THE MAP V ⟶ V//G NEED NOT BE SEPARABLE

BEN MARTIN AND AMNON NEEMAN

Abstract. We construct a vector space V with a linear action of a reductive group G such that the quotient map V ⟶ V//G (in the sense of geometric invariant theory) fails to be separable. This gives a counterexample to an assertion of Bardsley and Richardson.

0. Introduction

Let G be a reductive algebraic group, possibly nonconnected, and let X be an irreducible affine G–variety. Suppose that the ground field k has characteristic p > 0. In their paper on étale slices in characteristic p, Bardsley and Richardson claim ([1], Section 2, 2.1.9(b)) that the canonical projection from X to the quotient X//G is separable. We give a counterexample to show that this map need not be separable, not even when X is a vector space V and G acts linearly.

Bardsley and Richardson extend Luna’s important Étale Slice Theorem [2] from characteristic zero to characteristic p. At only one point (see [1], Section 4, 4.3) do they use the separability of the quotient map. There the group G is finite and their assertion is justified: for the function field k(X//G) is the whole of the field of invariants k(X)^{G} of the function field k(X), by [1], Section 4, 4.3.1, and whenever a group Γ acts on a field K, the extension K/K^{Γ} is separable [3], IV.1, Lemma 1.5. The main results of [1], therefore, remain valid.

Separability questions come up when one tries to generalise or strengthen the results of Bardsley and Richardson’s [1]. A major interest of the counterexample presented here is that it indicates limits on any possible Luna slice theorem in characteristic p > 0. The first author will explore this point further in a forthcoming paper.

Throughout this article, k will be an algebraically closed field. We denote by X//G the quotient of X by G in the sense of geometric invariant theory; see [1] for details.

Acknowledgements: The authors would like to thank the referee for several helpful expository improvements.

1. The counterexample

Notation 1.1. Let W be a finite dimensional vector space over k, of dimension d = dim(W) ≥ 2. Let G = GL(W) be the group of all linear automorphisms of W. Let W^{∗} be the dual of W, with the usual G–action. For any integer n > 0, we let W^{n} stand for the direct sum W ⊕ W ⊕ ⋯ ⊕ W of n copies of W. We first wish to consider the G–module W^{∗} ⊕ W^{n}.

Key words and phrases. Reductive group.

This G–module is naturally an affine variety. Let us choose a basis (x_{1}, x_{2}, …, x_{d}) for W, and take the natural dual basis (y_{1}, y_{2}, …, y_{d}) for W^{∗}. The polynomial functions on W^{∗} form the ring k[x_{1}, …, x_{d}], while the polynomial functions on W form the ring k[y_{1}, …, y_{d}]. The difference is that the G–actions on these rings are dual. The polynomial functions on W^{∗} ⊕ W^{n} form a ring R = k[x_{j}, y_{j}^{i}], with 1 ≤ j ≤ d and 1 ≤ i ≤ n. Note that in y_{j}^{i} the i is a superscript; y_{j}^{i} stands for the j^{th} component of the i^{th} vector. We are not raising anything to the i^{th} power. Our notation for raising to the p^{th} power, in this article, will be {y_{j}^{i}}^{p}.

Lemma 1.2. With the notation as in Notation 1.1, let I ⊂ R = k[x_{j}, y_{j}^{i}] be the ideal generated by all {x_{j}, y_{j}^{i} | j ≥ 2}. Then any G–invariant element of R that lies in the ideal I must vanish. In symbols, I ∩ R^{G} = 0.

