The Balmer spectrum of a tame stack. (English) Zbl 1374.14016

Ever since the pioneering work of M. J. Hopkins [Lond. Math. Soc. Lect. Note Ser. 117, 73–96 (1987; Zbl 0657.55008)] and A. Neeman [Topology 31, No. 3, 519–532 (1992; Zbl 0793.18008)] it has been clear that, for a commutative Noetherian ring \(A\), there is a strong connection between the geometry of \(\mathrm{Spec}(A)\) and the derived category of modules \(\mathbf{D}(A)\). A mayor breakthrough is the determination of thick \(\otimes\)-ideals of \(\mathrm{Perf}(X) = \mathbf{D}_{\mathrm{qc}}(X)_{\mathrm{parf}}\), the derived category of perfect complexes of sheaves of \(\mathcal{O}_X\)-modules with quasi-coherent homology for a quasi-compact and quasi-separated scheme \(X\) in terms of certain subsets of \(X\) by R. W. Thomason [Compos. Math. 105, No. 1, 1–27 (1997; Zbl 0873.18003)]. A thick \(\otimes\)-ideal of a monoidal triangulated category \(\mathbf{T}\) is a thick subcategory stable for tensoring with an object of \(\mathbf{T}\). The correspondence is with the now-called Thomason subsets, i.e. arbitrary intersections of open subsets of \(X\) with quasi-compact complement. This has opened the gate of a defintion of spectra of tensor triangulated category, as in P. Balmer’s [J. Reine Angew. Math. 588, 149–168 (2005; Zbl 1080.18007)]. In this paper the author proves the analogous theorem for a quasi-compact algebraic stack with quasi-finite and separated diagonal for the category \(\mathbf{D}_{\mathrm{qc}}(X)^{\mathrm{c}} \subset \mathrm{Perf}(X)\) of compact perfect complexes. Both categories agree on a scheme but may differ on a non tame algebraic stack. An algebraic stack is tame if its stabilizer groups at geometric points are finite linearly reductive group schemes, a notion due to D. Abramovich et al. [Ann. Inst. Fourier 58, No. 4, 1057–1091 (2008; Zbl 1222.14004)]. Note that every scheme and algebraic space is tame. Moreover, in characteristic zero, a stack is Deligne-Mumford if and only if is tame. In characteristic \(p > 0\), there are non tame Deligne-Mumford stacks. Another consequence is that for these stacks if they moreover have finite inertia and a coarse space then the Balmer spectrum of the stack is isomorphic as ringed space to the Balmer spectrum of its coarse space.


14F05 Sheaves, derived categories of sheaves, etc. (MSC2010)
13D09 Derived categories and commutative rings
14A20 Generalizations (algebraic spaces, stacks)
18G10 Resolutions; derived functors (category-theoretic aspects)
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