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 restrictionContinue reading “Presheaves that are not sheaves”

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 thatContinue reading “Proving that an isomorphism between varieties implies an isomorphism between their coordinate rings, and vice-versa”

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 vectorContinue reading “Exterior Algebra and Differential Forms I”

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.Continue reading “What does it mean for a group to split as a direct sum of 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 thisContinue reading “Ramblings on quasi-projective varieties”