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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An exact sequence of unit groups

In the answer of K. Conrad to this question, he mentions a "nice 4-term short exact sequence of abelian groups (involving units groups mod a, mod b, and mod ab)" proving the product formula for ...
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55 views

When does a short exact sequence of representations exist?

The context for this question is that I am trying to determine the Grothendieck group of finite-dimensional complex representations of $T = (\mathbb{C}^*)^n$, where $\mathbb{C}^*$ denotes the ...
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51 views

Action of the functor Ext$_1(-,-)$ on extensions

Suppose we have an exact sequence of $R$-modules \begin{array}{ccccccccc} 0 & \longrightarrow & L & \overset{f}{\longrightarrow} & M & \overset{g}{\longrightarrow} & E & ...
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If $\text{Ext}^1(\mathbb{Q}/ \mathbb{Z}, D ) = 0$ then $D$ is divisible

This is Exercise 7.15(ii) from Rotman's book, Introduction to homological algebra that I'm doing. If $D$ is an abelian group and $\text{Ext}^1(\mathbb{Q}/ \mathbb{Z}, D ) = 0$, prove that $D$ is ...
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44 views

If R and S are artinian and finite dimensional algebras respectively, then the tensor product of them is artinian.

Let $R$ be an artinian algebra and $S$ be a finite dimensional algebra over the field $k$. How can i show that $R\otimes_kS$ is artinian? I know that $S$ is also artinian since it is finite ...
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A Question About Notation (Homology with Local Coefficients)

I am currently reading A J Berrick’s An Approach to Algebraic K-Theory, and I am stuck at one of the propositions there because he does not define homology with local coefficients. Proposition: ...
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$mA = 0 = nC, \ \gcd(m,n) = 1 \Rightarrow $ every extension of $A$ by $C$ splits

This is Exercise 7.14(ii) from Rotman, Introduction to homological algebra, and I'm stuck on it. If $A$ and $C$ are abelian groups, with $mA = 0 = nC $ and $\gcd(m,n) = 1$ then every extension of ...
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25 views

Characterization of faithfully flat modules

This is an exercise from Rotman, introduction to homological algebra. A right $R$-module $B$ is called faithfully flat if : 1) $B$ is flat 2) If $X$ is a left $R$-module and $B \otimes_R X =0 $ ...
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47 views

Dual of Schanuel lemma

This is an exercise from Rotman, Introduction to homological algebra. Given exact sequences of $R$-modules \begin{array}{ccccccccc} 0 & \longrightarrow & M & \overset{i}{\longrightarrow} ...
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2answers
123 views

Examples of Noetherian local rings which are not Gorenstein

Can anyone give me an example of a Noetherian local ring which is not a Gorenstein ring?
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Injective dimension of $\mathbb Z_n$ as a $\mathbb Z$-module

What is the injective dimension of $\mathbb Z_n$ as a $\mathbb Z$-module? Can one use the well-known fact that $id(M)$ is less than or equal to $i$ iff $Ext^{i+1}(N,M)=0$ for all $N$? Thanks in ...
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42 views

Direct limits commute with $\mathrm{Tor}$ functor

How one could prove that direct limits commute with the functor $\mathrm{Tor}$? Of course, I know that $\mathrm{Tor}$ with its first $0$ index is the same as tensor product which does commute ...
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4answers
73 views

Proving that P/PJ is a projective right module over R/J

If P is a projective right module over a ring R and J is a two sided ideal of R. Prove that P/PJ is a projective right module over R/J . My idea was trying to proof that " $M$ is an $R$-module ...
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1answer
58 views

$\hom_{\mathbb{Z}}(\mathbb{Q}, C) = 0$ for every cyclic group $C$

This is part of an exercise I'm doing, exercise 2.22 Rotman, Introduction to homological algebra. Prove that $$\hom_{\mathbb{Z}}(\mathbb{Q}, C) = 0$$ for every cyclic group $C$. Any hint ?
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36 views

If a direct sum has a projective cover, must the summands have projective covers?

In “Cover of a direct summand” it is asked to show that if a direct sum has a projective cover, and if one of the summands has a projective cover, then so does the other. I gave a solution that works ...
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87 views

Finite projective dimension may lead to projectiveness!

Assume a ring $R$ is injective as an $R$-module. If the projective dimension of an $R$-module $P$ is finite could one conclude that $P$ is a projective $R$-module? Probably one should start with ...
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35 views

Homology Theory based on “linear symmetric differences”

Homology Theory is based on the idea that if you have a sequence $A \rightarrow B \rightarrow C$ of homomorphisms $f \colon A \rightarrow B$ and $g \colon B \rightarrow C$ such that their composition ...
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85 views

Total dimension of the cohomology of a homogeneous space (or of a graded Tor)

I want to calculate the cohomology ring with rational coefficients of a homogeneous space, but would be happy enough to know its total dimension. Let $G$ be a compact Lie group, $T$ a maximal torus, ...
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45 views

Intuition of higher push-forward constant sheaves.

