December 2016 – Fruits of procrastination

Prūfer Group

This is a short note on the Prūfer group. Let be a prime integer. The Prūfer group, written as , is the unique -group in which each element has different th roots. What does this mean? Take for example. Can we say that for any element in this group, there are mutually different elements which,Continue reading “Prūfer Group”

Sheafification

This is a blog post on sheafification. I am broadly going to be following Ravi Vakil’s notes on the topic. Sheafification is the process of taking a presheaf and giving the sheaf that best approximates it, with an analogous universal property. In a previous blog post, we’ve discussed examples of pre-sheaves that are not sheaves.Continue reading “Sheafification”

Toric Varieties: An Introduction

This is a blog post on toric varieties. We will be broadly following Christopher Eur’s Senior Thesis for the exposition. A toric variety is an irreducible variety with a torus as an open dense subset. What does a dense subset of a variety look like? For instance, in consider the set of integers. Or any infiniteContinue reading “Toric Varieties: An Introduction”

Birational Geometry

This is a blog post on birational geometry. I will broadly be following this article for the exposition. A birational map is a rational map such that its inverse map is also a rational map. The two (quasiprojective) varieties and are known as birational varieties. An example is , and . Varieties are birational ifContinue reading “Birational Geometry”

Filtrations and Gradings

This is going to be a blog post on Filtrations and Gradings. We’re going to closely follow the development in Local Algebra by Serre. A filtered ring is a ring with the set of ideals such that , , and . An example would be , where is the ideal generated by in . Similarly, aContinue reading “Filtrations and Gradings”

Invertible Sheaves and Picard Groups

This is a blog post on invertible sheaves, which form elements (over a fixed algebraic variety) of the Picard Group. The group operation here is the tensor product. We will closely follow the developments in Victor I. Piercey’s paper. We will develop invertible sheaves on algebraic varieties. However, instead of studying sheaves over varieties, we willContinue reading “Invertible Sheaves and Picard Groups”

Tight Closure

This is a small introduction on tight closure. This is an active field of research in commutative algebra, and this is essentially a survey article. This article will closely follow the paper “An introduction to tight closure” by Karen Smith. Definition: Let be a Noetherian domain of prime characteristic (not that in general, need notContinue reading “Tight Closure”