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Riemann-Roch theorems for Deligne-Mumford stacks. (Théorèmes de Riemann-Roch pour les champs de Deligne-Mumford.) (French) Zbl 0946.14004
The author establishes a Grothendieck-Riemann-Roch theorem for Deligne-Mumford stacks. More precisely: He defines “cohomology with coefficients in representations” of a Deligne-Mumford stack, constructs a Riemann-Roch transformation from \(K\)-theory to this new cohomology theory and proves a GRR theorem for this Chern character. The cohomology \(H^\bullet_{\text{rep}}(F, \ast)\) with coefficients in representations of a stack \(F\) is defined as the étale cohomology \(H^\bullet((I_F)_{\text{et}}, \Gamma(\ast))\) of the ramification stack \(I_F = F \times_{F\times F} F\) where \(\Gamma\) is a cohomology theory in the sense of H. Gillet [Adv. Math. 40, 203-289 (1981; Zbl 0478.14010)]. The main tool in the construction of the Riemann-Roch transformation is the so-called dévissage theorem for \(G\)-theory which may be viewed as a generalization of A. Vistoli’s decomposition theorem for equivariant \(K\)-theory [Duke Math. J. 63, No. 2, 399-419 (1991; Zbl 0738.55002)] and which is proved by generalizing the étale descent theorem [see R. W. Thomason and Th. Trobaugh, in: The Grothendieck Festschrift, Vol. III, Prog. Math. 88, 247-435 (1990; Zbl 0731.14001)] from schemes to stacks. The proof of the GRR theorem runs through the usual patterns (deformation to the normal cone, Chow envelopes, \(\ldots\)).

MSC:
14C40 Riemann-Roch theorems
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