## 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.

### MSC:

 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)

### Keywords:

derived categories; algebraic stacks
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