nLab Morita context

Idea

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.

Properties

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 as $B$-modules.

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

(v) The 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, in which case $f_P$ is an isomorphism.

(ii) $g_P$ is surjective iff $P$ is a generator of $Mod_B$, in which case $g_P$ is an isomorphism.

(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.

Related entries

• Morita equivalence, Eilenberg-Watts theorem

Literature

• 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

  doi

• 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

• Y. Doi, Generalized smash products and Morita contexts for arbitrary Hopf algebras, in: J. Bergen, S. Montgomery (Eds.), Advances in Hopf algebras, in: Lecture Notes in Pure and Appl. Math. 158, Dekker 1994 doi

• Stefaan Caenepeel, Joost Vercruysse, Shuanhong Wang, Morita theory for corings and cleft entwining structures, J. Algebra 2761 (2004) 210-235 doi

There are generalizations in more general bicategories:

• Laiachi 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