1
vote
1answer
61 views

Triviality of $H_3(G,\mathbb{Z})$

We know that the triviality of the Schur multiplier means that projective representations can be lifted to ordinary ones. The Schur multiplier is also a measure of the failure of how the commutator ...
5
votes
1answer
79 views

(Co)homology of free symmetric algebra

Let $V$ be a (co)chain complex, and let $Sym(V)$ be the free differential graded-commutative algebra generated by $V$. Definition and examples below in case you don't know what I mean. Question: ...
0
votes
1answer
37 views

Tor of submodule

Let $R$ be a $CRing$. If $i:A \rightarrow B$ is the inclusion of a $R$-subalgebra A into an $R$-algebra $B$, then what is ther relationship between: $Tor_{A^e}$ and $Tor_{B^e}$?
1
vote
0answers
32 views

Coproducts and Hochschild

I $\{X_i\}$ is a small family of associative $\mathbb{C}$-algebras and $X$ is their free product. Then I have two questions: 1) Why is $X$ their coproduct? 2) Is the Hochschild homology of X ...
3
votes
1answer
42 views

Is Cyclic cohomology a Weil cohomology?

Simply stated question... Is the cyclic homology theory of an algebra a Weil cohomology theory, given the appropriate reformulation of the Weil axioms?
-1
votes
1answer
40 views

Property of Homology and orientation [closed]

Show that the isomorphism type of homology groups does not depend on the orientations of simplical complex. I need help!!
1
vote
1answer
22 views

Show: if an abelian group is a $G$-module, it is also a $\mathbb{Z}G$-module.

Let $G$ be a finite group with identity $e$, $A$ an abelian group, and $\theta:G\to {\rm Aut}(A)$ a homomorphism. Show that (via $\theta$) $A$ is a $\mathbb{Z}G$-module: $(\sum\limits_{g\in ...
2
votes
0answers
89 views

Definition for a bar resolution for a module over a dg category

Let $ \mathcal{A}$ be a dg category and define a right $ \mathcal{A}$ module to be a dg functor $ M: \mathcal{A}^{op} \rightarrow dif\ k$ where $dif\ k$ is the category of differential $k$ modules ...
6
votes
0answers
137 views

Morita-invariance of Hochschild (co)homology.

Ok, I'm reading this paper by Christian Kassel on associative algebras and Hochschild (co)homology and on page 19 he says that Hochschild homology is Morita-invariant, by which he means that if $R$ ...
3
votes
1answer
67 views

Relative homology

Let $E$ be a real banach space if $E=Y\oplus Z$ and if $S^{m-1}$ is the sphere on $Y$ ($\dim Y =m $) why $H_{m-1}(E \setminus Z)\simeq H_{m-1}(S^{m-1})$ ? please thank you
0
votes
2answers
69 views

Question on relative homology

i have this: where $|\tau|$ is the support of the chain $\tau$, i don't understand the first part why $[\sigma]=0$ in $H_{m-1}(\phi^{c+\varepsilon},\emptyset)$ ??? Please, thank you.
4
votes
1answer
176 views

Finding example of quasi isomorphism that has no quasi inverse

Between differential graded algebra $V,W$, a chain map $f\colon V\to W$ induces homomorphism between its homology. If this becomes an isomorphism between the homology of $V,W$, call this quasi ...
0
votes
1answer
70 views

Why call them cycles and boundaries?

I have a small question. Why we have this designation: $n$-cycles for $Z_n$ and $n$-boundaries for $B_n$ ? Why they are called cycles and boundaries ? ...
1
vote
1answer
77 views

filtration on the (co)homology of a space from the filtration of a space

Fix $n\!\in\!\mathbb{N}$. Let $X$ be a topological space and $X_0\subseteq X_1\subseteq X_2\subseteq \ldots$ subspaces of $X$. Let $\iota_k:X_k\rightarrow X$ be the inclusion. Let ...
1
vote
1answer
71 views

Surjectivity in little diagram

Given the following commutative diagram of exact sequences $$ \begin{array} & & 0 & 0 & 0 &\\ & \downarrow & \downarrow & \downarrow &\\ 0 \rightarrow & A ...
40
votes
2answers
1k views

Algebraic Topology Challenge: Homology of an Infinite Wedge of Spheres

So the following comes to me from an old algebraic topology final that got the best of me. I wasn't able to prove it due to a lack of technical confidence, and my topology has only deteriorated since ...
4
votes
1answer
340 views

Category of isomorphism classes?

Is there such a thing as a category of isomorphism classes of, say, modules? First step in definining morphisms in such a category would be to identify two morphisms $f:M\rightarrow N$ and ...
2
votes
3answers
425 views

Acyclic vs Exact

I have a question about the words "acyclic" and "exact." Why does Brown use the term "acyclic" instead of "exact" in his book Cohomology of Groups? It seems to me that these two terms exactly ...