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Results tagged with homological-algebra
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user 102343
(Co)chain complexes, abelian Categories, (pre)sheaves, (co)homology in various (possibly highly generalized) settings, spectra, derived functors, resolutions, spectral sequences, homotopy categories. Chain complexes in an abelian category form the heart of homological algebra.
14
votes
How to prove that topological Hochschild homology of a smooth proper stable k-linear infinit...
$\newcommand{\THH}{\mathrm{THH}}
\newcommand{\Cat}{\mathrm{Cat}}
\newcommand{\perf}{\mathrm{perf}}
\newcommand{\Sp}{\mathrm{Sp}}
\newcommand{\Mod}{\mathrm{Mod}}$
If you ask about dualizability in $\TH …
- 15.7k
12
votes
What are abelian categories enriched over themselves?
To make sense of enrichment over a category $V$, you want $V$ to have a monoidal structure. Indeed, you want to be able to compose morphisms so you need a way to go from "something in $\hom(a,b)$ and …
- 15.7k
12
votes
Accepted
When does derived tensor product commute with arbitrary products?
This is true with no noetherian hypotheses.
Indeed, this condition implies that $X\otimes^L_R -$ commutes with all homotopy limits: it commutes with finite homotopy limits always (it commutes with fin …
- 15.7k
11
votes
How many tensor products of chain complexes are there?
tldr: (but also, spoilers !) If you impose the further condition that chain complexes be a $Gr$-algebra, then there is a unique monoidal structure - which one of your two options it is will depend on …
- 15.7k
9
votes
Accepted
Is the class of isomorphism types of a module category always a set?
Yes, because you're only considering finitely generated modules. No assumptions on $A$ are required.
Indeed, a finitely generated module is always a quotient of some $A^n$ by some submodule. For each …
- 15.7k
8
votes
Does $\mathbf R\text{Hom}_R(k, -)\otimes_R^{\mathbf L} k$ commute with co-products?
$\newcommand{\RHom}{\mathbf{R}\mathrm{Hom}_R} \newcommand{\Lotimes}{\otimes^\mathbf{L}_R} \newcommand{\fib}{\mathrm{fib}}$
I claim that this happens if and only if the same is true for $\RHom(k,-)$, i …
- 15.7k
7
votes
Accepted
Computing Ext groups in a functor stable $\infty$-category
$\newcommand{\Z}{\mathbb Z} \newcommand{\Ch}{\mathrm{Ch}} \newcommand{\Fun}{\mathrm{Fun}}$
Let $\Ch(\Z)$ be the projective model category of chain complexes. It is well known that it presents $D_\inft …
- 15.7k
7
votes
Accepted
derived tensor product and finite projective dimension
Let me preface this by saying that I don't know a reference - so if that's what you're really looking for, someone else will have to answer.
Let $k:=R/m$ denote the residue field (I'm assuming "commut …
- 15.7k
7
votes
Accepted
Does $\mathrm{Ind}(\mathcal{C})$ have enough injectives, if $\mathcal{C}$ is an abelian cate...
As Dan Petersen said in the comments, $Mod(k)$ isn't small. Note that even in this case, $Ind(Mod(k))$ only has small direct sums, and in particular you cannot take the direct sum that appears in poin …
- 15.7k
6
votes
Can not tell colimits from limits
EDIT: The main body of my answer is there because I didn't understand the question, so let me answer it in this edit (I'll leave the main body of the answer there, just in case)
No, there is no typo …
- 15.7k
6
votes
Difference between $K(1)$-local K theory and l-adic completion of etale $K$ theory
Yes, there is a map: by Thomason's work, $L_{K(1)}K$ satisfies étale descent and the canonical map $K\to L_{K(1)}K$ therefore induces a canonical map $K^{ét}\to L_{K(1)}K$.
This paper by Clausen and M …
- 15.7k
6
votes
Accepted
Hattori-Stallings trace
The answer to (1) is yes. In fact this is true more generally: for any $f: M\to N, g:N\to M$, you have $tr(fg) = tr(gf)$.
The point is that $\hom_R(M,N)\otimes \hom_R(N,M)\cong \hom_R(M,R)\otimes_R N …
- 15.7k
5
votes
Accepted
Derived functor of functor tensor product
The answer is yes if you assume enough things. In particular, the notion of a left flat object of $\mathcal A$ comes up :
Definition: An object $L\in\mathcal A$ is left flat if $-\otimes L$ is exact.
…
- 15.7k
5
votes
Accepted
Equivalences of categories of complexes of modules
The answer is yes by the same type of Morita theory, namely $Z(Ch(R))\cong R$, where $Z(A)$ is $End(id_A)$, the ring of endomorphisms of an abelian category
EDIT : sorry, I hadn't seen that you were a …
- 15.7k
5
votes
Accepted
Adjunctions and inverse limits of derived categories
A reference for exactly this type of problem in general is a paper by Horev and Yanovski called "On conjugates and adjoint descent".
Given a diagram $C \to D_i$ of left adjoints $f_i$ with right adjoi …
- 15.7k