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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In an abelian category, what are the projectives of the chain complex category?

In Prof. Weibel AIHA book exercise 2.2.1 it is asked to show that the projectives of the chain complex category of a category of modules are the split exact complexes of projectives. In Prof. Rotman ...
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Global dimension of matrix algebra

I want to calculate the global dimension of this algebra. $$ \quad A = \begin{pmatrix} k & 0 & 0& 0 \\ k & k & 0&0\\ k&0&k&0\\ k&k&k&k \end{pmatrix} ...
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Lifting the projective property through the affine centre

Let $\mathbb{k}$ be an algebraically closed field. There are many interesting examples of $\mathbb{k}$-algebras $R$ which admit a large central subalgebra $Z_0$ such that $R$ is a free $Z_0$-module ...
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Flat test lemma

How can I prove this result: Let $M$ be a right $A-$module, and $A$ a $k-$algebra. If $M\otimes I \cong MI$ for every finitely generated left ideal of $A$, then $M$ is flat.
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37 views

Tor amplitude of dual complex

Let $E^\bullet$ be a perfect complex of $R$-modules (where $R$ comm. ring). So $E^\bullet$ is quasi-isomorphic to a bounded complex of finitely generated projective R-modules. Now $E^\bullet$ has ...
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Connecting homomorphism in Galois homology using the standard resolution

Let $G$ be a finite group, although this may not be necessary for almost everything that follows. One of the ways of defining Galois homology groups is using the standard resolution for the ...
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$f_*$ induces an isomorphism in homology iff the mapping cone of $f_*$ is contractible.

Let $f_*:C_*\to D_*$ be a chain map. I'm stuck in the proof of the following statement: $f_*$ induces an isomorphism in homology iff the mapping cone of $f_*$, cone($f_*$), is contractible. (For ...
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Vanishing of a certain Tor

I am reading about the construction of the Affine Grassmannian in Dennis Gaitsgory's seminar notes and there are some commutative algebra facts that I am not able to figure out by myself apparently, ...
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How to use the Universal Coefficient Theorem to determine $H^i(M; \mathbb{Z}_p)$ from $H^i(M; \mathbb{Z})$? [on hold]

Let $M$ be a path-connected finite $CW$-complex. Suppose $$ H^2(M;\mathbb{Z})=\mathbb{Z}_{2k}, \text{ } k\geq 3; $$ $$ H^3(M;\mathbb{Z})=\mathbb{Z}\times\mathbb{Z}_{2}; $$ $$ ...
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22 views

universal coefficient theorem for mod p cohomology

In the book Algebraic Topology, Allen Hatcher, p. 266, Corollary 3A.6 (b): Question: I want to rewrite the above statement into a cohomology version. If I replace all homologies with cohomologies, ...
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71 views

Cellular homology of the real projective space $\mathbb R P^n$

I've been able to calculate the cellular homology of $\mathbb R P^2$ but I'm struggling to do the same for higher dimensions. My problem is that I don't exactly see how one get to the result $d_i: ...
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348 views

Long exact sequence for a triple follows from long exact sequence for a pair?

In homology theory, the long exact sequence for a pair $(X,Y)$ is just $H(Y)\to H(X)\xrightarrow{\partial(X,Y)}H(X,Y)\to H(Y)[-1]$. The long exact sequence for a triple $(X,Y,Z)$ is $H(Y,Z)\to ...
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87 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 = ... = ...
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43 views

Confusion regarding definition of adjoint functor - Hilton and Stammbach

While defining Adjoint functors in their book A Course in Homological Algebra, Hilton and Stammbach said the follwing: Let $F:\mathfrak{S}\rightarrow \mathfrak{M}_{\Lambda}$ be the free functor ...
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45 views

counterexample for $f_*:C_*\to D_*$ be a chain map such that $f_*$ induces an isomorphism in homology. Then $f_*$ is a chain homotopy equivalence

I want to understand a counterexample for: Let $f_*:C_*\to D_*$ be a chain map such that $f_*$ induces an isomorphism in homology. Then $f_*$ is a chain homotopy equivalence, because the statement ...
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49 views

Sheaf cohomology of long exact sequence of sheaves

Let $X$ be an algebraic variety and $F_1,...,F_n$ be a collection of coherent sheaves on $X$. Suppose we have a long exact sequence $$0\to F_n\to F_{n-1}\to...\to F_1\to0.$$ Knowing all sheaf ...
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53 views

Let $G$ be a group. Why is $ \operatorname{Ext}_{\mathbb{Z}G}^1(\mathbb{Z},\mathbb{Z})\cong \operatorname{Hom}_{Grp}(G,\mathbb{Z})$?