Proof. In the G–orbit of any point of W^{∗} there is a point (λ, 0, ⋯, 0). Therefore in the G–orbit of any point of W^{∗} ⊕ W^{n} there is a point whose coordinates are

( λ ; 0 ; ⋮ ; 0 ) ; ( μ_{1}^{1} ; μ_{2}^{1} ; ⋮ ; μ_{d}^{1} ) , ( μ_{1}^{2} ; μ_{2}^{2} ; ⋮ ; μ_{d}^{2} ) , ⋯, ( μ_{1}^{n} ; μ_{2}^{n} ; ⋮ ; μ_{d}^{n} )

Now the element of GL(W) given by the diagonal matrix

( \begin{matrix}1 & 0 & ⋯ & 0 \\ 0 & t & ⋯ & 0 \\ ⋮ & ⋮ & & ⋮ \\ 0 & 0 & ⋯ & t \end{matrix} )

takes the above to the point

( λ ; 0 ; ⋮ ; 0 ) ; ( μ_{1}^{1} ; tμ_{2}^{1} ; ⋮ ; tμ_{d}^{1} ) , ( μ_{1}^{2} ; tμ_{2}^{2} ; ⋮ ; tμ_{d}^{2} ) , ⋯, ( μ_{1}^{n} ; tμ_{2}^{n} ; ⋮ ; tμ_{d}^{n} )

Taking the limit as t ⟶ 0, we have that the closure of any G–orbit must contain a point of the form

( λ ; 0 ; ⋮ ; 0 ) ; ( μ_{1}^{1} ; 0 ; ⋮ ; 0 ) , ( μ_{1}^{2} ; 0 ; ⋮ ; 0 ) , ⋯, ( μ_{1}^{n} ; 0 ; ⋮ ; 0 )

Now any polynomial in the ideal I ⊂ R vanishes on the points above. Since G–invariant polynomials are constant on closures of orbits, any G–invariant polynomial in I must vanish on all closures of all orbits; in other words, it must be identically zero.

Definition 1.3. Let the notation be as in Lemma 1.2. For every 1 ≤ i ≤ n, we may form the polynomial

Y_{i} = x_{1}y_{1}^{i} + x_{2}y_{2}^{i} + ⋯ + x_{d}y_{d}^{i}.

The Y_{i}’s are obviously G–invariant.

Proposition 1.4. Let the notation be as in Definition 1.3. The subring R^{G} ⊂ R, of all G–invariant polynomials in R, is generated by the Y_{i}’s.

Proof. It is to prove Proposition 1.4 as a consequence of the First Main Theorem. Instead, we will give an equally easy, self-contained proof.

The center Z(G) of G, that is the set of non-zero scalar matrices, stabilises I. Therefore Z(G) acts on R/I. It is very obvious that the Z(G)–invariant polynomials in R/I are generated by the monomials x_{1}y_{1}^{i}, and Y_{i} is congruent mod I to x_{1}y_{1}^{i}.

Take any G–invariant polynomial f ∈ R^{G}. Then f gives a Z(G)–invariant polynomial modulo I, and by the above paragraph, there exists a polynomial P in n variables so that f is congruent to P (Y_{1}, …, Y_{n}) mod I. But then

f − P (Y_{1}, …, Y_{n})

is a G–invariant element of I, and by Lemma 1.2 it must vanish.

Lemma 1.5. With the notation as above, the Y_{i} ∈ R^{G} are algebraically independent. Even better: any monomial in {x_{j}, y_{j}^{i}} can occur in the expansion of at most one monomial Y_{1}^{M_{1}}Y_{2}^{M_{2}} ⋯ Y_{n}^{M_{n}}.

Proof. Just by checking the degrees of the monomials in the vectors (y_{1}^{i}, y_{2}^{i}, …, y_{d}^{i}) for different i.

Definition 1.6. Suppose now that k is of characteristic p > 0. Put X_{j} = {x_{j}}^{p}. The ring R = k[x_{j}, y_{j}^{i}] contains a subring S = k[X_{j}, y_{j}^{i}]. The ring S is not just a subring of R, it is also a G–submodule. In fact, S can be thought of as the ring of polynomial functions on the G–module π_{∗}W^{∗} ⊕ W^{n}. Here, π_{∗}W^{∗} is the Frobenius twist of W^{∗}. The matrices in GL(d) act by raising the entries to the p^{th} power.

