This homework is not due until May 11th, and as such I will be adding to it little by little as we cover more material.

1. Let 0 --> M' --> M --> M'' --> 0 be a short exact sequence of R-modules. Prove that one of the three following situations holds:

a) depth M' = depth M <= depth M'' or

b) depth M = depth M'' < depth M' or

c) depth M'' = depth M'-1 < depth M

2. Let M be a f.g. module of dimension n over a Noetherian local ring R. Suppose that for every part of a system of parameters of M (x_1,...,x_r) and for every p in Ass(M/(x_1,...,x_r)M), one has dim (R/p) = n-r. Show that M is Cohen-Macaulay.

3. Let R be a Noetherian local ring, M a f.g. R-module, and (x_1,...,x_r) an M-regular sequence. Show that pdim_{R} M/xM = pdim_R M + n (be careful with the infinite case!)

4. Give an example of a two dimensional ring that is not Cohen-Macaulay. (Note that it must necessarily not be normal)

5. Let R be a ring, and let M and N be R-modules. Suppose that (x_1,...,x_r) is an M-regular sequence in the annihilator of N. Show that Ext^r_R(N,M) \cong Hom_R(N,M/(x_1,...,x_r)M).

New Problems:

6. Let R be a Noetherian ring, M an R-module, N a f.g. R-module, and n > 0 an integer. Suppose that Ext^n_R(R/p,M) = 0 for all p \in Supp(N). Show that Ext^n_R(N,M) = 0.

7. Let (R,m,k) be a Noetherian local ring, \phi : M --> N a homomorphism of f.g. R-modules, and (x) = x_1,...,x_s an N-regular sequence. If \phi \otimes R/(x) is an isomorphism, then \phi is an isomorphism.

8. Let (R,m,k) be a Gorenstein local ring of dimension d, and M a f.g. module of finite projective dimension. Show that Tor_i^R(k,M) \cong Ext^d-i_R(k,M).

## Thursday, April 15, 2010

## Wednesday, March 10, 2010

### Homework #4

1. Problem #1 on the previous homework was meant to be an easy application of the characterization of Dedekind domains we did in class. However, I overlooked the crucial point of the problem, which is show that the integral closure of R in L is Noetherian. If you figured out a way around that, then great, but if you were unable to do it before, here is an outline of an easier proof of this fac, but only in the case that L is a separable extension of K. We will recall the following factoids from Galois theory:

Factoid: Let Tr denote the trace of the field extension L/K, which is a K-linear map from L to K that sends l \in L to the sum of its conjugates by the distinct embeddings L into the algebraic closure of K. By a theorem from field theory, the trace map is nonzero when the extension is separable. You may also assume you know the following fact: For any basis {u_1,...,u_n} of L over K, there exists another basis {v_1,...,v_n} of L over K such that Tr(u_iv_j) = \delta_{ij} (the Kronecker delta).

a) Show that Tr(s) \in R for all s \in S. (Hint: Show it is integral over R)

b) Show that the fraction field of S is L.

c) Let {u_1,...,u_n} be a basis for L over K. We may assume that the u_i are in S since we can clear denominators. Let {v_1,...,v_n} be the dual basis from the factoid. Show that S \subseteq Rv_1 + ... + Rv_n. (Hint: use a)

d) Conclude that S is Noetherian and that S is hence a Dedekind domain.

All graded rings below are assumed to be nonnegatively graded over Artinian base R_0

2. Let R be a Noetherian graded ring and M a finitely generated graded R-module. Then for any k >= 1, M_n \subseteq (R_+)^k M for n sufficiently large.

3. Let R be a Noetherian graded ring with R_0 artinian, and M a finitely generated graded R-module. An element x \in R_l is superficial for M if (0:_M x) has finite length (i.e. (0:_M x)_n = 0 for all but finitely many n). Show that for every M f.g., there exists an element x \in R_l that is superficial for M. (Hint: Use graded primary decomposition, as well as the graded version of prime avoidance to show that you can pick an element x \in R_+ homogeneous that is not in any of the associated primes of M. Then show that x is superficial.)

4. Let R be a Noetherian graded ring, and M a f.g. graded R-module. Let x \in R_n be a nonzerodivisor on M. Compute H(M/xM,t) in terms of H(M,t).

5. Let R be a Noetherian graded ring and M a finitely generated R-module. Show that if x is a homogeneous nonzerodivisor of M, then dim M/xM = dim M - 1.

6. Let R be a Noetherian graded ring and M a f.g. graded R-module of dimension d. Suppose that x_1,...,x_d is a regular sequence on M generated in degree 1. Show that the H(M,t)(1-t)^d is a polynomial with nonnegative coefficients.

7. Prove number 5 in the local case as well.

8. Let (R,m) be a Noetherian local ring and set G = gr^m(R) (the associated graded ring of R with respect to the m-adic filtration). Show that if G is a domain, then so is R. Show that the converse can fail.

