Contemporary Mathematics Rings of invariants and inseparable forms of algebras over the Steenrod algebra Clarence W. Wilkerson, Jr. Dedicated to the memory of my mother, Dorthea Gray Wilkerson, 1925-1999. Abstract.This paper establishes analogues of the the classical decomposi- tions of a normal field extension in the context of graded integral doma* *ins with an unstable action of the mod p Steenrod algebra. These decompositions h* *ave in later work proved to have interesting topological consequences. The concept of the maximal torus plays a key role in the classification of c* *om- pact connected Lie groups. Likewise the role of the cohomology of the classifyi* *ng space BT nof the n-torus in the study of characteristic classes has been long u* *n- derstood via the splitting theorems. With the work of Adams-Wilkerson, [1], it was realized that the cohomology rings H*(BT n, Fp) as algebras over the Steenr* *od algebra Ap were universal in a strong technical sense. This feature was presaged by (Serre, [18]), (Quillen, [15, 16]), and (Wilkerson, [19]), and is greatly ex* *tended by later work of Carlsson, Miller, Lannes, Zarati, Schwartz, and Dwyer-Wilkerso* *n. The Adams-Wilkerson work provides an algebraic analogue of the existence of a maximal torus, in a suitable category of algebras with an action of the Steen* *rod algebra. More precisely, it provides for each graded Fp- algebra R* which is an integral domain of finite transcendence degree n over Fp and equipped with an ü nstableä ction of the mod p Steenrod algebra Ap, an embedding tR* : R* -! H*(BT n, Fp) which is an Ap-map of graded algebras and is an invariant of R* and its Ap-acti* *on. Furthermore, H*(BT n, Fp) is algebraic and normal over R*, via the morphism tR*. In this paper, we denote H*(BT n, Fp) together with its Ap-action as S[V #], the symmetric algebra on the dual of the n-dimensional Fp-vector space V, for V concentrated in degree 2 (or degree 1 for p = 2). ____________ The author was supported in part by the National Science Foundation, the Wa* *yne State Fund, The Institute for Advanced Studies of Hebrew University, and Johns Hopkin* *s University. Oc0000 (copyright holder) 1 2 C. W. WILKERSON The embedding tR* has additional properties, depending on the properties of R* as an algebra over the Steenrod algebra: Properties of the Adams-Wilkerson embedding: a) S[V #] is integral over R* via tR if and only R* is a finitely generated Fp-algebra. b) tR* is a separable extension on the fraction field level if and only if t* *he Mil- nor primitives {P i} in Ap give n linearly independent derivations on the grad* *ed fraction field F R* of R* (as they do on S[V #]). Equivalently, (S[V #])p \ F R* ** = (F R*)p, if and only if the field extension is separable. * c) There is a subgroup W (R*) of GL(V ) such that R* = S[V #]W(R ), if and only if both conditions a) and b) hold, and, in addition, R* is integrally clos* *ed in its field of fractions. That is, under some general conditions, R* is a ring of invariants. Conversely, any ring of invariants has the properties properties a)-c).1 The three conditions required to ensure that R* is a ring of invariants may seem to lack topological motivation. However, the noetherian and integral closu* *re requirements together have an intrinsic meaning in terms of the Steenrod algebra action: Theorem 1. If R* is a graded integral domain of finite transcendence degree with an unstable Ap-action, then R* is integrally closed and finitely generated* * if and only if R* = Un(F R*), the set of unstable elements in the graded field of frac* *tions of R*. If we consider R* as a subring of S[V #] via tR*, then Un(F R*) = S[V #]\F * *R*. Interpretation of the separability condition is harder, but algebraic exampl* *es at all primes and topological examples at the prime 2 show that it is a necessa* *ry complication. The inseparable forms portion of the title refers