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 28
Speaker: Manav Batavia, Purdue University
Title: TBD
Abstract: TBD
Date: December 2
Speaker: Bradley Dirks, Stony Brook University
Title: TBD
Abstract: TBD
See Also: Ideal Conversations