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Questions tagged [morita-equivalence]

For questions related to Morita equivalence. $2$ rings like $R, S$ are Morita equivalent (denoted by $R\approx S$) if their categories of modules are additively equivalent (denoted by ${}_{R}M\approx {}_{S}M$).

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Dual to Eilenberg Watts

The Eilenberg-Watts theorem states: Theorem. Let $A, B$ be rings and let $F : A\textbf{-Mod} \to B\textbf{-Mod}$ be a right exact, coproduct preserving additive functor. Then there exists a unique (up ...
emilg's user avatar
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2 votes
0 answers
26 views

Let $R$ be a ring $P$ a projective generator of right $R$-modules. If $M$ is a left $A$-module, then $M \cong \text{Hom}_R(P, P \otimes_R M)$

Suppose $R$ is a ring and $P$ is a projective generator of right $R$-modules. If $M$ is a left $A$-module, show that the natural map $M \to \text{Hom}_R(P, P \otimes_R M)$ that sends $m \mapsto (p \...
love and light's user avatar
1 vote
1 answer
33 views

Rigidity of finitely generated projective modules

I am currently reading a lecture notes on module theory and more specifically on Morita Theory. Here is a porition of the lecture note that I do not understand. Finitely generated projective modules ...
Squirrel-Power's user avatar
1 vote
1 answer
63 views

Morita equivalent about group algebra and representation ring

Let $\mathbb{K}G$ be a group algebra and $R$ be the representation ring(Grothendieck Ring) of $\mathbb{K}G$, that means the product of $R$ is the tensor product of representations, add is the direct ...
popo's user avatar
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0 answers
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Asking for references on commutation and morita equivalence

I'm currently reading the chapter 8 of the book Algèbre of Bourbaki and I'm stuck on the sections 5 "Commutation" and 6 "Equivalence de Morita des modules et des algèbres". At that ...
newuser's user avatar
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1 vote
1 answer
65 views

Morita theory for presheaf (functor) categories

In this 2013 paper, in Proposition 3.14, the author notes $ \mathcal L(1) $ embeds in $ \mathbf L $, which embeds in $ \mathbf R $ the category of retracts of $ \mathcal L(1) $, so $ P \mathbf R \...
Tempestas Ludi's user avatar
3 votes
2 answers
170 views

Is it true that any algebra is Morita equivalent to oposite?

Let $A$ be an algebra with unit. Denote by $A^L$ the algebra $A$ considered as left module over itself. Then $A^{op} = End_A(A^L)$. So according to ncatlab.org/nlab/show/Morita+equivalence $A^{op}$ is ...
Galois group's user avatar
1 vote
1 answer
105 views

Why is the bicategory viewpoint useful?

In ring theory one often wants to think about bimmodules as being morphisms between rings using tensor product as composition. However, this composition is only associative if one uses isomphism ...
Mark's user avatar
  • 74
0 votes
0 answers
21 views

Isomorphisms of the tensor product of $\mathcal{A}^N$ [duplicate]

Let $\mathcal{A}$ be an algebra and define $\mathcal{A}^N:=\mathcal{A}\oplus\dots\oplus\mathcal{A}$. I need to show that $\mathcal{A}^N\otimes_\mathcal{A} \mathcal{A}^N\cong M_N(\mathcal{A})$ $\...
Schrödinger's cat's user avatar
4 votes
1 answer
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Is the nilpotent exponent of the Jacobson radical of algebra a Morita equivalent invariant?

I have known that two rings A and B are said to be Morita equivalent if the left module categories mod A and mod B are equivalent. Many properties of module category are Morita equivalent invariant, ...
Zhenxian Chen's user avatar
2 votes
0 answers
56 views

Condition for an equivalence of functor categories

Given small categories $\mathcal{C}$ and $\mathcal{D}$, we have that $[\mathcal{C}^\text{op},\textbf{Set}]\simeq[\mathcal{D}^\text{op},\textbf{Set}]$ if and only if the Cauchy-completions of $\...
Cameron's user avatar
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4 votes
1 answer
146 views

State-sum construction of the Drinfeld center of a fusion 2-category

If $\mathcal{C}$ is a fusion 1-category, the Turaev-Viro state-sum produces a 3d TQFT whose modular tensor category is the Drinfeld center of $\mathcal{C}$. In particular this means that the Turaev-...
Andrea Antinucci's user avatar
2 votes
1 answer
118 views

Reference request: Lindner's thesis on Morita equivalence

I would like to know the contents or at least the main results of the following thesis, which has been cited in other category theory papers: H. Lindner, Morita-Äquivalenz von Kategorien über einer ...
Martin Brandenburg's user avatar
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1 answer
79 views

Is it true that functors which are surjective on objects are obviously essentially surjective?

I am asking this as I have established a functor F between categories C and D such that F is faithful, full and surjective on objects. Can I say that F is an equivalence of categories? I think so but ...
Promit Mukherjee's user avatar
5 votes
1 answer
316 views

Are the quaternions Morita equivalent to the real numbers?

I'm wondering whether the quaternions are Morita equivalent to the real numbers. The characterisation in terms of full idempotents seems unwieldly. I can use the category-theoretic definition, but it ...
wlad's user avatar
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