Date of Award

Summer 2021

Document Type

Open Access Dissertation

Department

Mathematics

First Advisor

Matthew Ballard

Second Advisor

Andrew Kustin

Abstract

We produce a family of complexes called trimming complexes and explore applications. We first study ideals defining type 2 compressed rings with socle minimally generated in degrees s and 2s − 1 for s > 2. We prove that all such ideals arise as trimmings of grade 3 Gorenstein ideals and show that trimming complexes yield an explicit free resolution. In particular, we give bounds on parameters arising in the Tor-algebra classification and construct explicit ideals attaining all intermediate values for every s. This partially answers a question of realizability of Tor-algebra structures posed by Avramov. Next, we study how trimming complexes can be used to deduce the Betti table for the minimal free resolution of the ideal generated by subsets of a generating set for an arbitrary ideal I. In particular, explicit Betti tables are computed for an infinite class of determinantal facet ideals; previously, Betti numbers for anything more than the linear strand had not been computed explicitly. Next, we study certain classes of equigenerated monomial ideals with the property that the so-called complementary ideal has no linear relations on the generators. We then use iterated trimming complexes to deduce Betti numbers for such ideals. Furthermore, using a result on splitting mapping cones by Miller and Rahmati, we construct the minimal free resolutions for all ideals under consideration explicitly and conclude with questions about extra structure on these complexes. Finally, we consider the iterated trimming complex associated to data yielding a complex of length 3. We compute an explicit algebra structure in this complex in terms of the algebra structures of the associated input data. Moreover, it is shown that many of these products become trivial after descending to homology. We apply these results to the problem of realizability for Tor-algebras of grade 3 perfect ideals, and show that under mild hypotheses, the process of “trimming” an ideal preserves Tor-algebra class. In particular, we construct new classes of ideals in arbitrary regular local rings defining rings realizing Tor-algebra classes G(r) and H(p, q) for a prescribed set of homological data.

Included in

Mathematics Commons

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