Hello I'm starting to study algebraic geometry on my own with a book and I've been thinking on this problem a few days. I'd appreciate if someone could help me.
Let $V$ be an affine variety. We define diagonal map as $$\Delta:V\rightarrow \Delta (V)\subset V \times V $$ $$ v\longmapsto(v,v)$$ (a) Prove that $\Delta$ is a morphism.
(b) Let $V=\mathbb{A}^n(k)$ and fixed coordinates $x_1,...,x_n,y_1,...,y_n$ en $\mathbb{A}^n(k)\times \mathbb{A}^n(k)$. Show that $I(\Delta _V)= \langle x_1-y_1,...,x_n-y_n \rangle$.
(c) For general affine variety $V$, prove that $\Delta _V$ is closed in $V \times V$.
(d) Show that $\Delta:V \rightarrow \Delta _V$ is an isomorphism.
My attemp:
(a) I've proved that statement and I'm sure it's correct.
(b) I can't prove that, I tried many things but I don't get the solution.
(c) I don't know if this solution it's right.
A topological space $V$ is Hausdorff iff the diagonal set $\Delta _V$ is closed. So if $V$ is an affine variety, and $x,y \in V, x \neq y$. If I consider the lineal polinomial $f $ so that $f(x)=0$, and the paralel lineal polinomial $g$ that anihilates $y$ (It's possible since $x \neq y$), then $U,W$ (subset of points of $V$ such that anihilates f or g, respct.) are disjoint neigbourhoods of $x,y $ and hence, $V$ is Hausdorff.
(d) I've thinked about using projections $\pi_1,\pi_2: V \times V \rightarrow V$, but I didn't get the solution.
Thanks in advance
(a) The map is by definition the one induced from $ A \otimes_k A \to A$ and so is a morphism (by definition of a morphism of schemes). Here I am thinking of $A$ as $\Gamma(V,\mathcal{O}_V)$.
(b) If $V = \Bbb{A}^n$ then the diagonal map is coming from $k[x_1,\ldots,x_n]\otimes_k k[y_1,\ldots,y_n] \to k[t_1,\ldots,t_n]$ where we send $f(x_1,\ldots,x_n)\otimes g(y_1,\ldots,y_n) \mapsto f(t_1,\ldots,t_n)g(t_1,\ldots,t_n)$. The kernel is generated by $x_i \otimes 1 - 1\otimes y_i$.
(c) The ring map $A \otimes_k A \to A$ is surjective and so the map on Spec has to be a closed immersion.
(d) A closed immersion is also an isomorphism onto its image.