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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augmented algebras and their morphisms

Let $R$ be a commutative unital ring and $A$ an associative (unital) $R$-algebra. What is an augmented $R$-algebra? A (unital) $R$-algebra $A$, together with a (unital) ring morphism $\varepsilon: ...
7
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154 views

Derived category of certain ring

I'm interested in the structure of $D^b(R)$, where $R=k[x]/(x^n)$. How one can describe this category? What is the list of indecomposable objects in this category?
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1answer
39 views

Show that in any connected triangulation, a 0-cycle $\sum c_i[i]$ is a 0-Boundary $\iff$ $\sum c_i =0$.

Show that in any connected triangulation, a 0-cycle $\sum c_i[i]$ is a 0-Boundary $\iff$ $\sum c_i =0$. $(\Longrightarrow)$ Case 1: If the 0-cycle is the boundary of a 1-face, it must be of ...
10
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1answer
482 views

A direct product of projective modules which is not projective

I am looking for an elementary example of a family $\{M_\alpha\}_\alpha$ of projective $R$-modules whose direct product is not projective. The simplest example that I know is the $\Bbb{Z}$-modules, ...
4
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1answer
287 views

When does Hom commute with $\otimes$?

Suppose that $A$ is a finite dimensional algebra and $M,N$ are finitely generated $A$-module. Is it true that $Hom(A,M\otimes_A N)\cong Hom(A,M)\otimes_A Hom(A,N)$, as $A$-modules? This is definitely ...
4
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1answer
61 views

Are automorphisms of extensions trivial?

Here is a statement for abelian categories which seems so basic I'm feeling embarrassed to have to ask whether it's true in general. Suppose $0 \to A \to B \to C \to 0$ is an exact sequence with ...
2
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1answer
165 views

Understanding Equivalence of Categories

An equivalence of two categories $\mathcal{C},\mathcal{D}$ consists of a pair of functors $F:\mathcal{C} \rightarrow \mathcal{D}$, $G:\mathcal{D} \rightarrow \mathcal{C}$ and natural isomorphisms $FG ...
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31 views

Playing with the torus and semisimplicial sets (prove that $\phi$ and $\psi$ are not homotopic) [closed]

Recall that we can express the torus $|X.| \cong T$ as a square with edges $e$ and $f$, diagonal $g$, faces $T_1$ and $T_2$, and a single vertex $v$, and appropriate identifications. Let $Y.$ be the ...
2
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2answers
87 views

Prove that $H_n(A \sqcup B) \cong H_n(A) \oplus H_n(B)$ for all $n \in \mathbb{Z}$.

Let $A, B$ be topologyical spaces. Then, I want to prove that $H_n(A \sqcup B) \cong H_n(A) \oplus H_n(B)$ for all $n \in \mathbb{Z}$. I know how to prove this from the Mayer-Vietoris theorem, but I'm ...
6
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351 views

characterization of projective/injective/flat modules via $\operatorname{Hom}$ and $\otimes$

Let $R$ be a commutative unital ring and $M$ an $R$-module. Then $M$ is projective iff $\operatorname{Hom}(M,-)$ is exact, injective iff $\operatorname{Hom}(-,M)$ is exact, and flat iff $M\otimes-$ is ...
3
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1answer
76 views

$\mathrm{Ann}_RA+\mathrm{Ann}_RB\,\subseteq\mathrm{Ann}_R\,\mathrm{Ext}^n_R\!(A,B)$?

Let $R$ be a commutative unital ring and $r\!\in\!R$. Let $A$ and $B$ be $R$-modules. Does $rA\!=\!0$ or $rB\!=\!0$ imply $r\mathrm{Ext}^n_R(A,B)=0$ for all $n\in\mathbb{N}$? For $n=0$ it holds, but ...
3
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1answer
140 views

A short exact sequence of groups

I have a basic question from Rotman's 'An Introduction to Homological Algebra', Thm 5.3 pp 152. Let \begin{equation*} 0\xrightarrow{} A\xrightarrow{} E\xrightarrow{\pi} G\xrightarrow{} 1 ...
5
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1answer
268 views

Simplicial homology for n-simplex

I've just started to study homology theory. And I'm trying to calculate all $H_n(\Delta_N)$ for some $N$. I know that the number of $m$-simplex in $N$-simlex is $b_{N,m}={N+1 \choose ...
2
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1answer
62 views

A functor on commutative diagrams

As suggested by Daniel Rust I'll pose this as a separate question. Let $C$ be a category. Denote by $Ar(C)$ the following category: an object in $Ar(C)$ is a morphism $X_1 \rightarrow X_2$ in $C$. ...
2
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1answer
38 views

Morphisms as Objects, Comm. Diagrams etc.

Let $C$ be a category. Denote by $Ar(C)$ the following category: an object in $Ar(C)$ is a morphism $X_1 \rightarrow X_2$ in $C$. A morphism in $Ar(C)$ from $X_1 \rightarrow X_2$ to $Y_1 ...
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3answers
20 views

Choose a set that makes a sequence exact

What choice of $X \in \mathrm{Ab}$ and maps between the groups would make the following sequence exact? $$0 \rightarrow \mathbb{Z}/3 \rightarrow X \rightarrow\mathbb{Z}/2 \rightarrow 0$$ I'm thinking ...
2
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1answer
86 views

Right exactness on a dense subcategory

Let $F : C \to D$ be a $k$-linear functor between cocomplete $k$-linear categories, which preserves directed colimits (in particular arbitrary direct sums). Let $C' \subseteq C$ be a dense full ...
2
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1answer
74 views

Matrices over a ring: does $PAQ=A'$ imply $\mathrm{Coker}A\cong\mathrm{Coker}A'$?

In A Singular Introduction to Commutative Algebra by Greuel & Pfister, there is written on p. 127: Let $R$ be a commutative unital ring and $A\in R^{n\times k}$, $P\in R^{n\times n}$, $Q\in ...
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0answers
90 views

Extension of divisible abelian group

Let $A$, $G$ be abelian groups, $R \to F \to A$ be the standard free resolution of $A$, $\phi:R \to G$ be a morphism and $N:=\text{coker}(-\phi \times i:R \to G \otimes F)$. From MacLane - Homology ...
3
votes
2answers
152 views

Derived functors definition

I´m searching for a reference that defines $n^{th}$derived functors in an analogous way to the definition given in Mitchell´s "Theory of Categories" for the $0^{th}$ derived functor of $T$ covariant ...
0
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1answer
84 views

Diagram chasing, and more

1) Assume that $0 \rightarrow A_i \rightarrow B_i \rightarrow C_i \rightarrow 0$ and $0 \rightarrow C_1 \rightarrow C_2 \rightarrow D \rightarrow 0$ are exact, $i=1,2$. Show, using a diagram chase, ...
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1answer
178 views

$\mathrm{Tor}_1(R/a,M)$ and $\mathrm{Ext}^1_R(R/a,M)$, $a\in R$ a non-zero divisor

In Lecture Notes in Algebraic Topology, Davis & Kirk, it is written: Proposition $\mathbf{2.4.}\,\,$ Let $R$ be a commutative ring and $a\in R$ a non-zero divisor (i.e. $ab=0$ implies $b=0$). ...
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2answers
1k views

Homology groups of the Klein bottle

I've seen this but didn't really understand the answer. So here is what I tried: According to this picture we have one 0-simplex - $[v]$, two 1-simplices - $[v,v]_a,[v,v]_b$ and two 2-simplices - ...
8
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1answer
158 views

Is $R/N(R)$ a faithfully flat $R$-module?

