# Tagged Questions

Homological algebra studies homology in a general algebraic setting. The purpose is extraction of information about structures involved in terms of tangible objects like rings groups and modules.

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### A lemma from Hilton & Stammbach's book A Course in Homological Algebra

In orde to prove the set of equivalence classes of extensions of $A$ by $B$ is a contravariant functor of the first component and covariant functor of the second. The authors give us three lemma. I'm ...
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I need some help regarding an argument from the proof of Proposition 4.2.1 in John Rognes article Galois Extensions of Structured Ring Spectra. We are supposed to prove that $\text{Ext}_{R[G]}^s(R,T)=... 1answer 46 views ### References on completion and Tor/Ext I am currently working on a thesis that relates to the Functors$\text{Tor}$and$\text{Ext}$. I have found some work on localization with respect to them when it comes to information in my books but ... 1answer 65 views ### Is the cohomology ring (coefficients in a field) functor right adjoint to something? Or, why does it commute with products? Take coefficients in a field, so as to not have the correction from Tor. I am thinking about the functor sending a topological space$X$to its cohomology ring$H^*(X)$. So specifically, I am ... 0answers 29 views ### How to find a counter-example that the centralizer of differential graded algebras does not preserve quasi-isomorphism? Let$A^{\bullet}$,$B^{\bullet}$be two differential graded algebras (dga) and$f: A^{\bullet}\to B^{\bullet}$be a differential graded homomorphism between them. Now let$R$be another algebra ($R$... 1answer 62 views ### Homology with local coefficients as a functor from pointed, path-connected spaces and$\pi_1$-modules. A local system of coefficients on a space$X$is a functor$F\colon \Pi(X)\rightarrow Ab$from the fundamental groupoid to the category of abelian groups. From this, one can define the homology groups ... 1answer 42 views ### Higher self-extension$\text{Ext}^i_{\mathcal{O}}(L(\lambda), L(\lambda))$between two irreducible modules in BGG category$\mathcal{O}$Let$\mathfrak{g}$be a complex semisimple Lie algebra with Cartan subalgebra$\mathfrak{h}$. Let$\mathcal{O}$be the BGG category for$\mathfrak{g}$. It is well-known that the set of irreducible ... 0answers 40 views ### Calculate cohomology of the special double complex Let$B,B'$be finitely generated$R$-modules. So we have two short exact sequences: $$0\to A\to R^n\to B \to 0$$ $$0\to A'\to R^m\to B' \to 0$$ Using these triples i obtain double complex:$$\... 1answer 93 views ### Degeneracies of simplex$y$which appears as any face of some simplex$x$Let$K$be simplicial set and$d_i:K_n\rightarrow K_{n-1}$,$s_i: K_n\rightarrow K_{n+1}$($i = 0,...,n$) face and degeneracy maps respectively. Suppose we have some$x\in K_n$with$d_0x = ... = ...
I am working on functors and projective resolutions and of course the issue of "Enough projectives" comes up. I know $R$-modules have enough but I am curious about the category of rings in general? ...
Building up on a previous question, I am currently investigating in the properties of locally finite homology. Suppose that $X$ is a reasonably well-behaved space. I want to find out whether there is ...