Notes on Speyer’s paper titled “Some Sums over Irreducible Polynomials”

Let be the set of irreducible polynomials over . Then . The paper lists certain examples of below. These are all expanded as geometric series. As one can see only contribute to the coefficient of in the sum . Why don’t the other irreducible polynomials do the same? This is because these are the only

Introduction to Schemes

This is a short introduction to Scheme Theory, as modeled on the article by Brian Lawrence. A variety here is a zero set that can be covered by a finite number of affine varieties. Hence, a morphism between varieties can be considered to be a bunch of affine morphisms, as long as they agree on

Puiseux Series and Tropical Varieties

Puiseux series- This field is denoted by . Note that we have a double brace “[[ ]]” instead of “[]”. This implies that we have infinite series instead of finite ones (which would be polynomials). The Puiseux laurent series is denoted as . This means that is also allowed to have negative powers. Now ,

Adjoint Functors

Today we’re going to talk about adjoint functors. Definition: Let be a functor, and let be another functor. Then and are adjoint functors if for Obj and Ob, we have . Maclane has famously stated in his seminal book that “adjoint functors arise everywhere”. However, what is the utility of such functors? 1. Solutions to

De Rham Cohomology- I

This will be a rambling progression to the De Rham cohomology. First we have a map from an open set in the manifold to an open subset of . A collection of such charts over a cover of the manifold is known as an atlas. There can be many charts and hence many atlases. Do