Let $G$ be a Group and $\mathbb{Z}G$ is the ring of the formal sums $$\sum_{g\in G}n_gg$$ with multiplication $$(\sum_{g\in G}n_gg)(\sum_{h\in G}m_hh)=\sum_{g\in G}(\sum_{h\in ...
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1answer
40 views

Translate this theorem from Endliche Gruppen

In a paper I was doing a reference is given from "Endliche gruppen" by Huppert. I do not understand german and google translator was also not much helpful. Can some translate this theorem or much ...
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30 views

show that $H^1(Q,Z(P))=0$

Suppose $P$ is a normal sylow $2$-subgroup of $G$. Now let $\varphi$ is an automorphism of $G$ and $Q=G/P$. Now suppose we have this commutative diagram If $(|P|,|Q|)=1$ then I want to show that ...
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The Ext-functor and inverible modules

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 ...
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Getting the most general form of Mayer-Vietoris from the axioms of homology

I'd like to derive the most general form of the Mayer-Vietoris sequence from the Eilenberg-Steenrod axioms for homology (in particular: I do not want to use the definition of $H_\ast(X)$ in terms of ...
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Cohomology induces a functor

Let $\mathcal{A}$ be an abelian category and $(X^{\cdot}, d_{x}^{\cdot})$ a cochain complex in $\mathcal{A}$. Let $Ch(\mathcal{A})$ be the category of cochain complexes in $\mathcal{A}$. I define the ...
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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 ...
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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 ...
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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 ($R$ ...
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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 ...
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37 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 ...
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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: $$ ...
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459 views

Homology commutes with direct sum and product?

I'm looking at exercise 1.2.1 from Weibel's An Introduction to Homological Algebra. (I need to show that homology commutes with direct sum and direct product.) Is it possible to show that ...
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1answer
48 views

The category of (completable) rings has enough projectives in it

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? ...
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25 views

Relation between compactly supported cohomology and locally finite homology

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 ...
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$E_{p,q}^r$ spectral sequence. Find a l.e.s. $\cdots \to A_{p+1}\to E_{p+1,0}^2\xrightarrow{d_{p+1,0}^2} E_{p-1,1}^2\to A_p\to E_{p,0}^2\to\cdots$

Let $E_{p,q}^r$ be a spectral sequence which converges to $A_n$. Let $E_{p,q}^r=0$ for $q\ge 2$. How to construct a long exact sequence $$\cdots \to A_{p+1}\to E_{p+1,0}^2\xrightarrow{d_{p+1,0}^2} ...
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A general definition for a character of a (not necessarily associative) algebra

Let $A$ be a algebra over a algebraically closed field $k$. Is there certain definition of a "character" $f: A \rightarrow k$? That is, what is the common and useful condition for a linear map $f: A ...
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relations between homology and cohomology

Let $p$ be a prime number and $X$ a topological space. Are the following equivalent? (1) In the homology module $H_*(X;\mathbb{Z})$ there does not exist any element of order $p$. (2) In the ...
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Homology of Hom and Hom of homology

In 'Homological Algebra' by Cartan & Eilenberg: (page 203) For complexes $X$ and $Y$, consider the map $\alpha':H^{p+q}(\text{Hom}(X,Y))\rightarrow \text{Hom}(H_p(X),H^q(Y))$. Let $h_1\in ...
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Homomorphisms of modules over a corner ring

Let $R$ be a Noetherian ring and suppose that we can write $1 = e_1 + e_2 + \dots + e_n$ where the $e_i$ are pairwise orthogonal idempotents. Let $S = e_1 S e_1$, and consider the right $S$-modules ...
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Group cohomology of Z/2Z

