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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The injectivity of $f\mapsto f\circ v$ on $\hom(M'',N)$ implies that $v$ is surjective [duplicate]

I'm an undergrad getting familiar with some notions of commutative algebra by reading Atiyah-McDonald. On the exact sequences part, a part of the proof of (2.9) is proving that if ...
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10 views

When does homology commutes with arbitrary direct sums

Is it necessary to have the criteria that the direct sum of a collection of monics is a monic, to show that homology commutes with arbitrary direct sums? Because when I tried to prove the result, I ...
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44 views

Injective dimension and Krull dimension of a module

Let $R$ be a regular local ring and $M$ an $R$-module (not necessarily finite), then the injective dimension $\operatorname{id}_R(M)$ of $M$ is finite. When $M$ is finitely generated, we have ...
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Another description for the map $\text{Ext}^1_\mathbb{Z}(A,G)\to H^2(G,A)$

Group extensions of $G$ by $A$ $0\to A\to E\to G\to 0$ up to equivalence (where $G$ and $E$ may be nonabelian) are in bijection with the second group cohomology ...
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27 views

homology commutes with direct sum and product?

I'm looking at exercise 1.2.1 from Weibel's Intro to Homological Algebra. (I need to show that homology commutes with direct sum and direct product.) Is it possible to show that cokernels commute with ...
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On the definition of groups of multiplicative type

Let $k$ be a field of characteristic 0. The definition of a linear algebraic $k$-group of multiplicative type (m.t.) I've seen the most in the literature is that $G$ is of m.t. if it is a ...
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30 views

Basic computation of exact sequence

Given a long exact sequence of vector spaces: $$...\longrightarrow V_1 \overset{f}{\longrightarrow}V_2\overset{g}{\longrightarrow}V_3\longrightarrow...$$ Given another vector space $W$, is the ...
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118 views

Homology and topological propeties

i have this theorem with it's proof but i don't understand the last part They use this proposition: My question is Why $\varphi^c\cap U_i$ is closed and pairwise disjoint ? where ...
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Extension to rational and real chains

In the paper on stable commutator length, D. Calegari says that generalized $\operatorname{scl}$ function can be extended by linearity to rational group $1$-chains and by continuity to real chains ...
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Minimal injective resolution of a module

Let $R$ be a commutative Noetherian ring and $M$ an $R$-module. Let $0\rightarrow M \rightarrow E^{\bullet}$ be a minimal injective resolution of $M$ and $0\rightarrow M\rightarrow I^{\bullet}$ be an ...
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40 views

Additive, covariant functor commutes direct limits, then it commutes with direct sums?

Suppose $T:R-Mod \to R-Mod$ is an additive covariant functor that preserves direct limits. (R is commutative, unital. Noetherian if it suits you even). That is, if $(W_{\alpha})_{\alpha \in \Lambda}$ ...
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38 views

Question about a theorem from Chang's book: Methods in Nonlinear analysis

I have this this theorem from Chang's book: Methods in Nonlinear analysis, with it's proof, but i don't understand it, for example what it means $K(f_{\sigma_i})$ ? Please help me thank you
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Question about Homology from the Chang's book: Methods in Nonlinear analysis

In the K.C.Chang's book in page $336$ of the book this corollary without prove there is a theorem before it but I don't know if it is a corollary of this theorem, how I can prove this ...
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35 views

Question about Chains complexes

I have $\mathcal{U}=\lbrace X-U, A\rbrace$ such that $\overline{U}\subset \overset{º}{A}$ and $X=\overset{º}{(X-U)}\cup \overset{º}{A}$ where $X$ is a topological space, $A$ is a subset of $X$ and ...
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An example of short exact sequence which is not exact triangle.

Let $0 \to \mathbb Z/2\mathbb Z \to \mathbb Z/4\mathbb Z\to \mathbb Z/2\mathbb Z \to 0$ be a short exact sequence of complexes concentrated in degree $0$. How can I prove that it cannot be made into ...
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Derived Functors and nice Resolutions

Charles A. Weibel, like many other books I know, introduces the notion of (Left) dervied functros as following: "Let $\mathscr F:$$\mathscr A$$\rightarrow$$\mathscr B$ be a right exact functor ...
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Does the rank of homology and cohomology groups always coincide?

