189-570A: Higher Algebra I

Assignment 4

Due: Monday, November 11.

1. Let $R$ be a commutative ring and let ${\mathfrak p}$ be a prime ideal of $R$. Show that $$ V({\mathfrak p}) := \{ {\mathfrak q}\in {\rm Spec}(R) \mbox{ with } {\mathfrak q} \supset {\mathfrak p} \}$$ is the closure of the point ${\mathfrak p} \in {\rm Spec}(R)$.

2. Show that the spectrum of a ring $R$ is disconnected (as a topological space) precisely when $R$ is the direct product of two rings.

3. Let $\mathbb Z_{(p)}$ be the localisation of $\mathbb Z$ at a prime $p$, and let $R$ be the subring of ${\mathbb Z}_{(p)} \times {\mathbb Z}_{(p)}$ consisting of pairs $(a,b)$ satisfying $p| a-b$.

(a) Show that ${\rm Spec}(R)$ is a finite set, and enumerate the closed sets in this topological space.

(b) Describe the maps ${\rm Spec}(\mathbb Z_{(p)}) \rightarrow {\rm Spec}(R)$ arising from the two coordinate projections $R\rightarrow {\mathbb Z}_{(p)}$, and check that they are continuous.

(c) Show that the ring $R$ is isomorphic to $\mathbb Z_{(p)}[x]/(x(x-p))$. Show that the spectrum of this ring is connected, while ${\rm Spec}(\mathbb Q[x]/(x(x-p))$ is disconnected. Draw a picture of the map on spectra induced by the natural inclusion $\mathbb Z_{(p)}[x]/(x(x-p)) \rightarrow \mathbb Q[x]/(x(x-p))$.

4. Let $R$ be a commutative ring and let ${\tilde R} := R[x]/(x^2)$. Show that the natural homomorphisms $R\rightarrow {\tilde R}$ and ${\tilde R} \rightarrow R$ induce mutually inverse isomorphisms of topological spaces on the spectra. Describe the morphisms induced by these ring homomorphisms, in the category of locally ringed spaces.

5. Prove that a commutative ring $R$ is a local ring if and only if it has an ideal $I$ for which every element of $R-I$ is invertible.

6. Give examples of sheaves of rings on a tolological space $X$ for which
(a) The restriction maps are injective, but not surjective.
(b) The restriction maps are surjective, but not injective.
(c) The restriction maps are neither injective nor surjective.
(d) The restriction maps are isomorphisms.

You should strive to come up with the examples that are as simple and as natural as possible, in each case.

7. Problem (61), page 113 of Eyal Goren's Math 570 course notes.

8. Problem (62), page 113 of Eyal Goren's Math 570 Course notes.

9. Show that ${\rm Spec}(R)$ consists of a single point if and only if $R$ is a local ring whose maximal ideal consists of nilpotent elements.

10. Let $K$ be an algebraically closed field. Describe ${\rm Spec}(K[x,y]/(xy))$ as a topological space, and then as a locally ringed space.