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

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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)?
 
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, October 22:
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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.)
 
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) Show that for any Z-modules M and N we have Ext^i_Z(M, N) = 0 for i >=2. We say that Z (the integers) has homological dimension 1.
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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.)
  
c) 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  
 
 
d) If F: A -> B is a right exact functor between abelian categories, and if A has enough projectives (so that LF is defined) then we shall say that an object X of A is F-acyclic if it satisfies R^i F(X) = 0 for i>0. For example any projective in A is F-acyclic.
 
 
 
-- show that if F is in fact exact, then any X in A is F-acyclic.
 
 
 
-- an F-acyclic resolution of an object Y of A is a complex
 
 
 
... -> X_n -> X_{n-1} -> ... -> X_0 -> 0
 
 
 
such that the complex is exact except at the last spot, where the homology is Y, and each X_i is F-acyclic.
 
 
 
Show that the homology H_i(F(X.)) is naturally isomorphic to L_i F(Y). (Thus derived functors can be computed using F-acylic resolutions, not only using projective resolutions.) In some situations this allows us to construct left derived functors even when there are not enough projectives, if we can identify enough F-acyclics. (Hint: break up the resolution into short exact sequences.)
 
 
 
e) (More difficult) Consider a short exact sequence of R-modules  
 
  
 
0 -> M' -> M -> M" -> 0.
 
0 -> M' -> M -> M" -> 0.
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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.
 
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, Nov. 24:
 
Homework 3, due Tuesday, Nov. 24:
  

Revision as of 20:40, 15 March 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.