to a construct* *ion given here by which inseparable examples are made from separable examples. The goal is an algebraic reformulation of the separability condition so that it can* * be used more readily with topological techniques such as öm d p Hopf invariant one* *", for odd primes. Roughly stated, our aim in Theorem 2 below is to show that any R* with R* = Un(F R*) is a ring of invariants, not necessarily of S[V #] itself* *, but of a certain "diagonal" subalgebra D*(R*) of S[V #]. The definition of this D*(R*) is motivated by an analogy to elementary field theory. Recall that if K ! L is a finite normal field extension, then there are* * two ways of decomposing this into sub-extensions. First, one could find the maximal inseparable extension J of K in L, so K ! J ! L. Then J ! L is Galois, with Galois group W = AutK (L) = AutJ(L), so that J = LW . This is roughly the procedure in [1], with allowances made there for ambient Steenrod actions. A second method is more important for this paper - form the maximal separable extension I of K in L , where K ! I ! L. Here K ! I is separable, and J ! L ____________ 1We ask the reader to note that in Adams-Wilkerson, [1], the statement of T* *heorem 1.2 should include the hypothesis that H* is noetherian. This hypothesis was used but was * *inadvertently omitted. There is also an oversight in the statement of Theorem 1.9 - in order * *to prove that condition 1.2.2 is necessary, H* should be assumed also to be integrally closed* *. An alternate formulation is to assume that condition 1.2.2 holds for all elements of the fra* *ction field of H*, as stated above in part b). INSEPARABLE FORMS OF RINGS OF INVARIANTS 3 is purely inseparable. Notice that K is the intersection of LW with I, or K = * *IW , see for example, [Lang, Chapter VII, Section 7, Prop. 12, [10]]. Our Theorem 2 provides a ring level description of this process of taking a maximal separable extension. Somewhat surprisingly, the analogue of the maxi- mal separable field extension I above is greatly restricted by the existence of* * the Steenrod algebra action. This restricted structure of D*(R*) is the important f* *or applications, [2], [12]. Theorem 2. Let R* be a sub-Ap-algebra of S[V #] such that F R* ! F S[V #] is algebraic. Define a filtration {Ui(R*)} of V # by the rule that y is in Ui(R* **) if and only if ypi is separable over F R*. Let D*(R*) be the "diagonal " subalgebra of S[V #] generated by {Ui(R*)pi}. Let I* be the maximal separable graded field extension of F R* in F S[V #]. Then a) R* ! D*(R*) is separable and D*(R*) ! S[V #] is purely inseparable. b) D*(R*) = Un(I*) and Ui(R*) = {y|ypi2 I*}. c) D*(R*) i(S[Ui(R*)=Ui-1(R*)])pias Ap-algebras. d) AutR*(S[V #]) = AutR*(D*(R*)) = AutFR* (F S[V #]) e) For W (R)*= AutR*(S[V #]), W (R)(Ui(R*)) Ui(R*). f) D*(R*)W(R ) = Un(F R*). Note that although V # has a filtration by the W (R*)-vector spaces {Ui(R*)}, the filtration need not split over W (R*), so that V # is not necessarily isomo* *rphic to Ui+1=Ui as W (R*) vector spaces. In summary, Un(F R*) is explicitly determined by two pieces of data: 1) The subgroup W (R*) of GL(V #) 2) The W (R*) filtration {Ui(R*)} of V #, together with the implicit exponen* *ts pi. This leads to a constructive classification of all noetherian integrally clo* *sed unstable domains of a given trancendence degree. In Section Five we give some examples of this classification. These methods also answer a question of Mitchell and Stong: Theorem 3. If the polynomial algebra R* = Fp[y1, . .,.yn] has an unstable action of Ap, then the Adams-Wilkerson embedding tR* factors through D*(R*) Fp[z1, . .,.zn] where |zi| = 2pNi. Since R* ! D*(R*) is separable, the Jacobian | @yi=@zj | is nonzero for homogeneous algebra bases {zi} of D*(R*) and {yj} of R*. Theorem 2 does not directly answer the problem of realizing these rings as t* *he cohomology rings of topological spaces, but it