Let $G=Z/2Z=\{1, g\}$. Consider the ring of integer $Z$ with the alternating action of G, i.e., $g\cdot n=-n$ for $n\in Z$ and an abelian group $M$ on which $G$ acts. It is well-known that the group ...
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1answer
36 views

Intuition behind $Ext^1(A,\,C)$

So I recently asked a question concering $Ext^1(A,\,C)$ regarding the connection between isomorphism and the congruence '$\equiv$' (Where am I making a mistake with $Ext^1(A,C)$?). Suppose, for ...
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Research ideas in Homological algebra

I am planning to focus my research on Homological Algebra and related fields. I am on my first year, first semester and currently pursuing courses on Homological Algebra, Algebraic Geometry (first ...
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Computation of Maps Between Sheaves

Problem: Let $X$ be a locally compact topological space, $i:Z \hookrightarrow X$ inclusion of a closed subspace, and $j:U \hookrightarrow X$ inclusion of the complement. I want to compute: ...
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Bullet notation

I'm just trying to make acquaintance with homological algebra. I see there the notation $(A_\bullet,b_\bullet)$ as a short notation for $(\dots,A_{-1},A_0,A_1,\dots,\dots,b_{-1},b_0,b_1,\dots)$. ...
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For $x\in\operatorname{Ext}_R^1(C,A)$, how to construct an extension $0\to A\to B\to C\to 0$ such that $\partial (id_A)=x$? [duplicate]

Let $R$ be a ring, $C, A$ two $R$-modules. For all $x\in\operatorname{Ext}_R^1(C,A)$ I have to construct a short exact sequence $$0\to A\to B\to C\to 0$$ of $R$-modules such that $\partial(id_A)=x$, ...
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1answer
31 views

Where am I making a mistake with $Ext^1(A,C)$?

I am learning about $Ext^1(A,C)$ and how it forms a group under '$+$', the Baer sum and I am clearly missing the point somewhere. So, let us suppose for simplicity that $Ext^1(A,C)\cong\mathbb{Z}/3$. ...
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1answer
66 views

Tensor products of simple modules over algebras

Let $A$ and $B$ be $\mathbb{C}$-algebras. Suppose that $M$ and $N$ are respectively simple $A$ and $B$ modules. We can regard $M$ and $N$ as $A\oplus B$-modules in natural way, namely, $AN=0$ and ...
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When do we have $\operatorname{Hom}_{R}(L,M\otimes N)\cong \operatorname{Hom}_R(L,M)\otimes N$? [duplicate]

Let $R$ be a commutative ring and $L,M,N$ be $R$ modules. I would like to know that when is the natural map $$ \operatorname{Hom}_R(L,M)\otimes_R N\to \operatorname{Hom}_R(L,M\otimes_R N) $$ is an ...
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1answer
37 views

Adjoints to cofree modules tensor?

If $M$ is a cofree $R$-module and $A,B$ are arbitrary $R$-modules then, is there a left adjoint to the functor $M\otimes_R -$, i.e. is there an endofunctor $F$ on $_R \mathrm{Mod}$ such that ...
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1answer
55 views

Isomorphic Homology implies Isomorphic Cohomology

If two complexes have isomorphic integral homology, do the dual complexes have isomorphic integral cohomology? I can also assume that the homology, cohomology are finitely-generated abelian groups. ...
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164 views

Question on adjoint functors

Can someone provide me an enlightenment on the following three statements? (I stumbled on them at the part dealing injective modules in a text of homological algebra.) 1) Let $F \dashv G \colon ...
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1answer
101 views

(Hopefully) Simple question about the exterior algebra functor

I have some (hopefully super) basic questions about the exterior algebra functor $$ \wedge:R\text{-Mod}\rightarrow R\text{-Alg}. $$ As I (think I) understand it, if one considers it as a functor ...
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1answer
9 views

The boundary formula and cohomology of finite groups

I've a very basic notational question on group cohomology. Let $G$ be a finite group and $M$ a $G$-module. For $i\geq 0$, let $P_i=\mathbb Z[G^{i+1}]$ be the free $\mathbb Z$-module on $G^{i+1}$, made ...