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# Homological Algebra - Science topic

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Questions related to Homological Algebra

I'm looking a good technical and university notes for commutative algebra and homological Algebra that have good examples for any subjects. Please hint and guide me if you have In my point of view.

I need to know whether the ring k[x]/(x^2) with k field is von Neumann regular ring or not.

The problem is:

Let Z denote the set of all integers.

Consider Z/nZ = Z

_{n}as trivial G-module.Show that there is a isomorphism between the First Homology Group H

_{1}(G, Z_{n}) and the factor G/G'G^{n};Where G' is the commutator (derived) subgroup G'=[G,G] and G

^{n}is the subgroup of G generated by all its n-th Powers.- With this isomorphism proved, we want to conclude that if G is a
*finite p-Group,*then H_{1}(G,**F**_{p}) is a**F**_{p}-module (vector space) with dimension equal the number of least generators of G.

Let Z denote the set of all integers, and let G be a finite cyclic Group.

For every ZG-module A, and n=1,2,3...

Show that:

H

_{n}(G,A) is isomorphic to H^{n+1}(G,A).I have got a problem where other conditions satisfy except the openness. Can I invoke the theorem where X is closed subset of R

^{n}. Detail question will follow the below link.In the light of the presentations exist in the literature for the general and special linear group GLn(q) , SLn(q) [based on the construction given by Chevalley and Steinberg as well as on the concept of BN-pair syetem] such as http://dl.dropbox.com/u/5188175/slnpresentations.pdf , is there a convenient way for listing the elements of the general (or special) linear group GLn(q) over finite field in general or at least for GL2(q) or SL2(q) ?

Let F denote a finite field and let A denote a n times n matrix over F. How can one compute efficiently all the invariant subspaces of A with dimension less than or equal to n_{0} (where n_{0}<<n). Also, is there any method to build (and cover all) these subspaces from smaller subspaces ?

How Tor_1(M,F) = 0 for all finitely presented modules F implies M is flat?

As the title suggests, how do i see that for any n, the covering map S^{2n} → RP^{2n} induces 0 in integral homology and cohomology, except in dimension 0?

Considering G a group and H its subgroup. <br />F a field, and F(G) a group ring. Since F(G) is a ring, we can consider also that F(H) is a subring, and then F(G) as a left or right F(H)-module.<br /><br /><strong>*</strong>I thought about writting an isomorphism between F(G)/F(H) and G/H. If one is finitely generated, I could say rank[F(G)/F(H)] = rank[G/H] = [G : H].<br />I am not sure about this proposal.

Knowing that R is a commutative ring and R[X] is a free R-module.

It seems obvious because any R-module can be written as quotient of a free R-module and any of its submodules.

As R[X²] is a submodule of R[X] over R, then it is an ideal of R.

It turn to simply show a bijective homomorphism, but I encounter problems describing the elements in this relations.

For what type of separable C* algebras, there is no any integer n and any nontrivial morphism from A to the Cuntz algebra O_{n}? In the other word for what type of C* algebras, Hom(A, O_{n}) is trivial for all n?

The motivation: I was thinking to consider a NC analogy for singular homology as follows:

A singular object is a continuous map from Delta^{n}\to X. So We have a natural morphism from C(X)\to C(\Delta^{n}).

With some (or a lot of) abuse of notations and equations, the equation \sum t_{i}=1, which defines the standard simplex \Delta^{n}, could be considered as a commutative picture of universal property of Cuntz algebra O_{n}. This is a motivation to construct a complex as followos: We put C_{n}(A)= Free abelian group generated by Hom(A,, O_{n}). Now we can define a complex

....... C_{n}(A) \to C_{n+1}(A) \to....

using n+1 embeddings O_{n} to O_{n+1}

Does this idea leads to triviality?For what type of C* algebras, this construction is useful?

Suppose X is a simplicial complex and A its subcomplex. So, we can consider relative cocycles, in particular, relative coboundaries in dimension 2, for concreteness. Also, we can consider

*absolute*cocycles in dimension 2, i.e. such that are not obliged to vanish on A.And if we factor the space of these absolute cocycles by the space of those relative coboundaries - what is the generally accepted name for the resulting space?