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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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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60 views

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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48 views

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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45 views

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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28 views

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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50 views

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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32 views

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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69 views

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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107 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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46 views

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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76 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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34 views

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
74 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 ...
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175 views

Homology and (co)Limits

I've looked around on MSE and online only to find scattered results, which confuse me. I want to understand how homology behaves with (co)limits. I want to know in particular about singular homology, ...
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1answer
56 views

Determinant of a coherent sheaf over a smooth projective variety

We know a coherent sheaf $E$ over a smooth projective variety $X$ admits a finite locally free resolution. $0\longrightarrow E_n\longrightarrow E_{n-1}\longrightarrow\cdots\longrightarrow ...
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1answer
116 views

Example that inverse limit is not exact

Its known that "inverse limit is not exact". Matsumura in his book Commutative Ring Theory, page 272, gives an example for this. I can not understand how he proves that inverse limit of $Z$ is zero. ...
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43 views

direct limit of finitely generated submodule

if $A$ is a module,then the family fin($A$) of all the finitely generated submodules of $A$ is a directed set and direct limit of$M_i$ is isomorphic to$A$. for prove this needed to define to injection ...
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66 views

Is there a surjective homomorphism from $\oplus_{i=0}^\infty $ $\Bbb Z_p$ into $\prod_{i=0}^\infty$ $\Bbb Z_p$?

Is there a surjective homomorphism from $\oplus_{i=0}^\infty $ $\Bbb Z_p$ into $\prod_{i=0}^\infty$ $\Bbb Z_p$? Can this be shown by the order of element? Can we say this homomorphism is one-one, ...
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38 views

Constructing chain homotopy equivalence related to mapping cones

Problem (Weibel, Introduction to Homological Algebra, Exercise 1.5.8) Given a map $f\colon B\to C$ of complexes, let $v$ denote the inclusion of $C$ into $\operatorname{cone}(f)$. Show that there is ...
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67 views

Example of a compact module which is not finitely generated

Let $R$ be a ring and $M$ be an $R$-module. Definition: $M$ is called compact if $\text{Hom}_R(M,-)$ commutes with direct sums, that is, if for any set $I$ and any $I$-indexed family of ...
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52 views

Localization of Minimal free Resolution

Let $(R,m)$ be a local ring and $p \in \operatorname{Spec}(R)$. Let $$\cdots \longrightarrow F_n \longrightarrow F_{n-1}\longrightarrow\dots\longrightarrow F_1\longrightarrow F_0 \longrightarrow ...
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53 views

cohomology ring of abelian Lie algebra

This is a continuation of this question. Let $\mathfrak{g}$ be a Lie $R$-algebra with module basis $e_1,\ldots,e_n$ and zero brackets. Then in the Chevalley (co)chain complex, ...
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cohomology ring of Lie algebras: multiplication

If $\mathfrak{g}$ is a Lie $R$-algebra, then the Chevalley-Eilenberg complex defines the cohomology modules $H^k(\mathfrak{g})$. If $H^\ast(\mathfrak{g})=\bigoplus_kH^k(\mathfrak{g})$, then the ...
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122 views

Help with diagram chasing

Given the diagram $\require{AMScd}$ \begin{CD} 0 @>>> A @>f>> B @>g>> C @>>> 0 \\ @. @V\alpha VV \#@V\beta V V\# @VV\gamma V @. \\ 0 @>>> {A'} ...
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70 views

Showing minimal graded free resolutions are isomorphic

I'm currently reading Rogalski's notes on noncommutative projective algebraic geometry (which can be found here) and I'm currently trying to fill out the details of Lemma 1.24 (2). The step which I ...
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74 views

Abstract homotopy invariance of homology

When topology is involved, we know (singular) homology is homotopy invariant. However, homology and homotopy can be discussed in much more general contexts. Living in ...
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1answer
37 views

Homotopy in $\mathsf{Ch}_\bullet(R\mathsf{Mod})$ induced by homotopy in $\mathsf{Top}$?

I'm trying to put together the relationships between homotopy in $\mathsf{Top}$, chain homotopy, and homotopy in $\mathsf{Ch}_\bullet(R\mathsf{Mod})$. I more-or-less understand the connection between ...
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1answer
160 views

Is the converse of Proposition 3.5.4 (c) of Bruns_Herzog true?

Question 1. Is the converse of Proposition $3.5.4 (c)$ of Bruns_Herzog true? I can see that $R$ is cohen-macaulay. so if one can prove that $r(R)=1$ , $R$ will be Gorenstein. ...
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34 views

Derived Category in terms of Torsion Theory?

It is known that there's a bijection between hereditary torsion theories on, and localizations of, a fixed abelian category. Is this bijection natural? How/why not? How can I think of the derived ...
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38 views

A form of Künneth formula?

Problem (Exercise 15.12 in Bott & Tu's Differential Forms in Algebraic Topology) If $X$ is a space having a good cover, e.g., a triangularizable space, and $Y$ is any topological space, prove ...
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62 views

Derived equivalences implying isomorphism of Hochschild cohomologies, question.

