Assigned homework problems S. L. Kleiman, fall 2008

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1 Assigned homework problems S. L. Kleiman, fall 2008 Problem Set 1. Due 9/11 Problem R 1.5 Let ϕ: A B be a ring homomorphism. Prove that ϕ 1 takes prime ideals P of B to prime ideals of A. Prove in particular that, if A B, then A P is a prime ideal of A. Problem R 1.6 Prove or give a counterexample: (a) the intersection of two prime ideals P 1, P 2 is prime; (b) the ideal P 1 + P 2 generated by two prime ideals P 1, P 2 is again prime; (c) if ϕ: A B is a ring homomorphism, then ϕ 1 takes maximal ideals of B to maximal ideals of A; (d) the map ϕ 1 of Proposition 1.2 takes maximal ideals of A/I to maximal ideals of A. Problem R 1.12(a) Let I and J be ideals, and P a prime ideal. Prove that IJ P iff I J P iff I or J P. * Problem SLK1 (The * means that this problem is to be presented in class.) Let B be a ring, I an ideal, and A := B[y] the polynomial ring. Construct an isomorphism from A/IA onto (B/I)[y]. Problem Set 2. Due 9/18 Problem SLK2 Let B be a UFD, and A := B[y] the polynomial ring. Let f be a polynomial that has a term by i with i > 0 such that b is not divisible by some prime element p in B. Prove that the ideal (f) is not maximal. * Problem R 1.18 Use Zorn s lemma to prove that any prime ideal P contains a minimal prime. Problem R 1.9 (a) Let a be a unit, and x a nilpotent. Prove that a + x is again a unit. [Hint: expand (1 + x/a) 1 as a power series to guess the inverse.] (b) Let A be a ring, and I nilrad A an ideal; if x A maps to an invertible element of A/I, prove that x is invertible in A. Problem R 1.4 Two ideal I and J are said to be (strongly) coprime if I + J = A. Check that this notion is the usual notion of coprime for A = Z or k[x]. (Recall that two elements in a UFD are said to be coprime, or relatively prime, if they share no common factors.) Prove that, if I and J are strongly coprime, then IJ = I J and A/IJ = (A/I) (A/J). Prove also that, if I and J are coprime, then so are I n and J n for n 1. Problem R 1.11 Find the nilpotent and idempotent elements of Z/(n) where n = 6, 12, pq, pq 2 or p n i i where p, q, p i are distinct prime numbers. * Problem R 1.10 Think through the equivalent conditions in the definition of local ring in In particular, let A be a ring and I a maximal ideal such that 1 + x is a unit for every x I, and prove that A is a local ring. Does the implication still hold if I is not maximal? 1

2 Problem Set 3. Due 9/25 Problem R 2.1 Let A be an integral domain with field of fractions K, and let f A. Prove that A[1/f] is not a finite A-module. [Hint: assuming A[1/f] has a finite set of generators, prove that 1, f 1, f 2..., f k also form a finite set of generators for some k > 0, so that f k+1 can be expresses as a linear combination of them. Use this statement to prove that f is a unit.] * Problem R 2.6 Let (A,m,k) be a local ring, M a finite A-module, and s 1,...,s n M. Set M := M/mM, and write s i M for the image of s i. Prove that s 1,...,s n form a basis of M if and only if s 1,...,s n form a minimal generating set of M, and prove that every minimal generating set of M has the same number of elements. Problem R 2.8 Let A be a ring, and I an ideal. Assume I is finitely generated and satisfies I = I 2. Prove that there exists a unique idempotent e such that (e) = I. * Problem R 2.9 Let 0 L M N 0 be a short exact sequence of A-modules. Prove that, if L and N are finitely generated, then so is M. Problem SLK3 Let L, M, N be A-modules, and α: L M, β: M N, σ: N M, ρ: M L homomorphisms. Prove that M = L N and α = i L, β = π N, σ = i N, ρ = π L if and only and if and only if the following relations hold: βα = 0, βσ = 1, ρσ = 0, ρα = 1, and αρ + σβ = 1. Problem R 3.2 Let k be a field, and A k an overring. Assume that A is finite dimensional as a k-vector space. Prove that A is Noetherian and Artinian. Problem R 3.5 Let 0 L M N 0 be a short exact sequence of A-modules, and M 1, M 2 two submodules of M. Decide whether the following implication holds (prove it, or give a counterexample): β(m 1 ) = β(m 2 ) and α 1 (M 1 ) = α 1 (M 2 ) = M 1 = M 2. Problem Set 4. Due 10/2 Problem R 3.3 Let A be a ring, and I 1,...,I k ideals such that each A/I i is a Noetherian ring. Prove that A/I i is a Noetherian A-module, and deduce that, if I i = 0, then A too is a Noetherian ring. Problem R 3.4 Let A be a Noetherian ring, and M a finite A-module. Prove that there exists an exact sequence A q α A p β M 0. (That is, M has a presentation as an A-module in terms of finitely many generators and relations.) Problem R 3.7 Consider the Z-module M := (Z[1/p]) / Z with p a prime. Prove that any submodue N M is either finite (as a set) or the whole of M. Deduce that M is an Artinian Z-module, and prove that it is not finitely generated (so not Noetherian). * Problem R 3.8 Let A be an Artinian integral domain (see 3.2 for the definition). Prove that A is a field. [Hint: for f A, the dcc applied to (f) (f 2 ) gives a relation f k = af k+1.] Deduce that every prime ideal of an Artinian ring is maximal. Problem R 4.1(a) The extension k[x 2 ] k[x] is finite, hence integral. Find the equation of integral dependence for an arbitrary f k[x]. 2

