Białynicki-Birula decomposition for reductive groups in positive characteristic
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 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 . Explicitly, for a k-scheme S which we endow with the trivial G-action, the functor is given by Intuitively, the functor parameterizes G-orbits in X which compactify to -orbits. Indeed, its k-points are G-equivariant morphisms . The restrictions and give maps and . The map f is a partial compactification of the G-orbit of and is a “limit” or the most degenerate point of this compactification. The evaluations at zero and one extend to maps of functors:
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the map that sends φ to .
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the limit map that sends φ to . The limit map has a section that sends to the constant family .
To obtain the classical positive (resp. negative) Białynicki-Birula decomposition [2] we take a smooth proper X and pairs and respectively. For a connected linearly reductive group G, the functor is represented by a scheme, as proven in [11] and is smooth for smooth X. The proof of representability proceeds in three steps
- (I)
introduce a formal version of the functor and prove its representability. The stage for this part is the formal neighborhood of , hence the question becomes essentially affine and the representation theory of G plays a central role,
- (II)
prove that the formalization map 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 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].
The aim of the present article is to prove that is representable for a large class of algebraic monoids with zero, so called Kempf monoids. An algebraic monoid with zero is a Kempf monoid if there exists a central one-parameter subgroup such that the induced map extends to a map that sends 0 to . We prove that every monoid 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 be a Kempf monoid with zero and with unit group G and X be a Noetherian G-scheme over k. Then the functor is representable and affine of finite type over .
In its ideas, the proof proceeds along the steps (I)-(III) above. However, there are two fundamental problems along the way:
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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 inside and the corresponding inside .
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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 mimicking the unit map . In this way, becomes a -scheme with a map to X. This induces a section of the formalization map and implies that it is an isomorphism.
In the smooth case, we can say a little more about the morphism . Proposition 1.2 Let be such that is smooth at . Then locally near the point x, the map is an affine space fiber bundle with an action of fiberwise.
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 together with an associative multiplication that has an identity element . The unit group of is the open subset consisting of all invertible elements. We denote it by or simply by G. The unit group G is open in , 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 that we will introduce in the next section. The main advantage of the formal functor over 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 is a Kempf monoid and that .
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 let be the n-th thickening of the k-point . Consider the set-valued functor We have a natural formalization map given by . Eventually, we will prove that it is an isomorphism. The crucial technical advantage of over
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Supported by Polish National Science Center, project 2017/26/D/ST1/00755.