Questions tagged [stable-category]

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3
votes
1answer
154 views

Invertible bimodules which are isomorphic in the stable module category

I'm in the following situation. I have a self-injective finite-dimensional basic algebra $\Lambda$ (hence Frobenius) over a perfect field and two finite-dimensional invertible $\Lambda$-bimodules $M$ ...
2
votes
0answers
151 views

stable (?) model category of simplicial monoids

If $\mathcal{C}$ is the category of commutative unitary monoids, one can endow the category of simplicial objects in $\mathcal{C}$, $s\mathcal{C}$, with the structure of a cofibrantly generated model ...
2
votes
1answer
235 views

Can tangent ($\infty$,1)-categories be described in terms of the higher Grothendieck construction?

Given a locally presentable ($\infty$,1)-category $C$, can the fibrewise stabilization of it's codomain fibration, also called its tangent category $TC$, be given in terms of the Grothendieck ...
1
vote
0answers
112 views

Some questions in a paper of derived categoires and stable equivalence

I am reading the paper "Derived categories and stable equivalence", the link is here:http://www.sciencedirect.com/science/article/pii/0022404989900819. At theorem 2.1, there is an equivalent functor $...
3
votes
1answer
218 views

Some places I don't know of the paper “On the stable module category of a self-injective algebra”

Recently I am reading the paper "On the stable module category of a self-injective algebra", the link is here: http://www.ams.org/journals/tran/2000-352-05/S0002-9947-00-02232-7/S0002-9947-00-02232-7....
4
votes
0answers
174 views

Short proof of the classification of representation-finite symmetric algebras up to stable equivalence

Assume $K$ is an algebraically closed field and $A$ a finite dimensional $K$-algebra. Assume additionally that $A$ is symmetric and representation-finite. Then one has the following classification of ...
9
votes
1answer
433 views

derived categories as presentable DG-categories

Let $A$ be a ring. Is it true that the DG category of unbounded complexes of $A$-modules, localized by quasi-isomorphisms, is cocomplete and compactly generated? What would be a reference for that and ...
6
votes
1answer
314 views

When was the word “stable” first used to describe stable homotopy theory?

The word "stable" has many uses in mathematics, but in the context of stable homotopy theory, one might take it to mean one of two things: Homotopy groups stabilize after taking suspensions (...
4
votes
1answer
366 views

Does Ind-completion commute with finite limits?

The broad and vague question is in the title. The more precise question is: Say $\{\mathcal{C}_i\}$ is a finite diagram of (essentially small) stable $\infty$-categories and exact functors with ...
19
votes
3answers
2k views

how to make the category of chain complexes into an $\infty$-category

I'd like to have some simple examples of quasi-categories to understand better some concepts and one of the most basic (for me) should be the category of chain complexes. Has anyone ever written down ...
9
votes
1answer
482 views

Heller operator without left adjoint?

Suppose given a noetherian ring $R$. On the stable category $R\text{-}\underline{\text{mod}} := R\text{-mod}/R\text{-proj}$, we have the Heller operator $$ \Omega : R\text{-}\underline{\text{mod}} \...