Commutative Algebra

Excercises 1

The Spectrum of a ring [Reference: EGA I, Chapter 0 §2] -2. Compute Spec ℤ as a set and describe its topology. Let A be any ring. Let X be any topological space. -1. For f ∈ A we define D(f) := Spec A \ V (f). Prove that the open subsets D(f) form

a basis of the topology of Spec A.

0. Prove that the map I ↦ V (I) defines a natural bijection

\{I ⊂ A with I = \sqrt{I}\} ⟶ \{T ⊂ Spec A closed\}

A topological space X is called quasi-compact if for any open covering X = ⋃_{i∈I} U_{i} there is a finite subset {i_{1}, …, i_{n}} ⊂ I such that X = U_{i_{1}} ∪ … U_{i_{n}}.

1. Prove that Spec A is quasi-compact for any ring A. A topological space X is said to verify the separation axiom T_{0} if for any pair of points x, y ∈ X, x ≠ y there is an open subset of X containing one but not the other. We say that X is Hausdorff if for any pair x, y ∈ X, x ≠ y there are disjoint open subsets U, V such that x ∈ U and y ∈ V.

2. Show that Spec A is not Hausdorff in general. Prove that Spec A is T_{0}. Give an example of a topological space X that is not T_{0}.

Remark: usually the word compact is reserved for quasi-compact and Hausdorff spaces. A topological space X is called irreducible if X is not empty and if X = Z_{1} ∪ Z_{2} with Z_{1}, Z_{2} ⊂ X closed, then either Z_{1} = X or Z_{2} = X. A subset T ⊂ X of a topological space is called irreducible if it is an irreducible topological space with the topology induced from X. This definitions implies T is irreducible if and only if the closure T̄ of T in X is irreducible.

3. Prove that Spec A is irreducible if and only if Nil(A) is a prime ideal and that in this case it is the unique minimal prime ideal of A.

4. Prove that a closed subset T ⊂ Spec A is irreducible if and only if it is of the form T = V (𝔭) for some prime ideal 𝔭 ⊂ A.

A point x of an irreducible topological space X is called a generic point of X if X is equal to the closure of the subset {x}.

5. Show that in a T_{0} space X every irreducible closed subset has at most one generic point.

6. Prove that in Spec A every irreducible closed subset does have a generic point. In fact show that the map 𝔭 ↦ \overline{\{𝔭\}} is a bijection of Spec A with the set of irreducible closed subsets of X.

7. Give an example to show that an irreducible subset of Spec ℤ does not neccesarily have a generic point.

A topological space X is called Noetherian if any decreasing sequence Z_{1} ⊃ Z_{2} ⊃ Z_{3} ⊃ … of closed subsets of X stabilizes. (It is called Artinian if any increasing sequence of closed subsets stabilizes.)

8. Show that if the ring A is Noetherian then the topological space Spec A is Noetherian. Give an example to show that the converse is false. (The same for Artinian if you like.)

A maximal irreducible subset T ⊂ X is called an irreducible component of the space X. Such an irreducible component of X is automatically a closed subset of X.

9. Prove that any irreducible subset of X is contained in an irreducible component of X.

10. Prove that a Noetherian topological space X has only finitely many irreducible components, say X_{1}, …, X_{n}, and that X = X_{1} ∪X_{2} ∪…∪X_{n}. (Note that any X is always the union of its irreducible components, but that if X = ℝ with its usual topology for instance then the irreducible components of X are the one point subsets. This is not terribly interesting.)

11. Show that irreducible components of Spec A correspond to minimal primes of A. A point x ∈ X is called closed if \overline{\{x\}} = \{x\}. Let x, y be points of X. We say that x is a specialization of y, or that y is a generalization of x if x ∈ \overline{\{y\}}.

12. Show that closed points of Spec A correspond to maximal ideals of A.

13. Show that 𝔭 is a generalization of 𝔮 in Spec A if and only if 𝔭 ⊂ 𝔮. Characterize closed points, maximal ideals, generic points and minimal prime ideals in terms of generalization and specialization. (Here we use the terminology that a point of a possibly reducible topological space X is called a generic point if it is a generic points of one of the irreducible components of X.)

14. Let I and J be ideals of A. What is the condition for V (I) and V (J) to be disjoint? A topological space X is called connected if it is not the union of two nonempty disjoint open subsets. A connected component of X is a (nonempty) maximal connected subset. Any point of X is contained in a connected component of X and any connected component of X is closed in X. (But in general a connected component need not be open in X.)

15. Show that Spec A is disconnected iff A ≅ B × C for certain nonzero rings B, C.

16. Let T be a connected component of Spec A. Prove that T is stable under generalization. Prove that T is an open subset of Spec A if A is Noetherian. (Remark: This is wrong when A is an infinite product of copies of 𝔽_{2} for example. The spectrum of this ring consists of infinitely many closed points.)

17. Compute Spec k[x], i.e., describe the prime ideals in this ring, describe the possible specializations, and describe the topology. (Work this out when k is algebraically closed but also when k is not.)

18. Compute Spec k[x, y], where k is algebraically closed. [Hint: use the morphism φ : Spec k[x, y] → Spec k[x]; if φ(𝔭) = (0) then localize with respect to S = {f ∈ k[x] | f ≠ 0} and use result of lecture on localization and Spec.] (Why do you think algebraic geometers call this affine 2-space?)

19. Compute Spec ℤ[y]. [Hint: as above.] (Affine 1-space over ℤ.)