# Zariski topology on Affine spaces, Name of Functor

I've been studying the Zariski topology in my free time.

So I found this functor between Polynomial Algebras and Affine Spaces.

First, we have this $T$ such that for any affine space $(F^n,\mathcal{Z})$ we have: $$T(F^n)=F[x_1,\cdots,x_n]$$ And for any morphism $\phi:F^n\to F^m$ (A function of polynomial coordinates) we have a $F-$algebra homomorphism $T(\phi):T(F^m)\to T(F^n)$ defined by $$T(\phi)(f)=f\circ \phi$$

I know this can be generalized defining morphisms of algebraic sets, not only affine spaces.

So... how is this functor called? I've seen that when we want to build a Zariski topology on the family of all the prime ideals of some commutative rings, we arrive to a similar functor called $\text{Spec}$. In this case, is it called $\text{Spec}$ too? Or does it receive another name?

• What you describe is the coordinate ring of an (affine) algebraic set. Commented May 7, 2016 at 22:43
• Yes. I know this can be generalized defining morphisms of algebraic sets, in particular, that the objects assigned by the functor are precisely the coordinate rings of the algebraic sets. But does the functor itself have a name? Commented May 7, 2016 at 23:35

The possible names are Spm or mSpec or similar, that is the maximal spectrum of a commutative ring with unit; if you don't working in the scheme setting, otherwise $T$ is Spec!