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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Homological proof issue

At page 421 of this book, about middle page, we found: "... for this will surely imply that $\theta_\ast \colon F_{i+2} \to G_{i+2}$ is bijective." My question is: should the indices be $i+1$ ...
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27 views

An identity between maps in a split exact sequence of complexes

I'm trying to understand where the identity $$ i^{n+1} \circ \pi ^{n+1} \circ d_B^{n} \circ s^n = d_B^n \circ s^n - s^{n+1} \circ d_C^n $$ comes from here. I've tried to make use of the commuting ...
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34 views

Viewing Koszul complex as an algebra

I keep coming across notes which says that the Koszul complex can be viewed as an algebra. Is it true that complexes can be viewed as an algebra. If the complex is not exact, can the homologies also ...
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45 views

Term models in group theory

Let $S_{Gr}$ be the language of groups, $Z$ an arbitrary set that does not contain elements of $\mathcal A_{S_{Gr}}$ (the corresponding alphabet). For each $z \in Z$ take a new constant symbol $c_z$ ...
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27 views

Projective dimension of an ideal generated by a regular sequence

Let $R$ be a commutative ring with $1$ and $I$ be an ideal of $R$ generated by an $R$-sequence of length $n$. I want a simple (if any) proof that the projective dimension of $I$ is $n-1$. I ...
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Mayer-Vietoris in reduced homology for a torus.

By using the Mayer-Vietoris sequence in reduced homology : I have to calculate the homology groups of : The torus $\mathbb{T}^2 :=[0;1]^2 /\mathcal{T}$ by using the following decomposition $X_1 := ...
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85 views

Tensor Product of Complexes and the definition of the differentials

Suppose we have the following complexes, $$0 \rightarrow R \xrightarrow{x_1} R \rightarrow 0$$ $$0 \rightarrow R \xrightarrow{x_2} R \rightarrow 0$$ $$0 \rightarrow R \xrightarrow{x_3} R \rightarrow ...
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39 views

Show that an $R$-module homomorphism $\alpha:A \to B$ is injective.

I am working on an exercise on injective modules: Show that an $R$-module homomorphism $\alpha:A \to B$ is injective if the induced map Hom$_R(B,Q)\to$ Hom$_R(A,Q)$ is surjective for all injective ...
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40 views

On the origins of Homological algebra

In Martin Krieger's book "Doing Mathematics: Convention Subject, Calculation, Analogy" (2003) I find the following statement (apparently, a quote from somone else) : "Homological algebra starts from ...
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57 views

Relation of $\operatorname{Ext}$ and projective dimension

I have some problem to understand the proof of proposition 8.38, page 473 from An Introduction to Homological Algebra by Rotman. Proposition: Let $x\in Z(R)$ be an element which is not a zero-divisor ...
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24 views

Prove that if $A$ is $R$-projective and $C$ is $S$-injective then $\operatorname{Hom}_R(A,C)$ is $S$-injective

In the situation $(_RA,_RC_S)$, prove that if $A$ is $R$-projective and $C$ is $S$-injective then $\operatorname{Hom}_R(A,C)$ is $S$-injective. I appreciate your help.
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30 views

Tensor product of homology equivalences

Let $f : C \to C'$ and $g : D \to D'$ be chain maps of non-negative chain complexes of $R$-modules, where $R$ is any commutative ring. Assume that $f$ and $g$ are homology equivalences. Is the same ...
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37 views

Computation of Ext as a cohomologies of certain complex

Let $R$ be a ring and $K^\bullet$ be a complex of $R$-modules such that $K^\bullet$ has only one nontrivial cohomology $H^0(K^\bullet)=M$. Suppose that $R$-module $N$ is such that ...
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47 views

Multiplicative spectral sequence

I have a simple question regarding the definition of a multiplicative spectral sequence, which I couldn't answer myself by looking at the definitions in various texts: Is the product assumed to be ...
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58 views

Injective module and the homology of a complex

If $K$ is a complex of $R$-modules and $J$ is an injective $R$-module, prove that \begin{equation*} H^n(\operatorname{Hom}_R(K,J))\cong \operatorname{Hom}_R(H_n(K),J). \end{equation*} Thank you ...
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An introduction to algebraic topology from the categorical point of view

I'm looking for a modern algebraic topology textbook from a categorical point of view. Basically, I'd like a textbook that uses the language of functors, natural transformations, adjunctions, etc. ...
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25 views

How to show the homotopy category is not abelian [duplicate]

Suppose $K^+(M)$ is the category, whose objects are bounded below complex of abelian groups, morphisms are chain maps modulo homotopy equivalence. How to show the category is not abelian? Exercise ...
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53 views

Split exact sequences: a basic question.