does express the maximum algebra* *ic content obtainable from the Ap-action. Dwyer, Miller, and Wilkerson have used Theorem 2 together with applications of the Sullivan Conjecture technology of Miller and Lannes to prove very strong uniqueness and non-realizability results* * for classifying spaces. Theorem 4 below from [2] is a sampler that points out that separability is forced by topological considerations. Theorem 4. [2] If X is a simply connected CW complex such that R* = H*(X, Fp) is a integrally closed finitely generated graded integral domain of t* *ran- cendence degree n, then for p > 2 *) H*(X, Fp) S[V #]W(R . 4 C. W. WILKERSON Furthermore, the homotopy type of X determines a lift of the inclusion of W (R** *) ! GL(V ) up to the general linear group of rank n over the p-adic integers. The proof of Theorem 4 is beyond the scope of this paper, but we give here just a brief hint as to the role that Theorem 2 plays in [2]. If R* = H*(X), th* *en by a fundamental result of Lannes,[11], the embedding tR* : R* ! S[V #] ! H*(BV ) can be topologically realized by a continuous function f : BV ! X. The component of the function space Map(BV, X)f has cohomology TfV*(R*) = D*(R* ), where TfV*(_) is a summand of the T -functor of Lannes. Accepting this, öm d p Hopf invariant one" shows that the exponents in the definition of D*(R*) are all zer* *o, since it is the cohomology of a space. Thus Map(BV, X)f BT n. That is, [2] carries out the Adams-Wilkerson program on the space level and obtains new information about separability and lifting of automorphism groups to characteri* *stic zero. There is another approach to the "inseparability" questions treated in this * *pa- per. In Quillen, [15], the concept of an F -isomorphism provided a convenient t* *reat- ment of p-th powers. Quillen's main theorems have been treated more abstractly (for unstable algebras rather than equivariant cohomology) in work of Rector, [* *17] and Lam, [9]. We recall the principal technical result of Lam, and reinterpret * *it in terms of Theorem 2. Theorem 5. 1) Let R* be a sub-Ap-algebra of S[V #] such that S[V #] is algebraic over R*. Suppose in addition R* is p-closed, that is (S[V #])p \ R* = (R*)p. Then the top Dickson invariant element Y c0 = y y2V #-{0} lies in R*. 2) If R* ! S[V #] is algebraic but not necssarily p-closed, then there exists N 0 such that (c0)pN 2 R*. N 3) If z = cp0 is inverted to form S-1zR*, then Un(F R*) = Un(S-1zR*). Part 3) of Theorem 5 combined with the öl calization" invariance properties * *of the Lannes T -functor established in [2], gives the computation that each compo- nent of Lannes T -functor corresponding to a monomorphism _ : R* ! H*(BV ) is D*(R*), even in the absence of the noetherian or integrally closed requireme* *nts on R*. Of course, if we add the topological input that R* = H*(X) for some space X, then by D-M-W, [2], in fact D*(R*) = H*(BT n) for some n. The author would like to thank S. Mitchell and R. Stong for their interest a* *nd correspondence on this problem. The statement of Theorem 3 is due to them, and their questions led to the formulation of Theorem 2 from an earlier version of * *the results of Section Two. The author would also like to thank J.F. Adams for help* *ful comments on the organization of the proofs. 1. The Maximal Separable Extension Our technique is to apply field level arguments, and then use the Un-functor of taking the subalgebra of unstable elements to recover ring level results. A * *form of Un-functor appeared in Wilkerson, [20]. It has also found use in [1], [3], a* *nd [4]. INSEPARABLE FORMS OF RINGS OF INVARIANTS 5 Proposition 1.1. Let M* be an evenly graded module over Ap. Define the graded vector space Un(M*) as Un(M*)2i= {m 2 M2i|P jm = 0, 8j > i} Then a) Un(M*) is closed under the Ap- action on M*, and is an unstable Ap-module. b) If M* is a localization of some unstable Ap-algebra R*, then Un(M*) is an