Max-spectrum of multivariate polynomial ring over an algebraically closed field
Let be an algebraically closed field. The max-spectrum of is described below.
As a set
Further information: weak Nullstellensatz for algebraically closed fields
As a set, the max-spectrum is identified with , where the identification is as follows:
The left side denotes a point in and the right side denotes the maximal ideal for that point. Equivalently, we can think of the maximal ideal as the kernel of the "evaluation map" which sends a polynomial to the value .
The above is one of the many formulations of the weak Nullstellensatz for algebraically closed fields.
As a topological space
The topology that we give to the max-spectrum coincides with the usual Zariski topology we put on .