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CARS

Commutative Algebra Reading Seminar

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Fall 2013

Christopher Evans and Nick Packauskas

Multiplicity of Regular Local Rings

November 12 and 19
We will quickly define multiplicity of a module and prove a well-known Theorem of Nagata concerning multiplicity and regular local rings. This proof will include an application of the result about superficial elements which was proved in the previous seminar.

Brittney Turner

Applications of the Artin-Rees Lemma

November 5
Last week, we saw the statement and a proof of the Artin-Rees Lemma. This week, I will discuss some applications of the lemma that highlight its usefulness.

Douglas Dailey

The Artin-Rees Lemma

October 29
The Artin-Rees Lemma is an extremely useful theorem in commutative algebra. I will give a nice proof of the Artin-Rees Lemma that does not get bogged down in technical mumbo-jumbo and is easily understandable. Anyone who understands the basics of Noetherian rings will understand the majority of this talk.

Eric Canton

Morphisms between Projective Spaces OR An exercise out of Hartshorne

October 8 and 15

The leading question of these talks will be to understand the statement and solution of the following problem (Ex. II.7.3 in Hartshorne) Let f: ℙkn → ℙkm be a morphism of schemes. If m < n, then f(ℙkn) = pt (i.e. f is constant if m < n).

I will try to give a general feel for the algebraic geometry associated to this question, focusing on how the objects and structures are used rather than burdening the audience with technical baggage. Along the way, we'll see some old friends like Spec of a commutative ring, as well as meet some new friends like Proj of a graded ring and invertible sheaves. I will be assuming a basic familiarity with rings, graded rings, and modules, as well as some point-set topology.

Marcus Webb

Hilbert Functions

October 1
We'll talk about some classical and recent results concerning Hilbert Functions.

Michael Brown

Affine Semigroup Rings Part 2

September 24
When we left off, we were in the middle of proving the following theorem: Theorem (Hochster): If M is a positive, normal, affine semigroup ring, and k is any field, k[M] is Cohen-Macaulay. I'll quickly finish this proof, and then sketch another proof of the same fact using a completely different (but hopefully illustrative) approach involving local cohomology.

Michael Brown

What Do Combinatorics, Semigroups, and Toric Varieties Have in Common?

September 17
I'll talk about the connection between toric varieties, affine semigroups, and convex polyhedra. (References: Polytopes, Rings, and K-Theory, a text by Bruns and Gubeladze; Normal Semigroup Rings, and Normal Semigroup Rings by Bruns)

Haydee Lindo and Kat Shultis

The Tools from Croll's 2013 Paper 'Periodic Modules over Gorenstein Local Rings'

Septenber 3 and 10
In this two part talk, we will present the tools and techniques used by Amanda Croll in her thesis work. In the first talk, we will provide some historical and mathematical background along with the major results of her thesis. In the second talk, we will discuss some of the well-known results Amanda used repeatedly and proved for lack of adequate references.