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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Short Exact Sequence from Single Arrow in Abelian Category

Work in an abelian category. I'm aware that given an exact sequence, one can break it into short exact sequences like so: I was wondering whether it was possible to do derive one of these short ...
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33 views

Derived Equivalences and Limits

Let $\{A_n\}_{n>0}$ be a family of $k$-algebras such that for each $n$ there is a $k$-algebra morphism $f_n:A_n \to A_{n+1}$ that induces a triangulated equivalence: (where $D(A_n)$ denotes the ...
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50 views

Homotopy equivalence, Eilenberg-Maclane space to connected CW complex.

Let $X$ be any connected CW complex whose only non-vanishing homotopy group is $\pi_n(X) \cong \pi$. How do I construct a homotopy equivalence $K(\pi, n) \to X$, where $K(\pi, n)$ is an ...
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Are there any resolutions for $\mathbb{Z}$ over group algebra without topological <<model>>?

Let $G$ be group. Each cell partition of the universal cover of $K(G, 1)$ delivers a (projective?) resolution of $\mathbb{Z}$ over group algebra $\mathbb{Z}G$. Can one construct a pair of ...
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26 views

Limits, Tilting and Derived Categories

Let $\{A_n\}_{n>0}$ be a family of algebras such that for every $n>0$ there exist a triangulated equivalence $D(A_n) \cong D(A_{n+1})$. So all these algebras are derived equivalent. Take ...
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52 views

CW complex such that action induces action of group ring on cellular chain complex.

Let $X$ be a space that satisfies the hypotheses used to construct a universal cover $\overline{X}$. Let $\pi = \pi_1(X)$ and consider the action of the group $\pi$ on the space $\overline{X}$ given ...
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37 views

Projective cover and epimorphism

Let $C$ be an abelian category and let $X$ be an object with finite length. Thus there is a composition series $0=X_0 \stackrel{\iota_0}{\rightarrow}X_1\stackrel{\iota_1}{\rightarrow}\cdots X_{n-1} ...
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25 views

Help/Verification of Ext group calculation

I have been looking at extensions of irreducible representation over fields of positive characteristic. Specifically at the moment, to get the hang of things, I'm looking at $S_3$ over $\mathbb F_3$, ...
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31 views

Infinite torsion group with K(G,1) of finite type

I am wondering whether any group $G$ that is torsion and has a $K(G,1)$ of finite type (i.e. there are finitely many cells in each dimension) is already finite. The condition of having a $K(G,1)$ of ...
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1answer
28 views

finite length object and direct sum in an abelian category

Let $C$ be an abelian category and let $X$ an object with finite length. Then $X$ has a composition series $$0=X_0<X_1< \cdots X_n=X$$ where $X_i/X_{i-1}$ is simple for $i=1, \dots, n$. ...
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28 views

Epimorphisms in an abelian category

Let $C$ be an abelian category. Let $f:X \to Y$ and $g: Y \to Z$ are morphisms of $C$ and suppose that $g\circ f$ is epimorphism. Question Is $g$ epi as well? If so, I want to know the proof.
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45 views

Horseshoe Lemma

Trying to understand the horseshoe lemma. Suppose I start with the usual exact sequence $0 \to \mathbb{Z}/2\mathbb{Z} \overset{a}{\to} \mathbb{Z}/4\mathbb{Z} \overset{b}{\to} \mathbb{Z}/2\mathbb{Z} ...
3
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1answer
64 views

Colimit of k-algebras.

Let {$A_i$}$_{i\in I}$ be a family of $k$-algebras, where k is a field and I is a partially ordered set. Does the following colimit exists? $$ \operatorname{colim}_{i\in I} A_i $$ How can it be ...
4
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124 views

What is $H_*(S^m \times S^n)$, without using Künneth theorem?

As the question suggests, what is $H_*(S^m \times S^n)$ for $m \ge 1$ and $n \ge 1$? I would like to see a way without using the Künneth theorem...
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43 views

Projectivity and semisimple in an abelian category

Let $C$ be a locally finite $k$-linear abelian category. Here locally finite means that the home sets are finite dimensional vector space and every object of $C$ has finite length. Suppose that each ...
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2answers
65 views

left exact & preserve product

The problem is as follows: Prove that if$\ \ \ T:{}_R\mathrm {Mod}\rightarrow \mathrm{Ab}$ is an additive left exact functor preserving direct products,then $T$ preserves inverse limits. Suppose ...
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172 views

Difference between homology and integral homology?

What is integral homology? And how does it relate to homology? I can't find a good answer anywhere, so I thought I would ask here.
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Connecting homomorphism is Bockstein operation, construction of natural long exact sequence.

