A NOTE ON RETRACTS AND LATTICES (AFTER D. J. SALTMAN)

Z. REICHSTEIN

Abstract. This is an expository note based on the work of D. J. Saltman. We discuss the notions of retract rationality and retract equivalence and reprove some of the results from [Sa, Section 1] using geometric language.

1. Retracts

Throughout this note G will be a finite group and k will be a base field. We shall say that a dominant rational map f: X −−→ Y is a model for a field extension i: F ↪ E if k(X) ≃ E, k(Y) ≃ F, and i is induced by f (i.e., i = f^{∗}).

Definition 1.1. Let f: X −−→ Y be a dominant rational map of irreducible algebraic varieties. We say that Y is a retract of X via f if for every (not necessarily irreducible) subvariety X_{0} ≠ X there exists a rational section s: Y −−→ X of f such that f(Y) ⊄ X_{0}. If f is not specified, then the expression Y is a retract of X means “via some dominant rational map X −−→ Y ”.

Remark 1.2. If we view X as a variety over k(Y) then the above definition can be restated as follows: Y is a retract of X if and only if k(Y)-points are dense in X.

Lemma 1.3. (a) X is a retract of itself (via the identity map).

(b) If Y is a retract of X via f: X −−→ Y and Z is a retract of Y via g: Y −−→ Z then Z is a retract of X via gf: X −−→ Z.

Proof. (a) Take s to be the identity map (for any X_{0}).

(b) Suppose X_{0} ≠ X is a subvariety of X. Choose a section s_{1} : Y −−→ X whose image is not contained in X_{0}. Now choose a section s_{2} : Z −−→ Y whose image is not contained in the union of the indeterminacy locus of s_{1} and s_{1}^{−1}(X_{0}). Now s = s_{2}s_{1} : Z −−→ X is the section of gf we are looking for. □

The following lemma shows that the property of being a retract depends only on the field extension k(X)/k(Y) and not on the specific model f: X −−→ Y for this extension. Equivalently, this property does not change if we replace X and Y by birationally isomorphic varieties.

Lemma 1.4. Consider the following commutative diagram of rational maps,

\xymatrix{X \ar@{-->}[r]^{f} \ar@{-->}[d]^{α} & Y \ar@{-->}[d]^{β} \\ X′ \ar@{-->}[r]^{f′} & Y′}

where f and f′ are dominant and α and β are birational isomorphisms. If Y is a retract of X via f then Y′ is a retract of X′ via f′.

Proof. Given a subvariety X′_{0} ≠ X′ of X′, we want to construct a section s′: Y′ −−→ X′ of f′ whose image is not contained in X′_{0}. Let X_{0} be the union of α^{−1}(X′_{0}) and the indeterminacy locus of α and let s: Y −−→ X be a section of f whose image is not contained in X_{0}. Then s′ = αsβ^{−1} has the desired properties: it is well defined and its image is not contained in X′_{0}. □

Example 1.5. (a) Suppose X is birationally isomorphic to Y × Z, and k-points are dense in Z. (Note that latter condition is always satisfied if k is algebraically closed.) Then Y is a retract of X, via the projection f: X −−→ Y to the first factor. Indeed, for every k-point z ∈ Z, consider the section s_{z}: Y ⟶ Y × Z of f given by s_{z}(y) = (y, z). Since the union of the images of such sections is dense in X, we conclude that Y is a retract of X.

(b) If X is rational over Y (i.e. k(X) is isomorphic to a purely transcendental extension of k(Y)) then Y is a retract of X. Indeed, in view of Lemma 1.4, we may assume that X = Y × 𝔸_{k}^{n}. Since k-points are dense in 𝔸_{k}^{n}, the desired conclusion follows from part (a).

Definition 1.6. Let i: F ↪ E be a field extension, where E and F are finitely generated over k. We shall say that F is a retract of E (via i) if Y is a retract of X (via f) for some (and thus for every) model f: X −−→ Y of E/F.

Equivalently, F is a retract of E if for some (and thus any) algebraic F -variety X such that F (X) = E, the F -points are dense in X.

