Chapter 2 - Localization Flashcards
Define: local ring
A ring with just one maximal ideal
Discuss motivation for localization
See pg. 57-58
Define: multiplicatively closed set
A set U is multiplicatively closed if any product of elements in U is in U and 1 is in U
Define: localization of M at U
pg. 59 and Borcherds pg 115 - …
When does an element localize to 0? Proof?
An element m in M goes to 0 in M[U^-1] iff m is annihilated by an element u in U
Examples of localizations?
- Quotient field of integral domain
- Total quotient ring of arbitrary ring
- For P prime localization at R - P written R_P - write k(P) for the residue class field of R at P
- Local ring of an affine variety X at a point x in X
- Z and C[x] Borcherds
What is the localization of a map of R-modules? Functorial properties?
If f: M -> N is a map of R-modules, then there exists a map of R[U^-1]-modules f[U^-1]: M[U^-1] -> N[U^-1] that takes m/u to f(m)/u - called the localization of f. Makes localization into a functor from the category of R-modules to the category of R[U^-1] -modules.pg. 60
Discuss the universal property of localization
If f: R -> S is any ring hom with elements in U going to units, then there is a unique extension to a hom f’:R[U^-1] ->S. Pg. 60
Discuss the relationship between the ideal structure of R and its localization R[S^-1]. Spec R vs Spec R[S^-1]
See Borchards Lecture 17 and Prop 2.2 in Eisenbud
Discuss Spec C[x,y] localize at (0), (f), (x-a, y-b) vs quotient
See Borchards Lecture 17
Discuss C[x,y]/(xy) localized at (x,y)
See Borchards Lecture 17
Discuss functions on Spec R for R = C(X) continuous functions on compact Hausdorff space, R = C[x], R = Z
See Borchards Lecture 18
Why do nilpotents cause issues with trying to represent R as functions on Spec R? Solution? Examples?
See Borchards Lecture 18
Destroy injectivity. See Borchards Lecture 18
Discuss Nilradical. Instead of f: Spec R –> R/P do f: Spec R –> Rp (local ring at P)
Try C[x], Z with local rings
Define: Sheaf of rings on Spec R
See Borchards Lecture 18
What is the motivation for def of Affine Scheme of RIng?
Pretend R is a ring of functions on Spec R
Define: Affine Scheme of Ring
See Borchards Lecture 19
Discuss the algebra geometry dictionary
See Borchards Lecture 19
Define: tensor product of modules over a ring
Defined in terms if universal property i.e. universal module for bilinear maps from MxN. See Borchards Lecture 20.
Prove existence and uniqueness of tensor product
- Unique using universal
- Existence using massive free module construction
See Borchards Lecture 20.
Examples of tensor product - vector space and f.g. abelian groups-Z-modules?
Approach?
The def and construction is hard to compute with. Use 2 properties:
- (M1 + M2) x N = M1xN + M2xN
- R x_R M = M
Borcherds lec20
What problems can occur when taking tensor or hom of s.e.s.?
No problems if working over a field.
Otherwise lists 3 problems all using same s.e.s. 0–>Z–>Z–>Z–>0 where the second map is x2. Universal counterexample to everything
See Borchards Lecture 21.
Define: Hom_R(M,N)
If M and N R-modules, this is the abelian group of all homomorphisms from M to N. Actually an R-module
Prove: Hom_R(R,N) isomorphic to N
Pg 62
In what sense is Hom functorial? Left-exact functor? Right-exact?
pg 62-63 Borcherds Lec 21
What is the relationship between Hom and tensor? Discuss properties of tensor in this context
64
In what sense is tensor product functorial? Impact on s.e.s?
Pg 64-65 borcherds 21
How can you calculate tensor product?
Borcherds 21
What are direct limits? Impact of tensor product?
End of Borcherds 21
Define: Flat module
M is flat if M tensor preserves exactness
Lec 22
66
Prove: localization M[S^-1] preserves exactness.
Lecture 22
66
What is M_p? Stalk? How to think about?
Lec 22
Show R[S^-1] is flat over R
Lec 22
Show vanishing is local
Lec 22
Show exactness is local
Lec 22
Show flatness is local
Lec 22
How can you describe localization of module in terms of tensor product? Proof?
Pg 66
Define: support of M
Supp M - the set of prime ideals st M_p != 0
Exercise 2.1
Exercise 2.4
Exercise 2.6
Exercise 2.19
Exercise 2.20
Given an extension S of R, discuss the relationship between R-modules and S-modules. What does this have to do with localization?