Let $0 \to \pi \overset{f}{\to} \rho \overset{g}{\to} \sigma \to 0$ be an exact sequence of Abelian groups and let $C$ be a chain complex of flat Abelian groups. Write $H_*(C; \pi) = H_*(C \otimes ...
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28 views

If $X$ is a finite CW complex, does it follow that $\chi(X) = \chi(H_*(X; k))$ for any field $k$? [closed]

If $X$ is a finite CW complex, does it follow that $\chi(X) = \chi(H_*(X; k))$ for any field $k$?
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1answer
43 views

Half Exact Functor

A half exact functor is a functor F (between abelian categories) such that for every short exact sequence: $$ 0 \to A \to B \to C \to 0$$ then $$F(A) \to F(B) \to F(C)$$ is exact. Does anyone has an ...
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Category of Chain Complexes over Chain Complexes

I have a few questions. Let A be an abelian category, and Ch(A) the category of chain complexes over A. Now let $$Ch^2(A) = Ch(Ch(A)) $$ and inductively $$Ch^n(A) = Ch^{n-1}(Ch(A)) $$ Is there a ...
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85 views

Why a free resolution rather than merely a projective one?

Suppose we have a module over a ring. We choose a projective resolution which allows us to define derived functors. In particular, sometimes we can choose our resolution to be free. I always hear ...
2
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1answer
69 views

Finding Homology of Koszul Complex

Let $R = \mathbb{Q}[x,y,z]$ and $I = \langle x,y,z\rangle$. I have the free resolution of $R/I$ as $K : 0 \rightarrow R \rightarrow R^3 \rightarrow R^3 \rightarrow R \rightarrow 0$ I need to find ...
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1answer
34 views

Conditions for free/projective/flat module over a groupring

Let $H \subset G$ be a subgroup of the group $G$. When is $\mathbb{Z}[G]$ a free/projective/flat $\mathbb{Z}[H]$-module? If $\mathbb{Z}[G]$ is a free $\mathbb{Z}[H]$-module then there is a $n \in ...
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52 views

Trouble calculating projective modules

I've been trying to calculate projective modules in an effort to eventually classify extensions of modules of $S_3$ over a field of three elements. So I know that projective modules can be found by ...
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57 views

Tensor product of $k$-algebras, center, isomorphism.

Let $A$, $B$ be two $k$-algebras of finite dimension, where $k$ is a field. Here, $A$ and $B$ are not necessarily commutative. Do we have that$$Z(A \otimes_k B) \cong Z(A) \otimes_k Z(B),$$where ...
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36 views

Reference on Weibel's Homological Algebra: “$G/H$ acts by conjugation in LHS-spectral sequence”

I'm studying the Lyndon-Hochschild-Serre spectral sequence for $H\triangleleft G$: $$ H_p(G/H;H_q(H;A))\Rightarrow H_{p+q}(G;A) $$ where $A$ is a $G$-module. I was told (w/o giving a proof) that ...
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1answer
15 views

Cohomological dimension of $G$ equals $0$ iff $G=\{0\}$

I want to prove this statement with elementary considerations about group cohomology (started studying it today) If $G$ is a group such that its cohomological dimension is $0$, then $G$ is the ...
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1answer
31 views

Calculating Hom in derived category

I got stuck calculating $Hom^* (\mathcal O, \mathcal O(k)) \in D(Coh(\mathbb P^n))$. On one hand, $Ext^i (\mathcal O, \mathcal O(k)) = H^i (\mathcal O^* \otimes \mathcal O(k)) = H^i (\mathcal O(k))$, ...
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Euler characteristic, what is the precise relationship between $\chi(V)$, $\chi(V')$, and $\chi(V'')$? [duplicate]

For a graded vector space $V = \{V_n\}$ with $V_n = 0$ for all but finitely many $n$ and with all $V_n$ finite dimensional, define the Euler characteristic $\chi(V)$ to be $\sum (-1)^n \dim V_n$. Let ...
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35 views

Underlying abelian group of coproduct of commutative rings

If $A$ and $B$ are commutative rings (we assume) we can construct their coproduct $A \sqcup B$ with the inclusion maps $i_a$ and $i_b$. I'm trying to show that this coproduct has underlying abelian ...
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1answer
23 views

Subcategory that is not an abelian subcategory?

Weibel defines an abelian subcategory of an abelian category $A$ to be a subcategory $B$, which is an abelian category, such that a sequence of two maps in $B$ of is short exact iff it is short exact ...
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41 views

Exactness of the lie algebra functor

A question that occurs naturally but of which I (and, apparently, Google too) does not know an answer is the following: is the functor that associates to a lie Group its lie Algebra an exact functor? ...
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0answers
33 views

Uniqueness of the connecting homomorphism in theory of abelian categories

It is well known that two connecting homomomorphisms each belonging to a diagram are combined via naturality (if the two diagrams are commutatively related). But what about the uniqueness of the ...
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1answer
42 views

how to prove the commutativity of union and pullback without element?

