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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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 ...
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78 views

Can we produce a long exact sequence in cohomology from more than just short exact sequences?

It is well known that given a short exact sequence $$0\rightarrow A \rightarrow B \rightarrow C \rightarrow 0,$$ we can form a long exact sequence in cohomology. (Example: the proof of the ...
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70 views

What is base change, how does it work, and why is it important?

I'm familiar with some bare-bone basics of homological algebra - projective/injective resolutions, derived functors, sheaf cohomology. I constantly come across the term base change, but I don't really ...
2
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79 views

Homology/cohomology for the uninitiated [closed]

I have heard of (co)homology occurring in many mathematical contexts and I vaguely suspect that it non-trivially relates different subjects. Also that it somehow relates to category/topos theory, ...
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1answer
52 views

Quasi isomorphism and pullback

Let $A,B,C$ be chain complex, $f : A \to B$ a quasi isomorphism and $g : C \to B$ a cochain map. Consider the pullback $A \times_B C$ with its two projections $p_A$ and $p_C$. Is it possible to show ...
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1answer
48 views

Projective resolutions of modules in a short exact sequence (Dummit & Foote Proposition 17.1.7)

This proposition is as follows: Let $$ 0\rightarrow L\rightarrow M\rightarrow N\rightarrow 0 $$ be a short exact sequence of R-modules. Let the following be projective resolutions of L and N ...
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52 views

Generalization for Leray Hirsch theorem for Principal $G$-bundle

This is a general question: Is there a generalized Leray Hirsch theorem for Principal $G$-bundle? with $G$ finite group with discrete topology. I know it does not make sense to compare with original ...
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140 views

Galois cohomologies of an elliptic curve

I am studying basic theory of elliptic curves. I encountered Galois cohomology. But two introductory textbooks I read used only $H^0$ and $H^1$. I am curious why higher cohomologies did not appear. I ...
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41 views

Obstruction to be conjugated by an automorphism for subgroups of an abelian group

Let $A$ be a finite abelian p-group( p being a prime number). Let $M,N$ be subgroup such that $M \simeq N$ and $A/M \simeq A/N$ as groups. Can I conclude that there is $\phi \in Aut(A)$ such that ...
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40 views

Are there right-deformations for abelian sheaves?

A sufficient condition for the existence of a point-set derived functor is the existence of a deformation of the corresponding functor. For modules, such a deformation always exists (see section 2.3). ...
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1answer
71 views

Cohomology groups of the bar construction of free graded algebras

Let $A= \bigwedge \langle x_n \rangle $ be the free graded commutative algebra with differential zero ($x_n$'s are elements with positive grading )over a field of characteristic zero. I want a ...
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1answer
26 views

Functor between chain complexes $\overset{?}{\implies}$ preserves zero-objects?

Let $\mathsf A,\mathsf B$ be two abelian categories. Any functor $F:A\longrightarrow B$ which preserves zero objects lifts to a functor $\mathsf{Ch}(A)\longrightarrow \mathsf{Ch}(B)$. Furthermore, if ...
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18 views

Quotient of tensor product

I am doing some work on $Tor$, and maybe Odin too (joke!). Been calculating it for some and it's interesting to say the least. Well I have reached a step I feel intuitively is the case but I am not ...
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1answer
34 views

$2\mathbb{Z}_8\otimes \mathbb{Z}_4$ isomorphic to $\mathbb{Z}_4\otimes \mathbb{Z}_4$

I am working on a problem and after much work I have gotten out an answer $2\mathbb{Z}_8\otimes \mathbb{Z}_4$ which I decided to take on a detour and work with in my mind, mostly to see if it is ...
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1answer
24 views

How does the zero functor fail to be exact/additive?

Just learning some homological algebra, in particular the fact that an additive, exact functor between abelian categories will preserve homology. A question is in my head whose answer I'm sure is ...
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1answer
49 views

Calculating a free resolution of $\mathbb Q[x,y,z]/I$ where $I = (x,y,z)$

Let $R=\mathbb Q[x,y,z]$ and $I = (x,y,z)$. I am trying to find the minimal free resolution of $R/I$. This is what I have got: $R \rightarrow R/I$ whose kernel is $I$, which is generated by 3 ...
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1answer
40 views

Using the Snake lemma to prove an Extension

I am trying to prove that $E'$ is an extension of $Q$ by $N'$ \begin{array} 00 &\longrightarrow & N & \overset{i_1} \longrightarrow & E & \overset{\pi_1} \longrightarrow& Q ...
2
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72 views

Ext modules over different rings

Let $A$ be a commutative ring and $I$ be an ideal of $A$. Suppose $M, N$ are $A$-modules such that $IM=IN=0$, meaning they are canonically $A/I$-modules as well. I want to compare $Ext^{q}_{A}(M, N)$ ...
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61 views

Homology sequence of the pair $(X,A)$

So I know that the following sequence is exact $$\cdots\rightarrow H_q(A)\rightarrow H_q(X)\rightarrow H_q(X,A)\rightarrow H_{q-1}(A)\rightarrow\cdots$$ And I am confused about a particular ...
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1answer
36 views

Homological algebra and exact sequences [closed]

Let \begin{matrix}0\to&L&\stackrel{f'} \to& M'&\stackrel{g'}\to &N' & \to 0\\ ...
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1answer
36 views

Does exact sequence remain exact in opposite category?

