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MODULES OVER ÉTALE GROUPOID ALGEBRAS AS SHEAVES

Part of: Rings and algebras arising under various constructions Topological and differentiable algebraic systems Categories and geometry

Published online by Cambridge University Press:  09 September 2014

BENJAMIN STEINBERG
BENJAMIN STEINBERG*
Affiliation:
Department of Mathematics, City College of New York, Convent Avenue at 138th Street, NY, New York 10031, USA email [email protected]
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Abstract

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The author has previously associated to each commutative ring with unit $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\Bbbk $ and étale groupoid $\mathscr{G}$ with locally compact, Hausdorff, totally disconnected unit space a $\Bbbk $-algebra $\Bbbk \, \mathscr{G}$. The algebra $\Bbbk \, \mathscr{G}$ need not be unital, but it always has local units. The class of groupoid algebras includes group algebras, inverse semigroup algebras and Leavitt path algebras. In this paper we show that the category of unitary$\Bbbk \, \mathscr{G}$-modules is equivalent to the category of sheaves of $\Bbbk $-modules over $\mathscr{G}$. As a consequence, we obtain a new proof of a recent result that Morita equivalent groupoids have Morita equivalent algebras.

Keywords

étale groupoidsgroupoid algebrassheaves

MSC classification

Secondary: 16S99: None of the above, but in this section 16S10: Rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.) 22A22: Topological groupoids (including differentiable and Lie groupoids) 18F20: Presheaves and sheaves
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 97 , Issue 3 , December 2014 , pp. 418 - 429
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

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