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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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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24 views

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

Let $K$ be a field and $R=K[x_1,\ldots,x_n]$ the multigraded polynomial ring, where $\deg x^a=a$ for $a\in\mathbb{N}^n$. Have finitely-generated multigraded $R$-modules been classified? Are they of ...
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Decide two indices of Ext functor

This question is from the proof of Theorem 11.34 in the book: Twenty-four Hours of Local Cohomology. Let R and S be CM local ring and R$\to$S a local homomorphism such that S is a finite generated ...
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32 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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37 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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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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27 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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$p_k \colon M_k \to N_k$ 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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41 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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32 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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30 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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20 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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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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28 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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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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34 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 $\phi(f+B^n(G,Hom_{ZH}(ZG, A) ...
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$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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32 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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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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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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32 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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37 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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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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17 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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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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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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30 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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28 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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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 ...
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24 views

Bicomplexes-reference request

I'm not an expert in homological algebra, I would say that I have gathered only preliminary knowledge. I would like to learn more in particular about bicomplexes and homology and cohomology of such ...
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31 views

The hyper-derived functors $\mathbb L_\bullet F$ are just derived functors of $H_0F$?

Problem (Weibel's Introduction to Homological Algebra, Exercise 5.7.4,2) Let $\mathbf{Ch}_{\ge0}(\mathcal A)$ be the subcategory of complexes $A$ with $A_p=0$ for $p<0$. Then the hyper-derived ...
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A quasi-isomorphism between the total complex of a Cartan-Eilenberg resolution and the complex per se.

Problem (Weibel's Introduction to Homological Algebra, Exercise 5.7.1) Suppose $A$ is a (not necessarily bounded below) chain complex over an abelian category $\mathcal A$ where axiom (AB4) holds, ...
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How to compute cyclic/dihedral powers of modules?

The $n$-th cyclic power of an $R$-module $M$ is $$T^n_C(M)=M^{\otimes n}/\langle m_1\!\otimes\!m_2\!\otimes\!\ldots\!\otimes\!m_n-m_2\!\otimes\!\ldots\!\otimes\!m_n\!\otimes\!m_1;\, ...
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2answers
102 views

example of hom of direct sum

I have a question . Can anyone give me examples for $Hom(B,\oplus A_j)$ not isomorphic to $\oplus Hom(B,A_j)$ or $\prod Hom(B,A_j)$ as abelian groups? Here $A_j$ and B are both modules. I have read ...
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24 views

Injective Resolution in Abelian Categories

Let $\mathcal{C}$ be an Abelian category. There is a fact that if $\mathcal{C}$ has enough injective objects, then any object in $\mathcal{C}$ has an injective resolution. By the definition of ...
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A simpler definition of the snake map?

I would like to ask whether the following definition of the connecting morphism in the long exact sequence in homology of a pair $(X,A)$ is correct. First, define relative cycles and boundaries via ...
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Tensor product of $C^*$- algebras

We know from the paper of Douglas and Howe (enter link description here) that the commutator ideal $\mathcal{I}$ of $\mathcal{A}(C(T^2))$, the $C^*$-algebra generated by Toeplitz operators with ...
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Good Pair corollary of the excision theorem

I have problem with understanding the following proof $q_*$ is isomorphism as q is a quotient map and so outside A, it is a homeomorphism implies that $q_*$ induces isomorphism. Given the above ...
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A category is locally finitely presented if the relative purity and purity coincide

Let $A$ be a locally finitely presented additive category, $X$ an additive subcategory. A sequence $0\rightarrow A_1\rightarrow A_2\rightarrow A_3\rightarrow 0$ in $A$ is pure exact if it is ...
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Ext and extensions

There are two abelian groups up to isomorphism of order $p^2$, where $p$ is a prime. But Ext$(\mathbb{Z}/p,\mathbb{Z}/p)$ is cyclic of order $p$. I can embed $\mathbb{Z}/p$ into $\mathbb{Z}/p^2$ in ...
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Some questions about Chain and Homological groups

Consider an oriented complex $K$ and it's chain group; that's the set $C_p(K)$ of it's $p-$chains endowed with the point-wise addition. If $T_1$ and $T_2$ are two triangulation of the same polyhedron ...
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On a property of split short exact sequences

Let $A_{\bullet}, B_\bullet$ and $C_\bullet$ be three short exact sequences of groups (not necessarily abelian) out of which $A_\bullet$ and $B_\bullet$ are split. Assume that there is again a short ...
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79 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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A short exact sequence of chain complexes with null-homotopic chain maps

Problem Suppose $0\to K'\xrightarrow iK\xrightarrow pK''\to 0$ is an exact sequence of chain complexes of modules over $R$, say. If chain maps $i,p$ are null-homotopic, then $K$ is contractible. ...
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66 views

Hochschild (co)homology and derived functors

Suppose that $A$ is (complex) unital algebra. We will consider $A-A$ bimodules $M$: such a bimodule is the same as (say) left $A \otimes A^{op}$ module. Let us define $C_n(A,M)$ as $M \otimes ...
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The functor Tor for $r_R$

Suppose $R$ is commutative ring and $r \in R$. Show that if $r$ is a zero divisor, then $$\text{Tor}^R_n(R/(r),M) \cong \text{Tor}^R_{n-2}(r_R,M)$$ for $n\geq 3$, where $r_R =\{s \in R \ |\ rs =0 \}$. ...
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1answer
66 views

Is tilting theory extended also to arbitrary derived categories?

I was reading papers by Rickard ("Morita theory for derived categories") and Keller ("Derived categories and tilting") on tilting theory in derived categories, they seem to focus mostly on module ...