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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tensor of two vector space [on hold]

I don't know how to show this problem please help me. let $R$ be a domain and $Q=Frac(R)$ if either $C$ or $A$ is a vector space over $Q$,prove that both $C\otimes_RA$ and $Hom_R(C,A)$ are also vector ...
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26 views

thorough proof of Mayer-Vietoris implies Excision

Where could I find a very complete proof of how the Mayer-Vietoris sequence implies the Excision theorem? I've read a few proofs, but they always leave out the details! Thank you!
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29 views

How do we get this quotient $\textrm{Ext}^1(N,M)/\textrm{Hom}(N,M)$?

If $0\longrightarrow M\longrightarrow E\longrightarrow N\longrightarrow 0$ is a short exact sequence of vector bundles on a surface. Here $M$ and $N$ are line bundles, and so rank $ E$=2. Also, if ...
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2answers
48 views

Exact sequence - proof

Let $R$ be a ring. Prove that a sequence of left $R$-modules and homomorphisms $$0 \to N_1 \xrightarrow{f} N_2 \xrightarrow{g} N_3$$ is exact if and only if for all left $R$-modules $M$ sequence ...
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1answer
98 views

Elementary proof that the category of modules is not self-dual

If $R,S$ are rings such that ${}_R \mathsf{Mod}$ is equivalent to ${}_S \mathsf{Mod}^{\mathrm{op}}$, then $R$ and $S$ are trivial. This is well-known. The usual proof uses of the notions of limit and ...
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30 views

Reference for proof of Hochschild-Kostant-Rosenberg for Hochschild cohomology

Is there a place where there is a full proof of the Hochschild-Kostant-Rosenberg Theorem for Hochschild cohomology? I am aware of many places where the result is proven for Hochschild homology i.e. ...
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33 views

Direct limit of quotient groups

For a subgroup $H$ of $G$, we denote $\langle H\rangle $ be the smallest normal subgroup of $G$ containing $H$. (That is, the normal closure of $H$ in $G$.) Suppose that ...
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39 views

Colimits in $Ch_R$, help with a step of the proof

I want to prove that the category of chain complexes of R-modules admits small colimits. I was told to try proving that the chain complex defined degree wise as the colimit of the modules of the same ...
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1answer
58 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
37 views

Cohomology of a split cochain complex

I have $A$ and $B$ two graded vector spaces, and $D: A \oplus B \to A \oplus B$ with $D(a + b) = d_0(a) + d_1(a) + d_0(b)$ for $a \in A$ and $b \in B$, where $d_0 : A \to A$, $d_0 : B \to B$ and $d_1 ...
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39 views

Why a chain morphism can be factorized into a composition of a monomorphism with retraction and a homotopy equivalence?

Let $\mathscr{A}$ be an additive category and $f:X\rightarrow Y$ be a morphism of complexes in $\mathscr{A}$. The question is are there chain morphisms $h,g$ such that $f=gh$ where $h^{n}$ is a ...
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28 views

Computation of $Ext^*_R(k,k)$ as an algebra using a dga-resolution

There is a theorem (VIII.2.3) in Mac Lane's Homology that reads: Let $k$ be a commutative ring. Let $R,S$ be $k$-algebras, and let $U$ be a $k$-differential graded algebra. Suppose there is a ...
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1answer
23 views

Two short exact sequences with projective objects in the middle

Problem: Prove that for two short exact sequences $$ 0\rightarrow A \xrightarrow{f} B \xrightarrow{g} C \rightarrow 0 $$ $$ 0\rightarrow A' \xrightarrow{f'} B' \xrightarrow{g'} C \rightarrow 0, $$ ...
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7 views

How do I write the BRST-BV differential if I start with a dg Lie algebra and module?

The usual BRST-BV implements a Lie algebra and its module in terms of ghosts, etc. Where is there written a corresponding formula incorporating the differential of a dg Lie algebra and module? answer ...
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52 views

Question about Poincare duality and homology of a cylinder.

I am reading the paper. I have some questions about Poincare duality and homology of a cylinder. On page 9, example 2.6. Let $X = \mathbb{R} \times S^1$ be a cylinder and $Y = X/(0 \times S^1 )$, ...
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1answer
26 views

Projective resolution of ideal $\langle a,b\rangle $ in the $K=R[a,b]$

I saw a resolution as $$K\to K^2 \to \langle a,b\rangle \to 0,$$ but I can't figure out why, and can't figure out the maps. Could you give me some ideas? Thank you.
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15 views

Simplicial Complexes of Graphs notation question

I'm studying Jakob Jonsson's book Simplicial Complexes of Graphs very rigorously and in depth. I've been okay so far with the intensity and notation, but on page Chapter 3, section 2, page 30, I'm ...
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90 views

Values of the Herbrand quotient

For a finite cyclic group $G$, there is the Herbrand quotient in the theory of group cohomology. I calculated some of those quotients and I always came up with an Integer as solution. I failed at ...
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1answer
118 views

commutes homology

I am trying to prove the following: Let $R,A$ be rings and $\mathrm T:$$\mathscr M_R$ $\to $$\mathscr A_R$ such that $\mathscr M_R$ is category of left R modules and $\mathscr A_R$ is category of ...
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1answer
29 views

How to show if M is projective, then $Tor_i$(M,K)=0, for i>0?

