Tagged Questions

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Global dimension of endomorphism rings

Does anybody have an idea on how the global dimension of the endomorphism ring of a module over a (nice enough) ring is related to the global dimension of the endomorphism ring of its projective ...
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Extending monics in a commutative diagram

Given a commutative diagram in a Grothendieck category $\mathscr{A}$ \begin{array}{ccccccccc} 0 & \longrightarrow & A' & \overset{i}{\longrightarrow} & A & ...
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Different definitions of connectedness of commutative cochain algebras

Let $(M,d)$ be a commutative cochain algebra over the rationals, that is a differential graded, graded commutative Algebra over $\mathbb{Q}$ concentrated in nonnegative degrees. In the literature, ...
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Exercise in an abelian category

Supose we have an exact sequence $$A\overset{f}\longrightarrow B\overset{g}\rightarrow C\overset{h}\rightarrow D$$ in an abelian category $\mathcal{A}$. Is it true that $f$ is an epimorphism if and ...
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Showing that an epimorphism to a free module of finite rank splits

Let $M$ be an $R$-module and let $F$ be a free $R$-module of finite rank. Let $\phi : M \to F$ be an epimorphism. Then show that $M$ has a submodule $F' \cong F$ such that $M=F' \oplus \ker\phi$. ...
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Exact sequence induces exact sequences for free parts and torsion parts?

Let $A$ be a PID and consider the exact sequence of finitely generately modules over$A$: $$0\longrightarrow M' \overset{f}{\longrightarrow}M\overset{g}{\longrightarrow}M''\longrightarrow 0 \tag{1}.$$ ...
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Faithfully flat checkable on finitely generated modules

A left $R$-module $_RM$ is said to be faithfully flat if it is flat and, for any $N_R$, $N \otimes_R M = 0$ implies $N = 0$. I would like to show that $M$ is faithfully flat if it is flat and, for any ...
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Localization of an additive category which is no longer additive

Is there a nice example of an additive category $C$ and a family of morphisms $S\subset Mor(C)$ such that $C[S^{-1}]$ is no longer additive? I know that in general localization of categories ...
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Represent localization as a direct limit

Let $A$ be a commutative ring with identity, $S\subset A$ a multiplicatively closed subset and $1\in S$. Does the equation $$S^{-1}A=\varinjlim_{s\in S}A_s$$ make sense? Here $A_s$ is the ...
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Identifying some cyclic subgroup

Is there a fast way to argue that (for $a,b>1$ integers) the set of all $x\in\mathbf{Z}/b\mathbf{Z}$ with $ax=0$ is isomorphic to $\mathbf{Z}/{gcd(a,b)}\mathbf{Z}$? Maybe by counting the elements, ...
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Cartan and Eilenberg Homological Algebra

OK, I am looking at Cartan and Eilenberg Homological Algebra book (1956, 1973 printing). Chapter V.9, p97 they define functors T(-,-) of type L$\Sigma$ and R$\prod$. T is of type L$\Sigma$, if T(A,C) ...
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$\text{Ext}^1(-,B)$, independence of projective resolution.

If we want to compute the group $\text{Ext}^1(A,B)$ we take a projective resolution of $A$ $$\cdots\to P_2 \to P_1 \to P_0 \to A \to 0,$$ apply the contravariant functor $\text{Hom}(\cdot,B)$ to it ...
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Endomorphism rings of MCM Modules

Let $k$ be a field (algebraically closed of characteristic not equal to two, if you like) and let $R = k[[t^2, t^{2n+1}]]$. It is well known $R$ has finite type and the MCM (maximal Cohen-Macaulay) ...
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Definition for a bar resolution for a module over a dg category

Let $\mathcal{A}$ be a dg category and define a right $\mathcal{A}$ module to be a dg functor $M: \mathcal{A}^{op} \rightarrow dif\ k$ where $dif\ k$ is the category of differential $k$ modules ...
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Hochschild homology with trivial coefficients: how to make $K$ an $M_n(K)$-module

Let $R$ be a ring, $A$ an associative $R$-algebra, and $M$ an $A$-$A$-bimodule. Then the Hochschild homology of $A$ with coefficients in $M$, denoted $HH_\ast(A)$, is the homology of the chain complex ...
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Tensor product of modules preserve injectiveness and surjectiveness or not?

Let $R$ be a commutative ring with identity and $M$ an $R$-module. If $N_1\longrightarrow N_2$ is injective (resp. surjective), is the induced map $M\otimes_R N_1\longrightarrow M\otimes N_2$ ...
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Exactness of the Tensor Functor

This might turn out to be a very stupid question, so I apologize in advance, but it is confusing me a little bit. I know in general that if $$M'\rightarrow M\rightarrow M''\rightarrow 0$$ is an exact ...
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is the pullback of the cohomology of a group to the cohomology of a subgroup surjective?

If $H$ is a subgroup of $G$, is $i^*(H^*(G)$) surjective onto the cohomology of $H$? $i$ is the inclusion of $H$ in $G$.
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Ext of an $\mathfrak{m}$-primary ideal

Let $(A,\mathfrak m,k)$ be a Noetherian local ring, $M$ a finitely generated $A$-module, and $I$ an $\mathfrak{m}$-primary ideal. If $\operatorname{Ext}^{i}_{A}(A/\mathfrak{m},M)=0$ then ...
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Morita-invariance of Hochschild (co)homology.

Ok, I'm reading this paper by Christian Kassel on associative algebras and Hochschild (co)homology and on page 19 he says that Hochschild homology is Morita-invariant, by which he means that if $R$ ...
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Proof of the five lemma

How to do this using the snake lemma? this is an exercise in Lang's Algebra book. It should somehow be obvious, but I don't see it
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Group structure on module extensions

I'm looking for the proof of the fact that Baer sum give group structure on set of extensions of module $A$ by module $B$. The only proof I know (from Weibel's book) uses an isomorphism with ...
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Prove that a module is projective or not

Let $R=\left(\begin{array}{cc}\mathbb{Q}&\mathbb{Q}\\0&\mathbb{Q}\end{array}\right), J=\left(\begin{array}{cc}0&\mathbb{Q}\\0&0\end{array}\right)$. Prove that $R/J$ is not a ...
I have a small question. Why we have this designation: $n$-cycles for $Z_n$ and $n$-boundaries for $B_n$ ? Why they are called cycles and boundaries ? ...