I am a bit confused regarding the definition of a split exact sequence, whose definition is for example available here (http://ncatlab.org/nlab/show/split+exact+sequence). Let's work in an abelian ...
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22 views

Confusion about cohomology [duplicate]

Cohomology is a contravariant functor. It's easy to see that with singular cohomology, because if we have a map between cell complexes, we take $Hom(-,\mathbb R)$ (which is contrvariant) on the chain ...
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68 views

Correspondence between Ext group and extensions (from Weibel's book)

I am trying to understand the proof of Theorem 3.4.3 from Weibel's book Introduction to homological algebra. The statement is the following. Let $R$ be a ring. Given $R$-modules $A$ and $B$, an ...
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18 views

First cohomology of direct product (in the coefficients)

Let $k$ be a field and let $G = A \times B$ be the product of two algebraic groups over $k$ ($G$ is not necessarily finite nor abelian). Is there a nice way to express $H^1(Gal(k^s/k), G(k^s))$ in ...
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Connecting homomorphism of exact sequence of Ext groups

Let $0\to M\to L\to N\to0$ be an exact sequence of modules over a ring $A$. Having an $A$-module $K$ we obtain the exact sequence of Ext groups $$0\to Hom_A(N,K)\to Hom_A(L,K)\to ...
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40 views

Injective dimension $n$ implies $Ext^n$ does not vanish with an injective

Let $M$ be a finitely generated module and suppose that the injective dimension of $M$ is $n$. I want to show that there exists an injective module $I$ such that $Ext^n(I,M)\neq 0$ (and if the ...
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Derived functors - homotopical vs homological approach

In a first course in homological algebra, the lecturer introduced derived functors as universal $\delta$-functors, whose universal property is splicing short exact sequences into long ones. It so ...
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51 views

An short exact sequence of $\mathfrak{g}$ of which head and tail are in category $\mathcal{O}$.

Let $\mathfrak{g}$ be a finite-dimensional, semisimple Lie algebra over $\mathbb{C}$. Let $$ 0\rightarrow X \rightarrow Y \rightarrow Z \rightarrow 0 $$ be a short exact sequence of ...
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Whether a functor is exact?

I am stuck with exercise $1$ of section $3$ of chapter $1$ in the book Cohomology of number fields by Neukirch. The exercise is to show that the functor from $A \rightarrow C^n(G,A)$ is exact, where ...
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If $\mathfrak{m}\otimes M\rightarrow A\otimes M$ is injective, what else has to be injective?

Let $A$ be a local (not necessarily noetherian) ring with maximal ideal $\mathfrak{m}$ and residue field $k$. Let $M$ be a finitely generated $A$-module such that $\mathfrak{m}\otimes_A M\rightarrow ...
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1answer
26 views

Short exact sequence of abelian groups implies long exact sequnce of cohomologies

I am trying to compute cohomologies $H^i(\mathbb{Z}/n\mathbb{Z}\times\mathbb{Z}/m\mathbb{Z}, \mathbb{Z})$. Actually it is not a big deal, because I have already computed $H^i(\mathbb{Z}/n\mathbb{Z}, ...
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86 views

The homotopy category of complexes

I have some trouble in proving Exercise A3.51 of Eisenbud's book "Commutative Algebra with a view toward Algebraic Geometry", pag. 688. The solution is sketched at pag. 754 at the end of the book. The ...
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Question about Yoneda product

Let $A$ be a ring and $M,N,K$ are modules over $A$. Let $\xi\in\text{Ext}_A^1(N,M)$ and $\eta\in\text{Ext}_A^1(K,N)$ are given by $$\xi:\,\,\,0\to M\to X\to N\to0,$$ $$\eta:\,\,\,0\to N\to Y\to ...
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Cohomologies of Galois group of field extension

Let $k\subset K$ be a finite Galois extension with Galois group $G=\text{Aut}_k\,K$. How to prove that $H_i(G,K)=H^i(G,K)=0$ for all $i>0$?
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24 views

Naturality of connecting homomorphisms

Let $\mathcal{F}$ be a right-exact additive functor on the category of R-modules (R a fixed ring). Proposition A3.17(d.) in Eisenbud's Commutative algebra with a view towards algebraic geometry states ...
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Induced homology morphism of invertible linear transformation

I'm doing some excercises from Hatcher. I'm dealing with excercise 7 in section 2.2 (page 164 in PDF file): For an invertible linear transformation $f : \mathbb{R}^{n} \to \mathbb{R}^{n}$ show that ...
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33 views

exact sequence in directed limit

I want to show that proposition$5.33$ in introduction to homological algebra rotman :let $I$ be a directed set , and let $\{A_i,\alpha_j^i\}$, $\{B_i,\beta_j^i\}$, and $\{C_i,\gamma_j^i\}$ be directed ...
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Comments on Eilenberg and Steenrod's “Foundations of algebraic topology” and other similar books for recomendation

The biggest obstacle for me to learn geometry and topology is the haziness of textbooks. I took algebraic topology last semester and the textbook we used in class was Rotman's "An introduction to ...
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The relation between homotopy equivalence and contractible mapping cone?

