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Questions tagged [morita-theory]

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Categorical Morita equivalence implies equivalence of module categories?

Classically, two rings $R$ and $S$ are Morita equivalent if and only if any of the following is true ($R$-Mod) $\simeq$ ($S$-Mod). $S \simeq Hom_R(M,M)$, where $M$ is a finitely generated projective ...
Student's user avatar
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11 votes
3 answers

Does Morita theory hint higher modules for noncommutative ring?

Two possibly noncommutative rings are called Morita equivalent if their left-module categories are equivalent. In the commutative case, Morita equivalence is nothing more than ring isomorphism. ...
Student's user avatar
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0 votes
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Questions in the paper "Morita endomorphism algebras of generators"

I am reading this paper "Morita endomorphism algebras of generators", the link is here: There are two quesions I can't understand: on page ...
Xiaosong Peng's user avatar
3 votes
1 answer

Whether Morita equivalence holds the following properties?

Let $A,B$ be two K-algebras over a field K. $A$ and $B$ are said to be $Morita $ $equivalent$ if the category $Mod A$ and $Mod B$ are equivalent. $A$ and $B$ are said to be $derived$ $equivalent$ ...
Xiaosong Peng's user avatar
3 votes
0 answers

Non-homotopic cdga maps with same effect on cohomology

Let $f, g : \mathcal{A} \to \mathcal{B}$ be maps of commutative differential graded algebras. An easy way to tell if they are homotopic is by looking at their effect on cohomology. However, if $f$ ...
CDGAmateur's user avatar
8 votes
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Non-Standard Derived Equivalences of Non-Flat Algebras

I read that for algebras $R$ and $S$ (over a commutative ring), assuming that $R$ or $S$ is flat, the existence of a derived equivalence $\mathcal{D}(R) \to \mathcal{D}(S)$ implies the existence of an ...
Aras Ergus's user avatar
7 votes
1 answer

Can a finite von Neumann algebra be strongly morita equivalent to a properly infinite von neumann algebra?

Can a finite (by finite I mean when the projection $1$ is finite) von Neumann algebra be strongly morita equivalent to a properly infinite von Neumann algebra? (Strong morita equivalence is the same ...
Louis A's user avatar
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15 votes
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In what generality does Eilenberg-Watts hold?

In homological algebra, the Eilenberg-Watts theorem states that if $F\colon\text{Mod}_R\to\text{Mod}_S$ is a right-exact coproduct preserving functor of modules, then $F\cong-\otimes_R F(R).$ The ...
ziggurism's user avatar
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5 votes
1 answer

Generators of the derived category

For a ring $R$, which is a finite-dimensional algebra over a field, the category of finite-dimensional, projective, right $R$-modules, $\mathcal{P}_R$ is generated by the indecomposable projective ...
Eitan Chatav's user avatar
10 votes
2 answers

Hochschild (co)homology of A and of Mod_A

Let A be an algebra (or dg algebra). Where can I find a proof of HH_*(A) = HH_*(Mod_A) and HH^*(A) = HH^*(Mod_A)? (And does this hold for any A?) Here Mod_A is, e.g., the category of left A-modules. ...
Kevin H. Lin's user avatar
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