric determinantal rings. This is ongoing work with Mel Hochster, Jack Jeffries, and Anurag Singh.

For G a finite group and k a field of characteristic dividing |G| there is a rich history of studying the geometry and commutative algebra of the group cohomology ring of G with coefficients in k. These rings are complicated, but some of their geometric features can be described in group theoretic terms. These geometric features can often be phrased in terms of local cohomology. I will survey some of the results around local cohomology modules of group cohomology rings, and describe how to study these modules using a topological perspective. I will also mention some open problems.

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Let A be a polynomial ring and I_1,..., I_r be a collection of ideals in A. The multi-Rees algebra of I_1,..., I_r encode many algebraic properties of these ideals, their products, and powers. Additionally, the multi-Rees algebras arise in successive blowing up of Spec(A) at the subschemes defined by I_1,..., I_r. Due to this connection, Rees and multi-Rees algebras are also called blow-up algebras in the literature.

In this talk, we will focus on Rees and multi-Rees algebras of strongly stable ideals. In particular, we will discuss the Koszulness of these algebras through a systematic study of these objects via three parameters: the number of ideals in the collection, the number of Borel generators of each ideal, and the degrees of Borel generators. In our study, we utilize combinatorial objects such as fiber graphs to detect Gröbner bases and Koszulness of these algebras. This talk is based on a joint work with Kuei-Nuan Lin and Gabriel Sosa.