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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tensor of two vector space [on hold]

I don't know how to show this problem please help me. let $R$ be a domain and $Q=Frac(R)$ if either $C$ or $A$ is a vector space over $Q$,prove that both $C\otimes_RA$ and $Hom_R(C,A)$ are also vector ...
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26 views

thorough proof of Mayer-Vietoris implies Excision

Where could I find a very complete proof of how the Mayer-Vietoris sequence implies the Excision theorem? I've read a few proofs, but they always leave out the details! Thank you!
11
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2answers
468 views

Vanishing of a certain Tor

I am reading about the construction of the Affine Grassmannian in Dennis Gaitsgory's seminar notes and there are some commutative algebra facts that I am not able to figure out by myself apparently, ...
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29 views

How do we get this quotient $\textrm{Ext}^1(N,M)/\textrm{Hom}(N,M)$?

If $0\longrightarrow M\longrightarrow E\longrightarrow N\longrightarrow 0$ is a short exact sequence of vector bundles on a surface. Here $M$ and $N$ are line bundles, and so rank $ E$=2. Also, if ...
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2answers
48 views

Exact sequence - proof

Let $R$ be a ring. Prove that a sequence of left $R$-modules and homomorphisms $$0 \to N_1 \xrightarrow{f} N_2 \xrightarrow{g} N_3$$ is exact if and only if for all left $R$-modules $M$ sequence ...
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1answer
52 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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136 views

Example direct sum of injective modules is not injective

We knew that a direct product of injective modules is injective but a infinite direct sum need not be. I searched many article about this problem, and all most given a counter-example is: the ...
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1answer
97 views

Elementary proof that the category of modules is not self-dual

If $R,S$ are rings such that ${}_R \mathsf{Mod}$ is equivalent to ${}_S \mathsf{Mod}^{\mathrm{op}}$, then $R$ and $S$ are trivial. This is well-known. The usual proof uses of the notions of limit and ...
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153 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, ...
4
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1answer
90 views

Values of the Herbrand quotient

For a finite cyclic group $G$, there is the Herbrand quotient in the theory of group cohomology. I calculated some of those quotients and I always came up with an Integer as solution. I failed at ...
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2
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1answer
153 views

Computing Betti numbers using Macaulay program ??

Let $k$ be a field and $R=k[x,y,z]$, let $M=R/\langle x^2,xy,yz^2,y^4\rangle$ be $R$-module, how can we compute the left free resolution of $M$, and also the betti numbers of this resolution?
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Reference for proof of Hochschild-Kostant-Rosenberg for Hochschild cohomology

Is there a place where there is a full proof of the Hochschild-Kostant-Rosenberg Theorem for Hochschild cohomology? I am aware of many places where the result is proven for Hochschild homology i.e. ...
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39 views

Colimits in $Ch_R$, help with a step of the proof

I want to prove that the category of chain complexes of R-modules admits small colimits. I was told to try proving that the chain complex defined degree wise as the colimit of the modules of the same ...
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33 views

Direct limit of quotient groups

For a subgroup $H$ of $G$, we denote $\langle H\rangle $ be the smallest normal subgroup of $G$ containing $H$. (That is, the normal closure of $H$ in $G$.) Suppose that ...
3
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1answer
58 views

(Hopefully) Simple question about the exterior algebra functor

I have some (hopefully super) basic questions about the exterior algebra functor $$ \wedge:R\text{-Mod}\rightarrow R\text{-Alg}. $$ As I (think I) understand it, if one considers it as a functor ...
2
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28 views

Computation of $Ext^*_R(k,k)$ as an algebra using a dga-resolution

There is a theorem (VIII.2.3) in Mac Lane's Homology that reads: Let $k$ be a commutative ring. Let $R,S$ be $k$-algebras, and let $U$ be a $k$-differential graded algebra. Suppose there is a ...
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1answer
37 views

Cohomology of a split cochain complex

I have $A$ and $B$ two graded vector spaces, and $D: A \oplus B \to A \oplus B$ with $D(a + b) = d_0(a) + d_1(a) + d_0(b)$ for $a \in A$ and $b \in B$, where $d_0 : A \to A$, $d_0 : B \to B$ and $d_1 ...
3
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1answer
108 views

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

Let $K$ be a field and $R=K[x_1,\ldots,x_n]=\bigoplus_{a\in\mathbb{N}^n}Kx^a$ the multigraded polynomial ring. Have finitely-generated multigraded $R$-modules been classified? Are they of the ...
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1answer
59 views

Quillen groupoid of a groupoid.