Let $(C_i)_{i \in \mathbb{Z}}$ be a chain complex of free abelian groups. Does the rank of the homology and cohomology groups of $(C_i)_{i \in \mathbb{Z}}$ always coincide, i.e. is ...
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85 views

Composition of bicartesian squares

A commutative square is called bicartesian when it is both pull-back and push-out. In an abelian category, consider two pull-back squares $(X)$ and $(Y)$: $$ \begin{array}{ccccc} A & ...
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35 views

Example of Tor-Rigid Module

Let $R$ be a commutative ring (with 1) and $M$ a finitely generated $R$-module. We say that $M$ is rigid if for every finitely generated $R$-module $N$ whenever Tor$_i^R(M,N)=0$ then Tor$_j^R(M,N)=0$ ...
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Generalisation of cochain complexes and “curvature”

Someone has mentioned to me that generalizations of co-chain complexes and their cohomology have been studied, where instead of $d^2 = 0$ we have something like $d^2 \alpha = q \alpha $, which is ...
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37 views

First cohomology group of direct products

Let $p$ be a prime number and H be a finite group with $|H|=p-1$ and consider $\varphi: H \times Z_{p^k} \rightarrow Aut(Z_{p^k})$ as a non-trivial action of $H \times Z_{p^k}$ on $Z_{p^k}$ such that ...
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Hypercohomology: finding a resolution for the de Rham complex of $\mathbb{CP}^1 $

Let $\mathbb{P}^1 $ be the complex projective line. Using the standard affine cover, $\mathcal{U} = \lbrace U,U' \rbrace, \ \ $ we can define some quasi-coherent sheaves on $\mathbb{P}^1 $. We can ...
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Derived categories of curves equivalent then the curves are isomorphic

I am a beginner at derived categories and I'm looking for a proof of the following fact: If $X$ and $Y$ are smooth projective curves such that $D^b(Coh\,X)$ is equivalent to $D^b(Coh\,Y)$ then $X$ ...
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Action of $G/H$ on $H_n(H;M)$

I'm currently studying group cohomology and have trouble with the Hochschild-Serre spectral sequence. My problem is this: Given a short exact sequence of groups $$ 0 \to H \to G \to G/H \to 0$$ how ...
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The semidirect product as a deformation of the direct product

The way I think of the semidirect product is as a "deformation" of the direct product. Is there a way of making this intuition precise? Perhaps using some certain (co-) homology theory of groups?
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Question about the Betti numbers

can someone explain me this definition from :http://en.wikipedia.org/wiki/Betti_number The $n^{th}$ Betti number represents the rank of the $n^{th}$ homology group, denoted $H_n$ "which tells us the ...
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Simplicial homology of the skeleton of a simplex

Let $n$ and $k$ two natural numbers. We consider the (abstract) simplicial complex $K$ on $n$ vertices $v_1,\dots,v_n$ and such that a subset of $\{v_1,\dots,v_n\}$ is a face of $K$ if and only if it ...
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23 views

Universal property of tensor product of dg algebras

Does the tensor product of dg algebras have a universal property? I have not seen anything about this in the literature.
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36 views

Global dimension regular rings of finite type

Have I made an error in my reasoning? If $k$ is a field, $A$ is a commutative regular $k$-algebra of finite type and ${\mathfrak{m}}$ is a maximal ideal in $A$ then since $Ext_{A_{\mathfrak{m}} ...
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1answer
40 views

Weibel “Introduction to homological algebra” Main Theorem 4.4.16

I can't understand the proof of Main Theorem 4.4.16 from Weibel's book "An Introduction to homological algebra". The Theorem states Let $R$ be a local noetherian commutative ring, then $R$ is ...
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Question about the proof of the universal coefficient theorem

When deriving the universal coefficient theorem, in class we proceeded as follows: We have the SES: $$0\to ...
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Calculating the intersection product in CH(X)

Let CH$(X)$ be the Chow-Ring of a projective,smooth variety with cycles modulo rational equivalence. Lets assume Kunneth-Formula holds. There is an intersection product CH$^a(X) \otimes $ CH$^b(X) ...
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Ext functor commutes with connecting homomorphisms?

Suppose we have an exact sequence $0 \to L \to M \to N \to 0$ and a morphism $f \colon A \to B$ of $R$-modules. If $\delta \colon \text{Ext}^{i}_{R}(B,N) \to \text{Ext}^{i+1}_{R}(B,L)$ and $\delta' ...
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51 views

Hom and $\otimes$ functors on chain complexes.

I can't solve the exercise $2.7.3$ from Weibel's book "An Introduction to homological algebra": Let $P,Q$ be right and left $R$-module chain complexes, $I$ be a cochain complex of abelian groups. ...
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$i^{-1} F$ a sheaf if and only if $\varinjlim_{ U \subseteq X \text{ open}, ~ x,y \in U } F(U) \to F_x \times F_y$ is an isomorphism.

