Chapter 2 - Localization

Question 1

Define: local ring

Answer 1

A ring with just one maximal ideal

Question 2

Discuss motivation for localization

Answer 2

See pg. 57-58

Question 3

Define: multiplicatively closed set

Answer 3

A set U is multiplicatively closed if any product of elements in U is in U and 1 is in U

Question 4

Define: localization of M at U

Answer 4

pg. 59 and Borcherds pg 115 - …

Question 5

When does an element localize to 0? Proof?

Answer 5

An element m in M goes to 0 in M[U^-1] iff m is annihilated by an element u in U

Question 6

Examples of localizations?

Answer 6

1. Quotient field of integral domain
2. Total quotient ring of arbitrary ring
3. For P prime localization at R - P written R_P - write k(P) for the residue class field of R at P
4. Local ring of an affine variety X at a point x in X
5. Z and C[x] Borcherds

Question 7

What is the localization of a map of R-modules? Functorial properties?

Answer 7

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

Question 8

Discuss the universal property of localization

Answer 8

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

Question 9

Discuss the relationship between the ideal structure of R and its localization R[S^-1]. Spec R vs Spec R[S^-1]

Answer 9

See Borchards Lecture 17 and Prop 2.2 in Eisenbud

Question 10

Discuss Spec C[x,y] localize at (0), (f), (x-a, y-b) vs quotient

Answer 10

See Borchards Lecture 17

Question 11

Discuss C[x,y]/(xy) localized at (x,y)

Answer 11

See Borchards Lecture 17

Question 12

Discuss functions on Spec R for R = C(X) continuous functions on compact Hausdorff space, R = C[x], R = Z

Answer 12

See Borchards Lecture 18

Question 13

Why do nilpotents cause issues with trying to represent R as functions on Spec R? Solution? Examples?

Answer 13

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

Question 14

Define: Sheaf of rings on Spec R

Answer 14

See Borchards Lecture 18

Question 15

What is the motivation for def of Affine Scheme of RIng?

Answer 15

Pretend R is a ring of functions on Spec R

Question 16

Define: Affine Scheme of Ring

Answer 16

See Borchards Lecture 19

Question 17

Discuss the algebra geometry dictionary

Answer 17

See Borchards Lecture 19

Question 18

Define: tensor product of modules over a ring

Answer 18

Defined in terms if universal property i.e. universal module for bilinear maps from MxN. See Borchards Lecture 20.

Question 19

Prove existence and uniqueness of tensor product

Answer 19

1. Unique using universal
2. Existence using massive free module construction

See Borchards Lecture 20.

Question 20

Examples of tensor product - vector space and f.g. abelian groups-Z-modules?

Approach?

Answer 20

The def and construction is hard to compute with. Use 2 properties:

1. (M1 + M2) x N = M1xN + M2xN
2. R x_R M = M

Borcherds lec20

Question 21

What problems can occur when taking tensor or hom of s.e.s.?

Answer 21

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.

Question 22

Define: Hom_R(M,N)

Answer 22

If M and N R-modules, this is the abelian group of all homomorphisms from M to N. Actually an R-module

Question 23

Prove: Hom_R(R,N) isomorphic to N

Answer 23

Pg 62

Question 24

In what sense is Hom functorial? Left-exact functor? Right-exact?

Answer 24

pg 62-63 Borcherds Lec 21

Question 25

What is the relationship between Hom and tensor? Discuss properties of tensor in this context

Answer 25

64

Question 26

In what sense is tensor product functorial? Impact on s.e.s?

Answer 26

Pg 64-65 borcherds 21

Question 27

How can you calculate tensor product?

Answer 27

Borcherds 21

Question 28

What are direct limits? Impact of tensor product?

Answer 28

End of Borcherds 21

Question 29

Define: Flat module

Answer 29

M is flat if M tensor preserves exactness Lec 22 66

Question 30

Prove: localization M[S^-1] preserves exactness.

Answer 30

Lecture 22 | 66

Question 31

What is M_p? Stalk? How to think about?

Answer 31

Lec 22

Question 32

Show R[S^-1] is flat over R

Answer 32

Lec 22

Question 33

Show vanishing is local

Answer 33

Lec 22

Question 34

Show exactness is local

Answer 34

Lec 22

Question 35

Show flatness is local

Answer 35

Lec 22

Question 36

How can you describe localization of module in terms of tensor product? Proof?

Answer 36

Pg 66

Question 37

Define: support of M

Answer 37

Supp M - the set of prime ideals st M_p != 0

Question 38

Exercise 2.1

Question 39

Exercise 2.4

Question 40

Exercise 2.6

Question 41

Exercise 2.19

Question 42

Exercise 2.20

Question 43

Given an extension S of R, discuss the relationship between R-modules and S-modules. What does this have to do with localization?

Answer 43

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

Question 44

Prove Hom_S(S (x) M, S (x) N) = S (x) Hom_R(M,N) if S is FLAT R-module

Answer 44

B163-166 Include proof of 5-lemma See also pg69

Question 45

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?

Answer 45

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.

Question 46

Define: Artinian ring/module, chain of submodules, composition series, length of module, module of finite length

Answer 46

1. Descending chain condition on ideal/submodules 2. 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.

Question 47

Examples displaying the relationship between Artinian and Noetherian modules?

Answer 47

``` 1. Artinian and Noetherian 0, Z/nZ (over Z), any module with FINITE number of elements. Any f.d. vector space over a field ``` 2. Noetherian, not Artinian Z (over Z) Z > 2Z > 4Z > ... Z_(2) = {a/b : b odd} 3. Neither Q over Z ... > 1/4 Z > 1/2 Z > Z > 2Z > 4Z > ... 4. Artinian NOT Noetherian Z[1/2]/Z

Question 48

Examples displaying the relationship between Artinian and Noetherian rings?

Answer 48

Noetherian, not Artinian Z Neither Z[x1, x2, ...} Artinian and Noetherian

Question 49

Discuss (with proof) the relationship between modules of finite length and Artinian, Noetherian conditions

Answer 49

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

Question 50

What conditions are equivalent to R being an Artinian ring? Proof?

Answer 50

1. R is Noetherian and all all the prime ideals of R are maximal … Pg 75

Question 51

Say A(X) is the coordinate algebra of an affine algebraic set and A(X) is Artinian. Discuss what this implies about the set

Answer 51

The following are equivalent: 1. X is finite 2. A(X) is a f.d. vector space over k, whose dimension is the number of points of X 3. A(X) is Artinian pg 76

Question 52

Structure theorem for Artinian rings?

Answer 52

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

Question 53

Characterize modules of finite length over Noetherian rings

Answer 53

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.

Question 54

How can we turn a finitely generated module into a module of finite length via localization?

Answer 54

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.