For any category $\mathcal{C}$ we can define its Quillen's groupoid, denoted $\mathcal{Q}(\mathcal{C})$, as the category which have the same objects than $\mathcal{C}$ and the arrows between two ...
3
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1answer
49 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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39 views

Why a chain morphism can be factorized into a composition of a monomorphism with retraction and a homotopy equivalence?

Let $\mathscr{A}$ be an additive category and $f:X\rightarrow Y$ be a morphism of complexes in $\mathscr{A}$. The question is are there chain morphisms $h,g$ such that $f=gh$ where $h^{n}$ is a ...
5
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1answer
179 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 ...
3
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1answer
148 views

Duality between Tor and Ext?

Let $A$ be a $\mathbb{N}$-graded, locally finite $\Bbbk$-algebra, $\Bbbk$ being a field, $A=\oplus_{n \geq 0} \ A^n$, each $A^i$ being finitely dimensional as a $\Bbbk$ vector space. Assume also that ...
3
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1answer
23 views

Two short exact sequences with projective objects in the middle

Problem: Prove that for two short exact sequences $$ 0\rightarrow A \xrightarrow{f} B \xrightarrow{g} C \rightarrow 0 $$ $$ 0\rightarrow A' \xrightarrow{f'} B' \xrightarrow{g'} C \rightarrow 0, $$ ...
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7 views

How do I write the BRST-BV differential if I start with a dg Lie algebra and module?

The usual BRST-BV implements a Lie algebra and its module in terms of ghosts, etc. Where is there written a corresponding formula incorporating the differential of a dg Lie algebra and module? answer ...
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52 views

Question about Poincare duality and homology of a cylinder.

I am reading the paper. I have some questions about Poincare duality and homology of a cylinder. On page 9, example 2.6. Let $X = \mathbb{R} \times S^1$ be a cylinder and $Y = X/(0 \times S^1 )$, ...
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1answer
26 views

Projective resolution of ideal $\langle a,b\rangle $ in the $K=R[a,b]$

I saw a resolution as $$K\to K^2 \to \langle a,b\rangle \to 0,$$ but I can't figure out why, and can't figure out the maps. Could you give me some ideas? Thank you.
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15 views

Simplicial Complexes of Graphs notation question

I'm studying Jakob Jonsson's book Simplicial Complexes of Graphs very rigorously and in depth. I've been okay so far with the intensity and notation, but on page Chapter 3, section 2, page 30, I'm ...
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1answer
118 views

commutes homology

I am trying to prove the following: Let $R,A$ be rings and $\mathrm T:$$\mathscr M_R$ $\to $$\mathscr A_R$ such that $\mathscr M_R$ is category of left R modules and $\mathscr A_R$ is category of ...
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1answer
29 views

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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1answer
34 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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1answer
30 views

$p_k \colon M_k \to N_k$ 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 ...
2
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1answer
104 views

Tor for graded modules over a graded ring

I am confused about how this Tor is defined. Suppose $R$ is a graded ring, $M,N$ graded modules over $R$. What is $\operatorname{Tor}_{st}^R(M,N)$? I am confused about the subscripts. I realize ...
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42 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 ...
2
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2answers
50 views

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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32 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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2answers
108 views

Show that Q, as a Z-module, is a direct summand in a direct product of copies of Q/Z.

Prove that $\Bbb Q$, as a $\Bbb Z$-module, is a direct summand in a direct product of copies of $\Bbb Q/\Bbb Z$. This is a problem from Hilton and Stammbach's Homological Algebra. If this is ...
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1answer
44 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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2answers
3k views

Proving that the tensor product is right exact

Let $A\stackrel{\alpha}{\rightarrow}B\stackrel{\beta}{\rightarrow}C\rightarrow 0$ a exact sequence of left $R$-modules and $M$ a left $R$-module ($R$ any ring). I am trying to prove that ...
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0answers
35 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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32 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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1answer
44 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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21 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: ...
2
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1answer
84 views

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 ...
7
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1answer
59 views

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 ...
2
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1answer
43 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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1answer
23 views

$M' \to M \to M'' \to 0$ 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} ...
5
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3answers
119 views

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 ...