Let us consider the higher phsh-forward sheaves $R^if_*\mathbb{R}$ of a map $f:X\rightarrow Y$ between two compact manifolds. We assume that the fibers has a constant dimension, say $n$. I think ...
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13 views

The product of $E_2$-degenerate spectral sequences also $E_2$-degenerates?

Assume the Leray spectral sequence of a map $f_i:X_i\rightarrow B_i$ $E_2$-degenerates for $i=1,2$. Is it true that the Leray spectral sequence of the map $f_1\times f_2:X_1 \times X_2 \rightarrow B_1 ...
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induced sequence exact

If $D$ is a multiplicatively closed subset of $R$. I'm trying to come up with an example where $$0\to L \to M \to N \to 0$$ is not exact, but the induced sequence $$0 \to D^{-1}L \to D^{-1}M \to ...
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1answer
127 views

showing exact functors preserve exact sequences (abelian categories, additive functors, and kernels)

I'm working through Vakil's algebraic geometry text and I've been stuck on Exercise 1.6.E (page 52 on http://math.stanford.edu/~vakil/216blog/FOAGjun1113public.pdf.) Suppose that $F$ is an exact ...
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30 views

Comparing injective dimensions in a short exact sequence

If $0→A→B→C→0$ is an exact sequence in the category of $R$-modules ($R$ commutative having unity) with injective dimensions of $A$ and $C$ both $≤n$, is that of $B$ also $≤n$? It seems to me that ...
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59 views

In the Universal Coefficient Theorem, how does the cohomology generator relate to the homology generators?

Consider homology and cohomology of some space $X$ where the homology groups are finitely generated. Consider $tor(H^i(X))$, the torsion part of $H^i(X)$. How do the generators of $tor(H^i(X))$ ...
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19 views

How does one make cochain complex of sphere into an associative DGA?

Given the singular chain complex of the sphere $S^n$, $S^*(S^n)$, a reference says that one can use the Alexander Whitney product to make $S^*(S^n)$ into an associative differential graded algebra. ...
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1answer
68 views

if $R$ is a noetherian local ring, then every 2-generated ideal has finite projective dimension iff $R$ is a UFD

This question is about m zcn's comment on my question Projective dimension of all principal ideals is finite. Is R an integral domain?. It's a good point. so i ask it for use of everybody: if ...
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32 views

Why is $S_{\ast}\left(X,A\right)$ free? [duplicate]

Why is $S_{\ast}\left(X,A\right)$ free? it is the quotient of two free groups $S_{\ast}\left(X\right)$ & $S_{\ast}\left(A\right)$
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117 views

When does the tensor product consist of elementary tensors only?

The question is: Assume that $R$ is a (commutative) ring. Under what conditions on $R$-modules $M,N$ does the tensor product $M\otimes_RN$ consist of elementary tensors only? That is, every ...
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93 views

Localization of Gorenstein ring

Let $R$ be a Gorenstein local ring and $S=R \setminus Z(R)$. I want to prove $S^{-1}R =⊕_{ht\ p=0} R_p$ and $S^{-1}R$ is injective $R$-module. I can see the above $p$'s are minimal, $id_{R_p} R_p=0$ ...
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Topology on an Ext group

One can show that the group $\text{Ext}^1(\mathbf Q, \mathbf Z)$ (calculated in $Ab$) identifies naturally with $\mathbf A_f/\mathbf Q$, where $\mathbf A_f$ is the additive group of finite adèles. ...
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Exercise 2.2.1 in Charles A. Weibel's book An Introduction To Homological Algebra

How to prove that a chain complex is a projective object in $ {Ch} $ (chain complexes of $R$-modules) iff it is a split exact complex of projectives? A chain complex of projectives means a chain ...
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Left and Right minimal homomorphisms.

In the literature on representation theory of finite dimensional algebras, a left (and similarly right) minimal homomorphism is defined as the following: For a pair of modules $L $ and $M$ in ...
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How to prove this equivalent condition to that B is an injective R-module?

B is an injective R-module iff $ Ext^{1}_{R}(A,B) $ vanishes for all A. I know how to prove this statement from left to right, but don't know the opposite direction. Please help.
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Are all cyclic groups $\mathbb{Z}/n\mathbb{Z}$ cotorsion?