In a category C which is exact and and we the following two exact sequences 0-->P-->Q-->R 0-->P'-->Q'-->R' We also have the maps u:P-->P' v:Q-->Q' w:R-->R' Here u,v,w are such that the diagram ...
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1answer
61 views

Find free resolution of ideal $(x,y)$ in $\mathbb{Q}[x,y]$

If we consider the ideal $I=(x,y)$ as a module over the ring $\mathbb{Q}[x,y]$, how can we find a free resolution of $I$? That is, a sequence $$...P_2 \to P_1 \to P_0 \to I \to 0$$ with $P_i$ ...
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1answer
46 views

Find free resolution of module

Let $R=\mathbb{Q}[x,y]$ and let $M=\mathbb{Q}$ be an $R$-module where $x \cdot a = 0 = y \cdot a$ for all $a \in \mathbb{Q}$. Is the following a free resolution of $M$? Define $\mathbb{Q}[x,y]^{(1)}= ...
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1answer
40 views

Definition of chain homotopic.

Let $C_{\bullet}, D_{\bullet}$ be two nonnegatively graded chain complexs of $R$-modules with maps $d^C,d^D$ respectively($d^C_n: C_{n+1} \to C_n$), and let $f,g: C_{\bullet} \to D_{\bullet}$ be two ...
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1answer
56 views

Extensions of sheaves with isomorphic middle terms

Let $\mathcal{F}$ and $\mathcal{G}$ be two coherent sheaves on a variety $X/k$. If I know that $\dim_k \operatorname{Ext}^1(\mathcal{F}, \mathcal{G})=1$, and I have two different nontrivial (not ...
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1answer
31 views

Injectivity of a Hom space

(This question from the proof of Corollary 10.65 in Rotman's homological algebra.) Let $R$ and $S$ be rings, let $_S B _R$ be an $(S,R)$-bimodule and let $_S E$ be an injective left $S$-module. If ...
3
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73 views

Why homotopy category is not abelian?

Let A denote an abelian category, Ch(A) denote the corresponding category of chain complex. Then let HoCh(A) denote the category whose objects are the same of Ch(A), but the map between objects are ...
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0answers
36 views

Any acyclic complex of vector space over a field is split exact?

So how to prove there exists a split map, i.e. a map s s.t. d=dsd, where d is the differential. What I know is there exists some decomposition of a vector space, but I still can not find such a map s. ...
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28 views

Proof of the Künneth Formula

I would like to proof the following version of the Künneth formula: $0\to \oplus_{p+q=n} H_p(C_*)\otimes H_q(C_*')\to H_n(C_*\otimes C_*')\to \oplus_{p+q=n-1}Tor(H_p(C_*),H_q(C_*'))\to 0$ is an short ...
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1answer
29 views

$M\oplus N$ is free, is $M\oplus 0_N$ free

Just a quick question, as the title says let $M,N$ be $R$-modules, if $M\oplus N$ is free, is then $M\oplus 0$ also free? My initial feeling is "no" because it would be isomorphic to $M$ then which ...
2
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1answer
49 views

Question on tensor product.

Let $\text{CAlg}_R$ denote the category of commutative $R$-algebras and $R$-algebra homomorphisms. How can I show that if $A,B \in \text{CAlg}_R$, the tensor product $A \otimes_R B$ can be given the ...
3
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1answer
49 views

Let $\mathscr{A}$ be a chain complex. Show that the kernel of the map $A_n/B_n \rightarrow Z_{n-1} $ is isomorphic to $H_{n}(A)$.

Let $A_{n+1} \xrightarrow{p_{n+1}} B_n \xrightarrow{r_{n+1}} Z_n \xrightarrow{k_{n+1}} A_n$, where $k_{n+1} \circ r_{n+1}=i_{n+1}$ is the monomorphism in the image factorization of $d_{n+1}$ the ...
3
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0answers
73 views

Ext over the group ring $\mathbb{Z}[D_{2n}]$

Let $D_{2n}$ be the dihedral group of order $2n$ ($n$ odd) generated by $x,y$ with $x^n = y^2 =1.$ Let $H = \langle xy\rangle$. Now we can see that $D_{2n}$ acts on $D_{2n}/H$ and the action is $x.k = ...
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86 views

Basic questions about convergence of spectral sequences.

I have a fairly rudimentary understanding of spectral sequences. I have a couple questions though. In Algebra, Lang states what seems to me, to be two slightly different notion of convergence of a ...
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1answer
37 views

Conjugation action of $G$ on $N$ induces an action of $G/N$ on $H_{\ast}N$.