If the field extension i is not specified then the expression “F is a retract of E” means ”via some field extension E/F.

2. Generators of tori

In this section we make some simple observations about algebraic tori. This material will be used in the proof of Theorem 3.1 in the next section (but not anywhere else in this paper).

Let T = (k^{∗})^{r} be an algebric torus. We will say that t ∈ T generates T, if the cyclic subgroup <t> = {t^{n} | n ∈ ℤ} is Zariski dense in T. We will say that t_{1}, …, t_{r} ∈ k^{∗} are multiplicatively independent, if t_{1}^{a_{1}} …, t_{r}^{a_{r}} = 1 implies a_{1} = ⋯ = a_{r} = 0. (Here a_{1}, …, a_{r} are integers.)

Lemma 2.1. Let t = (t_{1}, …, t_{r}) ∈ T for some t_{1}, …, t_{r} ∈ k^{∗}. Then the following conditions are equivalent:

(a) {t^{n} | n ∈ ℤ − {0}} is dense in T,

(b) t is a generator of T,

(c) t does not lie in any proper closed subgroup of T,

(d) χ(t) ≠ 1 for any non-trivial character of T,

(e) t_{1}, …, t_{r} are multiplicatively independent. □

Proof. (a) ⇒ (b): Obvious.

(b) ⇔ (a): The set C = {t^{n} | n ∈ ℤ − {0}} differs from <t> in at most one point, i.e.,

<t> \ {id} ⊂ C ⊂ <t> .

Hence, C is dense in T if and only if so is <t>.

(b) ⇒ (c): If t lies in a closed subgroup S then the closure of <t> is contained in S.

(c) ⇒ (b): The Zariski closure of <t> is a closed subgroup of T.

(c) ⇔ (d): Immediate from [Sp, Corollary 2.5.3].

(d) ⇔ (e): Every character χ of T = (k^{∗})^{n} has the form χ(t_{1}, …, t_{r}) = t_{1}^{a_{1}} … t_{r}^{a_{r}} for some integers a_{1}, …, a_{r}; see, e.g. [Sp, Exercise 2.5.12]. □

Lemma 2.2. If t ∈ T is a generator of T then t^{n} is also a generator for any integer n ≠ 0.

Proof. Assume the contrary: t^{n} is not a generator. Then by Lemma 2.1, there is a non-trivial character χ: T ⟶ k^{∗} such that χ(t^{n}) = 1. Since χ^{n} is also a non-trivial character of T, and χ^{n}(t) = χ(t^{n}) = 1, we conclude that t is not a generator of T, contradicting our assumption. □

Lemma 2.3. Let T_{gen} be the set of generators of T. Assume either char(k) = 0 or char(k) = p but k is not algebraic over its prime field 𝔽_{p}. Then T_{gen} is Zariski dense in T.

Proof. First we claim that T_{gen} ≠ ∅. If char(k) = 0 then k contains the field of rational numbers. By Lemma 2.1 any point of the form t = (p_{1}, …, p_{r}), where p_{1}, …, p_{r} are distinct prime integers, is a generator of T (distinct primes are clearly multiplicatively independent). If char(k) = p, then, by our assumption, k contains a copy of the polynomial ring 𝔽_{p}[x], and we use the same argument as above, with ℤ and ℚ replaced by 𝔽_{p}[x] and 𝔽_{p}(x), respectively. That is, if f_{1}(x), …, f_{r}(x) are distince irreducible polynomials then t = (f_{1}, …, f_{r}) is a generator of T. This proves the claim.

Now suppose t ∈ T_{gen}. Lemma 2.2 tells us {t^{n} | n ∈ ℤ − {0}} lies in T_{gen}. By Lemma 2.1, this set is dense in T. Hence, T_{gen} is also dense in T, as claimed. □

3. Lattices and retracts

As usual, by a G-lattice L we shall mean ℤ^{n}, with a linear G-action, i.e., an integral representation G ⟶ GL_{n}(ℤ). We will say that L is faithful if this representation is faithful, i.e., if every non-trivial element of G acts non-trivially on L. We will say that L is permutation, if it has a ℤ-basis permuted by G.