3 * Problem R 4.5 Set A := k[x,y ] / (Y 2 X 2 X 3 ), and let x, y A be the residues of X, Y. Set t := y/x Frac(A). As in 4.4, Example (iii), prove that k[t] is the normalization of A. Problem AM 5.6, p. 67 Let B 1,...,B n be integral A-algebras. Show that their product B i is an integral A-algebra. Problem Set 5. Due 10/9 Problem R 4.9 Let k be an arbitrary field, and set A := k[x,y,z] / (X 2 Y 3 1, XZ 1). Find α,β k such that A is integral over B := k[x + αy + βz], and for your choice, write down a set of generators of A as a B-module. * Problem R 4.10 Let k := F q be the finite field with q elements, and consider the polynomial f := XY q 1 X q in k[x,y ]. Set A := k[x,y ] / (f). For every α k, show that A is not module-finite over B α := k[x αy ]. (In other words, the alternative proof of Section 4.8 does not work in this case.) [First examine the case α = 0.] Problem AM 5.9, p. 68 Let B be a ring, A a subring, and C the integral closure of A in B. Prove that the polynomial ring in one variable C[x] is the integral closure of A[x] in B[x]. [Hint: Let f B[x] be integral over A[x], say f m + g 1 f m g m = 0 with g i A[x]. Let r be an integer larger than m and the degrees of g 1,...,g m. Set f 1 := f x r. Then or say (f 1 + x r ) m + g 1 (f 1 + x r ) m g m = 0, f m 1 + h 1 f m h m = 0 where h m := (x r ) m + g 1 (x r ) m g m A[x]. Apply [AM 5.8, p.67] to the polynomials f 1 and g := f1 m 1 + h 1 f1 m h m 1 (do not work [AM 5.8], as it was done in class).] * Problem R 6.3(a) Let A and A be rings, and set A := A A. Prove that A and A are rings of fractions of A. Problem R 6.1 Find a ring A and a multiplicative set S such that the relation (a,s) (b,t) at = bs is not an equivalence relation. (Show that the ring A of Exer. 6.3(a) works.) Problem R 6.5 Find all intermediate rings Z A Q. (Describe each A as a localization of Z.) [Hint: as a starter, consider the subring Z[2/3] Q. Is 1/3 Z[2/3]?] 3

