Algebraic Geometry 4: A short note on Projective Varieties

What is a variety? It is the set of common zeroes for a set of polynomials. For example, for the set of polynomials , the variety is . Now what is a projective variety? Simply put, it is the common set of zeroes of polynomials in which a one-dimensional subspace is effectively considered one point.Continue reading “Algebraic Geometry 4: A short note on Projective Varieties”

Algebraic Geometry 3: Some more definitions

Index categories: These are categories in which the objects are essentially elements of a partially ordered set, and there exists at most one morphism between two objects. One example would be , where Hom iff . Let be an index category. A functor is said to be indexed by . What does such a functorContinue reading “Algebraic Geometry 3: Some more definitions”

Algebraic Geometry 2: Philosophizing Categories

Faithful functor: Let be a functor between two categories. If the map Mor Mor is injective, then the functor is called faithful. For example, the functor between the category of sets with only bijective mappings qualified to be morphisms between objects, to the category of sets with all kinds of mappings allowed to be morphisms,Continue reading “Algebraic Geometry 2: Philosophizing Categories”

Algebraic Geometry Series 1: An introduction to Category Theory

This is intended to be a series of articles, that fleshes out the bare skeleton of Algebraic Geometry. I will be closely following Ravi Vakil’s treatment of the topic, supplemented with other treatments. A category consists of a “collection” of objects, and for each pair of objects, a set of morphisms (or arrows) between them.Continue reading “Algebraic Geometry Series 1: An introduction to Category Theory”