Algebra Commons^™

Homology Of Artinian Modules Over Commutative Noetherian Rings, Micah J. Leamer

Department of Mathematics: Dissertations, Theses, and Student Research

This work is primarily concerned with the study of artinian modules over commutative noetherian rings.

We start by showing that many of the properties of noetherian modules that make homological methods work seamlessly have analogous properties for artinian modules. We prove many of these properties using Matlis duality and a recent characterization of Matlis reflexive modules. Since Matlis reflexive modules are extensions of noetherian and artinian modules many of the properties that hold for artinian and noetherian modules naturally follow for Matlis reflexive modules and more generally for mini-max modules.

In the last chapter we prove that if the Betti …

Hilbert–Samuel And Hilbert–Kunz Functions Of Zero-Dimensional Ideals, Lori A. Mcdonnell

Department of Mathematics: Dissertations, Theses, and Student Research

The Hilbert-Samuel function measures the length of powers of a zero-dimensional ideal in a local ring. Samuel showed that over a local ring these lengths agree with a polynomial, called the Hilbert-Samuel polynomial, for sufficiently large powers of the ideal. We examine the coefficients of this polynomial in the case the ideal is generated by a system of parameters, focusing much of our attention on the second Hilbert coefficient. We also consider the Hilbert-Kunz function, which measures the length of Frobenius powers of an ideal in a ring of positive characteristic. In particular, we examine a conjecture of Watanabe and …

Annihilators Of Local Cohomology Modules, Laura Lynch

Department of Mathematics: Dissertations, Theses, and Student Research

In many important theorems in the homological theory of commutative local rings, an essential ingredient in the proof is to consider the annihilators of local cohomology modules. We examine these annihilators at various cohomological degrees, in particular at the cohomological dimension and at the height or the grade of the defining ideal. We also investigate the dimension of these annihilators at various degrees and we refine our results by specializing to particular types of rings, for example, Cohen Macaulay rings, unique factorization domains, and rings of small dimension.

Adviser: Thomas Marley

Formalizing Categorical And Algebraic Constructions In Operator Theory, William Benjamin Grilliette

Department of Mathematics: Dissertations, Theses, and Student Research

In this work, I offer an alternative presentation theory for C*-algebras with applicability to various other normed structures. Specifically, the set of generators is equipped with a nonnegative-valued function which ensures existence of a C*-algebra for the presentation. This modification allows clear definitions of a "relation" for generators of a C*-algebra and utilization of classical algebraic tools, such as Tietze transformations.

On The Betti Number Of Differential Modules, Justin Devries

Department of Mathematics: Dissertations, Theses, and Student Research

Let R = k[x₁, ..., x_n] with k a field. A multi-graded differential R-module is a multi-graded R-module D with an endomorphism d such that d² = 0. This dissertation establishes a lower bound on the rank of such a differential module when the underlying R-module is free. We define the Betti number of a differential module and use it to show that when the homology ker d/im d of D is non-zero and finite dimensional over k then there is an inequality rank_R D ≥ 2ⁿ. This …

The Cohomology Of Modules Over A Complete Intersection Ring, Jesse Burke

Department of Mathematics: Dissertations, Theses, and Student Research

We investigate the cohomology of modules over commutative complete intersection rings. The first main result is that if M is an arbitrary module over a complete intersection ring R, and if one even self-extension module of M vanishes then M has finite projective dimension. The second main result gives a new proof of the fact that the support variety of a Cohen-Macaulay module whose completion is indecomposable is projectively connected.

Vanishing Of Ext And Tor Over Complete Intersections, Olgur Celikbas

Department of Mathematics: Dissertations, Theses, and Student Research

Let (R,m) be a local complete intersection, that is, a local ring whose m-adic completion is the quotient of a complete regular local ring by a regular sequence. Let M and N be finitely generated R-modules. This dissertation concerns the vanishing of Tor(M, N) and Ext(M, N). In this context, M satisfies Serre's condition (S_{n}) if and only if M is an nth syzygy. The complexity of M is the least nonnegative integer r such that the nth Betti number of M is bounded by a polynomial of degree r-1 for all sufficiently large n. We use this notion of …

Fan Cohomology And Equivariant Chow Rings Of Toric Varieties, Mu-Wan Huang

Department of Mathematics: Dissertations, Theses, and Student Research

Toric varieties are varieties equipped with a torus action and constructed from cones and fans. In the joint work with Suanne Au and Mark E. Walker, we prove that the equivariant K-theory of an affine toric variety constructed from a cone can be identified with a group ring determined by the cone. When a toric variety X(Δ) is smooth, we interpret equivariant K-groups as presheaves on the associated fan space Δ. Relating the sheaf cohomology groups to equivariant K-groups via a spectral sequence, we provide another proof of a theorem of Vezzosi and Vistoli: equivariant K …

Fan Cohomology And Its Application To Equivariant K-Theory Of Toric Varieties, Suanne Au

Department of Mathematics: Dissertations, Theses, and Student Research

Mu-Wan Huang, Mark Walker and I established an explicit formula for the equivariant K-groups of affine toric varieties. We also recovered a result due to Vezzosi and Vistoli, which expresses the equivariant K-groups of a smooth toric variety in terms of the K-groups of its maximal open affine toric subvarieties. This dissertation investigates the situation when the toric variety X is neither affine nor smooth. In many cases, we compute the Čech cohomology groups of the presheaf K_q^T on X endowed with a topology. Using these calculations and Walker's Localization Theorem for equivariant K-theory, we give explicit formulas …