Is it correct that all cyclic groups $\mathbb{Z}/n\mathbb{Z}$ are cotorsion, i.e., $Ext^1(\mathbb{Q},\mathbb{Z}/n\mathbb{Z})=0$? I believe yes, since ...
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Is $\operatorname{Hom}_\mathbb{Z}(\mathbb{Q},\mathbb{Q}/\mathbb{Z})\cong\bigoplus_p\mathbb{Q}_p$?

Is $\operatorname{Hom}_\mathbb{Z}(\mathbb{Q},\mathbb{Q}/\mathbb{Z})\cong\bigoplus_p\mathbb{Q}_p$? Or maybe $\prod_p\mathbb{Q}_p$? I know $\mathbb{Q}/\mathbb{Z}\cong\bigoplus_p \mathbb{Z}_{p^\infty}$, ...
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Flatness of $\mathbb{Z}$-modules

I have to prove that : 1) For every positive integer $n$, $\mathbb{Z}_{n}$ is not a flat $\mathbb{Z}$-module 2) Every torsion free abelian group is a flat $\mathbb{Z}$-module What I have done: 1) ...
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What is $\operatorname{Hom}_\mathbb{Z}(\mathbb{Q}/\mathbb{Z},\mathbb{Q}/\mathbb{Z})$?

Is $\operatorname{Hom}_\mathbb{Z}(\mathbb{Q}/\mathbb{Z},\mathbb{Q}/\mathbb{Z})$ isomorphic to any "known" group? I suppose what I mean is, is it isomorphic to a group that isn't a Hom group? If such ...
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RHom and Koszul complexes.

Let $A$ be a Koszul algebra over a field $k$ that is both left and right finite. One can consider the its Koszul complex $X$ as a dg $ A^!$-$A$ bimodule. I want to show: For all $ i \in \mathbb{Z}$, ...
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2answers
92 views

The Relationship Between Cohomological Dimension and Support

Let $ R $ be a commutative unital ring, $ I $ an ideal of $ R $, and $ M $ an $ R $-module. The cohomological dimension of $ M $ with respect to $ I $ is defined as $$ \operatorname{cd}(I,M) ...
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28 views

Extending morphism between derived functors

Let N,M be objects in a left exact functor $F:A\rightarrow B$ between abelian categoire's source, most importantly say there is an isomorpihsm $\psi: F(M)\rightarrow R^dF(N)$ is it possible to extend ...
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42 views

Inequality amongst projective dimensions!?

Assume $φ : R\to S$ is a ring homomorphism between commutative rings sending unity to unity. Taking any $S$-module $M$ as an $R$-module, is it true that always $$\operatorname{pd}_R (M)\le ...
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Exercise 2.27 Atiyah-Macdonald, absolute flatness

A commutative ring $R$ is absolutely flat if every $R$-module is flat. Prove that the following are equivalent: 1) $R$ is absolutely flat 2) Every principal ideal of $R$ is idempotent 3) Every ...
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Limit of inverse system where morphisms are non-zero products

I have a rather complicated inverse limit which I am trying to compute. I will try to distil the question to its barest form, but if more details are necessary I can supply them. Let $R$ be an ...
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1answer
57 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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Truncation of inverse systems

I'm trying to ascertain when (if ever) it is acceptable to ``truncate'' terms of an inverse system and arrive at an isomorphic limit. To simplify matters, assume that our directed system is $\mathbb ...
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1answer
86 views

Exercise 2.26 Atiyah-Macdonald, flatness

I'm stuck on this exercise. $A$ is a commutative ring with unit. $N$ is an $A$-module. Then $N$ is flat $\Longleftrightarrow $ $\text{Tor}_{1}(A/a, N ) = 0 $ for every finitely generated ideal $a$ of ...
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Has this variation of Hochschild cohomology been studied?

Let $k$ be a field, and let $A$ be a commutative $k$-algebra. Let $M$ be an abeliean group, and assume that it an $n$-$A$-module. That is: it has $n$ different $A$-module structures, and they are ...
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53 views

Exercise from Atiyah about flatness

This is an exercise from Atiyah. Let $N$ be a flat $B$-module, and $B$ a flat $A$-algebra where $A$ is a commutative ring with unit. Then $N$ is flat as $A$-module Any hint ?
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1answer
68 views

Derived functor vs. spectral sequence

I heard many times that because of introducing derived category, we can avoid cumbersome spectral sequence. However, I don't quite understand its meaning. Here is a precise example people talking ...
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Cohomology-Homology bilinear form of Seifert surfaces

Let $C_\ast$ be any chain complex of $R$-modules. Then for any $k\in\mathbb{Z}$ we obtain a $R$-bilinear map $$\langle-,-\rangle:H^k\!C_\ast\times H_kC_\ast\longrightarrow R, ...