4 Problem Set 6. Due 10/16 * Problem R 6.13 Let S T be multiplicative sets in a ring A. Set A := S 1 A, and let ϕ: A A be the canonical map. Set T := ϕ(t). Prove that T 1 A = T 1 A. In other words, a composite of two localizations is a localization. [Hint: the easiest way to proceed is to use the UMP of 6.2 (c) and Ex ] Problem SLK 4 Let k be a field, and K an algebraically closed field containing k. (Recall that K contains a copy of every algebraic extension of k.) Let A be the polynomial ring in n variables over k, and f,f 1,...,f r polynomials in A. Suppose that, for any n-tuple a := (a 1,...,a n ) of elements a i of K such that f 1 (a) = 0,...,f r (a) = 0, also f(a) = 0. Prove that there are an integer N and polynomials g 1,...,g r in A such that f N = g 1 f g r f r. Problem SLK5 Let A be a ring, and P a module. Then P is called projective if the functor N Hom(P, N) is exact. (1) Prove that P is projective if and only if, given any surjection ψ: M N, every map ν : P N lifts to a map µ: P M; that is, ψµ = ν. (2) Prove that P is projective if and only if every short exact sequence 0 L φ M ψ P 0 is split. (3) Prove that P is projective if and only if P is a direct summand of a free module F; that is, F = P L for some L. (4) Assume that A is local and that P is finitely generated; then prove that P is projective if and only if P is free. Problem SLK 6 Let A be a Noetherian ring, and P a finitely generated A-module. Prove that the following three conditions are equivalent: (1) P is projective; (2) P p is free over A p for every prime ideal p; and (3) P m is free over A m for every maximal ideal m. * Problem AM 3.1, p.43 Let S be a muliplicative set of a ring A, and M a finitely generated A-module. Prove that S 1 M = 0 if and only if there exists an s S such that sm = 0. Problem SLK7 Let A be a ring, M an arbitrary A-module, and I the annihilator of M. Prove that the support Supp(M) is always contained in the set V(I) of primes containing I. Problem SLK8 Let Z be the ring of integers, Q the rational numbers, and set M := Q/Z. Find the support Supp(M), and show that it s not Zariski closed (that is, it does not consist of all the primes containing any ideal). Problem Set 7. Due 10/23 * Problem R 7.2 Set M := Z/(2) Z and view M as a Z-module. Prove (i) AssM = {(0), (2)} and (ii) there exist two submodules M 1,M 2 M such that M 1,M 2 = Z and M = M1 + M 2. Problem R 7.3 If M = M 1 + M 2, then Ass M = Ass M 1 Ass M 2 : true or false? (If true, prove it; if false, give a counterexample.) Problem SLK 9 Let A be a Noetherian ring, M a finitely generated module. Prove that the intersection of all the associated primes of M is equal to the radical of the annihilator Ann(M). Problem Set 8. Due 10/30 Problem R 7.4 Give an example of a ring A and an ideal I that is not primary, but satisfies the condition fg I f n I or g n I for some n. [Hint: that is, find a nonprimary ideal whose radical is prime; there s only one counterexample in primary decomposition!] 4

5 Problem R 7.6 Let A be a ring, S a multipicative set, and ϕ: A S 1 A the natural map. Let P A be a prime disjoint from S, and Q a P-primary ideal. Prove that S 1 P is a prime of S 1 A. (This statement was proved in class via a direct computation. Give an alternative proof: show that S 1 A/S 1 P is a domain by using the exactness of localization.) Prove also that S 1 Q is S 1 P-primary and that ϕ 1 (S 1 Q) = Q. * Problem R 7.7 Let ϕ: A B be a ring homomorphism, and Q B a P-primary ideal. Show that ϕ 1 Q A is ϕ 1 P-primary. Problem R 7.8 Let A be the polynomial ring k[x,y,z], and I the ideal (XY, X Y Z). Find a primary decomposition I = Q 1 Q n, and determine P i := Q i. Hint: to guess the result, draw the variety V(I). To prove it, note that the variety V(X Y Z) is the graph of a function, so isomorphic to the (Y, Z)-plane; consider the homomorphism A k[y, Z] sending X to Y Z, and use Problem 7.7.] Problem R 7.10 Let A := k[x,y,z]/(xz Y 2 and P := (X,Y ) (as in 7.10, Example 2), and set M := A/P 2. (a) Determine Ass M. [Hint: use 7.10, Example 2, and Theorem 7.12 (The First Uniqueness Theorem).] (b) Find elements of M annihilated by each assasin. (c) Find a chain of submodules M 1 M n := M as in Theorem 7.6. * Problem SLK10 Let A be a Noetherian ring, I and J ideals. Assume JA P is contained in IA P for all associated primes P of A/I. Prove J is contained in I. Problem Set 9. Due 11/6 * Problem SLK11 Let A be a Noetherian ring, x A. Assume x lies in no associated prime of A/I. Prove the intersection of the ideals (x) and I is equal to their product (x)i. Problem R 8.1 Using Nakayama s lemma, show that, if (A, m) is a Noetherian local ring, then the maximal ideal m is principal if and only if m/m 2 is 1-dimensional over k := A/m. Deduce from Proposition 8.3 that A is a DVR if and only if A is Noetherian and local with SpecA = {0,m} and m/m 2 is 1-dimensional over k. Problem R 8.2 DVRs and nonsingular curves Let k be an algebraically closed field and f k[x,y ] an irreducible polynomial of the form f = l(x,y ) + g(x,y ) with l(x,y ) = ax + by and g (X,Y ) 2. Set R = k[x,y ]/(f) and P = (X,Y )/(f) and (A,m) = (R P,m P. Prove that A is a DVR if and only if l 0. [Hint: use Exercise 8.1.] (The result says that A is a DVR if and only if the plane curve C : f = 0 k 2 is nonsingular at (0,0).) Problem R 8.4 Let A be an intermediate ring between the polynomial ring and the formal power series ring in one variable, k[x] A k[[x]], and suppose that A is local with maximal ideal (x). Prove that A is a DVR, and in particular A is Noetherian. [Hint: use Proposition 8.3.] (Local rings of this type include the localization of very complicated rings, but they are all covered by the theory of DVRs. They can be viewed from an anyalytic point of view as rings of power series with curious convergence conditions.) 5

