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.
Thursday, February 25, 2010
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.
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