Current Session: Fall 2026
Organizers: Giulio Caviglia and Bernd Ulrich
When: 1.30 to 2.20 pm on Wednesdays
Where: LWSN 1106
Date: August 26
Speaker: Kaito Kimura, Nagoya University
Title: Gorensteinness and Gorenstein Homological Dimensions
Abstract: A basic problem in homological commutative algebra is to understand how much information about a local ring is encoded in the existence of modules of finite homological dimension. One of the classical results in this direction is Foxby’s characterization of Gorenstein rings: the existence of a nonzero module, (or more generally a complex,) with finite depth, finite flat dimension, and finite injective dimension already forces the ring to be Gorenstein. Christensen, Foxby, and Holm raised the corresponding question for Gorenstein homological dimensions. They showed that the answer is affirmative if either flat dimension or injective dimension is replaced by its Gorenstein analogue, while the case in which both are replaced remained open. Recently, Azerˆedo, Miranda-Neto, and Souza provided an important partial answer, proving the result for Cohen–Macaulay rings under some additional assumptions. Their work provides a key foundation for the approach presented here. In this talk, I will discuss a complete solution of this question and explain a stronger version formulated for complexes.
Date: September 9
Speaker: Dharm Veer, Purdue University
Title: Complements and Complementary Homologies
Abstract: In a finite lattice L, two elements are said to be complements if their meet is the minimal element 0̂ of L and their join is the maximal element 1̂ of L. For x ∈ L, let C(x) denote the set of all complements of x in L. It is known that if L̄ = (0̂, 1̂) is non-acyclic, then every element of L has a complement.
In this talk, we show that if H̃ₖ(L̄, K) ≠ 0 for some k > 0 and some field K, then for every x ∈ L̄, there exist y ∈ C(x) and z ∈ (0̂, x] ∩ C(y) such that y and z have “complementary homologies”. We apply the above result to derive consequences for the Betti numbers of monomial ideals.
This is an ongoing joint project with Sara Faridi and Volkmar Welker.
Date: October 21
Speaker: Anurag Singh, University of Utah
Title: The b-invariant of a normal graded ring
Abstract: The b-invariant of a normal graded ring was defined recently by Okuma, Watanabe, and Yoshida as the least degree of a nonzero homogeneous element of the anticanonical module. It is readily seen that the b-invariant is greater than or equal to the a-invariant, and that equality holds precisely when the ring is quasi-Gorenstein.
We will discuss some examples and results in the setting of invariant rings of finite groups. We will also describe a construction of standard-graded Cohen-Macaulay rings of characteristic zero, with isolated singularities and negative b-invariant, that are not F-regular; this disproves a conjecture of Okuma-Watanabe-Yoshida.
This is joint work with Aryaman Maithani and Kei-ichi Watanabe.
Date: October 23 (Friday)
When: 1.30-2.20 pm
Where: LWSN 1106
Speaker: Juan Ramirez, University of Notre Dame
Title: The Rees algebra of a monomial ideal
Abstract: The Rees algebra of an ideal is a classical object in commutative algebra that encodes asymptotic properties of an ideal I such as the syzygies of powers I^k. Finding an explicit description of the defining ideal of the Rees algebra is a fundamental and difficult problem. In this talk, we will consider the Rees algebra of a monomial ideal in a polynomial ring over a field. I will present an approach to constructing the defining equations of the Rees algebra for any monomial ideal. We will then discuss some applications where we explicitly determine the defining ideal of the Rees algebra of some families of monomial ideals.
Date: October 28
Speaker: Manav Batavia, Purdue University
Title: The arithmetic rank and cohomological dimension of some residual intersections
Abstract: The arithmetic rank of an affine variety is the minimal number of equations needed to define it set-theoretically, that is, the smallest number of polynomials generating the defining ideal up to radical. Despite this elementary definition, it is often difficult to compute: the minimal generators up to radical often bear little relation to the given ideal generators, and their optimal number can depend on the characteristic. Local cohomological dimension provides a lower bound for arithmetic rank, but is itself a subtle invariant.
Residual intersections generalize the classical notion of algebraic linkage. In this talk, we determine the arithmetic rank and cohomological dimension of generic residual intersections of generic Gorenstein ideals of height three in any characteristic. In characteristic zero, our methods also reveal structural properties of the top local cohomology modules supported at these ideals.
This is joint work with Vaibhav Pandey and Uli Walther.
Date: November 4
Speaker: Benjamin Baily, University of Michigan
Title: Mixed multiplicities and extremal F-pure and log canonical thresholds
Abstract: Let R be a polynomial ring, J ⊆ R an ideal, and m a maximal ideal containing J. We consider invariants of the pair (R, J) which measure the singularities of the embedding Spec(R/J) ⊆ Spec(R) at m: the log canonical threshold (lct) in characteristic zero and the F-pure threshold (fpt) in positive characteristic. A smaller value of the lct/fpt means that the embedding is “more singular”; we seek to classify pairs which are as singular as possible.
In 1972, Skoda showed that lctₘ(R, J) ≥ 1/ordₘ(J), where ordₘ denotes the order of vanishing at m. Skoda’s bound has been generalized and refined many times since; among these improvements is a 2014 result by Demailly and Pham using mixed multiplicities of J and m. We extend Demailly and Pham’s lower bound to positive characteristic and study the pairs (R, J) for which lctₘ(R, J) (or fptₘ(R, J)) equals the lower bound. We classify these “extremal pairs” in two cases: homogeneous ideals in arbitrary characteristic, and arbitrary ideals in characteristic zero. The latter case implies a conjecture of Bivia-Ausina.
Date: November 11
Speaker: Linquan Ma, Purdue University
Title: TBD
Abstract: TBD
Date: November 18
Speaker: Siva Somasundaram, Purdue University
Title: TBD
Abstract: TBD
Date: November 23 (Monday; special venue)
When: 1.30-2.20 pm
Where: LWSN B134
Speaker: Alessandra Costantini, Tulane University
Title: TBD
Abstract: TBD
Date: December 2
Speaker: Bradley Dirks, Stony Brook University
Title: TBD
Abstract: TBD
Date: December 9
Speaker: Benjamin Mudrak, Purdue University
Title: TBD
Abstract: TBD
See Also: Ideal Conversations