20
votes
Accepted
Does Morita theory hint higher modules for noncommutative ring?
Yes. The trick is to use not just categories, but pointed categories, which are categories equipped with a choice of object (the "pointing"). Given any ring $R$, the category $\mathrm{Mod}(R)...
- 54.6k
6
votes
Accepted
Morita equivalence and connectivity
Suppose $A$ is any nontrivial connective ring spectrum, with right module $P = A \oplus A[1]$, and let $B = End_A(P)$.
The ring $B$ satisfies
$$B \simeq A \oplus A[1] \oplus A[-1] \oplus A$$
and in ...
- 52.6k
4
votes
Categorical Morita equivalence implies equivalence of module categories?
Yes, it is Theorem 7.12.16 in [1]. In fact these are 2-equivalent (as the categories of modules are 2-categories).
Theorem 7.12.16. Let $M$ be a faithful exact module category over $C$. The $2$-...
- 639
3
votes
Does Morita theory hint higher modules for noncommutative ring?
Although I think my answer "pointed categories" is an important one, there is another way that the question could be interpreted: What is an interesting class of rings which are recoverable ...
- 54.6k
3
votes
Accepted
Whether Morita equivalence holds the following properties?
1) Yes, Morita equivalence trivially implies derived equivalence. Note that two algebras over an algebraically closed field are Morita equivalent iff their quiver algebras are isomorphic. So compared ...
- 26.5k
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