unstable Ap-algebra. c) Un(F S[V #]) = S[V #] d) If R* is an unstable integral domain, then u 2 F R* and u integral over R* implies that u 2 Un(F R*). Lemma 1.2. If R* is a graded unstable integral domain of finite transcendence degree with an unstable Ap-action, then R* is integrally closed and finitely ge* *nerated as an algebra if and only if R* = Un(F R*), the set of unstable elements in the graded field of fractions of R*. Proposition 1.3. Let R* be an Ap-sub-algebra of S[V #] such that S[V #] is algebraic over R*. Define S*Ras the set of elements in S[V #] which are separab* *le over R*. Then S*Ris a Ap-sub-algebra of S[V #] which contains R*. In addition, a) The fraction field F S*Rof S*Ris the maximal subfield I* of the fraction * *field of S[V #] which is separable over R*. b) R* ! S*Ris separable, and S*Ris maximal for this property. c) S*R= Un(F S*R) = Un(I*) and hence is integrally closed and noetherian. d) S*R! S[Vs#] is purely inseparable ( for each x in S[V #] there exists an s such that xp 2 S*R). Proof. Proof of Prop. 1.3 Let I* be the maximal separable extension of F R* in F S[V #]. Then S*R= Un(I*), since Un(F S[V #]) = S[V #]. It follows that S*Ris maximal among subal- gebras of S[V #] separable over R*. By I.1, S*Ris integrally closed and noether* *ian, since S*R= Un(F S*R). Now if u 2 S[V #], there exists N 0 so that upN 2 I*. Since this power of u is unstable, it is actually in S*R. It remains to show that F S*R= I*. Let u in F S[V #] be separable over R* , with minimal separable equation g(u) = 0 = a0uN +. .+.aN with aiin R*. Multiply by aN-10to obtain an integral equation for a0u = x over R*. Then x is in S[V #], since it is integral over R*. Also, x is separable over R*. Hence x 2 Un(I*) = * *S*R. Thus putting y = a0, we obtain u = x=y, with x and y in S*R. That is, F S*R= I*. Proof. Proof of Lemma 1.2 The favor of these arguments is similar to those in Wilkerson, [20]. From [1], R* is noetherian if and only if S[V #] is integral over R* via tR . Also, * *since Un(F S[V #]) = S[V #], for any subfield L*, Un(L*) = L* \ S[V #]. Now assume R* = Un(F R*). If u 2 F R* is integral over R*, then u 2 Un(F R*) = R*. That is, R* is integrally closed. Next if y 2 V # has x = vpN separable over F R*, x satisfies a minimal equation of the form Xr + c1Xr-1 . .+.cr = 0 with the coefficients {cj} in F R*. But since the extension is normal, and has one root x in S[V #] , it factors completely, with all roots in S[V #]. Hence * *the 6 C. W. WILKERSON coefficients {ci} are unstable, since they are polynomials in the roots. Thus x* * is integral over Un(F R*) = R*, V # is integral over R*, and so is S[V #]. From property a) of the introduction, R* is finitely generated. Conversely, assume that R* is noetherian and integrally closed. We need to show that R* = Un(F R*). Let u 2 Un(F R*). Then u 2 S[V #], and hence is integral over R* since R* is noetherian. Hence u is in the integral closure of * *R* in its field of fractions. But we assumed that R* is integrally closed, so u 2 R*.* * That is, Un(R*) R*. Proof. Proof of Proposition 1.1 If the action of the Bockstein were non-zero, the appropriate definition of * *an ü nstable" element would be more complicated. The concept of ü nstable" involves recursion, since if u is unstable, one also wants `u to be unstable, for any ` * *2 Ap. However, Lemma 2.6 from [1], shows that this recursion is automatic if only the reduced power Steenrod operations are considered. If M* is an algebra over Ap, then Un(M*) is also an algebra over Ap, and of course, Un(M*) is also an unstab* *le module. However, Un(M*) need not be an unstable algebra, since this requires also the condition P im2i= mp for each m. Although this is true for case b) , s* *ee [20], it is not true in general. Finally, let u = x=y in F R*. Let uN + c1uN-1 . .c.0= 0 be the monic equation of integrality, so that ci 2 R*. Apply the total Steenrod operation PT to the equation. Multiply through by PT(y)N , and compare coeffi- cients of powers of T . One sees that the coefficients for PT(u) vanish above h* *alf the dimension of u. A form of this argument appears in Wilkerson, [20]. 