I'm studying recently faithfully flat modules and I'd like to know the following: Is $R/N$ faithfully flat as $R$-module, where $R$ is a commutative ring with unit and $N$ is the subset of ...
0
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1answer
71 views

Show that $C$ is a chain complex

I suppose it's a common exam question to show that a certain sequence actually is a chain complex. What is it that has be shown, minimally? A chain complex is a sequence of modules and module maps, ...
7
votes
1answer
143 views

$H_0(X)\simeq\Bbb{Z}^k$, where $k$ is the number of path components

I want to prove that $H_0(X)\simeq\Bbb{Z}^k$, where $k$ is the number of path components of $X$. What I tried Since $∂_0=0$, ...
5
votes
1answer
91 views

Does $\varprojlim\ ^1$ vanish whenever it doesn't have to account for non-right exactness of $\varprojlim$?

The projective limit functor is not right-exact: if $G_\bullet\rightarrowtail H_\bullet\twoheadrightarrow K_\bullet$ is a projective system of extensions, then there is a long exact sequence $$ ...
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168 views

Computing ext over graded rings

This question came up as I was reading Beilinson, Ginzburg, Soergel paper Koszul Duality Patterns in Representation Theory. Suppose that $A$ is a Koszul ring (for the definition of Koszul ring ...
2
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1answer
91 views

Cycles and Closed Paths

$\newcommand{\Im}{\operatorname{Im}}$ Show that if $\Delta$ is a triangulation, then $[a_1, b_1]+[a_2, b_2]+\cdots+[a_n,b_n]$ is a $1$-cycle precisely when the indicated oriented edges $1$-faces ...
2
votes
1answer
83 views

On the element $2\in\mathrm{Ext}(\mathbb Z/4,\mathbb Z)$.

There is a canonical isomorphism $\mathrm{Ext}(\mathbb Z/4,\mathbb Z)\cong\mathbb Z/4$ based on the fact that an extension $$0\to\mathbb Z\xrightarrow i G\xrightarrow{p}\mathbb Z/4\to 0$$ is ...
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2answers
135 views

A simple question on Ext groups

Let $G$ be a finite abelian group. Is $\mathrm{Ext}_{\mathbb{Z}}(G,\mathbb{R}/\mathbb{Z})$ trivial? If not, under what condition is it trivial?
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0answers
23 views

Deriving a boundary operation from three homomorphism which compose to 0.

Let $C_n$, $C_{n-1}$, $C_{n-2}$, $C_{n-3}$ are modules with the homomorphisms $$\partial_n: C_{n} \to C_{n-1}$$ $$\partial_{n-1}: C_{n-1} \to C_{n-2}$$ $$\partial_{n-2}: C_{n-2} \to C_{n-3}$$ such ...
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Is there an analog of Cech complex for local cohomology over noncommutative rings?

Let $A$ be a noetherian ring, and let $I\subseteq A$ be an ideal. Suppose $I$ is generated by $a_1,\dots,a_n$. Let $M$ be a left $A$-module. If $A$ is commutative, then one can compute the derived ...
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177 views

A corollary of Grothendieck’s Finiteness Theorem

Well-known Theorem: Grothendieck’s Finiteness Theorem. Assume that $R$ is a homomorphic image of a regular (commutative Noetherian) ring. Let $\mathfrak a$ be an ideal of $R$, and let $M$ be a ...
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139 views

filtration on the cohomology of a complex

Let $K^\bullet$ be a complex and let $F_I$ and $F_{II}$ be two filtrations on it. suppose $F_I^i K^n$ intersects $F_{II}^i K^n$ trivially. It then follows that in the induced filtration on the ...
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0answers
120 views

When do equivariant quasi-isomorphisms of chain complexes induce a quasi isomorphism on the tensor product

Suppose I have chain complexes $A,B,C,D$ where $A$ and $C$ have right $R$-module structures and $B$ and $D$ have left $R$-module structures, and that I have maps $f:A\to C$ and $g:B\to D$ which ...
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0answers
34 views

$SF_1 (f_1 + f_2) = SF_1(f_1) + SF_1(f_2)$

Let F be a right exact functor. Prove $SF_1 (f_1 + f_2) = SF_1(f_1) + SF_1(f_2)$ where $f_1, f_2 :M \rightarrow M'$. I have just started reading homological algebra so please help me to solve this ...
4
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1answer
91 views

Complete abelian categories with projectieve generators are fully abelian.