Given an S-module, easily get and R-module by restricting ring action. Given R-module, can induce an S-module by tensoring with S over R. B lecture 23. Pg 69
Prove Hom_S(S (x) M, S (x) N) = S (x) Hom_R(M,N) if S is FLAT R-module
B163-166
Include proof of 5-lemma
See also pg69
Prove: If R is any commutative ring, U < R a multiplicatively closed subset, and I < R an ideal maximal among those not meeting U, then I is prime. Corollaries?
It is a surprisingly general phenomenon that ideals maximal with respect to some property are prime
Pf. If f, g in R are not in I, then, by the maximality of I, both I + (f) and I + (g) meet U…pg71
Also do proof in terms of localization
Corollary. If I is an ideal in R, then rad I = intersection of all primes containing I. In particular, the intersection of all primes is the radical of (0), which is the set of all nilpotent elements of R.
Define: Artinian ring/module, chain of submodules, composition series, length of module, module of finite length
- Descending chain condition on ideal/submodules
- Every set of ideals/submodules has a minimal element
If M is a module, a CHAIN of submodules of M is a sequence of submodules with strict inclusions:
M = M0 > M1 > … > Mn
Such a chain is said to have LENGTH n. The chain is sub a COMPOSITION SERIES if each Mj/Mj+1 is a nonzero simple module (ie. has no nonzero proper modules)
The LENGTH of M is the least length of a composition series for M or inf if M has no finite composition series.
Examples displaying the relationship between Artinian and Noetherian modules?
1. Artinian and Noetherian 0, Z/nZ (over Z), any module with FINITE number of elements. Any f.d. vector space over a field
- Noetherian, not Artinian
Z (over Z) Z > 2Z > 4Z > …
Z_(2) = {a/b : b odd} - Neither
Q over Z … > 1/4 Z > 1/2 Z > Z > 2Z > 4Z > … - Artinian NOT Noetherian
Z[1/2]/Z
Examples displaying the relationship between Artinian and Noetherian rings?
Noetherian, not Artinian
Z
Neither
Z[x1, x2, …}
Artinian and Noetherian
Discuss (with proof) the relationship between modules of finite length and Artinian, Noetherian conditions
Thm. Let R be a ring, and let M be an R-module. M has a finite composition series <=> M is Artinian and Noetherian.
If M has a finite composition series M = M0 > M1 > … > Mn = 0 of length n, then:
a. Every chain of submodules of M has length <= n, and can be refined to a composition series
b. The sum of the localization maps M --> M_p, for P a prime ideal, gives an isomorphism of R-modules M = (+) M_p where the sum is taken over all maximal ideals P s.t. some Mi/Mi+1 = R/P. The number of Mi/Mi+1 isomorphic to R/P is the length of Mp as a module over Rp, and thus independent of the composition series chosen
c. We have M = Mp <=> M is annihilated by some power of P.
Pf pg 72-74
What conditions are equivalent to R being an Artinian ring? Proof?
- R is Noetherian and all all the prime ideals of R are maximal
…
Pg 75
Say A(X) is the coordinate algebra of an affine algebraic set and A(X) is Artinian. Discuss what this implies about the set
The following are equivalent:
1. X is finite
- A(X) is a f.d. vector space over k, whose dimension is the number of points of X
- A(X) is Artinian
pg 76
Structure theorem for Artinian rings?
Any Artinian ring is a finite direct product of local Artinian rings.
Pf. This is essentially the same as (b) from our main theorem 2.13 about properties of modules of finite length. We just recognize that R has finite length as an R-module over itself and then show that the isomorphism of R with a product of local modules is actually a ring isomorphism
Characterize modules of finite length over Noetherian rings
Cor. 2.17
Let R be a Noetherian ring, and let M be finitely generated R-module. TFAE:
1. M has finite length
2. Some finite product of maximal ideals annihilates M
3. All the primes that contain the annihilator of M are maximal
4. R / ann(M) is an Artinian ring
Combine this with Theorem 2.13
Let R be a ring and M an R-module. M has finite length (finite composition series) <=> M is Artinian and Noetherian.
1. Every chain of submodules has length <= length n and can be refined to composition series
2. M = (+) Mp where the sum is taken over all maximal ideals P s.t. some Mi/Mi+1 = R/P
3. M = Mp <=> M is annihilated by some power of P.
How can we turn a finitely generated module into a module of finite length via localization?
Every f.g. module M over a Noetherian ring R can be made into a module of finite length by localization at a prime minimal over its annihilator.
Prop. Let R be a Noetherian ring, 0 != M a f.g. R-module, I the annihilator of M, and P a prime ideal containing I. The Rp-module Mp is a nonzero module of finite length <=> P is minimal among primes containing I.