I can't find any clue about proving that, and the so called proof online is too short to understand. Can you offer some clear clues? Thank you!
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1answer
108 views

Classification of finitely generated multigraded modules over $K[x_1,\ldots,x_n]$?

Let $K$ be a field and $R=K[x_1,\ldots,x_n]=\bigoplus_{a\in\mathbb{N}^n}Kx^a$ the multigraded polynomial ring. Have finitely-generated multigraded $R$-modules been classified? Are they of the ...
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1answer
34 views

About a chain homotopy

Assume that $C=\lbrace C_{q},d_{q}\rbrace$ is a chain complex with each $C_{q}$ a free $R$-module. Let $C^{'}$ be another chain complex. Furthermore, assume that each $H_{q}$ is also free and that we ...
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42 views

Spectral sequences and Ext between extension of modules

Suppose $A$ is a commutative ring, $M_1,M_2,N_1,N_2$ are $A$-modules and we have two exact sequences of $A$-modules $$0\to M_1\to M\to M_2\to 0,$$ $$0\to N_1\to N\to N_2\to 0.$$ I want to write a ...
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2answers
50 views

diagram of short exact sequence

I have this commutative diagram of vector complex spaces where all the sequences that appear are short and exact. is there a way to say that $H$ is the intersection between $W1$ and $W2$?
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32 views

Tensor product of flat modules - proof verification

Let $A$ be a commutative ring, and let $B,C$ be commutative $A$-algebras. Let $M$ be a flat $B$-module and $N$ a flat $C$-module. I want to show that $M\otimes_A N$ is a flat $B\otimes_A C$-module. ...
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1answer
30 views

$p_k \colon M_k \to N_k$ is onto for $k>0$, if $p_0$ induces an iso on homology level, prove that $p_0$ is onto

We are working in $\textbf{Ch}_R$, chain complexes of $R-$modules. As the title suggest, I'm given a map (of chain complexes) $p\colon M \to N$ which is onto for $k>0$. It is known that ...
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1answer
44 views

Prove that $Ker(g \otimes k)= Im(f \otimes 1_{N}) + Im (1_{M} \otimes h)$

Suppose we have two short exact sequences: $$0 \to M' \mathrel{\overset{f}{\to}} M \mathrel{\overset{g}{\to}} M'' \to 0 $$ in Mod-R $$0 \to N' \mathrel{\overset{h}{\to}} N \mathrel{\overset{k}{\to}} ...
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1answer
45 views

Why is this Hilbert's Syzygy theorem?

In Lang's Algebra, chapter XXI, §4, on p. 861 he describes the standard construction of a graded (in principle infinite) free resolution of a finite graded module $M$ over the polynomial ring $A = ...
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32 views

If $Ext_A^n(M,N) \ne 0$, then $Ext_A^n(M,N') \ne 0$, for every indecomposable summand N' of N?

Let $A$ be an artin algebra and $M$ and $N$ finitely generated modules over $A$. Suppose that $Ext_A^n(M,N) \ne 0$, is it possible to conclude that $Ext_A^n(M,N') \ne 0$, for each indecomposable ...
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21 views

Reference for derived functor

I'm following a course in algebraic geometry and in 2-3 month we will see the cohomology of schemes using derived functors. I don't know anything about it, (and about category theory in general), ...
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34 views

Counterexample to exactness of functor from group representations to fixed points

I recently asked this question. Now, the answer there claimed that the functor $()^G:Rep_G\to Vect_{\mathbb{C}}$, where $Rep_G$ are complex representations of a group $G$, and $V^G=\{v\in V: ...
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1answer
59 views

Intuitive explanation of Four Lemma

In the Short Five Lemma where the rows are exact, it is a fact that $$\alpha \text{ and }\gamma \text{ injective (surjective) }\implies \beta \text{ injective (surjective)}.$$ I've heard this fact ...
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1answer
43 views

(An arbitrary direct product of free modules need not be free)

For each positive integer $i$ let $M_i$ be the free $\Bbb Z$-module $\Bbb Z$, and let $M$ be the direct product $\prod _{i \in \Bbb Z^+} M_i$. Each element of $M$ can be written uniquely in the form ...
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1answer
84 views

If the cohomology of two objects in the derived category are equal, are the objects isomorphic?

