During the semester of Fall 2024, I organized a seminar to run in parallel to our commutative algebra course, taught by Professor Andrew Kustin. Below, you can find the speakers, titles and abstracts of the talks that we had.
Title: Completion an Exact Functor??? (two part talk)
Speaker: Dinesh Limbu
Abstract: This presentation will be in two parts and the end goal will be to prove that the I-adic completion is an exact functor on the category of the finitely generated module over a Noetherian ring. First, I will give some motivation for this talk, and then introduce some basic definitions from Category-Theory and then we will look at completion as an Inverse limit and then briefly go over the topological construction of completion. At the end we will go over p-adic completion of integers and (x)-adic completion of a polynomial ring.
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Title: " On the Weak Lefschetz Property for Ideals Generated by Powers of General Linear Forms "
Speaker: Matthew Booth
Abstract: Let F be a field of characteristic 0. Take the polynomial ring P = F[x1, . . . , xn], and let I = (ℓ a 1 , . . . , ℓa n+1) be an ideal generated by general linear forms raised to a uniform power a. It was conjectured in 2011 that the algebra R := P/I does not enjoy the Weak Lefschetz Property (WLP) if the number of variables n is sufficiently large. Following incremental progress on this conjecture, a 2023 result of Boij & Lundqvist completely classified when the WLP holds for such algebras in terms of n and a. We say that WLP fails in degree d if the map ×L : Rd−1 → Rd does not have maximal rank for a general linear form L. The results of Boij & Lundqvist (along with earlier partial classifications) show failure of surjectivity in a certain degree, but this requires dimF(Rd−1) ≥ dimF(Rd). A clear necessary condition for WLP to hold in any algebra is that the Hilbert function of that algebra be unimodal, so failure of surjectivity may not detect the earliest degree where WLP fails. Of course, unimodality means that examining WLP in earlier degrees requires checking injectivity of multiplication by a general linear form, and that is the central theme of this talk. For R = P/I as above, we show that multiplication by a general linear form is injective at a fixed degree d provided n is sufficiently large in comparison. This is joint work with Pankaj Singh and Adela Vraciu.
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Title: Grobner Basis
Speaker: Dinesh Limbu
Abstract: In this talk, I will briefly introduce the Grobner basis and then define an algorithm to compute the Grobner basis. We will also look at syzygies and how to compute it. At the end, we will compute the resolution of the Twisted Cubic curve using the Grobner basis and determine its Hilbert series.
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Title: On structure and symmetry in covering systems
Speaker: Jonah Klein
Abstract: A covering system of $\mathbb{Z}$ is a finite set of arithmetic progressions, with the property that every integer belongs to at least one of them. Covering systems were first introduced by Erd\H{o}s in 1950. For a general ring, a (left) covering system is a finite set of (left) translates of ideals, with the property that every element of the ring belongs to at least one of them. In this talk, we will go over the basics of covering systems over $\mathbb{Z}$, and see how products of symmetric groups act on the set of all covering systems with a fixed set of moduli. We will see how this group action acts "nicely" when we add additional assumptions on the set of moduli in question. We will then introduce some key properties of Dedekind domains, and look at how these group actions generalize to covering systems of Dedekind domains. This is work in progress.
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Title: Lending Hom a Helping Hand
Speaker: Matthew Booth
Abstract: Some setup: let R be a (not necessarily commutative) ring with unity and fix any left R-module M. Suppose we are given a short exact sequence of left R-modules 0 → A → B → C → 0. Recall in this situation that A may be identified as a submodule of B and C may be identified as the corresponding quotient module (i.e. C ∼= B/A). Algebra is often more concerned with maps between its objects than with the objects themselves. (Indeed, homomorphisms encode information about the objects they map.) With this philosophy in mind, suppose we have an R-module homomorphism between M and A (a submodule of B) or between M and C (a quotient module of B). The question is: can such an R-module homomorphism be used to obtain an R-module homomorphism between M and B? The answer is “no” in general, an answer closely tied to the failure of the Hom functor to be exact. Our goal in this talk will be to briefly examine this failure before seeing that certain types of modules “fix” Hom so that it is exact. This will culminate in providing several equivalent descriptions of projective and injective modules: the category-theoretic description, the Hom functor description, the splitting description, and the direct summand description