## All Dissertations

8-2021

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

School of Mathematical and Statistical Sciences

#### Committee Member

Keri Sather-Wagstaff

Kevin James

James Coykendall

Hui Due

#### Abstract

This dissertation is the culmination of my work in [16-18]. These papers add to a body of work focused on rings known as fiber products. A ring $F$ is said to be a fiber product if there exist ring homomorphisms $S \xrightarrow{\pi_S} W \xleftarrow{\pi_T} T$ and we have $F \cong S \times_W T := \{(s,t) \in S \times T: \pi_S(s) = \pi_T(t)\}$. In this set-up, we say that $F$ is the fiber product of $S$ and $T$ over $W$.

For our purpose we only consider rings that are commutative, noetherian rings with identity. We further consider the case where $R$ is a regular local (or standard graded polynomial) ring and study fiber products that can be realized as homomorphic images of $R$. To this end, we study quotients $R/\langle \mathcal{I}', \mathcal{I} \mathcal{J}, \mathcal{J}' \rangle$ where $\mathcal{I}'$, $\mathcal{I}$, $\mathcal{J}'$, and $\mathcal{J}$ are ideals in $R$. In Chapter 3 we impose conditions as these ideals that allow us to realize these quotients as fiber products (see Proposition 3.2.1). We then explicitly construct a minimal resolution of the quotient over $R$. From this, we recover formulas for the Betti numbers of the quotient as well as the Poincar\'e series.

We further establish sufficient conditions on the four ideals above to impose a differential graded structure on our minimal resolution. In Chapter 4, we address the case $\mathcal{I}' = 0 = \mathcal{J}'$ and obtain a minimal differential graded algebra resolution. We then use the techniques of [2,27] to establish further homological properties of the fiber product. In particular, we show that these fiber products are Golod and are Tor-friendly.

In Chapter 5, we allow for $\mathcal{I}' \neq 0$ but maintain $\mathcal{J}' = 0$. We obtain minimal differential graded modules and then establish sufficient conditions on $\mathcal{I}'$ to extend the module structure to that of an algeba. We again apply techniques from [2,27] to obtain the Golod and Tor-friendly properties for this larger class of fiber products.

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