This homework is due on March 30th (the Tuesday after spring break).

Factoid: Let Tr denote the trace of the field extension L/K, which is a K-linear map from L to K that sends l \in L to the sum of its conjugates by the distinct embeddings L into the algebraic closure of K. By a theorem from field theory, the trace map is nonzero when the extension is separable. You may also assume you know the following fact: For any basis {u_1,...,u_n} of L over K, there exists another basis {v_1,...,v_n} of L over K such that Tr(u_iv_j) = \delta_{ij} (the Kronecker delta).

a) Show that Tr(s) \in R for all s \in S. (Hint: Show it is integral over R)

b) Show that the fraction field of S is L.

c) Let {u_1,...,u_n} be a basis for L over K. We may assume that the u_i are in S since we can clear denominators. Let {v_1,...,v_n} be the dual basis from the factoid. Show that S \subseteq Rv_1 + ... + Rv_n. (Hint: use a)

d) Conclude that S is Noetherian and that S is hence a Dedekind domain.

All graded rings below are assumed to be nonnegatively graded over Artinian base R_0

2. Let R be a Noetherian graded ring and M a finitely generated graded R-module. Then for any k >= 1, M_n \subseteq (R_+)^k M for n sufficiently large.

3. Let R be a Noetherian graded ring with R_0 artinian, and M a finitely generated graded R-module. An element x \in R_l is superficial for M if (0:_M x) has finite length (i.e. (0:_M x)_n = 0 for all but finitely many n). Show that for every M f.g., there exists an element x \in R_l that is superficial for M. (Hint: Use graded primary decomposition, as well as the graded version of prime avoidance to show that you can pick an element x \in R_+ homogeneous that is not in any of the associated primes of M. Then show that x is superficial.)

4. Let R be a Noetherian graded ring, and M a f.g. graded R-module. Let x \in R_n be a nonzerodivisor on M. Compute H(M/xM,t) in terms of H(M,t).

5. Let R be a Noetherian graded ring and M a finitely generated R-module. Show that if x is a homogeneous nonzerodivisor of M, then dim M/xM = dim M - 1.

6. Let R be a Noetherian graded ring and M a f.g. graded R-module of dimension d. Suppose that x_1,...,x_d is a regular sequence on M generated in degree 1. Show that the H(M,t)(1-t)^d is a polynomial with nonnegative coefficients.

7. Prove number 5 in the local case as well.

8. Let (R,m) be a Noetherian local ring and set G = gr^m(R) (the associated graded ring of R with respect to the m-adic filtration). Show that if G is a domain, then so is R. Show that the converse can fail.

This homework is due on March 30th (the Tuesday after spring break).

## Saturday, March 6, 2010

### HW #3 correction

Note that I had the containment backwards in the hint in problem #5.

It should have been that if M is a finitely generated torsion free module of rank n over a Noetherian domain R, then R^n embeds in M.

It should have been that if M is a finitely generated torsion free module of rank n over a Noetherian domain R, then R^n embeds in M.

## Monday, March 1, 2010

## Thursday, February 25, 2010

### Homework #3

1. Let R be a Dedekind domain with fraction field K. Let L be a finite field extension of K, and let S be the integral closure of R in L. Prove that S is a Dedekind domain.

2. Let Q be a prime ideal of R, and let Q^(n) be the n^th symbolic power of Q, i.e., the Q-primary component of the ideal Q^n (we saw in class that powers of prime ideals are not necessarily primary). Since Q is minimal over Q^n, we know that the Q-primary component of Q^n is unique.

a) Prove that Q^(n) := {r \in R | sf \in Q^n for some s \in R, s \not\in Q}

b) Prove that if \phi : R --> R_Q is the localization map, then Q^(n) = \phi^(-1)(Q_Q^n)

c) Prove that (Q^(n))_Q = (Q_Q)^n.

d) Prove that elements outside Q are nonzerodivisors mod Q^(n).

3. Complete the proof of Krull's Principal ideal theorem.

4. Prove that any ideal in a Dedekind domain can be generated by at most two elements.

5. Prove that a finitely generated torsion-free module over a Dedekind domain is projective, and show it is isomorphic to a direct sum of ideals. Note that this direct sum decomposition is not unique (but there is a uniqueness here - it is formulated in terms of the ideal class group of R).

Hint for #5: Suppose R is a domain. For a f.g. torsion free module M of R, define the rank of M to be the dimension over Q(R) of M \otimes_R Q(R). Show that if M is a torsion free module over a Noetherian domain of rank n, then R^n embeds in M.

Next show that rank 1 torsion free modules are isomorphic to ideals over a Dedekind domain, and then argue by induction on the rank of M to get the general case.