Let $X$ be a topological space containing two closed points $x,y$ and let $i : \{x,y\} \to X$ denote the inclusion map. Notice that $\{x,y\}$ carries the discrete topology. Let $F$ be a sheaf on $X$. ...
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Homotopy limits

Let $\mathfrak C$ be a Grothendieck category and let ${\bf D}=\mathrm{D}(\frak C)$ be its derived category, that is, consider the injective model structure on the category $\mathrm{Ch}(\frak C)$ of ...
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Leray's theorem for cech and derived sheaf cohomology.

My question is about the hypothesis of Leray's theorem. This theorem says that if $\mathcal{U}$ is an open cover of a topological space $X$, and $\mathcal{F}$ is a sheaf over $X$ and if ...
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The inverse image of a sheaf

By definition, the inverse image of the sheaf $ \mathcal{F} : \mathrm{Ouv} (Y) \to \mathrm {Set} $ is the sheaf associated to the presheaf $ f^{-1} \mathcal{F} : \mathrm{Ouv} (X) \to \mathrm{Set} $ ...
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Definition of Hochschild homology in terms of Tor functor (bar resolutions)

I had 2 kind of dumb questions about the definition of Hochschild homology in terms of the Tor functor: 1 - Let $R$ be a $k$-algebra and $M$ an $R$-bimodule, let $H_*(R,M)$ be the Hochschild homology ...
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Global dimension of translation algebra

What is the Hochschild cohomological dimension of the "translation algebra": $\mathbb{C}\langle x,y\rangle/(xy-yx-x)$? I expect it to be $2$, but I haven;t found a serious argument as to why this ...
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Motive of Pfisterforms spectral sequence

In this famous paper http://www.math.uni-bielefeld.de/~rost/data/motive.pdf Rost constructs the motive of a Pfister-Form/Pfister-Quadric. In the last proof on page 13 he writes: "By a spectral ...
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Exact sequence from Serre spectral sequence

let me say first that I don't know homological algebra very well, so I apologize in advance if my question is stupid.. It regards the Serre spectral sequence associated to a fibration $0\rightarrow F ...
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Kunneth formula for group homology

I'm trying to prove Kunneth formula for group homology. $$ 0 \to \bigoplus_p H_p(G,M)\otimes H_{n-p}(G',M') \to H_n(G\times G',M \times M') \to \bigoplus_p Tor_1^{\mathbb Z}(H_p(G,M),H_{n-p-1}(G',M')) ...
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Computing the order of the first cohomology group $|H^1(S_n, \mathbb F_p^n)|$

Assume $n\geq 3$, $p$ is a prime, and that $S_n$ acts on $V=\mathbb F_p^n$ by permuting the basis vectors $v_1,\ldots, v_n$. I want to compute the order of the first cohomology group of this action. ...
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Why does tensoring a projective resolution with a flat module give another projective resolution?

This question came up in this thread: Proving that tensoring a projective module with a flat module gives a projective module? Assume $\left\{P_i\right\}$ is a projective resolution of an $R$-module ...
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Projective modules over $kG$ equivalent to injective.

Let $k$ be a field and $G$ is finite group. I want to prove that a $kG$ module $P$ is projective iff it's injective. I proved that if module is projective then it's injective. 1) $kG$ is injective ...
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$L\otimes_{\Delta}\text{Hom}_{\Delta}(M,\Delta)\cong \text{Hom}_{\Delta}(M,L)$

This is exercise 5 in maximal orders by I.Reiner. This is not homework though. Let $\Delta$ be a ring $L_{\Delta}$ be any module, and let $M_{\Delta}$ be a finitely generated and projective. ...
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What is the injective envelope of $\mathbb{Z}/n\mathbb{Z}$?

In the category of $\mathbb{Z}$-modules, what is the injective envelope of $\mathbb{Z}/n\mathbb{Z}$? I was hoping to find a divisible group containing $\mathbb{Z}/n\mathbb{Z}$ such that it is ...
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1answer
17 views

Using Tor to find the Torsion submodule

Say $R$ is an integral domain with field of fractions $F$. I need to show that, for any $R$-module $B$, $Tor_1^R(F/R, B)\cong t(B)$, where $t(B)$ is the torsion submodule of $B$. So say $$\cdots\to ...
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68 views

Finite pushforward commute with taking cohomology

Let $f: X \to Y$ be a finite morphism of schemes. How one can show that $f_*H^i(G) \cong H^i(f_* G)$ for any $G \in D(X)$ and any $i \in \mathbb{Z}$? In english, $G$ is a complex of quasi-coherent ...