Given a G-lattice L, we will write k[L] for its group ring, k(L) for the fraction field of k[L], and X_{L} for the spectrum of k[L]. The G-action on L induces G-actions on k[L], k(L) and X_{L}. Ignoring these G-actions for a moment, we see that since L ≃ ℤ^{n}, k[L] is isomorphic to the Laurent polynomial ring k[x_{1}^{±1}, …, x_{n}^{±1}], k(L) is isomorphic to the purely transcendental field extension k(x_{1}, …, x_{n}) of k, and X_{L} is isomorphic to the n-dimensional torus (k^{∗})^{n}.

For the rest of this paper we will assume that char(k) = 0 or char(k) = p and k is not algebraic over the prime field 𝔽_{p}, so that we can use Lemma 2.3.

Theorem 3.1. Let G be a finite group and L ⊂ M be G-lattices. Then k(M)^{G} is a retract of k(L ⊕ M)^{G}.

Proof. The inclusion i: L ↪ M gives rise to a dominant morphism i_{∗}: X_{M} −−→ X_{L}. Consider the maps

X_{M} ⟶^{α} X_{L} × X_{M} ⟶^{β} X_{M}

of G-varieties, where α = (i_{∗}, id) and β is the projection to the second component. Passing to rational G-quotients, we obtain

X_{M}/G −−→^{α/G} (X_{L} × X_{M})/G −−→^{β/G} X_{M}/G .

The function fields of X_{M}/G and (X_{L} × X_{M})/G are, by definition, k(M)^{G} and k(L ⊕ M)^{G}. Hence, it suffices to show that (X_{L} × X_{M})/G is a retract of X_{M}/G via β/G.

First observe that since β ∘ α = id_{X_{M}} (and hence, β/G ∘ α/G = id_{X_{M} /G}), α/G is a section of β/G. Note also that β (and thus β/G) does not depend on i_{∗}, but α (and thus α/G) does. Thus if we modify i_{∗}: X_{M} ⟶ X_{L} by composing it with a G-equivariant dominant map σ: X_{M} ⟶ X_{M}, we will create another section of β/G. In the sequel I will take σ = σ_{n} to be the dominant map X_{M} ⟶ X_{M} induced by mult_{n}: M ⟶ M, where mult_{n} is multiplication by a fixed integer n ≠ 0. (Note that X_{M} is, by definition, a torus, and σ_{n}(t) = t^{n} for any t ∈ X_{M}.) It remains to show that the union of the images of the sections obtained in this way is dense in (X_{L} × X_{M})/G. In fact, it suffices to show that the union Z of the images of

α_{n} = (i_{∗}σ_{n}, id): X_{M} ⟶ X_{L} × X_{M}

is dense in X_{L} × X_{M}, as n ranges over non-zero integers. More concretely, the k-points of Z are of the form (i_{∗}(x^{n}), x), where n ranges over the nonzero integers and x ranges over X_{M} (and x^{n} denotes the nth power of x in the torus X_{M}).

Let (X_{M})_{gen} be the set of generators of X_{M} (viewed as a torus (k^{∗})^{rank(M)}). By Lemma 2.3 (X_{M})_{gen} is Zariski dense in X_{M}, and by Lemma 2.1

D_{x} = {x^{n} |n ∈ ℤ − {0}}

is Zariski dense in X_{M} for any x ∈ (X_{M})_{gen}. Since i_{∗}: X_{M} ⟶ X_{L} is dominant, i_{∗}(D_{x}) is Zariski dense in X_{L}. To sum up, Z contains i_{∗}(D_{x})×{x} for every x ∈ (X_{M})_{gen}. We thus conclude that the closure \overline{Z} of Z in X_{L}×X_{M} contains X_{L} × (X_{M})_{gen}. Since (X_{M})_{gen} is dense in X_{M}, this implies \overline{Z} = X_{L} × X_{M}, as claimed. □

4. Stable retracts

Definition 4.1. Let X and Y be irreducible varieties defined over k.

