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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.

5114 questions
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Exercise from Assem-Simson-Skowronski

I'm having trouble with this exercise from Elements of the Representation Theory of Associative Algebras I: Techniques of Representation Theory. The exercise in question is from chapter IV. So, let $k$ be an algebraically closed field and let $A$ be…
Samantha Y
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Is complex exact if its Euler characteristic is zero?

For a bounded complex $M$ of finite-dimensional $k$-vector spaces we define its Euler characteristic as $$ \chi=\sum_{n\in \mathbb{Z}} (-1)^n\dim(M_n) $$ In particular, if complex is exact then its Euler characteristic is zero. Is the converse true?
Jimmy R
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Free DG modules

Let $A$ be a DG algebra and $f : F \to M$ a morphism of DG $A$-modules such that $F$ is free and the induced map $H^{\bullet}F \to H^{\bullet}M$ vanishes. Does it follow that $f$ is nullhomotopic? My first thought: since $f$ maps $Z^{\bullet}F$ into…
Justin Campbell
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Essential extension.

I'm trying to solve this question. My TA told me that it was easy and the information/assumption given is useless. Question We have the following inclusions of $R$-modules $M\subseteq N \subseteq L$ Prove that if N and L are both essential…
postguest12
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Acyclic resolutions

Hallo, I have to worry you one more time with these acyclicity problems, but as I am currently working on derived functors in a.g., I really need to understand derived functors in a very general form. So my question is just: I have an exact functor…
Descartes
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Acyclic Objects and cohomologically finite functors

let's start with a left exact functor $F: A\longrightarrow B$ of abelian categories, where the derived functor $RF: D^{+}(A)\longrightarrow D^{+}(B)$ exists. Furthermore the class of F-acyclic objects be adapted to F and furthermore F of finite…
Descartes
  • 650
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Inverse of Natural Projection?

May this be a silly question, but can I construct an inverse of a natural projection $p$ from a module $M$ to its quotient module $M/A$? Of course more than one element can be assigned for each coset in $M/A$, but if we limit the inverse's range to…
generic properties
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Showing that the mapping cone is a chain complex

Let $\alpha \colon \mathcal{A} \to \mathcal{D}$ be a morphism of chain complexes. Let $$ d^{C(\alpha)}_n = \begin{bmatrix} -d^{\mathcal{A}}_{n-1} & 0 \\ \phantom{-}\alpha_{n-1} & d^{\mathcal{D}}_{n-1} \end{bmatrix} $$ …
user58289
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Morphisms of complexes send $n$-cycles (resp. $n$-boundaries) to $n$-cycles (resp. $n$-boundaries)

Let us denote $$ \cdots \xrightarrow{\enspace d^A_n \enspace} A_n \xrightarrow{\enspace d^A_{n+1} \enspace} A_{n+1} \xrightarrow{\enspace d^A_{n+2} \enspace} \cdots $$ by $A_\bullet$ , and $$ \cdots \xrightarrow{\enspace d^B_n…
user58289
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Morphisms as Objects, Comm. Diagrams etc.

Let $C$ be a category. Denote by $Ar(C)$ the following category: an object in $Ar(C)$ is a morphism $X_1 \rightarrow X_2$ in $C$. A morphism in $Ar(C)$ from $X_1 \rightarrow X_2$ to $Y_1 \rightarrow Y_2$ is a commutative diagram. 1) Show that…
Erik Vesterlund
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$SF_1 (f_1 + f_2) = SF_1(f_1) + SF_1(f_2)$

Let F be a right exact functor. Prove $SF_1 (f_1 + f_2) = SF_1(f_1) + SF_1(f_2)$ where $f_1, f_2 :M \rightarrow M'$. I have just started reading homological algebra so please help me to solve this one.
Germain
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Exactness of $\varprojlim$ on short exact sequence of Mittag-Leffler systems

Let $I$ be a directed set. A prjective system $\{V_i, f_{ij}\}$ of abelian groups is called Mittag-Leffler if for each $i\in I$, the family $f_{ij}(V_j)\subseteq V_i$ for j≥i stabilizes. Let $0\to \{U_i\}\to \{V_i\}\to \{W_i\}\to 0$ be an exact…
G.-S. Zhou
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Error in Rotman's AIHA book

There is the statement on page $652$ in Chapter $10.5$ on the subject of Cartan Eilenberg Resolution: "($C$ is a chain complex) given an object $A'$, let $Q^n (A')$ be the complex with $A'$ concentrated in degree $n$; given a morphism $f:C_n…
fufufufuf
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Proof of 1.18.18 of Linckelmann's "Block Theory of Finite Group Algebras"

I'm trying to understand this one step in the proof of Theorem 1.18.18 (i), the statement of the theorem is as follows: Let $\mathscr{C}$ be an abelian category and let $$0 \longrightarrow X \overset{f}{\longrightarrow} Y…
Naweed G. Seldon
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Does $\operatorname{Ext}_{\Lambda}^i (D\Lambda,X)=0$ for any $i\ne 0$ imply that $X$ is injective?

Let $K$ be a field, let $\Lambda$ be a finite-dimensional $K$-algebra with global dimension $\le n$, let $D=\operatorname{Hom}_K(-,K)$. Assume that $X\in \bmod \Lambda$ satisfies $\operatorname{Ext}_{\Lambda}^i (D\Lambda,X)=0$ for any $i\ne 0$, is…
Ryze
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