This is my first time on stackexchange so if you need more detail from me , please ask. I was reading the book "Abelian Categories : An Introduction to the Theory of Functors" by Peter Freyd , and I ...
5
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1answer
146 views

Is the t-truncation a functor?

One of the axioms of a t-structure in a triangulated category is that any object $X$ can be embedded inm a distingueshed triangle $$ X_0\to X\to X_1\to^+ $$ The original work by ...
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1answer
134 views

Compute $\operatorname{Tor}_n^R(I,R/I)$

The problem is as follows: Let $I=\langle x^2,y\rangle\subset R=\mathbb{Q}[x,y]$. Compute $\operatorname{Tor}_n^R(I,R/I)$ for all $n\geq 0$. Thoughts: Usually when I see these types of ...
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2answers
57 views

Do $\operatorname{Hom}( - , R)$ and $ - \otimes_R B$ commute when applied to $A\cong R^d?$

Let $R$ be a commutative ring with identity. Let $A$ and $B$ are $R$-modules, and further suppose that $A$ is free with finite rank. Is it true that $$ \operatorname{Hom} (A \otimes_R B , R) \cong ...
3
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0answers
62 views

The Simplicial Flabby Resolution of a Sheaf

I study sheaf cohomology by Demailly's book and I have a trouble. Am I right, that inductive formula at end of page 198 $$\mathcal{A}^{[q]}=(\mathcal{A}^{[q-1]})^{[0]}$$ is wrong? I think that ...
3
votes
1answer
86 views

Proof that derived functors don't depend on choice of resolution.

Can somebody help me out with this? Let $X$ be an object in an abelian category $A$ with enough injectives, let $0 \rightarrow X \rightarrow M^{\bullet}$ be an injective resolution , let $0 ...
5
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1answer
119 views

Images in a short exact sequence

Suppose $$ 0\to V\to W\to X\to 0\\ \downarrow\quad\quad\downarrow\quad\quad\downarrow\\ 0\to V'\to W'\to X'\to 0\\ $$ is a commutative diagram of vector spaces, with the top and bottom rows short ...
2
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1answer
109 views

Existence of injective hull

Is that true that every module over a ring has an injective hull? The term "has an injective hull" appears in several different contexts, some of them say that modules over a ring has this property ...
3
votes
1answer
572 views

Local-global properties (localization): free, projective, injective, flat, torsion-free, etc?

Let $R$ be a commutative unital ring. We say that a property $(\ast)$ of modules is local-global when the following conditions are equivalent for any $R$-module $M$: $M$ is a $(\ast)$ $R$-module; ...
3
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0answers
102 views

free dimension of a module?

For any module $M$ there is defined the projective/injective/flat dimension, which is the length of the shortest projective/injective/flat resolution of $M$. Why isn't there defined a free dimension ...
4
votes
1answer
118 views

Compute $Tor_{n}^{\mathbb{Z}_{8}}(\mathbb{Z}_{4},\mathbb{Z}_{4})$

Let $\mathbb{Z}_{4}$ as a $\mathbb{Z}_{8}$ module. How can I prove this: $Tor_{n}^{\mathbb{Z}_{8}}(\mathbb{Z}_{4},\mathbb{Z}_{4})=\mathbb{Z}_{4}$? Pleas see this
2
votes
2answers
146 views

Is taking cokernels coproduct-preserving?

Let $\mathcal{A}$ be an abelian category, $A\,A',B$ three objects of $\mathcal{A}$ and $s: A\to B$, $t: A' \to B$ morphisms. Is the cokernel of $(s\amalg t): A\coprod A'\to B$ the coproduct of the ...
8
votes
3answers
198 views

Is $G$ a semidirect product of $Z(G)$ and $\operatorname{Inn}(G)$?

The title pretty much sums it up. $\operatorname{Inn}(G)$ is the group of inner automorphisms, $Z(G)$ is the center. I know that $\operatorname{Inn}(G)$ is isomorphic to $G/Z(G)$. This means that ...