Existence of slices on a tame context

European Journal of Mathematics (2015) 1:54–77 DOI 10.1007/s40879-014-0022-4 RESEARCH ARTICLE Existence of slices on a tame context Sophie Marques Received: 11 May 2014 / Accepted: 27 October 2014 / Published online: 9 January 2015 © Springer International Publishing AG 2015 Abstract We study the ramification theory for actions involving group schemes, focusing on the tame ramification. We consider the notions of tame quotient stack introduced in Abramovich et al. (Ann Inst Fourier (Grenoble) 58(4):1057–1091, 2008) and of tame action introduced in Chinburg et al. (Duke Math J 82(2):269–308, 1996). We establish a local slice theorem for unramified actions, prove interesting lifting properties for linearly reductive group schemes, and establish a slice theorem for actions by commutative group schemes inducing tame quotient stacks. We show that these actions are induced from an action of an extension of the inertia group on a finitely presented flat neighborhood. Finally, we consider the notion of tame action and its relation to tame quotient stacks. Keywords Group schemes · Ramification · Freeness · Tameness · Slice theorem · Linearly reductive group schemes · Lifting · Quotient stack Mathematics Subject Classification 14L15 · 14A20 Originally considered by the Central European Journal of Mathematics but withdrawn due to imposition of publishing fees and resubmitted to the European Journal of Mathematics. The paper is an extract of the PhD thesis jointly supervised by Boas Erez at Université of Bordeaux and Marco Garuti at Università degli studi di Padova. S. Marques (B) Courant Institute of Mathematical Sciences, New York University, 251 Mercer St., New York, NY 10012, USA e-mail: 123 Existence of slices on a tame context 55 1 Introduction We consider an action, denoted by (X, G), of a flat group scheme G, of finite type over S, on a scheme X , locally of finite type over S, where the base scheme S is the spectrum of a commutative, unitary noetherian ring R. Throughout, we suppose that any orbit of the action is contained in an affine scheme. A notion of slices for the action (X, G) was introduced in [8], generalizing the one defined previously in [15]. Definition 1.1 We say that the action (X, G) over S admits étale (resp. finitely presented flat) slices if: • The quotient Q = X/G exists (the quotient X/G stands for the categorical quotient for the action (X, G) in the category of algebraic spaces). • For any topological point q ∈ Q, there exist: – an S-scheme Q  and an étale (resp. finitely presented flat) S-morphism Q  → Q such that there is q  ∈ Q  which maps to q via this morphism; – a closed subgroup G q of the group scheme G Q  = G × Q Q  over Q  which stabilizes some point p  of X × Q Q  above q  . That is, there is a group scheme isomorphism over the residue field k( p  ) at p , G q × Q  Spec(k( p  ))  IG Q  ( p  ), where IG Q  ( p  ) denotes the inertia group at p  of the base changed action (X × Q Q  , G Q  ); – a Q  -scheme Z q with a G q -action such that the quotient Z q /G q exists and equals Q  . The group G q acts on Z q × Q  G Q  on the right by (z, g)h = (zh, h −1 g) (on points), we require also that the quotient (Z q × Q  G Q  )/G q exists and that the base changed action (X × Q Q  , G Q  ) is induced by (Z q , G q ). That is, there exists a Q  -scheme G Q  -equivariant isomorphism, called balanced product: X × Q Q   (Z q × Q  G Q  )/G q , where G Q  acts on (Z q × Q  G Q  )/G q via the factor G Q  (so that, the quotient for the action (X × Q Q  , G Q  ) exists and equals Q  ). We call the subgroup G q the slice group and Z q the slice scheme. The action (X × Q Q  , G Q  ) should be thought of as a G Q  -stable neighborhood of an orbit. Such a neighborhood induced from an action (Z , H ), where H is a lifting of an inertia group of a point, is called a tubular neighborhood. Roughly speaking, for the étale (resp. fppf) topology, an action which admits slices can be described by the action of a lifting of an inertia group at some given point. It is well known that locally, for the étale topology, actions of constant group schemes (and slightly more generally étale group schemes) are induced from actions of their inertia groups: one can reconstruct the original action from the action of an inertia group at a point, Lemma 4.1 (this is a direct extension of the classical result on decomposition of finite extensions of valuation field after moving to the completion [20, II, Section 3, 123 56 S. Marques Théorème 1, Corollaire 4 and Proposition 4]). This statement is an instance of a slice theorem, Theorem 4.2. Here, we study the existence of slices, Theorem 5.5, under a tameness hypothesis on the action (X, G). One motivation for studying slices in this generality is the theory of tame covers, in the sense of Grothendieck and Murre: these admit étales slices, see Abhyankar’s lemma in [7,12]. Another motivation is the fundamental theorem of Luna [15] which states that actions of linearly reductive algebraic groups on affine varieties are tame and admit slices. In the simple case, when the action by a finite flat group scheme over S admits a trivial inertia group at some topological point x, we establish a local slice theorem proving that in this case the action is locally free on the fppf topology, Theorem 5.5. Tame stacks were introduced by Abramovich, Olsson and Vistoli in [1, Section 3] in their study of certain ramification issues arising in moduli theory. [1, Theorem 3.2], Theorem 7.4, characterized the tameness applied to the quotient stack [X/G] for the action (X, G) by the property that all the inertia groups at topological points of X are linearly reductive (it suffices to require this only at geometric points). As a consequence, this notion of tameness on [X/G] generalizes the classical notion of tameness in algebraic number theory, Remark 7.5. Moreover, it allows one to define a notion of “local tameness” on topological points of X , Definition 7.7. As a slice theorem, we manage to characterize tameness for quotient stacks for actions of finite, flat commutative group schemes over S via the existence of finitely presented flat slices with linearly reductive slice groups. Theorem 1.2 Let S be the spectrum of a noetherian ring R and X a scheme of finite type over S. Assume that the group scheme G is flat, commutative and finite over S and that any orbit of the action of G on X is included in an affine scheme. The following assertions are equivalent: (a) The quotient stack [X/G] is tame. (b) The action (X, G) admits finitely presented flat slices at any topological point x ∈ X such that the slice group at x ∈ X is linearly reductive and the slice scheme is a closed subscheme of some flat, finitely presented base change of X . Furthermore, this lifting of the inertia group can be constructed also in the noncommutative case, as a subgroup of the initial group and as a flat and linearly redu (...truncated)