2. Let Q be a prime ideal of R, and let Q^(n) be the n^th symbolic power of Q, i.e., the Q-primary component of the ideal Q^n (we saw in class that powers of prime ideals are not necessarily primary). Since Q is minimal over Q^n, we know that the Q-primary component of Q^n is unique.

a) Prove that Q^(n) := {r \in R | sf \in Q^n for some s \in R, s \not\in Q}

b) Prove that if \phi : R --> R_Q is the localization map, then Q^(n) = \phi^(-1)(Q_Q^n)

c) Prove that (Q^(n))_Q = (Q_Q)^n.

d) Prove that elements outside Q are nonzerodivisors mod Q^(n).

3. Complete the proof of Krull's Principal ideal theorem.

4. Prove that any ideal in a Dedekind domain can be generated by at most two elements.

5. Prove that a finitely generated torsion-free module over a Dedekind domain is projective, and show it is isomorphic to a direct sum of ideals. Note that this direct sum decomposition is not unique (but there is a uniqueness here - it is formulated in terms of the ideal class group of R).

Hint for #5: Suppose R is a domain. For a f.g. torsion free module M of R, define the rank of M to be the dimension over Q(R) of M \otimes_R Q(R). Show that if M is a torsion free module over a Noetherian domain of rank n, then R^n embeds in M.

Next show that rank 1 torsion free modules are isomorphic to ideals over a Dedekind domain, and then argue by induction on the rank of M to get the general case.

## Thursday, February 11, 2010

### Homework #2

Here is the second homework assignment. It is due February 25th.

1. Prove that for a finitely generated module M over a Noetherian ring R, that Min Supp(M) = Min Ass(M)

2. If R is Noetherian, and M and N are finitely generated R-modules, show that Ass(Hom(M,N)) = Supp M \cap Ass N

3. Let f : M --> N be a surjective map of R-modules, and L a proper submodule of N. Show that L is primary if and only if f^(-1)(L) is primary, and that in this case, they are primary for the same prime P of R.

4. Our goal will be to prove Krull's Intersection Theorem: Let R be Noetherian, I a proper ideal, and M a finitely generated R-module. Then there is an s \in I such that (1-s)\bigcap_{n=1}^\infty I^nM = (0).

a) Let N = \bigcap_{n=1}^\infty I^nM. Show that IN = N. To do this, consider a primary decomposition of IN = Q_1 \cap ... \cap Q_l, and show that N \subseteq Q_i for each i.

b) Deduce the existence of s \in I that satisfies the theorem.

c) Deduce that if R is a local Noetherian ring, and M a finitely generated R-module, then \bigcap_{n=1}^\infty I^nM = (0).

5. Let R be a ring. Prove that if R is a UFD, then R is normal.

6. Let R be an integral domain, and K its field of fractions. We say that x \in K is almost integral over R if there exists an element r \in R nonzero such that rx^n \in R for all n > 0. Show that if x is integral over R, then it is almost integral, and if R is Noetherian, the converse holds.

7. Let R be a subring of a ring S, with S integral over R. Show that dim R = dim S.

1. Prove that for a finitely generated module M over a Noetherian ring R, that Min Supp(M) = Min Ass(M)

2. If R is Noetherian, and M and N are finitely generated R-modules, show that Ass(Hom(M,N)) = Supp M \cap Ass N

3. Let f : M --> N be a surjective map of R-modules, and L a proper submodule of N. Show that L is primary if and only if f^(-1)(L) is primary, and that in this case, they are primary for the same prime P of R.

4. Our goal will be to prove Krull's Intersection Theorem: Let R be Noetherian, I a proper ideal, and M a finitely generated R-module. Then there is an s \in I such that (1-s)\bigcap_{n=1}^\infty I^nM = (0).

a) Let N = \bigcap_{n=1}^\infty I^nM. Show that IN = N. To do this, consider a primary decomposition of IN = Q_1 \cap ... \cap Q_l, and show that N \subseteq Q_i for each i.

b) Deduce the existence of s \in I that satisfies the theorem.

c) Deduce that if R is a local Noetherian ring, and M a finitely generated R-module, then \bigcap_{n=1}^\infty I^nM = (0).

5. Let R be a ring. Prove that if R is a UFD, then R is normal.

6. Let R be an integral domain, and K its field of fractions. We say that x \in K is almost integral over R if there exists an element r \in R nonzero such that rx^n \in R for all n > 0. Show that if x is integral over R, then it is almost integral, and if R is Noetherian, the converse holds.

7. Let R be a subring of a ring S, with S integral over R. Show that dim R = dim S.

## Friday, February 5, 2010

### Error in HW

In problem number 6, the definition of saturated should be that if ab \in S, then a *and* b are in S. Sorry for the error, I had prime ideals on the brain.

Frank

Frank

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