Date of Award

8-22-2025

Date Published

September 2025

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Advisor(s)

Graham Leuschke

Keywords

Artinian ring;Betti Number;Homological Algebra;Tor and Ext

Subject Categories

Mathematics | Physical Sciences and Mathematics

Abstract

We study the behavior of Betti growth of modules over Artinian rings under some $\Tor$ vanishing conditions. Let $(R,\fm)$ be a local Artinian ring with a fixed ideal $I$, and $M$ a finite module. We extend the framework developed in \cite{huneke2014vanishing} by introducing and analyzing a new invariant, $\gamma_I(M)$ (see Definition \ref{definition}), associated with the ideal $I$ of the ring $R$ and a finitely generated $R$-module $M$. It turns out that this invariant $\gamma_I(M)$ provides an upper bound for the ratio of certain consecutive Betti numbers of the finitely generated modules $N$ such that $\Tor_i(M,N) = 0$ for some $i>0$. The invariant was introduced by Craig Huneke, Liana \c Sega, and Adela Vraciu in \cite{huneke2014vanishing} for the case $I = \fm$. It turns out that a critical assumption to extend the result of Huneke et al. is that for any module $M$, $M/\fm M$ is always free over $R/\fm$. Hence, their results can be generalized to modules such that $M/IM$ is free over $R/I$ for a fixed ideal $I$. For convenience, we call such modules \textit{$I$-free}. The bounds involving $\gamma_I(M)$ are often sharper than the original ones given in \cite{huneke2014vanishing}. In Chapter \ref{ch:3}, we prove some generalizations of results in \cite{huneke2014vanishing}, as well as some different applications. In particular, we first explore some key properties of the invariant $\gamma_I(M)$. Then we show that for nonzero modules $M$ and $N$ such that two consecutive syzygies of $N$ are $I$-free (say $N_{i}$ and $N_{i+1}$) and $\fm IM = 0$, if $\Tor_{i}(M,N) = \Tor_{i+1}(M,N) = 0$, the ratio $\dfrac{b_{i+1}(N)}{b_{i}(N)}$ is given by $\gamma_I(M)$. Moreover, if $N_{i-1}$ is also $I$-free, the ratio $\dfrac{b_{i}(M)}{b_{i-1}(M)}$ is given by $b_0(I) - \gamma_I(M)$. In particular, the condition $\fm IM = 0$ inspires us to look at the rings such that $\fm^2I = 0$. In such rings, all syzygy modules are killed by $\fm I$. Using these two lemmas, we can show that if there are sufficiently many vanishing of the modules $\Tor_i(M,N)$, then $\gamma_I(M)$ and $\gamma_I(N)$ must satisfy some special numerical properties. Then we show that in certain Artinian rings or for certain modules $M$ and $N$, sufficiently many vanishing of $\Tor_i(M,N)$ modules will imply that at least one of them is free. Lastly, we apply our results to the canonical module $\omega$, providing conditions under which the invariant characterizes Gorenstein rings. In particular, we address a question posed in \cite{jorgensen2007growth}: given a ring $R$ with a canonical module $\omega$, does $b_0(\omega) \leq b_1(\omega)$ imply that $R$ is Gorenstein? We obtain a partial answer to this question (Corollaries \ref{cor3.4} and \ref{cor3.5}). In Chapter \ref{ch:4}, we show more general results by relaxing the condition that $\fm I M = 0$ to $IJ M = 0$ where $J$ is another fixed ideal. In the previous section, we assume all modules are $I$-free. However, it turns out that the modules do not have to be $I$-free for the same ideal $I$. One module just needs to be $(I+J)$-free, which is a weaker assumption. The other module needs to be $I \cap J$-free. In this case, we can extend the results in Chapter \ref{ch:4} to more general situations. In Chapter \ref{ch:5}, we include some results generalizing \cite{kimura2022maximal}. In particular, we extend the results in \cite{kimura2022maximal} from Cohen-Macaulay rings to Noetherian local rings by assuming the existence of a dualizing complex instead of a canonical module. Applying similar techniques as in \cite{kimura2022maximal}, we show that the Auslander-Reiten Conjecture is true for modules that are locally free in codimension $1$ over Noetherian rings.

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