6 Problem SLK 12 Let A be a Noetherian ring, M a finitely generated module, Q a submodule. Set P := Ann(M/Q). Prove the equivalence of these two conditions: (1) Q is P-primary; that is, Ass(M/Q) = {P }; and (2) every zero divisor on M/Q is nilpotent on M/Q; in other words, given an a A for which there exists an x M Q such that ax Q, necessarily a P. Problem SLK13 Let A be a domain, K its fraction field. Show that A is a valuation ring if and only if, given any two ideals I and J, either I lies in J or J lies in I. * Problem R 8.6 Let A be a general valuation ring, m its maximal ideal, and P m another prime ideal. Prove that there is a valuation ring B A such that P is the maximal ideal of B, and prove that A/P is a valuation ring of the fields B/P. Problem Set 10. Due 11/13 * Problem SLK14 Let v be a valuation of a field K, and x 1,...,x n nonzero elements of K with n > 1. Show that (1) if v(x 1 ) and v(x 2 ) are distinct, then v(x 1 + x 2 ) = min{v(x 1 ),v(x 2 )} and that (2) if x x n = 0, then v(x i ) = v(x j ) for two distinct indices i and j. Problem SLK15 Prove that a valuation ring is normal. Problem R 8.7 Let k be a field, and K := k(y ) the field of rational functions in one variable. Form the valuation ring C := k[y ] (Y ) of K. Form the field K(X) = k(x,y ) of rational functions, and form its valuation ring B := K[X] (X). Finally, let A B be the inverse image of C under the canonical surjection of B onto K = B/(X). Prove that A is a valuation ring whose value group K(X) /A is equal to Z Z with the lex order (compare the first components, and break ties with the second). Problem SLK16 Let A be a Dedekind domain. Suppose A is semilocal (that is, A has only finitely many maximal ideals). Prove A is a PID. Problem AM 9.7, p.99 Let A be a Dedekind domain, and I, J two nonzero ideals. (1) Prove every ideal in A/I is principal. (2) Prove J can be generated by two elements. Problem AM 9.8, p.99 Let A be a Dedekind domain, and I, J, K ideals. By first reducing to the case that A is local, prove that I (J + K) = (I J) + (I K), I + (J K) = (I + J) (I + K). Problem Set 11. Due 11/20 Problem SLK17 Let A be a Noetherian ring, and suppose A P is a domain for every prime P. Prove the following four statements: (1) Every associated prime of A is minimal. (2) The ring A is reduced. (3) The minimal primes of A are pairwise coprime. (4) The ring A is equal to the product of its quotients A/P as P ranges over the set of all minimal primes. 6

