The November 3, 2014 O.M.N.I. Talk: Daniel Juda presents

Algebraic Sets and the Zariski Topology

Abstract: Let f be a polynomial in C[x,y]. The algebraic set associated to f is the set of all points where f = 0, denoted by V(f). For example if f(x,y) = y - x^2, the "real" points of V(f) form a parabola. We can define a topology on C^2, called the Zariski topology, by defining V(f) to be a closed set. This gives us a correspondence between a topological space, C^2, and a commutative ring, C[x,y]/(f) by drawing a correspondence between algebraic sets V(f) in C^2 and prime ideals (f) in C[x,y]. We will show that for an arbitrary commutative ring R we can use the Zariski topology to construct a space using the prime ideals of R where a similar correspondence holds. Interestingly, this allows us to use many of the topological ideas that are applicable to C^2 in the more general setting of an arbitrary commutative ring.