Suppose F -> E and {U(i) -> E :a monomrphism} , how to prove the fibre product of F and the union of {U(i)} is the union of {the fibre products of F and U(i)} ? I think the union of {U(i) -> E} can ...
2
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1answer
67 views

Serre quotient category

Let $\mathcal{A}$ be an abelian category. A Serre subcategory of $\mathcal{A}$ is a nonempty full subcategory $\mathcal{C}$ of $\mathcal{A}$ such that given an exact sequence $$A \longrightarrow ...
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23 views

Special Case of Universal Coefficient theorem

Show that if $C_* :$ $ ...\to C_2 \to C_1 \to C_0 \to 0$ be a complex of vector spaces over a field $k$, then $H^n$($Hom_k(C_*,k)) \cong$ $Hom_k(H_n(C_*),k)$.Does this result holds if $k$ is not a ...
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17 views

Behavior $\otimes$-Triangulated Subcategory under Inverse

I am reading Thomason's "The Classification of Triangulated Subcategories". There we learn that for a given $\otimes$-triangulated category $\mathcal T$ and a subset of objects $E\subseteq \mathcal T$ ...
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30 views

Mapping Lemma for Spectral Sequences in Weibel's book - Help with the proof

We start by recalling the definition of a morphism in the category of Spectral Sequences: a morphism $f \colon A \to E$ is a family of maps $f^r_{pq}\colon A^r_{pq}\to E^r_{pq}$ in the abelian ...
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1answer
22 views

Chain Complex operation

In my homework, it asks me to show that the operation $-\otimes_{k} V$ sends exact sequence to exact sequence. What does the operation mean in terms of the map? For example if the map from $C_{n}$ to ...
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1answer
44 views

Infinite product of a short exact sequence

I was trying to show that in an abelian category satisfying (AB4)* the product of a short exact sequence is a short exact sequence. Given $0 \rightarrow A_i \rightarrow B_i \rightarrow C_i ...
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1answer
58 views

What is the necessary and sufficient condition for abelian categories to have arbitrary direct limit?

As a beginner to learn homological algebra, I have just learned about the direct system and its direct limit.As R-mod categories have arbitrary coproducts indexed ...
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Variations on the definition of “chain homotopy”

Suppose I have two chain complexes $(C_k, \partial_k^C)_{k \in \mathbb{Z}^{\geq0}}$ and $(D_k, \partial_k^D)_{k \in \mathbb{Z}^{\geq0}}$ of modules over a ring $R$, and a chain homotopy ...
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1answer
63 views

Short Five Lemma for Nonabelian Groups?

Is there a version of the "short five lemma" that applies to nonabelian groups? Perhaps it would incorporate some extra information about splittings. I have in mind the following kind of statement: ...
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1answer
42 views

Finding generators of a kernel in a free resolution

I am trying to find out the kernel (syzygy) in a free resolution. Here $R = K[x,y,z]$ where $K$ is a field. I am trying to resolve the ideal $M=(x,y,z)$. I have the following resolution. $ \phi_0 : ...
2
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1answer
108 views

What is Ext in the case of polynomial ring?

Let $R=k[x_1,x_2]$ where $k$ is a field and consider the $R$-module $M=R/(x_1,x_2) \cong k$. What is $\text{Ext}^n_R(M,M)$? I am also wondering if the result can be generalized to find ...
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37 views

Group (co)homology with coefficients in a tensor product

This question concerns the (co)homology of groups. How well does the functor $H^\ast(G;-)$ play with tensor products of $G$-modules? Are there nice general statements one can make when one's ...
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1answer
38 views

Compute $\text{Tor}^R_n(M,M)$ in the following specific case.

Let $R=\mathbb{Z}/ 8 \mathbb{Z}$ and let $M=\mathbb{Z} / 4 \mathbb{Z}$ be an $R$-module. How can I compute $\text{Tor}^R_n(M,M)$? I was just introduced to the theory of Tor, and I am having ...
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1answer
48 views

What is an additive functor?

I am a little confused with the definition of an additive functor. In my notes, it says that if $R,S$ are two rings and $F: \text{Mod}_R \to \text{Mod}_S$ is a covariant functor, then we call $F$ ...
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56 views

Affine smooth variety has only trivial first-order deformations

Does someone have a quick and direct argument that infinitesimal deformations of an affine smooth variety over a field $k$ are only the trivial infinitesimal deformations? (without previous ...