Let $k$ be a field, let $A$ and $B$ be associative unital $k$-algebras, in this paper (http://webusers.imj-prg.fr/~bernhard.keller/publ/dih.pdf) Keller says that if there's an equivalence of derived ...
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47 views

Homotopy and chain homotopy determine each other

In continuation of this previous question, I'm having problems with the following proposition from Kamps and Porter's Abstract Homotopy and Simple Homotopy Theory: Proposition (3.7). [page 210] ...
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25 views

Does homotopy invariance of induced maps on homology of pairs follow algebraically from the basic homotopy invariance result?

If $f, g$ are homotopic maps between chain complexes $C(X) \to C(Y)$, that restrict to homotopic maps between the subcomplexes $C(A) \to C(B)$, then are the induced chain maps $C(X,A) \to C(Y,B)$ ...
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42 views

Intuition for chain homotopy via tensor products

An approach to chain homotopies, alternative to the usual boundary relation, uses the monoidal (closed) structure of $\mathsf{Ch}_\bullet(R\mathsf{Mod})$ with $R$ a commutative ring. In particular, a ...
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42 views

Equivalences between categories $\mathcal{K}^b(\text{Injectives})$ and $\mathcal{D}^b(\mathcal A)$ if $\mathcal{A}$ has enough injectives

I have the following question: Let $\mathcal{A}$ be a abelian category and $\mathcal{I}$ be the full subcategory of injective objexts of $\mathcal{A}$. Assume that $\mathcal{A}$ has enough ...
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1answer
57 views

Homotopy category of chain complexes as a localization

For an abelian category $\mathcal{A}$, define the homotopy category of chain complexes $\mathcal{K}(\mathcal{A})=\mathcal{C}(\mathcal{A})/\mathcal{I},$ where $\mathcal{C}(\mathcal{A})$ denotes the ...
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74 views

On finite generation of certain $\operatorname{Ext}$'s

All rings below are commutative. I have the following situation: $A$ is a commutative ring, $B=A/I$, and I know that $B$ is noetherian. I have a $B$-module $M$ which is finitely generated as a ...
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44 views

Reference for closed categories and monoidal categories

I'm looking for a book that: Defines closed categories separately from monoidal categories, and then proves in detail that the structure induced by a left adjoint to the internal hom is closed ...
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1answer
57 views

The internal hom in $\mathsf{Ch}_\bullet(\mathsf{Ab})$ as a coproduct?

The internal hom in $\mathsf{Ch}_\bullet(\mathsf{Ab})$ is defined grading-wise by $$(A\Rightarrow B)_n=\prod_{i\in \mathbb Z} \text{Hom}_R(A_i, B_{i+n})$$ Intuitively, I would have defined the ...
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26 views

Functoriality of group homology

I understand that group homology $H_*(-)\colon \mathsf{Grps} \to \mathsf{Ab} $ is a functorial. In Weibel's homological algebra, there is an argument in 6.7 show this by using that $H_*(G;-)\colon ...
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1answer
76 views

Are there any theorems about functors that reflect exactness?

Suppose $F:\mathbf{A}\to \mathbf{B}$ is an additive functor between two abelian categories, we say $F$ is exact iff it preserves short exact sequences. Is there a name for a functor $F$ that ...
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78 views

A seemingly wrong definition of convergence of spectral sequences in Bott & Tu?

After introducing exact couples, Bott & Tu defines spectral sequences as follows: A sequence of differential groups $\{E_r,d_r\}$ in which each $E_r$ is the homology of its predecessor ...
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70 views

Extensions of short exact sequences and second cohomology group

Let $G=\mathbb{I}_{p}=<g>$ be the cyclic group of order $p$, where $p$ is a prime and $A=\mathbb{Z}_{p}\oplus\mathbb{Z}_{p}$ a $G-$ module with the action $g^{n}(x,y)=(x+ny,y)$. I want to show ...
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65 views

Non-split chain complex which is chain-homotopy equivalent to its homology sequence

This is exercise 1.4.4 from Weibel. Consider the homology $H_*(C)$ of chain complex $C$ as a chain complex with zero differentials. It is easy to show that if C is split, then there is a chain ...
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50 views

Show an $R$-module is a direct limit

This is a scenario I've encountered in my class on $p$-adic L functions. Let $G$ be a profinite group which is the inverse limit of a system $(G_i, f_{ij})$ of discrete finite topological groups. ...
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1answer
48 views

Group homology with coefficients vanish

Say $G$ is a group and $M$ is a $\mathbb ZG$-module with the property that $H_i(G;M)=0$ for all $i\ge 0$. Does this happen besides $M=0$?
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35 views

Existence of non-split sequence

Let $G$ be an abelian group such that $G$ contains non-zero elements of finite order. Why there exists some short exact non-split sequence: $0 \rightarrow \mathbb{Z} \rightarrow H \rightarrow G ...
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21 views

Syzygies in geometry or topology?

I am interested in knowing about the application of Hilbert's Syzygy Theorem (or, for that matter, of the concept of syzygy itself) in geometry or topology, that is, in the fields that have to do with ...
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Merge operation in homological algebra?

I provide you with a definition for the Merge operation in one standard textbook on the minimalist program in linguistics: Merge: "basic structure-building mechanism. Merge takes two elements A and B ...
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36 views

Hochschild-Serre spectral sequence for not normal subalgebra

I am trying to understand lemma 2.26 from http://www.math.ru.nl/~solleveld/scrip.pdf I am coserned about calculation of $E^{p, q}_1$. If $\mathfrak{h}$ is Lie ideal than everything is fine. But here ...