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Questions tagged [homological-algebra]

Homological algebra studies homology and cohomology groups in a general algebraic setting, that of chains of vector spaces or modules with composable maps which compose to zero. These groups furnish useful invariants of the original chains.

A chain complex is a sequence of abelian groups, vector spaces, or modules, with linear maps connecting them which compose to zero.

Homological algebra is the study of chain complexes and their homology groups.

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What's the correct sign regarding Hom complexes and shifts?

Let $M$ be a complex and $K$ be a module which is viewed as a complex concentrated in degree 0. I'm wondering what sign should a canonical map $\Phi\colon\text{Hom}(M, K)[a-b]\to \text{Hom}(M[b], K[a])$ impose on $f\in\text{Hom}(M^n, K)$. As shown…
Noto_Ootori
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Rotman Homological Algebra Theorem 3.69

Theorem Question My only question is: why is the diagrammatic condition equivalent to the Remark? I'm hoping to see a rigorous proof of this claim.
IsaacR24
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Equivalent condition to Essential Extension

Problem (Rotman) Definition Partial proof ($\Rightarrow$) Assume that $E$ is an essential extension and let $e \in E$ be nonzero. There exists an $\alpha : M \hookrightarrow E$ such that $re \in \langle e \rangle \cap \alpha(M)$ for some $r \in…
IsaacR24
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Rotman's Homological Algebra Problem 3.13(iii)

Problem Corollary 3.13 Partial Proof If $M'$ and $M$ are finitely presented (and therefore finitely generated) it's obvious that $M''$ is finitely presented. If $M$ is finitely presented, then $M'$ is isomorphic to a submodule of $M$, which is…
IsaacR24
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Rotman's Homological Algebra Problem 3.42 (without using direct limits)

This is Problem 3.42 from Rotman's Homological Algebra text. Claim Let $ 0 \to B' \to B \to B'' \to 0$ be an exact sequence of left $R$-modules. If the sequences remains exact after tensoring with all finitely presented right $R$-modules then the…
IsaacR24
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$Hom(B,R) \otimes C \cong Hom(B,C)$ if $B$ is finitely generated free

This is problem 3.34(ii) from Rotman's Homological Algebra text. Let $B$ and $C$ be a left $R$-modules. Definite $\nu : Hom(B,R) \otimes C \rightarrow Hom(B,C)$ by $f \otimes c \mapsto \hat{f}$ where $\hat{f}(b) = f(b)c$ for $b \in B$. Claim Prove…
IsaacR24
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An equivalent statement of split exactness

Let $0\to A\to B\to C\to 0$ be a short exact sequence of finite abelian groups and $Hom(-,A)$ functor induces an exact sequence $0\to Hom(C,A)\to Hom(B,A)\to Hom(A,A)$. How to prove the equivalence bewteen the split exactness of the original…
Visible Wings
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Ring with IBN and isomorphic free modules

This is problem 2.26(iii) in Rotman's Homological Algebra. 1. Problem 2.26(iii) (R a ring with IBN) 2. Partial Proof Let $W \subset X$ be finite and define $$W' = \{b \in B : \text{supp}(b) \subset W \} $$ By construction, if $b \in W'$ then $b \in…
IsaacR24
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Homology of union of circles sharing a point

Given $n$ circles $C_1, \dots, C_n$ glued by a common point $p$. Find the homology of the union $\cup_{i=1}^n C_i$. I am defining $c_0 = \langle p \rangle = \mathbb{Z} \cdot p$ and $c_1 = \langle l_1, \dots, l_n \rangle = \mathbb{Z} \cdot l_1 +…
Paolo Jove
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Research ideas on Homological Algebra

I am planning on doing some personal studies and some poster research for some future conferences in Homological Algebra, does there exist a current list of outstanding problems in the field of Homological Algebra?
Mathematician
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Frobenius Regularity and Cohen-Macauly

In the proof of Theorem 4.27 in the paper (Characteristic p Techniques in Commutative Algebra and Algebraic Geometry Math 732 - Winter 2019 Karen E Smith1), it is written that Without loss of generality, we can assume that (R,m,K) is a complete…
Kinan
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Prove that $(\mathbb{Z}/n\mathbb{Z})/m(\mathbb{Z}/n\mathbb{Z})=(\mathbb{Z}/n\mathbb{Z})/(\gcd(m,n)\mathbb{Z}/n\mathbb{Z})$

I assume it's very easy to explain. I want to prove that: $(\mathbb{Z}/n\mathbb{Z})/m(\mathbb{Z}/n\mathbb{Z})=(\mathbb{Z}/n\mathbb{Z})/(\gcd(m,n)\mathbb{Z}/n\mathbb{Z})$. Here $m,n$ are integers $>0$ and $\gcd(m,n)$ is the greatest common divisor of…
user
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How to compute connecting homomorphism?

Let $A_.$ be the complex $\dots\to\mathbb{Z}^2\to\mathbb{Z}^2\to\mathbb{Z}^2\to\dots$ with morphisms $d_{A,n}((a,b))=(b,0)$. Let $B_.$ be the complex $\dots\to\mathbb{Z}^3\to\mathbb{Z}^3\to\mathbb{Z}^3\to\dots$ with morphisms…
Shean
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Computing $\mathrm{Ext}_{\mathbb{Z}}(\mathbb{Z}, \mathbb{Z})$

I'm trying to find $\mathrm{Ext}_\mathbb{Z}^n(\mathbb{Z},\mathbb{Z})$, which involves computing the homologies of \begin{equation} 0\leftarrow \mathrm{Hom}_\mathbb{Z}(\mathbb{Q},\mathbb{Z})\leftarrow…
ChrisWong
  • 507
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Definition of quasi-isomorphic chain complexes

If we say that two chain complexes in an abelian category are quasi-isomorphic, does this mean that there exists a quasi-isomorphism between them (in some direction) or that there exists a zigzag of quasi-isomorphisms?
Simba
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