The polynomials

\overline{Y}_{i} = X_{1}\{y_{1}^{i}\}^{p} + X_{2}\{y_{2}^{i}\}^{p} + ⋯ + X_{d}\{y_{d}^{i}\}^{p}
= x_{1}^{p}{y_{1}^{i}}^{p} + x_{2}^{p}{y_{2}^{i}}^{p} + ⋯ + x_{d}^{p}{y_{d}^{i}}^{p}
= Y_{i}^{p}

are clearly G–invariant elements of the ring S.

Proposition 1.7. Let the notation be as in Definition 1.6. The subring S^{G} ⊂ S, of all G–invariant elements of S, is generated by the \overline{Y}_{i}’s.

Proof. The ring S = k[X_{j}, y_{j}^{i}] is a subring and G–submodule of R = k[x_{j}, y_{j}^{i}]. By Proposition 1.4 we know that R^{G} is generated by

Y_{i} = x_{1}y_{1}^{i} + x_{2}y_{2}^{i} + ⋯ + x_{d}y_{d}^{i}.

The ring S^{G} is nothing more than the intersection of R^{G} with S.

By Lemma 1.5, the elements Y_{1}^{M_{1}}Y_{2}^{M_{2}} ⋯ Y_{n}^{M_{n}} ∈ R^{G} have disjoint monomial expansions. A linear combination of Y_{1}^{M_{1}}Y_{2}^{M_{2}} ⋯ Y_{n}^{M_{n}}’s will lie in S if and only if every term does. Suppose therefore that some Y_{1}^{M_{1}}Y_{2}^{M_{2}} ⋯ Y_{n}^{M_{n}} belongs to S.

In the expansion of the product, there is a term

x_{2}^{M_{1}}{y_{2}^{1}}^{M_{1}} ∏_{i=2}^{n} x_{1}^{M_{i}}{y_{1}^{i}}^{M_{i}}

and since this lies in S, it follows that p must divide M_{1}. By symmetry, p must divide M_{i} for every i. That is, our monomial is really a monomial in Y_{i}^{p} = \overline{Y}_{i}.

Theorem 1.8. There exists a vector space V, and a reductive group G acting on V, so that the geometric invariant theory map V ⟶ V//G is not separable.

Proof. Put V = π_{∗}W^{∗} ⊕ W^{n} as above, with n > d = dim(W). We assert that the map V ⟶ V//G is not separable. The map corresponds to the inclusion S^{G} ⊂ S. We know that S^{G} is the polynomial algebra k[\overline{Y}_{1}, …, \overline{Y}_{n}]. The derivative of the inclusion S^{G} ⊂ S takes d\overline{Y}_{i} to

{y_{1}^{i}}^{p}dX_{1} + {y_{2}^{i}}^{p}dX_{2} + ⋯ + {y_{d}^{i}}^{p}dX_{d}

which is in the linear span of {dX_{1}, dX_{2}, …, dX_{d}}. The image is therefore contained in a d–dimensional vector subspace of the 1–forms on V. Since the dimension of V//G is n > d, the map Ω_{V//G}^{1} ⟶ Ω_{V}^{1} cannot be generically injective.

References

1. Peter Bardsley and Roger W. Richardson, Étale slices and transformation groups in characteristic p, Proc. London Math. Soc. 51 (1985), 295–317.

2. Domingo Luna, Slices étales, Sur les groupes algébriques, Soc. Math. France, Paris, 1973, pp. 81–105. Bull. Soc. Math. France, Paris, Mémoire 33.

3. Tonny A. Springer, Aktionen reduktiver Gruppen auf Varietäten, Algebraische Transformationsgruppen und Invariantentheorie, Birkhäuser, Basel, 1989, pp. 3–39.

Department of Mathematics, Division of Information and Communication Sciences, Macquarie University, Sydney NSW 2109, AUSTRALIA

   E-mail address: bmartin@ics.mq.edu.au

Center for Mathematics and its Applications, School of Mathematical Sciences, John Dedman Building, The Australian National University, Canberra, ACT 0200, AUSTRALIA

   E-mail address: Amnon.Neeman@anu.edu.au