I need some help in proving this result from Weibel or Brown (page $48$ corollary $6.3$). If $G$ is a group and $N$ a normal subgroup of it, then the conjugation action of $G$ on $N$ induces an ...
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37 views

Right derived derivations

Let $k$ be a field and $A$ a graded algebra (If it simplifies things, we can assume that $A$ is graded commutative, too). The Lie algebra of derivations is the linear subspace $Der(A)\subset End_k(A)$ ...
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2answers
91 views

Notation in book homological algebra

In A Course in Homological Algebra, Hilton and Stammbach. Can anyone please explain me this notation $\{\varphi, \psi\}$ in the theorem: Proposition 9.3: Given $$A\xrightarrow{\{\varphi, ...
4
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1answer
62 views

Is a kernel in a full additive subcategory also a kernel in the ambient abelian category?

Setting: Let $\mathscr{C}\subset \mathscr{A}$ be a full additive subcategory of an abelian category. Let $C,C'$ be objects of $\mathscr{C}$ and let $f\in ...
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22 views

How to explicitly find the class of a short exact sequence in the extension group using the injective resolution?

For a short exact sequence of abelian groups $\xi:0\to A\to B\to C\to 0$ we know that there is a long exact sequence $$ 0\to Hom(C,A)\to Hom(C,B)\to Hom (C,C)\overset{\partial}{\to} Ext^1(C,A)\to ...
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1answer
146 views

Jeffrey Strom, ''Modern Classical Homotopy Theory'', prerequesites and recommended knowledge

This is mostly adressed to those who has studied the book. I've heard a lot about this particular book and browsed it's contents. However, I would like to be certain whether I will encounter some ...
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15 views

Equivalence of cycles in homology group

Let $X$ be $\mathbb{R}^2/\mathbb{Z}^2$.$\sigma_1:t\mapsto (t,0), \sigma_2:t\mapsto (0,t),\sigma:t\to(t,t)$. We know that $H_1(X) = \langle [\sigma_1],[\sigma_2]\rangle$. I want to prove that ...
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1answer
57 views

Computing $\operatorname{Tor}_1^R(R/I,R/J)$

I am trying to convince myself that for any ring $R$ (commutative, so I don't have to bother with left-or-right modules) and ideals $I$, $J$ we have $\operatorname{Tor}_1^R(R/I,R/J)=I\cap J/IJ$. I ...
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1answer
36 views

Is there any chain complex $A_\bullet$ such that $H_n(A_\bullet)$ is $\mathbb{Z}/2\mathbb{Z}$ for all $n\in\mathbb{Z}$? [closed]

Is there any chain complex $A_\bullet$ such that $H_n(A_\bullet)$ is $\mathbb{Z}/2\mathbb{Z}$ for all $n\in\mathbb{Z}$ and $H_n(A_\bullet/2A_\bullet)=\mathbb{Z}/2\mathbb{Z}$?
2
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1answer
256 views

What is “Ext” short for in “Ext Functor”? [duplicate]

Strangely, I've never heard Ext functors referred to by any other name, and so I'm not sure what "Ext" actually means. The only thing I can think of that "Ext" might stand for is "Exterior", which ...
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15 views

Can I determine the groups and homomorphisms?

Let $C_n$ denote the cyclic group of order $n$, and let $X_1,\dots,X_5$ denote some abelian groups. Assume that the following six sequences are exact: $$0\to C_3\to X_1\to C_2\to0$$ $$0\to X_5\to ...
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23 views

square zero extensions, abelian group objects in a category, module categories

Let $A$ be a commutative ring, and $\mathcal{C}$ the category of commutative rings over $A$. Let $\mathcal{C}_{ab}$ denote the category of abelian group objects in $\mathcal{C}$, its elements $(B,p)$ ...
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47 views

Show that homology is a functor in a pure categorical way.

Let $\mathscr{A}$ be an abelian category i want to show that $\mathcal{H^i}$ ( the i-th homology group) is a functor from the category of complexes of $\mathscr{A}$ to $\mathscr{A}$. I showed this for ...
5
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1answer
60 views

Find entries to commutative diagram.

Assume that the following diagram of abelian groups has exact rows and columns. Can you determine the missing entries and maps? Give short reasoning. This is what I tried: Where for example I ...
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2answers
47 views

Example of Short exact Sequence of chain complexes

I am working on some homological algebra and I struggle to find an example of a short exact sequence of chain complexes. That is if $$0\to A.\to B. \to C.\to 0$$ then what can $A.$,$B.$, $C.$ be ...
0
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0answers
52 views

Injective resolutions of a complex

Let $\mathcal{A}$ be an abelian category, $M\in\mathcal{A}$. An injective resolution of $M$ is a quasi-isomorphism $M\longrightarrow I$, where $I$ is a complex of injective objects. This can be made ...