Presheaves that are not sheaves

This is a post about pre-sheaves that are not sheaves. The two properties that a sheaf satisfies that a presheaf does not, are the “Gluability” axiom and the identity axiom. Gluability- Over the open set , consider the set , which is the set of all bounded continuous functions. Sections which have the same restriction

Projective varieties

Today I will try and study projective varieties and their ideals. Although my understanding of these objects has improved over time, there are still a lot of chinks that need to be filled. Something that has defied complete understanding is what kinds of polynomials are projective varieties the zeroes of? Do these polynomials have to

The Jacobian of a linear map

This is a small blog post. What is the Jacobian of a linear map? Say I have an matrix- call it . Also, I have a linear map which is given by . What is the Jacobian of ? This is a question that has confused me before. The function takes a vector , and

Proving that an isomorphism between varieties implies an isomorphism between their coordinate rings, and vice-versa

Let and be two algebraic sets in . Today I will try and prove that . Not writing a full proof of this before has eaten away at my existence for way too long. First let us assume that . This means that there exist polynomial maps from to . Hence, for any polynomial that

Exterior Algebra and Differential Forms I

This is going to be a post about exterior algebra and differential forms. I have studied these concepts multiple times in the past, and feel that I have an idea of what’s going on. However, it would be good to straighten the chinks, of which there are many, once and for all. For a vector

Sweating out the homology

This is going to be a rather long post on homology. I hope I do manage to understand it. It will ultimately go up in a polished form on my blog. The reason why it is difficult to understand homology and cohomology without typing it all out is that the information given is so little.

What does it mean for a group to split as a direct sum of subgroups

Today I shall be talking about what it means for a group to split into a direct sum. In other words, if , then what does it mean for the structure of the group? is obviously not the union . It is a much bigger set than that in general. But it contains as subgroups.

Ramblings on quasi-projective varieties

This blog post is mean to be an exposition on quasi-projective varieties, something that I am having problems understanding. A quasi-projective variety is a locally closed projective variety. What does that mean? It means that it is the intesection of a Zariski open set and a Zariski closed set in some projective space. Does this

Notes from my last meeting with Pete

(by Serre duality) Estimate (we can refine this later) (Riemann’s result, ) (Riemann-Roch) As described in Hirzebruch’s book, from , we get , and then There are two approaches that we can take: 1. Tensor by and compute 2. Riemann-Roch for vector bundles: we have instead of . Here, is the rank.