In this MO thread, the OP claimed that it is obvious that homotopy equivalence implies the mapping cone contractible, whereas the converse proposition is wrong. I hate to admit that it's not obvious ...
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What is higher kernel explicitly?

Let $\mathcal{A}$ be an abelian category (for simplicity you can think that $\mathcal{A}$ is the category of modules over ring $R$). Let $[1]$ be the category with two objects and one arrow between ...
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28 views

Direct limit of modules: a property.

Suppose $A$ to be a ring and $M_i$ the indexed $A$-modules used to build the direct limit of modules $M \doteq \lim{M_i}$. Let $f_{ij}: M_i \to M_j$ the transition maps and $\phi : M_i \to M$ the ...
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58 views

Definition of (left) resolution

Let $\mathsf C$ be an abelian category. A (left) resolution of an object $A$ is a nonnegative chain complex $$\cdots \rightarrow P_2\rightarrow P_1\overset{\partial_1}\rightarrow P_0\rightarrow ...
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“Stable model categories are categories of modules” - Clarification about a few things

I was reading Schwede and Shipley's "Stable model categories are categories of modules", I needed clarification about a few things: 1 - When they say that stable model categories are categories of ...
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36 views

Question about computing cohomology of trivial action on $\mathbb{Z}_{4}$

I'm currently considering the trivial action of the group $G = \mathbb{Z}_{2}$ on the group $A = \mathbb{Z}/4\mathbb{Z}$. It is easy to show that $|C^{2}(G,A)|$ = $2^{8}$ and that $|B^{2}(G,A)| \leq ...
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Hochschild homology of a free commutative algebra

Let $V$ be a graded vector space over $k$. Let $Com(V)$ be the free commutative algebra over V. Let $HH_*(-,k)$ be the Hochschild homology with coefficients in $k$ functor. My questions are : $$ ...
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A question about the definition of tensor product

Let $M$ and $N$ be modules over a ring $R$. Generally, the tensor product $M\otimes N$ is defined to be an abelian group with a balanced map $j:M\times N\to M\otimes N$ such that for any abelian group ...
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every projective module has a free complement.

I have to prove that every projective module has a free complement. Now Rotman ask it to first do for $R=Z/6Z$ and $P=Z/2Z$, We know $Z/6Z \cong Z/2Z \oplus Z/3Z$. Now $Z/2Z$ is projective as it a ...
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Calculating Hodge numbers by means of locally free resolutions

In this paper the author considers a smooth $3$-fold $X$ in $\Bbb{CP}^6$ with the following locally free resolutions of its structure sheaf and squared ideal sheaf: $$0\to \mathcal O_\Bbb {P^6}(-7) ...
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92 views

example of inverse limit and direct limit

Does a direct limit of projective need to be projective? And is the inverse limit of injectives injective? I guess they need not, but I can't find an example. Can you help please?
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40 views

Prove that $I_k \otimes_k \Omega \rightarrow I$ is injective

Let $\Omega$ be an algebraically closed field, $k$ a subfield of $\Omega$, $I$ an ideal of $\Omega[X_1, ... , X_n]$, and $I_k = I \cap k[X_1, ... , X_n]$. Then $I_k$ is an ideal of $k[X_1, ... , ...
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Let $R$ be a domain. Then $\operatorname{Tor}_n^R(A,B)$ is a torsion module

I have some problem to understanding the proof of this problem. This theorem is on page $414$ introduction to homological algebra Rotman. The theorem says: If $R$ is a domain, then ...
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67 views

How to show fraction field is flat (without localization)

Here I asked that if one can prove the field of fraction of a domain is flat. The answers used localization, which I am not familiar with. Can anyone prove it without using localization?
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Why field of fractions is flat?

I want to show this lemma: Let $R$ be a domain. If $A$ is a torsion $R$-module, then $\operatorname{Tor}_1^R (K,A)\cong A$ where $\operatorname{Frac}(R)=Q$ and $K=Q/R$. When I was reading ...