2. Jacobson Differential Correspondence The proof of Theorem 2 requires a study of the subfields intermediate between the fraction field F S[V #] of S[V #] and some ps-th power of this field. The s* *etting of inseparable field extensions has has a rich algebraic structure, and there i* *s a large established theory, see Winter, Chapters 5 and 6 for a survey. Let K ! L * *be an inseparable extension - the theory revolves around the correspondence between the subfield K and the subring EndK (L) of additive functions from L into itself which are K-linear. In the case of our interest L = F S[V #], the L-span of the Steenrod operations form a large part of the endomorphism ring. One in principle could work through the general theory, and prove Theorem 2 by characterizing the endomorphisms of S[V #] which are linear over R* as those which are linear over the diagonal algebra D*(R*). This would be in the spirit of section 5 of [1] and indeed was the original intent of this paper. However, we can apply a simpler theory. The general approach to inseparable Galois theory sketched above was designed to generalize to larger exponents the exponent 1 correspondence of Jacobson, ([7], Volume III, Chapter 4, page 186): Jacobson Differential Correspondence Let L be a field of characteristic p. There is a 1-1 correspondence between 1) subfields K of L which contain Lp and which have finite codimension in L ( dimK (L) < 1 ), INSEPARABLE FORMS OF RINGS OF INVARIANTS 7 and 2) finite dimensional L-subspaces of Der(L) which are closed under commutators and p-th powers (L-restricted Lie subalgebras of Der(L)). The correspondence is K _ DerK (L) and B _ LB , the constant field of B. Here dimK (L) = pdimLDerK (L) The present proof of Theorem 2 uses an induction step provided by the treat- ment of p-th powers by the Steenrod algebra, together with a version of the res* *ults of section 5 of [1]], interpreted by the Jacobson Differential Correspondence. * *Recall that the Milnor primitive {P (i)} from Ap acts as a derivation of degree 2(pi-* * 1) on any algebra over Ap, [1]. Proposition 2.1. Suppose S[V #]pN ! R* ! S[V #], are monomorphisms of unstable Ap-algebras, for some N 0, and that R* = Un(F R*). Define M* = R* \ (S[V #])p, so (F R*)p ! F M* ! F R*. Then a) The Milnor primitives {P (i)} span the graded derivations of F R* into F R* which vanish on F M*, DerFM* (F R*). b) RankFR* SpanFR* {P (i)} = rankFp(V \ R*) = nR . c) dimFM* (F R*) = pnR The Ap-action is crucial at this point. For example, consider the elements x = t21, y = t22, u = t1t2 in F2[t1, t2]. Let R* be the sub-algebra generated * *by these elements. Then R* is not closed under the Steenrod operations, but it is finitely generated and integrally closed. However, V \ R* = 0 even though rankFR* DerFM* (F R*) = 1. In general, there are many more intermediate rings between S[V #]p and S[V #] than those predicted by Proposition 2.1. The Steenrod action forces a more "linear" structure than the general theory can see. Lemma 2.2. If p = 2 or we restrict to evenly graded objects, then given an Fp-restricted graded Lie algebra L and an action of L on a commutative graded field K, the tensor product K FpL is a graded K-Lie algebra closed under bracke* *ts and p-th powers, and it inherits an action on K. Proof. Proof of Prop. 2.1.a By 2.2, the F R*-span of {P (i)} in Der(F R*) is a F R*-restricted Lie subalgebra. Since by [1] the Milnor primitives span t* *he derivations of F S[V #], an element of F S[V #] is a p-th power in F S[V #] if * *and only if it is annihilated by each P (i). That is, the constant field of the F* * R*- span of {P (i)} acting on F R* is F M*, the intersection of F R* with the p-th powers from F S[V #]. By the Jacobson Correspondence, the Milnor primitives span DerFM* (F R*). Proof. Proof of Proprositions 2.1.b and 2.1.c By 5.1 of [1], there exists for R* an nR such that any distinct nR of the {P (i)} are linearly independent over F R* in Der(F R*), and any nR + 1 are linearly dependent. This holds even if the "grading" derivation P~0is included. Here ~Px02n= nx. Hence 8 C. W. WILKERSON RankFR* SpanFR* {P (i)} = nR . For any non-trivial equation of linear dependence with nR + 1 terms ffi = c0P~0+ . .