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Jan
6
awarded  Good Question
Jan
2
comment A module over an algebra. Is it a vector space?
Another thought: one could think that one could make a "relative" version of an $A$-module, as in that post: an $A$-module relative to $k$ should be a $k$-module $M$ with an action of $A$ such that the action of $k$ it induces coincides with the original one. But it's a boring notion, unlike the bimodule case, where it makes more sense: in that scenario, you have two induced actions of $k$, so it makes sense to require that they're equal (and moreover, equal to a given one).
Jan
2
comment A module over an algebra. Is it a vector space?
A small caveat. Let $M$ be a $k$-module. Suppose moreover that $M$ is an $A$-module. Then this last $A$-action induces another $k$-action that need not coincide with the original one. This reminds me of this: math.stackexchange.com/questions/889130/…
Jan
2
accepted Hochschild homology: change of ground ring
Jan
2
comment Hochschild homology: change of ground ring
Thanks a lot! I didn't know this convention: as you suspected, I was only aware of the "ring"-version of a bimodule... It's nice to learn it by making a small tedious routine verification and realizing that something was missing.
Jan
1
revised Hochschild homology: change of ground ring
edited tags
Jan
1
comment How do you break up an exact sequence of any length to a “succession of short exact sequences”?
Duplicate? math.stackexchange.com/questions/207551/…
Jan
1
asked Hochschild homology: change of ground ring
Dec
8
awarded  Popular Question
Dec
4
comment Verify proof that $f:X\to Y$ a homeomorphism implies $f_*:H_p(X)\to H_p(Y)$ is an isomorphism
Do you know what a functor is? Functors preserve isomorphisms. Homology is a functor.
Dec
4
asked Two-sided bar construction for algebras: $B_*(M,R^e,R)$ is quasi-isomorphic to $M\otimes_{R^e}B(R,R,R)$
Dec
1
comment Some questions about the Tor functor as a two-variable functor related to the arbitrary character of the choice of projective resolutions
If $A$ is a $k$-algebra and $N$ is a left $A$-module, then the bar construction $B_*(A,A,N)$ gives an $A$-resolution of $N$ as with augmentation given by the action map. Moreover, if $A$ and $N$ are $k$-flat, this is a flat resolution. This gives a nice way to compute Tor under some hypotheses.
Nov
30
revised If $p:(E,e_0) \to (X, x_0)$ is a simply connected covering space with group of covering transformations $G$ then $G \cong\pi_1(X,x_0)$.
edited title
Nov
29
awarded  Announcer
Nov
26
comment Reference for the Universal Coefficient Spectral Sequence
Essentially, theorem 10.90 in Rotman's Intro to homological algebra, second edition.
Nov
26
comment Reference for the Universal Coefficient Spectral Sequence
mathoverflow.net/questions/165714/kunneth-spectral-sequence has a reference to Rotman. Künneth theorems are generalizations of universal coefficient ones.
Nov
17
comment why $HF_p$(Eilenberg Mac Lane spectrum) smash X (connective spectrum with finite type cohomology groups) is a wedge of suspensions of $HF_p$?
For a ring $R$, there is a Quillen equivalence between $HR$-modules and chain complexes over R. For $R=K$ a field, chain complexes split. You could think of the result you state as the spectra-side of this chain complex well-known fact.
Nov
15
comment What's the point of spectra?
Related question: math.stackexchange.com/questions/415787/…
Nov
10
comment Definition of symmetric product in Milnor's paper
Is it perhaps the join?
Nov
9
awarded  Enlightened