Difference between revisions of "Math 750 -- Homological algebra -- Homeworks"

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The homework assignments refer to the following file, which also contains easier and harder exercises.
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<!-- The homework assignments refer to the following file, which also contains easier and harder exercises.
  
[[Media:Math 750 exercises and problems]]
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[[Media:Homological_algebra_problems.pdf|Math 750 problems and exercises]]
  
;Homework 1: TBA, Due Thursday, February 6th.
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;Homework 1: Section 1.1: 4, 5, 8, 10, 20, 21, 22, due Thursday, February 6th.
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;Homework 2: Section 1.2: 2, 4, 6, 7, 8, 14, 15, due Thursday, February 20th. -->
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Homework 1, due Tuesday, February 27:
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a) Give an example of a bounded chain complex in Ch(Z-mod) which is acyclic but not split exact. (So both conditions of the Theorem we proved in class are needed.)
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b) Prove that any acyclic complex of vector spaces over a field is split exact. (Do not assume that the complex is bounded!)
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c) Prove that the full subcategory F of Ch^-(R-Mod) consisting of bounded above complexes of free R-modules satisfies the properties below:
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-- the cone of a morphism between two objects in F is again in F;
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-- if G : R-Mod -> A is an additive functor to an abelian category A, then G takes quasi-isomorphisms in F to quasi-isomorphisms in Ch(A);
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-- if M is an R-module, a resolution of M is an object C of F together with a map C -> M[0] which is a quasi-isomorphism. Prove that maps M -> N of R-modules lift to maps of resolutions of the modules, uniquely up to homotopy.
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d) Let f: M' -> M be an injective map of R-modules, and let M" denote the cokernel of f. Show that cone(f) is quasi-isomorphic to M"[0]. Are they always isomorphic in HoCh(R-mod)?
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Homework 2, due Thursday, April 5:
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a) Compute Ext^i_{Z/4Z}(Z/2Z, Z/2Z) and Tor_i^{Z/4Z}(Z/2Z, Z/2Z) for all i. (For Ext use projective resolutions in the first variable.)
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b) Compute Ext^i_R(R, k) and Ext^i_R(k,k) where R = k[x1,...,xn] is the polynomial ring in n variables, k is a field, and k is regarded as an R module by the identification k = R/(x1, ..., xn). (Hint: you may want to look up the following commutative algebra topics on Wikipedia or in any standard textbook: regular sequence, Koszul resolution.) So the homological dimension of R is >=n. (It is in fact n.)
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c) Consider a short exact sequence of R-modules
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0 -> M' -> M -> M" -> 0.
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Let eta in Ext^1_R(M", M') be the image of the identity in Hom(M', M') under the map
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Hom_R(M', M') -> Ext^1_R(M", M')
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obtained by applying the Hom_R(--, M') functor to the above short exact sequence. Prove that eta=0 if and only if the short exact sequence we started with is split.
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d) Prove directly from the definitions (without using Baer's criterion) that Ext^1_Z(M, I) = 0 for every finitely generated Z-module M if and only if I is divisible.
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Homework 3, due Tuesday April 24:
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Do the following exercises from Gelfand-Manin, "Methods of Homological Algebra":
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a) Exercise 1 on p. 163
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b) Exercises 1 and 3 in the section beginning on p. 183
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c) Exercises 1 (parts a and b only) and 5 in the section beginning on p. 214
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Homework 4, due Thursday, May 3:
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a) Exercises 5.1.2 and 5.1.3 from Weibel's book.
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b) Exercise A3.45 from Eisenbud's Commutative Algebra book.

Latest revision as of 15:06, 26 April 2018


Homework 1, due Tuesday, February 27:

a) Give an example of a bounded chain complex in Ch(Z-mod) which is acyclic but not split exact. (So both conditions of the Theorem we proved in class are needed.)

b) Prove that any acyclic complex of vector spaces over a field is split exact. (Do not assume that the complex is bounded!)

c) Prove that the full subcategory F of Ch^-(R-Mod) consisting of bounded above complexes of free R-modules satisfies the properties below:

-- the cone of a morphism between two objects in F is again in F;

-- if G : R-Mod -> A is an additive functor to an abelian category A, then G takes quasi-isomorphisms in F to quasi-isomorphisms in Ch(A);

-- if M is an R-module, a resolution of M is an object C of F together with a map C -> M[0] which is a quasi-isomorphism. Prove that maps M -> N of R-modules lift to maps of resolutions of the modules, uniquely up to homotopy.

d) Let f: M' -> M be an injective map of R-modules, and let M" denote the cokernel of f. Show that cone(f) is quasi-isomorphic to M"[0]. Are they always isomorphic in HoCh(R-mod)?

Homework 2, due Thursday, April 5:

a) Compute Ext^i_{Z/4Z}(Z/2Z, Z/2Z) and Tor_i^{Z/4Z}(Z/2Z, Z/2Z) for all i. (For Ext use projective resolutions in the first variable.)

b) Compute Ext^i_R(R, k) and Ext^i_R(k,k) where R = k[x1,...,xn] is the polynomial ring in n variables, k is a field, and k is regarded as an R module by the identification k = R/(x1, ..., xn). (Hint: you may want to look up the following commutative algebra topics on Wikipedia or in any standard textbook: regular sequence, Koszul resolution.) So the homological dimension of R is >=n. (It is in fact n.)

c) Consider a short exact sequence of R-modules

0 -> M' -> M -> M" -> 0.

Let eta in Ext^1_R(M", M') be the image of the identity in Hom(M', M') under the map

Hom_R(M', M') -> Ext^1_R(M", M')

obtained by applying the Hom_R(--, M') functor to the above short exact sequence. Prove that eta=0 if and only if the short exact sequence we started with is split.

d) Prove directly from the definitions (without using Baer's criterion) that Ext^1_Z(M, I) = 0 for every finitely generated Z-module M if and only if I is divisible.

Homework 3, due Tuesday April 24:

Do the following exercises from Gelfand-Manin, "Methods of Homological Algebra":

a) Exercise 1 on p. 163

b) Exercises 1 and 3 in the section beginning on p. 183

c) Exercises 1 (parts a and b only) and 5 in the section beginning on p. 214

Homework 4, due Thursday, May 3:

a) Exercises 5.1.2 and 5.1.3 from Weibel's book.

b) Exercise A3.45 from Eisenbud's Commutative Algebra book.