Let $\mathcal{A}$ be an abelian category. Given objects $A^\bullet,B^\bullet$ in the derived category $D(\mathcal{A})$. Assume that $H^n(A^\bullet)=H^n(B^\bullet)$ for all $n\in\mathbb{Z}$. Can we ...
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35 views

Shapiro's Lemma-Finding the inverse of an isomorphism.

Consider the isomorphism $\phi: H^n(G, Hom_{ZH}(ZG, A))\cong H^n(H,A)$ of shapiro's lemma. I would like to describe this via cochains. So the obvious map is $\phi(f+B^n(G,Hom_{ZH}(ZG, A) ...
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1answer
23 views

$M' \to M \to M'' \to 0$ exact $\implies 0\to \text{Hom}(M'',N) \to \text{Hom}(M,N) \to \text{Hom}(M',N)$ is exact.

Let, $M', M'', M, N$ be $A$-modules. $M' \stackrel{u}{\to} M \stackrel{v}{\to} M'' \to 0$ exact $\implies 0\to \text{Hom}(M'',N) \stackrel{\bar{v}}{\to} \text{Hom}(M,N) \stackrel{\bar{u}}{\to} ...
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1answer
38 views

Why is the torus not a boundary of a 3-chain?

I'm learning about homology right now and the author simply states that the torus $T^2$ does not have a boundary (I understand this) and also is not a boundary of a 3-chain. This is not at all obvious ...
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1answer
36 views

Extensions of quasicoherent sheaves are quasicoherent.

Harts theorem 5.7: Given an exact sequence $0 \to \mathscr F_1 \to \mathscr F_2 \to \mathscr F_3 \to 0 $ of sheaves on $X = \mathrm{spec} A$, if $\mathscr F_1$ and $\mathscr F_3$ are quasicoherent, ...
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Properties characterized by a vanishing Ext or Tor module

While reading Weibel's "An introduction to homological algebra'', I've noticed that many properties of a module are characterized by the vanishing of some Tor or Ext. Fix a (commutative) ring $R$ and ...
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1answer
33 views

what is the inclusion map for $Y$ to $Y$ x $Y$?

I am studying homotopy and homology and one map we have been using is the left and right inclusion maps $i_L$, $i_R$, for example from the space $Y$ to the cartesian product $Y$ x $Y$. Whilst I ...
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1answer
33 views

Modules in Morita Equivalence

In Method of Homological Algebra by Gelfand and Manin (Exercise 2.2.3). How are $\mathrm{Hom}_A(P,X)$ and $\mathrm{Hom}_B(P^*,Y)\,$ regarded as a $B$-module and $A$-module respectively?
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38 views

Characterization of Projective Objects

In which categories is an object $P$ projective if and only if every short exact sequence ending with it splits? $$0\longrightarrow A\longrightarrow B\longrightarrow P \longrightarrow 0$$
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1answer
28 views

Direct limit and constant are adjoint functors

I have a question. Why $(\varinjlim, | |)$ is an adjoint pair of functors? Here the definition of constant direct system || is: For any I, fix a module A and set $A_i=A$, all $i\in I$, and ...
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1answer
41 views

Defining a Map Between Two Chain Complexes

I would like someone to check my reasoning here and, if my reasoning is correct, help me define a map to make a short exact sequence. I am given a short exact sequence of chain complexes $$ ...
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19 views

Closure properties for classes of modules that form a cotorsion pair

A torsion theory is a pair of classes of $R$-modules (where $R$ is an associative ring with identity) $({\mathbb T},{\mathbb F})$, such that $r({\mathbb T})={\mathbb F}$ and $l({\mathbb F})={\mathbb ...
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3answers
95 views

Does a long exact sequence of flat modules remain exact after tensoring with an arbitrary module?

In Liu's Algebraic Geometry and Arithmetic Curves, Proposition 1.2.6 states that given any short exact sequence $0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0$ with $M''$ flat, taking ...
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1answer
42 views

Projective dimension of module over local ring

This question arose reading the well known article by Buchsbaum Lectures on regular local rings. He states without proof that, given $(R,m)$ a local ring and an $R$-module $M$ over $R$, we have the ...
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33 views

The kernel of an antiderivation on an exterior algebra

This is a simple algebraic question I feel I should be obvious, but maybe isn't. Let $d'\colon V \twoheadrightarrow W$ be a surjective linear map of finite-dimensional vector spaces over a field of ...
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31 views

Derived projection formula

In the MO question here two "versions" of the projection formula are stated. The projection formula in algebraic geometry is, given a (quasicompact, quasiseparated) map of schemes $f: X \rightarrow ...
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1answer
59 views

Topological modules and relative homological algebra.

This question might be a bit dumb but I'm tired right now and this is just going over my head at the moment, in "The homology of Banach and topological algebras" Helemskii said that relative ...