The algebra-geometry connection скачать в хорошем качестве

The algebra-geometry connection

A natural source of commutative algebras is the algebra of complex-valued functions on a set X. If X happens to have some extra geometry associated to it, then one can restrict to functions that preserve that structure. For example, if X is a topological space, then one can just look at continuous functions, whilst if X is an affine variety, one considers algebraic (i.e. polynomial) functions. What is surprising is that such algebras are highly stereotypical, so that commutative algebra can be studied via geometry as long as you pick an appropriate class of algebras to match the geometry. For example, for affine varieties, one considers finitely generated reduced algebras whilst for topological spaces, one considers C*-algebras. In this video, we introduce the common elements in this algebra-geometry connection. In particular, we discuss how to concoct a set (from which one can potentially build a geometry) from a commutative algebra. This is the notion of a spectrum.