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

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;Homework 2: Section 1.2: 2, 4, 6, 7, 8, 14, 15, due Thursday, February 20th. -->
 
;Homework 2: Section 1.2: 2, 4, 6, 7, 8, 14, 15, due Thursday, February 20th. -->
  
Homework 1, due Thursday, October 1:
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Homework 1, due Tuesday, February 27:
  
a) Show by example that HoCh(A) is not always an abelian category.
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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.)
  
b) 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!)
  
c) 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:
 
 
d) 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;
 
-- the cone of a morphism between two objects in F is again in F;
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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.
 
-- 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.
  
e) 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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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, October 22:
  
 
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.)
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c) Exercises 1 and 5 in the section beginning on p. 214
 
c) Exercises 1 and 5 in the section beginning on p. 214
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-->

Revision as of 23:49, 12 February 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)?