7 Problem SLK18 Let A be a UFD, and M an invertible fractional ideal. Prove M is principal. * Problem SLK 19 Let A be a domain, K its fraction field, L a finite extension field, and B the integral closure of A in L. Show that L is the fraction field of B. Show that, in fact, every element of L can be expressed as a fraction b/a where b is in B and a is in A. Problem SLK20 Let A B be domains, and K, L their fraction fields. Assume that B is a finitely generated A-algebra, and that L is a finite dimensional K-vector space. Prove that there exists an f A such that B f is a finite generated A f -module. Problem E13.2 Let G be a finite group acting on a domain T, and R := T G the ring of invariants. Show that every element b T is integral over R, in fact, over its subring S generated by the elementary symmetric functions in the conjugates gb for g G. * Problem E 13.3 Prove the following celebrated theorem of E. Noether (1926): in the setup of the preceding problem, if T is a finitely generated algebra over the field k, then so is R. Problem SLK 21 Let A be a ring, P a prime ideal, and B an integral extension ring. Suppose B has just one prime Q over P. Show (a) that QB P is the only maximal ideal of B P, (b) that B Q = B P, and (c) that B Q is integral over A P. Problem SLK22 Let A be a ring, P a prime ideal, B an integral extension ring. Suppose B is a domain, and has at least two distinct primes Q and Q over P. Show B Q is not integral over A P. Show, in fact, that if x lies in Q, but not in Q, then 1/x B Q is not integral over A P. Problem SLK23 Let k be a field, and x an indeterminate. Set B := k[x], and set y := x 2 and A := k[y]. Set P := (y 1)A and Q := (x 1)B. Is B Q is integral over A P? Explain. Problem Set 12. Due 12/4 * Problem SLK24 Let A be a ring (possibly not Noetherian), P a prime ideal, and B a modulefinite A-algebra. Show that B has only finitely many primes Q over P. [Hint: reduce to the case that A is a field by localizing at P and passing to the residue rings.] Problem E9.4 Let A be a ring, S a multiplicative set, Q a prime of the localization S 1 A, and P its contraction in A. Prove that Q and P have the same height: ht(q) = ht(p). Problem SLK25 Let k be a field, A a finitely generated k-algebra, and f a nonzero element of A. Assume A is a domain. Prove that A and its localization A f have the same dimension. Problem SLK26 Let A be a DVR, and f a uniformizing parameter. Show that A and its localization A f do NOT have the same dimension. Problem SLK27 Let L/K be an algebraic field extension. Let X 1,...,X n be indeterminates, and A and B the corresponding polynomial rings over K and L. (1) Let Q be a prime of B, and P its contraction in A. Show ht(p) = ht(q). (2) Let f and g be two polynomials in A with no common factors in A. Show f and g have no common factors in B. * Problem SLK28 Let k be a field, and A a finitely generated k-algebra. Prove that A is Artin if and only if A is a finite-dimensional k-vector space. 7

8 Problem SLK 29 Let A be an r-dimensional finitely generated domain over a field, and x an element that s neither 0 nor a unit. Set B := A/(x). Prove that B is equidimensional of dimension r 1 (that is, dim(b/q) = r 1 for every minimal prime Q); prove that, in fact, r 1 is the length of any maximal chain of primes in B. * Problem SLK30 Let A,m be a Noetherian local ring. Assume that m is generated by an A-sequence x 1,...,x r. Prove that A is regular of dimension r. Problem SLK31 Let A,m be a Noetherian local ring of dimension r, and B := A/I a factor ring of dimension s. Set t := r s. Prove that the following three conditions are equivalent: (1) A is regular, and I is generated by t members of a regular sop; (2) B is regular, and I is generated by t elements; and (3) A and B are regular. Problem SLK32 (a) Let A be a Noetherian local ring, and P a principal prime ideal of height 1. Prove that A is a domain. (b) Let k be a field, and x an indeterminate. Show that the product ring k[x] k[x] is not a domain, yet it contains a principal prime ideal P of height 1. Problem Set 13. Due 12/19 Problem SLK33 (a) Let A be a ring, S a multiplicative set, and M an A-module. Prove that S 1 M = S 1 A M by showing that the two natural maps M S 1 M and M S 1 A M enjoy the same universal property. (b) Show that (1,1,... ) is nonzero in Q ( i Z/(i)). * Problem SLK34 Let A be a ring, I and J ideals, and M an A-module. (a) Use the right exactness of tensor product to show that (A/I) M = M/IM. (b) Show that (A/I) (A/J) = A/(I + J). (c) Assume that A is a local ring with residue field k, and that M is finitely generated. Show that M = 0 if and only if M k = 0. (d) Let R be the real numbers, C the complex numbers, and X an indeterminate. Using the formula C = R[X]/(1 + X 2 ), express C R C as a product of Artin local rings, identifying the factors. Problem SLK35: Let A be an arbitary ring, M and N A-modules, and k a field. (a) Assume M and N are free of ranks m and n. Prove that M N is free of rank mn. (b) Given nonzero k-vector spaces V and W, show that V W is also nonzero. (c) Assume A is local, and M and N are finitely generated. Prove that M N = 0 if only only if M = 0 or N = 0. (d) Assume M and N are finitely generated. Prove Supp(M N) = Supp(M) Supp(N). 8

ALGEBRA EXERCISES, PhD EXAMINATION LEVEL

ALGEBRA EXERCISES, PhD EXAMINATION LEVEL 1. Suppose that G is a finite group. (a) Prove that if G is nilpotent, and H is any proper subgroup, then H is a proper subgroup of its normalizer. (b) Use (a)