prime spectrum

Grothendieck's revolutionary idea was to treat any commutative ring as if it were a ring of functions on some space, and then to manufacture that space from the ring itself. The points of this space are not just the classical solution points but all the prime ideals, which lets the construction work for every commutative ring, even ones with nilpotents or no classical points at all.

For a commutative ring R, the prime spectrum Spec R is the set of all prime ideals of R, equipped with the Zariski topology whose closed sets are V(I) = { primes p containing I } for ideals I of R. The maximal ideals are the closed points and correspond, over an algebraically closed field, to the classical points of a variety; the non-maximal primes are the extra generic points that fatten the space to remember irreducible subvarieties.

Spec is a contravariant functor: a ring homomorphism R to S induces a continuous map Spec S to Spec R by pulling back primes, since the preimage of a prime ideal is prime. Together with its structure sheaf, Spec R becomes a locally ringed space, the affine scheme attached to R, and these affine pieces are the building blocks glued together to form general schemes. This is the precise sense in which 'rings are functions on spaces'.