Skip to main content Accessibility help




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.



Hide All
[1]Abrams, G. D., ‘Morita equivalence for rings with local units’, Comm. Algebra 11(8) (1983), 801837.
[2]Abrams, G. D. and Aranda-Pino, G., ‘The Leavitt path algebras of arbitrary graphs’, Houston J. Math. 34 (2008), 423442.
[3]Anh, P. N. and Marki, L., ‘Morita equivalence for rings without identity’, Tsukuba J. Math. 11(1) (1987), 116.
[4]Brown, J., Clark, L. O., Farthing, C. and Sims, A., ‘Simplicity of algebras associated to étale groupoids’, Semigroup Forum 88(2) (2014), 433452.
[5]Clark, L. O. and Edie-Michell, C., ‘Uniqueness theorems for Steinberg algebras’, 2014,
[6]Clark, L. O., Farthing, C., Sims, A. and Tomforde, M., ‘A groupoid generalisation of Leavitt path algebras’, Semigroup Forum (to appear).
[7]Clark, L. O. and Sims, A., ‘Equivalent groupoids have Morita equivalent Steinberg algebras’, 2013,
[8]Crainic, M. and Moerdijk, I., ‘A homology theory for étale groupoids’, J. reine angew. Math. 521 (2000), 2546.
[9]Exel, R., ‘Inverse semigroups and combinatorial C -algebras’, Bull. Braz. Math. Soc. (N.S.) 39(2) (2008), 191313.
[10]García, J. L. and Simón, J. J., ‘Morita equivalence for idempotent rings’, J. Pure Appl. Algebra 76(1) (1991), 3956.
[11]Johnstone, P. T., Sketches of an Elephant: A Topos Theory Compendium. Vol. 1, Oxford Logic Guides, 43 (Clarendon Press and Oxford University Press, New York, 2002).
[12]Johnstone, P. T., Sketches of an Elephant: A Topos Theory Compendium. Vol. 2, Oxford Logic Guides, 44 (Clarendon Press and Oxford University Press, Oxford, 2002).
[13]Kališnik, J., ‘Representations of étale Lie groupoids and modules over Hopf algebroids’, Czechoslovak Math. J. 61(3) (2011), 653672.
[14]Lawson, M. V., Inverse Semigroups, The Theory of Partial Symmetries (World Scientific, River Edge, NJ, 1998).
[15]Mitchell, B., ‘Rings with several objects’, Adv. Math. 8 (1972), 1161.
[16]Moerdijk, I., ‘The classifying topos of a continuous groupoid. I’, Trans. Amer. Math. Soc. 310(2) (1988), 629668.
[17]Moerdijk, I., ‘Toposes and groupoids’, in: Categorical Algebra and its Applications (Louvain-La-Neuve, 1987), Lecture Notes in Mathematics, 1348 (Springer, Berlin, 1988), 280298.
[18]Moerdijk, I., ‘The classifying topos of a continuous groupoid. II’, Cah. Topol. Géom. Différ. Catég. 31(2) (1990), 137168.
[19]Mrcun, J., ‘Stability and invariants of Hilsum–Skandalis maps’, PhD Thesis, Utrecht University, 1996, http:/arxiv:org/abs/math/0506484 (2005).
[20]Paterson, A. L. T., Groupoids, Inverse Semigroups, and their Operator Algebras, Progress in Mathematics, 170 (Birkhäuser, Boston, MA, 1999).
[21]Pierce, R. S., ‘Modules over commutative regular rings’, Mem. Amer. Math. Soc. 70 (1967).
[22]Renault, J., A Groupoid Approach to C -algebras, Lecture Notes in Mathematics, 793 (Springer, Berlin, 1980).
[23]Renault, J., ‘Représentation des produits croisés d’algèbres de groupoïdes’, J. Operator Theory 18(1) (1987), 6797.
[24]Resende, P., ‘Étale groupoids and their quantales’, Adv. Math. 208(1) (2007), 147209.
[25]Steinberg, B., ‘A groupoid approach to discrete inverse semigroup algebras’, 2009,
[26]Steinberg, B., ‘A groupoid approach to discrete inverse semigroup algebras’, Adv. Math. 223(2) (2010), 689727.
MathJax is a JavaScript display engine for mathematics. For more information see


MSC classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed