Białynicki-Birula decomposition for reductive groups in positive characteristic

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Abstract

We prove the existence of Białynicki-Birula decomposition for Kempf monoids, which form a large class that contains all monoids with reductive unit group in all characteristics. This extends the existence statements from [11], [1].

Résumé

On prouve l'existence de la décomposition de Białynicki-Birula pour les monoïdes de Kempf, qui forment une large classe contenant tous les monoïdes avec groupe des unités réductif dans toutes les caractéristiques. Ça étends les énoncés d'existence dans [11], [1].

Introduction

Let k be a field. By a linear group G we mean a smooth affine group scheme of finite type over k. When talking about a G-action we always mean a left G-action. Let G be a linear group and G be a geometrically integral affine algebraic monoid with zero and with unit group G. For a fixed G-scheme X over k, its Białynicki-Birula decomposition is a set-valued functor X+:=Map(G,X)G. Explicitly, for a k-scheme S which we endow with the trivial G-action, the functor is given byX+(S):={φ:G×SX|φ is G-equivariant}. Intuitively, the functor X+ parameterizes G-orbits in X which compactify to G-orbits. Indeed, its k-points are G-equivariant morphisms f:GX. The restrictions ff(1)X and ff(0)XG give maps X+(k)X(k) and X+(k)XG(k). The map f is a partial compactification of the G-orbit of f(1) and f(0) is a “limit” or the most degenerate point of this compactification. The evaluations at zero and one extend to maps of functors:

  • the map iX:X+X that sends φ to φ|1×S:SX.

  • the limit map πX:X+XG that sends φ to φ|0×S:SXG. The limit map has a section sX that sends φ0:SXG to the constant family φ=φ0pr2:G×SX.

Altogether, we obtain the diagram
see Figure 1.

To obtain the classical positive (resp. negative) Białynicki-Birula decomposition [2] we take a smooth proper X and pairs (G,G)=(Gm,A1=Gm{0}) and (G,G)=(Gm,A1=Gm{}) respectively. For a connected linearly reductive group G, the functor X+ is represented by a scheme, as proven in [11] and X+ is smooth for smooth X. The proof of representability proceeds in three steps

  • (I)

    introduce a formal version Xˆ of the functor X+ and prove its representability. The stage for this part is the formal neighborhood of XG, hence the question becomes essentially affine and the representation theory of G plays a central role,

  • (II)

    prove that the formalization map X+Xˆ is an isomorphism for affine schemes,

  • (III)

    for a G-scheme X find an affine G-equivariant étale cover of fixed points of X and use an easy descent argument to show that the natural map X+Xˆ is an isomorphism. The existence of such a cover is proven in [1], which crucially depends on the linear reductivity of G, see [1, Prop 3.1].

In positive characteristic the only connected linearly reductive groups are tori. Thus it is a natural question whether one could extend those existence results to reductive groups in positive characteristic. This seems also interesting from the point of view of geometric representation theory, similarly to how the Gm-case is used in [6].

The aim of the present article is to prove that X+ is representable for a large class of algebraic monoids G with zero, so called Kempf monoids. An algebraic monoid G with zero is a Kempf monoid if there exists a central one-parameter subgroup Gm,kZ(G) such that the induced map Gm,kG extends to a map Ak1G that sends 0 to 0G. We prove that every monoid G with zero and with reductive unit group is a Kempf monoid. We stress that we make no assumptions on the characteristic. Reductive means as usual that the unipotent radical of G is trivial. There are plenty of Kempf monoids with non-reductive unit group as well, such as the monoid of upper-triangular matrices. The main result of this paper is the following representability result.

Theorem 1.1

Let G be a Kempf monoid with zero and with unit group G and X be a Noetherian G-scheme over k. Then the functor X+ is representable and affine of finite type over XG.

This directly extends the previous results for the one-dimensional torus [7], [1] and linearly reductive groups [11], [1]. This extension is particularly far-reaching in positive characteristic, where tori are the only geometrically connected linear groups. It also clarifies the situation in general in that the complicated representation theory for G poses no obstructions to representability.

In its ideas, the proof proceeds along the steps (I)-(III) above. However, there are two fundamental problems along the way:

  • the representation theory for G is complicated and thus step (I) requires much more care. We introduce finitely generated Serre subcategories of representations and heavily employ the Kempf torus (Gm)k inside Gk and the corresponding Ak1 inside Gk.

  • the analogue of (III) is not known and it is clear that the current ideas are insufficient (the problems arising are similar to the ones with smoothness discussed below). We overcome this by employing Tannakian formalism of Hall-Rydh [9] to get a map iXˆ:XˆX mimicking the unit map iX. In this way, Xˆ becomes a G-scheme with a map to X. This induces a section of the formalization map X+Xˆ and implies that it is an isomorphism.

The language of stacks is most appropriate for (III). We delegate this part to the appendix in order to make the paper more accessible. The Tannaka duality in the required generality follows from the results of [9]. However [9] needs to be applied with care: the main results of that article require additional assumptions, so to deduce our claim we go into details of their argument.

In the smooth case, we can say a little more about the morphism πX.

Proposition 1.2

Let xXG be such that πX:X+XG is smooth at sX(x)X+. Then locally near the point x, the map πX is an affine space fiber bundle with an action of G fiberwise.

It would be desirable to have Proposition 1.2 for every smooth X, without any assumptions on X+. However, this is still open. The main problem is that the linear map mxmx/mx2 may not have a G-equivariant splitting, so the regularity of xX does not immediately imply the regularity of xX+, see Example 2.16. This is the same issue which implies that XG may not be smooth for smooth G-schemes X. Indeed, for example SLp acting on itself by conjugation has fixed points μp, which is non-reduced. In general [8] shows that smoothness of fixed points for actions on smooth varieties characterizes linearly reductive groups. Very curiously, it seems that very few examples of non-smooth fixed points are known and all known examples seem to have smooth underlying reduced schemes. Also, by Lemma 2.1 below and affineness of πX we have XkG=(X+)kG=(X+)kGm, so we cannot hope for Xk+ to be smooth in general, as Gm-fixed points would be smooth while XkG can be singular. However, we do not have examples of smooth X such that πX:X+XG is not smooth.

Section snippets

Acknowledgments

We thank Torsten Wedhorn and Andrew Salmon for inquiring about extending the Białynicki-Birula decomposition to positive characteristic that encouraged us to write this paper. We also thank Michel Brion and Andrew Salmon for helpful remarks on the preliminary version of this paper. We thank the anonymous referee for careful reading and pointing out several imprecise points.

Monoids and affine schemes

Throughout, we fix a base field k. We do not impose any characteristic, algebraically closed, perfect or other assumptions on k. An algebraic monoid is a geometrically integral affine variety G together with an associative multiplication μ:G×GG that has an identity element 1G(k). The unit group of G is the open subset consisting of all invertible elements. We denote it by G× or simply by G. The unit group G is open in G, so it is dense and connected. The monoid is reductive if G is

Formal G-schemes

In this section we prove the main results for the formal Białynicki-Birula functor Xˆ that we will introduce in the next section. The main advantage of the formal functor over X+ is that it is defined on the affine level; correspondingly in this section we speak the language of algebra rather than geometry. Throughout, we assume that G is a Kempf monoid and that 0N.

Formal Białynicki-Birula functors

In this section we introduce the formal version of Białynicki-Birula functors. The notation and general outline are consistent with [11, Section 6]. For an nZ0 let Gn=V(m0n+1)G be the n-th thickening of the k-point 0G. Consider the set-valued functorXˆ(S)={(φn)nZ0|φn:Gn×SX,φn is G-equivariant and (φn+1)|Gn×S=φn for all n}. We have a natural formalization map X+Xˆ given by φ(φ|Gn×S)n. Eventually, we will prove that it is an isomorphism. The crucial technical advantage of Xˆ over X+

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Supported by Polish National Science Center, project 2017/26/D/ST1/00755.

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