# Understanding morphisms of affine algebraic varieties

In class, we defined an affine algebraic variety to be a $$k$$-ringed space $$(V, \mathcal{O}_V)$$ where $$V$$ is an algebraic set in $$\bar{k}^n$$ defined by a system of polynomial equations over $$k$$, and the sheaf of regular functions $$\mathcal{O}_V$$ that assigns an open subset of $$V$$ to the set of regular functions on that open set.

We then defined a morphism of affine algebraic varieties $$(V, \mathcal{O}_V)$$, $$(W, \mathcal{O}_W)$$ to be given by a morphism of the underlying ringed spaces, i.e. a continuous map (which I'm assuming means continuous with respect to the Zariski topology) $$\phi: V \to W$$, together with a family of $$k$$-algebra homomorphisms $$\phi_U : \mathcal{O}_W(U) \to \mathcal{O}_V(\phi^{-1} (U))$$.

This definition is very abstract and I'm having a bit of a hard time wrapping my head around it. However, we were given some equivalent definitions of this, as well as the following correspondence:

For affine algebraic varieties $$(V, \mathcal{O}_V)$$, $$(W, \mathcal{O}_W)$$, there is a $$1-1$$ correspondence between the morphisms of ringed spaces between $$(V, \mathcal{O}_V)$$, $$(W, \mathcal{O}_W)$$, and the $$k-$$algebra homomorphisms $$k[W] \to k[V]$$, where $$k[W], k[V]$$ are the coordinate rings of $$W, V$$ respectively (so $$k[W] = k[x_1 , \dots , x_n]/I(W)$$).

So, just to make sure I am understanding this correctly. Let's say I am defining a morphism of the affine algebraic varieties of the circle $$x^2 + y^2 = 1$$ over $$\mathbb{C}$$ to the hyperbola $$xy = 1$$. Does the correspondence stated above mean that it is enough to just write a ring homomorphism $$\phi : \mathbb{C}[x,y]/(x^2 + y^2 - 1) \to \mathbb{C}[x,y]/(xy-1)$$?

• Ahh I see. So a morphism of affine algebraic varieties $(V, \mathcal{O}_V) \to (W, \mathcal{O}_W)$ corresponds to a $k$-algebra homomorphism in the opposite direction $k[W] \to k[V]$? Mar 8, 2021 at 20:59