# Tagged Questions

Two rings are said to be Morita equivalent if their categories of (left) modules are equivalent. The notion is also used in more general contexts when certain categories of representations are equivalent.

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### When are Morita classes represented by certain structured algebra objects?

Let $\mathcal{C}$ be a monoidal category. There is a notion of Morita equivalence of algebra objects internal to $\mathcal{C}$. Does each Morita class have a symmetric Frobenius representative? A Hopf ...
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### Morita equivalence base equivalence relation for discrete groups

In the context of "discrete groups", is there an equivalence relation that implies the Morita equivalence of their reduced group C*-algebras? We define $G \sim H$ for discrete groups $G$ and $H$, ...
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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 ...
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### A canonical representative in Morita equivalence class

Let $A$ be a finite dimensional algebra over an algebraically closed field $K$. If $A$ is semisimple, then $A$ is Morita equivalence with a commutative algebra, that is $A \backsim K^n$ where $n$ is ...
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### Reference request: Morita bicategory

I have two closely related questions: Who first recognized that Morita equivalence was equivalence in the bicategory of Rings, Bimodules, and Intertwiners? I've heard this bicategory called the "...
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### Morita equivalence of $K$-algebras

Given $K$ a unital commutative ring and $A$ a $K$-algebra different from $K$. Can $K$ be Morita equivalent to $A \amalg A$, where $A \amalg A$ is the coproduct in the category of unital associative $K$...
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### 2-periodic derived equivalence

Let $A$ and $B$ be finite-dimensional algebras with finite global dimension over some field (in fact I am thinking of rational incidence algebras of finite posets). Suppose we know that $A$ and $B$ ...
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### 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 ...
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### C*-bimodules: the mess with definitions

I used to participate in a seminar that taught students about foundations of non-commutative geometry. It isn’t very complicated to define a C*-module $\mathcal E$ (also known as C* Hilbert module) ...
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### Are two $W^*$-algebras $A$ and $B$ that are Morita equivalent (in the sense of Rieffel) also Morita equivalent in the algebraic sense (as rings)?

Let $A$ and $B$ be $C^*$-algebras, I had a question about Morita equivalence for $C^*$-algebras and $W^*$-algebras, I've come across 3 concepts: Morita equivalence for $C^*$-algebras: Equivalence ...
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### Is every separable algebra in a modular tensor category Morita equivalent to a commutative one?

Separable algebras in modular tensor categories are interesting algebraic structures, which have received significant attention because of their connection to conformal field theories. My ...
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### Morita equivalence for utrametric Banach algebras (reference needed)

Is there any descent description of Morita theory for ultrametric Banach algebras? To make this question more precise let $K$ be some completion of the field $\mathbb{Q}_p$ (I'm mostly interested in ...
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### Quick question about conjugate equivalence bimodules and inner products

Let $A$ and $B$ be $W^{*}$-algebras, let $X$ be an $A-B$-equivalence bimodule (according to the definition given in "Morita equivalence for $C^*$-algebras and $W^*$-algebras" by Rieffel, link:http://...
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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 ...
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### Which endomorphism algebras are not Morita-trivial?

Let $\mathcal C$ be a symmetric monoidal category — for example, $\mathcal C$ might be the category of $R$-modules for a commutative ring $R$. At a minimum, I request further that: $\mathcal C$ is ...
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### Strong Morita Equivalence and Morphisms Between $C^{*}$-Algebras

If $A$ and $B$ are $C^{*}$-algebras, then they are strongly Morita equivalent if there exist a $(B,A)$-bimodule $E$ and an $(A,B)$-bimodule $F$ such that  E \otimes_{A} F \cong B \quad ...
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### Morita equivalence for operator algebras and tensor products, question about proof

This is a bit of a dumb question I know, but I was reading "Morita equivalence for C*-algebras and W*-algebras" by Rieffel, in this section about Morita equivalences and how they relate to forming ...
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### For a ring $A$, is $A$ Morita equivalent to $M_\infty(A)$? [closed]

Let $A$ be a ring, let $M_n(A)$ be the ring of $n$-by-$n$ matrices with elements in $A$, $A$ is Morita equivalent to $M_n(A)$, I was wondering if this also applied to infinite matrices? That is, if $A$...
Definition and context: An (associative, unital, not necessarily commutative) ring $R$ is called classical if every regular element of $R$ is a unit. Equivalently, $R$ is its own classical ring of ...