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.
5,502 questions
0 votes
1 answer
58 views
Additive Right Adjoint functor F preserves left exact sequence? Question about the proof
$f:A\to B$ and $g: B\to C$ are two morphisms of R-Mod. $0\to A \to B\to C\to 0$ is exact and F is an additive right adjoint functor. I want to show $0\to F(A) \to F(B)\to F(C)\to 0$. I know right ...
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2 votes
2 answers
65 views
Why is $W_2(\hat{A})$ derived $I$-complete?
Lets say we have a $\delta$-ring $A$ and a finitely generated ideal $I \leqslant A$ containing the prime $p$ that appears in the definition of $A$ being a $\delta$-ring. I want to understand why the ...
- 169
1 vote
1 answer
92 views
$f: \mathbb{RP}^n \to \mathbb{RP}^m$ for $n > m$ induces trivial map on reduced cohomology
On a homework problem (Q8(i) from here) I am asked to show: Show that any map $f: \mathbb{RP}^n \to \mathbb{RP}^m$ induces a trivial map on reduced cohomology if $n > m$. Here's my attempt: ...
- 614
0 votes
1 answer
61 views
Grothendieck spectral sequence for right exact functors?
Hey Stack exchange community! I know that for two left exact functors $F: \mathcal{A} \to \mathcal{B}$ and $G: \mathcal{B} \to \mathcal{C}$ where $\mathcal{A}, \mathcal{B}$ and $\mathcal{C}$ are ...
- 169
4 votes
1 answer
240 views
Kernel zero but not monic?
Is there a category $\mathcal{C}$ with zero object and a morphism $f:A\to B$ such that $0:A\to A$ is a kernel but $f$ is not monic? If $\mathcal{C}$ is preadditive then $\ker f = 0 \iff f$ is monic. ...
- 7,076
4 votes
0 answers
166 views
Relationship between étale cohomology and group cohomology of the étale fundamental group
I am wondering to what extent and under which conditions there exists a generalization of the well-known relationship between the cohomology of the separable Galois group of a field and the étale ...
- 1,750
2 votes
0 answers
43 views
Splitting of the Chiral de Rham differential for affine space
I am currently reading through Malikov, Schechtman and Vaintrob's paper Chiral de Rham Complex. In the proof of Theorem 2.4, i.e. that the chiral de Rham complex extends the usual de Rham complex for ...
2 votes
1 answer
74 views
How are the Hyper-EXT modules defined?
In Weibel's Homological Algebra, the hyper-ext module $\text{Ext}(M, N^\bullet)$, where $M$ is an $R$ module and $N^\bullet$ is a co-complex of $R$-modules, is defined in two ways. Take any injective ...
1 vote
0 answers
69 views
Confusion regarding quivers with potential and cluster tilted algebras
I was looking at P. Plamondon's paper titled "GENERIC BASES FOR CLUSTER ALGEBRAS FROM THE CLUSTER CATEGORY". I'm confused about the calculation in Example 4.3. The example starts with a ...
- 857
3 votes
1 answer
77 views
Why can’t the universal covering map $\mathbb{R} \to S^1$ be made into an étale sheaf of abelian groups?
In Rotman’s An Introduction to Homological Algebra he defines an étale sheaf (of abelian groups) on p. 276 as follows: Definition. If $p: E \rightarrow X$ is continuous, where $X$ and $E$ are ...
- 591
2 votes
0 answers
31 views
Are flat modules acyclic w.r.t. the completion functor?
Let $(R,\mathfrak{m})$ be a noetherian ring, and $\hat{R} = \varprojlim_i R/\mathfrak{m}^iR $ its $\mathfrak{m}$-adic completion. We may define the completion functor $\Lambda\colon \text{$R$-Mod} \to ...
- 2,498
2 votes
1 answer
81 views
How can we show this "Small Wall-Kervaire Braid Lemma"?
Consider the following diagram of abelian groups: Assume that the $\color{blue}{\text{blue braid}}$ and the $\color{green}{\text{green braid}}$ are long exact sequences and that the $\color{red}{\...
- 4,970
0 votes
0 answers
33 views
Is there a groupoid structure on generalized elements in an abelian category?
Let $C$ be an abelian category and fix $A\in \operatorname{Obj}(A)$. Then an element of $A$ is an equivalence class of maps $[x:X\rightarrow A]$ under the equivalence relation $(x:X\rightarrow A)\sim (...
- 855
1 vote
0 answers
167 views
Literature on $\mathrm{Ext}_R(-,-)$ as simultaneous $\delta$-functor in each variable
This is a followup question to Balancing Ext as a δ-functor. In the original post, I realized that $\mathrm{Ext}^n_R$ is an additive bifunctor with $\delta$-functions for exact sequences in each ...
- 7,713
2 votes
0 answers
97 views
Is the natural isomorphism $\mathrm{Hom}_R(\bigoplus A_i, \prod B_j) \cong \prod_{i,j} \mathrm{Hom}_R(A_i, B_j)$ an isomorphism of modules?
It is a fundamental result in module theory that for any families of $R$-modules $\{A_i\}_{i \in I}$ and $\{B_j\}_{j \in J}$, there is a natural isomorphism of abelian groups: $$ \mathrm{Hom}_{R}\left(...
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