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Math 603, Spring 2003, HW 3, due 2/24/
Part A
AI) Consider the two rings A = R[T ] and B = C[T ]. Show that Max(B) is in one-to-one correspondence with the points of the complex plane while Max(A) is in one-to-one correspondence with the closed upper half plane: {ξ ∈ C | Im(ξ) ≥ 0 }. Since A is a PID (so is B) we can characterize an ideal by its generator. In these terms, which ideals of Max(A) correspond to points in Im(ξ) > 0, which to points on the real line? What about Spec B, Spec A?
AII) When X is compact Hausdorff and A = C(X), we identified X and Max(A) in class via x 7 → mx. Now Max(A) has the induced topology from Spec A.
(a) Show the induced topology on Max(A) is compact Hausdorff by proving x 7 → mx is a homeomor- phism. (b) Prove all finitely generated ideals of A are principal but that no maximal ideal is finitely generated.
AIII) (a) Given A → B a homomorphism prove that B is faithfully flat over A iff B is flat over A and the map Spec B → Spec A is surjective. (b) Say A → B is a homomorphism and B is faithfully flat over A. Assume A is noetherian. Show that the topology on Spec A is the quotient topology from Spec B.
AIV) Here A is a commutative ring, but not necessarily with unity. Let A#^ denote A × Z (category of sets) and addition componentwise and multiplication by
〈a, n〉〈b, q〉 = 〈ab + nb + qa, nq〉.
(a) Clearly, A#^ is a commutative ring with unity 〈 0 , 1 〉. A is a subring of A#, even an ideal. Suppose A has the ACC on ideals, prove that A#^ does too. (b) If you know all the prime ideals of A, can you find all the prime ideals of A#?
AV) Let B, C be commutative A-algebras, where A is also commutative. Write D for the A-algebra B ⊗A C.
(a) Give an example to show that Spec D is not Spec B × Spec A
Spec C (category of sets over Spec A).
(b) We have A-algebra maps B → D and C → D and so we get maps Spec D → Spec B and Spec D → Spec C (even maps over Spec A), and these are maps of topological spaces (over Spec A). Hence, we do get a map
θ : Spec D → Spec B Π Spec A Spec C (top. spaces).
Show ∃ closed sets in Spec D not of the form θ−^1 (Q), where Q is a closed set in the product topology of Spec B Π Spec A Spec C.
Part B
BI) Let A = Z[T ], we are interested in Spec A.
(a) If p ∈ Spec A, prove that ht(p) ≤ 2. (b) If {p} is closed in Spec A, show that ht(p) = 2. Is the converse true? (c) We have the map Z ↪→ Z[T ] = A, hence the continuous map Spec A −→π Spec Z. Pick a prime number, say p, of Z. Describe π−^1 (p), is it closed?
(d) When exactly is a p ∈ Spec A the generic point (point whose closure is everything) of π−^1 (p) for some prime number p? (e) Describe exactly those p ∈ Spec A whose image, π(p), is dense in Spec Z. What is ht(p) in these cases? (f) Is there a p ∈ Spec A so that the closure of {p} is all of Spec A? What is ht(p)? (g) For a general commutative ring, B, if p and q are elements of Spec B and if q ∈ {p} show that ht(q) ≥ ht(p) (assuming finite height). If p, q are as just given and ht(q) = ht(p) is q necessarily p? Prove that the following are equivalent: i. Spec B is irreducible (that is, it is NOT the union of two properly contained closed subsets) ii. (∃ p ∈ Spec B)(closure of {p} = Spec B) iii. (∃ unique p ∈ Spec B)(closure of {p} = Spec B) iv. N (B) ∈ Spec B. (h) Draw a picture of Spec Z[T ] as a kind of plane over the “line” Spec Z and exhibit in your picture all the different kinds of p ∈ Spec Z[T ].
BII) If A is a commutative ring, we can view f ∈ A as a “function” on the topological space Spec A as follows: for each p in Spec A, as usual write κ(p) for Frac(A/p); [note that κ(p) = Ap/its max. ideal] and set f (p) = image of f in A/p considered in κ(p). Thus, f : Spec A →
p∈Spec A
κ(p). Observe that
if f ∈ N (A), then f (p) = 0 all p, yet f need not be zero as an element of A.
(a) Let A = k[X 1 ,... , Xn]. We’ll prove soon that there are fields, Ω, containing k so that i. Ω has infinitely many transcendental elements independent of each other and of the Xj over k and ii. Ω is algebraically closed, i.e., all polynomials with coefficients in Ω have a root in Ω. An example of this is when k = Q or some finite extension of Q and then we can take Ω = C. In any case, fix such an Ω. Establish a set-theoretic map Ωn^ → Spec A so that f ∈ A = k[X 1 ,... , Xn] viewed in the usual way as a function on Ωn^ agrees with f viewed as a function on Spec A. We can topologize Ωn^ as follows: call a subset of Ωn^ k-closed iff ∃ finitely many polynomials f 1 ,... , fp from A so that the subset is exactly the set of common zeros of f 1 ,... , fp. This gives Ωn^ the k-topology (an honest topology, as one checks). Show that your map Ωn^ → Spec A is continuous between these topological spaces. Prove, further, that Ωn^ maps onto Spec A. (b) Show that Ωn^ is irreducible in the k-topology. (c) Define an equivalence relation on Ωn: ξ ∼ η ⇐⇒ each point lies in the closure (k-topological) of the other. Prove that Ωn/ ∼ is homeomorphic to Spec A under your map.
BIII) Let A be an integral domain and write K for Frac(A). For each ξ ∈ K, we set
dom(ξ) = {p ∈ Spec A | ξ can be written ξ = a/b, with a, b ∈ A and b(p) 6 = 0}.
(a) Show dom(ξ) is open in Spec A. (b) If A = R[X, Y ]/(X^2 + Y 2 − 1), set ξ = (1 − y)/x (where x = X and y = Y ). What is dom(ξ)? (c) Set A = C[X, Y ]/(Y 2 − X^2 − X^3 ) and let ξ = y/x. What is dom(ξ)? (d) Note that as ideals of A (any commutative ring) are A-modules, we can ask if they are free or locally free. Check that the non-zero ideal, a, of A is free ⇐⇒ it is principal and
a → (0)
The second condition is automatic in a domain. Now look again at A = R[X, Y ]/(X^2 + Y 2 − 1), you should see easily that this is a domain. Characterize as precisely as you can the elements m ∈ Max(A) which are free as A-modules. If there are other elements of Max(A), are these locally free? What is the complement of Max(A) in Spec A? Prove that A ⊗R C is a PID.