(a) We will say that Y is a stable retract of X (and write Y ⪯ X) if Y is a retract of X × 𝔸_{k}^{m}. Similarly, we will say that a field F is a stable retract of E (and write F ⪯ E) if F is a retract of a purely transcendental extension E(t_{1}, …, t_{m}) of F.

(b) We will say X and Y are stably retract equivalent (and write X ∼ Y) if X is a stable retract of Y and Y is a stable retract of X. Similarly, we will say that finitely generated field extensions F and E of k are stably retract equivalent (and write F ∼ E) if F is a stable retract of E and E is a stable retract of F.

(c) X is called retract rational if it is stably retract equivalent to a point (here by a point I mean Spec(k)). A finitely generated field extension F of k is retract rational if it is stably retract equivalent to k. Example 4.2. Clearly if F/k is stably rational then it is stably retract rational. Lemma 4.3. (a) ∼ is an equivalence relation on the set of varieties/k (or of finitely generated field extensions of k).

(b) ⪯ induces a partial order on the set of stable retract equivalence classes of varieties (or finitely generated field extensions F/k),

(c) If X and Y (or F and E) are stably isomorphic over k, then they are stably retract equivalent. Proof. (a) ∼ is symmetric by definition. It is reflexive and transitive by Lemma 1.3.

(b) Suppose ⪯ is a reflexive and transitive relation on a set S. Then the relation s ∼ t, defined by (s ∼ t if s ⪯ t and t ⪯ s), is easily seen to be an equivalence relation, and ⪯ induces a partial order on the set S/ ∼ of equivalence classes in S. This is precisely the construction we carried out, with S = set of irreducible varieties (or finitely generated field extension of k).

(c) Suppose Z is rational over both X and Y. Then X ∼ Z and Y ∼ Z (see Example 4.2), and since ∼ is an equivalence relation, we conclude that X ∼ Y. □

Theorem 4.4. Let M be faithful G-lattice. Then k(M)^{G} is a stable retract of k(L ⊕ M)^{G} for any G-lattice L.

This theorem is a variant of Theorem 3.1. The conclusion of Theorem 4.4 is a little weaker (“retract” has been replaced by “stable retract”), and M is assumed to be faithful. On the other hand, L is no longer required to be a sublattice of M.

Proof. Recall that L is contained in some permutation lattice P; see [Sa, Lemma 1.2]. Let M′ = M ⊕P. Then L is contained in M′ and thus k(M′)^{G} is a retract of k(L⊕M′)^{G} by Theorem 3.1. By the no-name lemma, k(M′)^{G} is stably equivalent to k(M)^{G}, and k(L ⊕ M′)^{G} = k(L ⊕ M ⊕ P)^{G} is stably equivalent to k(L ⊕ M)^{G}. This shows that k(M)^{G} is a stable retract of k(L ⊕ M)^{G}, as claimed. □

Corollary 4.5. (cf. [Sa, Lemma 1.5]) Let G be a finite group, L be a faithful G-lattice and G ↪ GL(V) be a faithful linear representation of G. Then k(V)^{G} is a stable retract of k(L)^{G}.

Proof. Use Theorem 4.4, with M = faithful permutation G-lattice. By the no-name lemma, k(M)^{G} is stably equivalent to k(V)^{G}, k(L ⊕ M)^{G} is stably equivalent to k(L)^{G} (here we use the fact that L is faithful!), and the corollary follows. □

Corollary 4.6. Let G ↪ GL(V) be a faithful linear representation of G and let N be a G-lattice such that k(N)^{G} is stably retract equivalent to k(V)^{G}.

Then k(M)^{G} is stably retract equivalent to k(V)^{G} for any faithful direct summand M of N.

