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
Tuesday, February 2, 2010
Homework #1
Here is the first homework:
1. Let M be a finitely generated R-module and P in Spec(R). Then M_p = 0 if and only if P does not contain the annihilator of M.
2. Suppose R is a domain. Then one can view R_m as a subset of the fraction field of R, for every maximal ideal m. Show that R is the intersection of all R_m where m ranges over all maximal ideals.
3. Let R be a ring, and S,T two m.c. subsets of R. Let U be the image of T in S^(-1)R. Show that (ST)^(-1)R = U^(-1)(S^(-1)R).
4. Show that if S is a m.c. subset of R, then S^(-1)R is isomorphic to R[{x_u}]/(ux_u - 1).
5. Let R be a ring and let M be an R-module. Suppose that {f_i} is a set of elements of R that generate the unit ideal. Prove that
a) If m \in M goes to 0 in each M_f_i, then m = 0, and
b) If m_i \in M_f_i are elements such that m_i and m_j go to the same element of M_(f_if_j), then there is an element m \in M such that m goes to m_i in M_f_i.
6. A m.c. subset S of R is called saturated if for all a,b in R, ab in S implies a or b is in S. Show that S is saturated if and only if the complement of S is a union of prime ideals.
7. Let S_0 be the set of nonzerodivisors of a ring R. Show that S_0 is a saturated m.c. subset. The ring S_0^(-1)R is called the total ring of fractions of R.
i) Show that S_0 is the largest m.c. subset for which R --> S_0^(-1)R is injective.
ii) Show that every element of S_0^(-1)R is either a zerodivisor or a unit.
iii) Show that every ring in which every non-unit is a zerodivisor is equal to its total ring of fractions.
Please email me if you have any questions, or if you see a typo in the questions.
Forgot to include the due date for the assignment: Feb 11th.
1. Let M be a finitely generated R-module and P in Spec(R). Then M_p = 0 if and only if P does not contain the annihilator of M.
2. Suppose R is a domain. Then one can view R_m as a subset of the fraction field of R, for every maximal ideal m. Show that R is the intersection of all R_m where m ranges over all maximal ideals.
3. Let R be a ring, and S,T two m.c. subsets of R. Let U be the image of T in S^(-1)R. Show that (ST)^(-1)R = U^(-1)(S^(-1)R).
4. Show that if S is a m.c. subset of R, then S^(-1)R is isomorphic to R[{x_u}]/(ux_u - 1).
5. Let R be a ring and let M be an R-module. Suppose that {f_i} is a set of elements of R that generate the unit ideal. Prove that
a) If m \in M goes to 0 in each M_f_i, then m = 0, and
b) If m_i \in M_f_i are elements such that m_i and m_j go to the same element of M_(f_if_j), then there is an element m \in M such that m goes to m_i in M_f_i.
6. A m.c. subset S of R is called saturated if for all a,b in R, ab in S implies a or b is in S. Show that S is saturated if and only if the complement of S is a union of prime ideals.
7. Let S_0 be the set of nonzerodivisors of a ring R. Show that S_0 is a saturated m.c. subset. The ring S_0^(-1)R is called the total ring of fractions of R.
i) Show that S_0 is the largest m.c. subset for which R --> S_0^(-1)R is injective.
ii) Show that every element of S_0^(-1)R is either a zerodivisor or a unit.
iii) Show that every ring in which every non-unit is a zerodivisor is equal to its total ring of fractions.
Please email me if you have any questions, or if you see a typo in the questions.
Forgot to include the due date for the assignment: Feb 11th.
Tuesday, January 26, 2010
Welcome!
Hello everyone!
In this course we will be covering the basic material in commutative algebra, a subject that has applications to algebraic geometry, number theory, combinatorics, and invariant theory. Below is the (tentative) list of topics I will cover:
Localization, support of a module
Local rings, NAK
Associated primes, primary decomposition
Integrality, Going up/down
Valuation rings, valuative criterion for normality
Noether normalization
Artinian rings
Discrete valuation rings, Dedekind domains
Completions (?)
Hilbert functions, dimension theory
Introduction to Tor/Ext
The Koszul complex
Depth
Cohen-Macaulay rings
Gorenstein rings
This seems quite ambitious on paper, and we may not get to cover everything, especially some of the things at the end.
Things that we may have to review along the way:
Exact sequences
Hom and tensor, exactness properties
Nilpotent elements and the radical
Nullstellenzatz
We will follow a combination of the books by Atiyah-Macdonald and Eisenbud, both standard texts in the area. Homeworks will be assigned and collected on a somewhat regular basis (about once every 1-2 weeks). I'll post what we have done and where we intend to go at the end of each class on the blog.
In this course we will be covering the basic material in commutative algebra, a subject that has applications to algebraic geometry, number theory, combinatorics, and invariant theory. Below is the (tentative) list of topics I will cover:
Localization, support of a module
Local rings, NAK
Associated primes, primary decomposition
Integrality, Going up/down
Valuation rings, valuative criterion for normality
Noether normalization
Artinian rings
Discrete valuation rings, Dedekind domains
Completions (?)
Hilbert functions, dimension theory
Introduction to Tor/Ext
The Koszul complex
Depth
Cohen-Macaulay rings
Gorenstein rings
This seems quite ambitious on paper, and we may not get to cover everything, especially some of the things at the end.
Things that we may have to review along the way:
Exact sequences
Hom and tensor, exactness properties
Nilpotent elements and the radical
Nullstellenzatz
We will follow a combination of the books by Atiyah-Macdonald and Eisenbud, both standard texts in the area. Homeworks will be assigned and collected on a somewhat regular basis (about once every 1-2 weeks). I'll post what we have done and where we intend to go at the end of each class on the blog.
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