Date of Award

Fall 2025

Language

English

Embargo Period

8-18-2025

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

College/School/Department

Department of Mathematics and Statistics

Program

Mathematics

First Advisor

Alexandre Tchernev

Committee Members

Antun Milas, Michael Lesnick, Anupam Srivastav

Keywords

Betti numbers, generic ideals, commutative algebra, homology, polynomial rings, Hilbert series

Subject Categories

Algebra | Algebraic Geometry

Abstract

We present results related to Betti numbers of so called generic ideals over a polynomial ring $T = \Bbbk[x_1, \ldots, x_n]$. For each $m$ define $T(m) = T/(x_{m+1}, \ldots, x_n)$ and for a homogeneous ideal $J$ we use the notation $J(m) = JT(m)$. Also we set $Q(m) = T(m)/J(m)$, and $L(m) = ann_{Q(m)}(x_m)$. The first main result is Theorem \ref{Long Exact Sequence} where we produce the following long exact sequence \begin{align*} \cdots \rightarrow &Tor_{k+1,j}^{T(m-1)}(Q(m-1),\Bbbk) \rightarrow Tor_{k-1,j+1}^{T(m-1)}(L(m),\Bbbk)_{j-1} \rightarrow Tor_{k,j}^{T(m)}(Q(m),\Bbbk) \rightarrow \\ &Tor_{k,j}^{T(m-1)}(Q(m-1),\Bbbk) \rightarrow Tor_{k-2,j-1}^{T(m-1)}(L(m),\Bbbk) \rightarrow \cdots. \end{align*} The second main result is Theorem \ref{Theorem F(j,m) equiv k(j,m)} where we explicitly describe a "small" chain complex $\mathcal{F}(j,m)$ such that \[H_k\mathcal{F}(j,m) \cong \Tor_{k,j}^{T(m)}(Q(m),\Bbbk).\] The third main result is Theorem \ref{Theorem new pure cases} which proves several new cases of the Migliore and Mir{\'o}-Roig Conjecture \ref{conj migl and mir}. Macaulay 2 code used for these computations is found at the end of Chapter 7. Also in Chapter 7 is Macaulay 2 code that can compute Betti diagrams in our examples faster than the standard Macaulay 2 commands.

License

This work is licensed under the University at Albany Standard Author Agreement.

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