We give an algorithm for removing stackiness from smooth, tame Artin stacks with abelian stabilisers by repeatedly applying stacky blow-ups. The construction works over a general base and is functorial with respect to base change and compositions with gerbes and smooth, stabiliser preserving maps. As applications, we indicate how the result can be used for destackifying general Deligne-Mumford stacks in characteristic zero, and to obtain a weak factorisation theorem for such stacks. Over an arbitrary field, the method can be used to obtain a functorial algorithm for desingularising varieties with simplicial toric quotient singularities, without assuming the presence of a toroidal structure

Abstract. We prove that the dg category of perfect complexes on a smooth, proper Deligne-Mumford stack over a field of characteristic zero is geometric in the sense of Orlov, and in particular smooth and proper. On the level of triangulated categories, this means that the derived category of perfect complexes embeds as an admissible subcategory into the bounded derived category of coherent sheaves on a smooth, projective variety. The same holds for a smooth, projective, tame Artin stack over an arbitrary field.

Motivated by the local flavor of several well-known semi-orthogonal decompositions in algebraic geometry, we introduce a technique called conservative descent, which shows that it is enough to establish these decompositions locally. The decompositions we have in mind are those for projectivized vector bundles and blow-ups, due to Orlov, and root stacks, due to Ishii and Ueda. Our technique simplifies the proofs of these decompositions and establishes them in greater generality for arbitrary algebraic stacks.

We prove that the class of the classifying stack BPGLn is the multiplicative inverse of the class of the projective linear group PGL n in the Grothendieck ring of stacks K0(Stack k ) for n = 2 and n = 3 under mild conditions on the base field k. In particular, although it is known that the multiplicativity relation {T } = {S} · {PGL n} does not hold for all PGLn-torsors T → S, it holds for the universal PGL n-torsors for said n.Definition 1.1. Let S be an algebraic stack and let G be an algebraic group over S. We say that G is special, provided that every G-torsor over any field K over S is trivial.Examples of special groups are GL n , SL n , Sp 2n , G a and G m and extensions thereof, whereas PGL n is not special. In particular, split tori are special since they are products of G m . Nonsplit tori need not be special in general, but quasi-split tori are, also when considered over a general base. Proposition 1.1. A quasi-split torus T over any base stack S is special.