Proof. By Corollary 4.5, we have k(V)^{G} ⪯ k(M)^{G}, where ⪯ stands for “is a stable retract of”, as before. On the other hand, by Theorem 4.4 k(M)^{G} ⪯ k(N)^{G} ∼ k(V)^{G}. This shows that k(M)^{G} and k(V)^{G} are retract equivalent. □

Example 4.7. If N is a quasi-permutation lattice then k(N)^{G} and k(V)^{G} are stably isomorphic; see [LL, Proposition 1.4]. Consequently, Corollary 4.6 tells us that k(M)^{G} and k(V)^{G} are stably retract equivalent for any faithful direct summand M of a quasi-permutation lattice.

Remark 4.8. Saltman has shown that there exist faithful G-lattices M such that k(M)^{G} and k(V)^{G} are not stably retract equivalent; see [Sa, Theorem 2.8].

Corollary 4.9. (cf. [Sa, Corollary 1.6]) If k(L)^{G} is stably retract rational for some faithful G-lattice L then k(V)^{G} is stably retract rational for every faithful linear representation G ↪ GL(V).

Proof. Our goal is to show that k ∼ k(V)^{G}, where ∼ denotes stable retract equivalence, as before.

Since k(V)^{G} ⪯ k(L)^{G} (by Corollary 4.5) and k(L)^{G} ∼ k (given), we conclude that k(V)^{G} ⪯ k. The opposite inequality, k ⪯ k(V)^{G} follows from Remark 1.2, since k(V)^{G} is unirational over k (and hence, k-points are dense on any model of k(V)^{G}/k). □

5. Appendix: Smooth retracts

Saltman’s terminology is a bit different from ours; see [Sa, p. 223]. Given a field extension E/F, he says that F is a retraction of E if there is a F - variety X, such that F (X) = E and X has a F -point. If F -points are dense in X (i.e., F is a retract of E in the sense of our Definition 1.6), Saltman calls F is a dense retraction of E. He remarks that the relationship between these notions, beyond the obvious implication (dense retraction) ⇒ (retraction), is obscure.

The problem, as I see it, is that the notion of retraction defined above is difficult to work with, because the existence of a F -point is, in general, not a birational invariant of X. The following ”intermediate” notion seems more natural to me (even though I have no applications for it at this point).

Definition 5.1. Suppose char(F) = 0. Given a field extension E/F, we will say that F is a smooth retract of E, if the following equivalent conditions hold.

(a) There exists a F -variety X such that F (X) = E and X has a smooth F -point.

(b) For every smooth proper F -variety X′ such that F (X′) = E, X′ has a F -point.

The implication (a) ⇒ (b), is a consequence of Nishimura’s Lemma [RY, Proposition A6] (see also [N]) to the birational isomorphism X −−→ X′. To prove the implication (b) ⇒ (a), we only need to show that there exists a smooth proper F -variety X′ such that F (X′) = E (then we can take X = X′). To construct such X′, embed some model of E/F in a projective space ℙ_{F}^{n}, take the closure and resolve its singularities (this requires the assumption char(F) = 0!).

Using Nishimura’s lemma one can reprove the results of this note (and in particular, Lemmas 1.3 and 4.3) for smooth retracts. I opted to use “dense retracts” instead, because this leads to stronger versions of Theorems 3.1, 4.4 and their corollaries.

References

[LL] N. Lemire, M. Lorenz, On certain lattices associated with generic division algebras, J. Group Theory 3 (2000), no. 4, 385–405.

[N] H. Nishimura, Some remarks on rational points, Mem. Coll. Sci. Univ. Kyoto. Ser. A. Math. 29 1955 189–192.

[P] V. L. Popov, Sections in invariant theory, in: The Sophus Lie Memorial Conference (Oslo, 1992), Scand. Univ. Press, Oslo, 1994, pp. 315–361.

[RY] Z. Reichstein, B. Youssin, Essential dimensions of algebraic groups and a resolution theorem for G-varieties, with an appendix by J. Kollár and E. Szabó, Canad. J. Math. 52 (2000), no. 5, 1018–1056.

[Sa] D. J. Saltman, Multiplicative field invariants, J. Algebra 106 (1987), no. 1, 221– 238.

[Sp] T. A. Springer, Linear Algebraic Groups, Birkhäuser, 1981.

Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada

   E-mail address: reichst@math.ubc.ca