Morita context

A **Morita context** or, in some authors (e.g. Bass) the **pre-equivalence data** is a generalization of Morita equivalence between categories of modules. In the case of right modules, for two associative k-algebras (or, in the case of $k = \mathbb{Z}$, rings) $A$ and $B$, it consists of bimodules ${}_A P_B$, ${}_B Q_A$, and bimodule homomorphisms $f: P\otimes Q\to A$, $g: Q\otimes P\to B$ satisfying mixed associativity conditions.

Theorem. (Bass II.3.4) If $f$ is surjective, then:

(i) $f$ is an isomorphism

(ii) $P$ and $Q$ are generators in the categories of $A$-modules

(iii) $P$ and $Q$ are finitely generated and projective

(iv) $g$ induces isomorphisms of bimodules $P\cong Hom_B(Q,B)$ and $Q\cong Hom_B(P,B)$

(v) homomorphisms of $A$-algebras $End_B(P)\leftarrow A\rightarrow End_B(Q)$ are isomorphisms

(Bass II.4.1) A Morita context can be constructed from an $A$/algebra $B$ and a right $B$-module $P$. Then set $A = End_B(P)$ and $Q=Hom_B(P,B)$. Then $f = f_P$ and $g = g_P$ are defined by $(b q) p = b (q p)$ and $(q a) p = q(a p)$.

(Bass II.4.4) (i) $f_P$ s surjective iff $P$ is finitely generated projective $B$-module. Then $f_P$ s iso.

(ii) $g_P$ is surjective iff $P$ s a generator of $mod_B$, then $g_P$ is iso

(iii) The Morita context $(A,B,P,Q,f,g)$ is a Morita equivalence iff $P$ is both projective and a generator. Then $\otimes_P : mod_A\to mod_B$ and its right adjoint $Hom_B(P,-)$ form the equivalence.

- Hyman Bass,
*Algebraic K-theory*, chapter 2 - Tomasz Brzeziński, Adrian Vazquez Marquez, Joost Vercruysse,
*The Eilenberg-Moore category and a Beck-type theorem for a Morita context*, Appl. Categ. Structures**19**(2011), no. 5, 821–858 MR2836546 - Bruno J. Müller,
*The quotient category of a Morita context*, J. Algebra**28**(1974), 389–407 MR0447336 doi - A. I. Kashu,
*On equivalence of some subcategories of modules in Morita contexts*, Discrete Math. 2003, no. 3, 46–53, pdf

There are generalizations in more general bicategories:

- L. El Kaoutit. Wide Morita Contexts in Bicategories. Arab. J. Sci. Eng.
**33**, (2008), 153–173 - Bertalan Pecsi,
*On Morita context in bicategories*, pdf

category: algebra

Last revised on February 24, 2014 at 12:36:34. See the history of this page for a list of all contributions to it.