+.crP (nR) all of the coefficients {ci} are non-zero, since otherwise the choicerof nR wou* *ld be contradicted. If v 2 V0 = V \R*, then ffiv = 0 = c0v +. .+.crvp for r = nR . H* *encer dimFpV0 nR . However, by 5.2.(i-ii) of [1], all solutions of {c0X + . .+.crXp* * } are in S[V #]. Hence dimFpV0 = nR , since the equation has no repeated roots. Proof. Proof of 2.2 One takes the natural bracket definition : [aD, bffi] = ab[D, ffi] + aD(b)ffi - bffi(a)D and extends by bilinearity, keeping the action of L on K in mind. The p-th power operation requires more work. In fact there are two non-trivial formulas which * *are needed: 1) Given a 2 K and D 2 L, there exist b, c 2 K so that (aD)p = bD + cDp. This is attributed to Hochshild in exercise E.5.10, page 125 of Winter, [21]. This allows us to define the p-th power on monomials from K L. 2) If a, b 2 K and D, ffi 2 L, we need a formula for (aD + bffi)p in terms of brackets and p-th powers. Then induction on the number of summands in aiDi provides a general definition of the p-th power map on K L. The needed formula is provided in Jacobson, [6], page 187: In the free associative algebra Fp < X, Y >, p-1X (X + Y )p = Xp + Y p+ Si(X, Y ) 1 where the Si are (non-commutative) polynomials expressible in terms of iterated commutators in X and Y . Notice that even if the original Lie algebra L had zero p-th powers and zero* * Lie brackets ( as in the case of application to the Lie algebra spanned by the {P * *(i)}), the same is not necessarily true for the semi-tensor product Lie algebra constr* *ucted via these formulas. Remark 2.3. If the reader prefers an absolutely minimal path to the proof of Theorem B, the crucial fact established in this section is that *) dimFM* (F R*) = pdimFp(V \R This can be deduced in the case above from a result of Gerstenhaber and Zaromp quoted in exercise E.5.9 of [21], using the derivations {P (i)} for the* *ir {Di}. The crucial property needed is closure under the p-th power map, and this is clear for the {P (i)} INSEPARABLE FORMS OF RINGS OF INVARIANTS 9 3. Proof of Theorem 2 in the purely inseparable case By the results of Section One, we can replace the arbitrary R* by its maximal separable closure SR*. This leaves some details about the automorphism groups to be sorted out in Section Four, but the harder work is here, to show that if the extension R* ! S[V #] is purely inseparable and R* = Un(F R*), then R* = D*(R*). The method of proof is an induction using the results of Section Two. Recall that the exponent of an inseparableeextension is the smallest integer* * e such that for any x in F S[V #], xp is in F R*. The induction will be on the exponent e(R) of the inseparable extension F R** * ! F S[V #]. The exponent 0 case is covered by the hypothesis R* = Un(F R*), while the exponent 1 case is essentially Proposition 2.1. Exponent 1 Case: (F S[V #])p ! F R* ! F S[V #]. The filtration is U0(R*) = V \ R* and U1(R*) = V . We have (S[V #])p ! D*(R*) ! R* ! S[V #] But by Proposition 2.1, the dimension of F *R* and that of F D*(R*) over F S[V * *#]p are each pnR. Hence the two fields coincide. Special Case: If R* is entirely containedpin_S[Vp#]p,_we can replacepR*_bypt* *he_ algebra of its p-th roots in S[V #], pR*p._e(_pR*p=_e(R)_- 1 andpp_R*_= UnF pR* so the induction hypothesispgives_that pR* = D*( pR*). Since ( pR*)p = R* as Ap-algebras, R* = (D*( pR*))p = D*(R*). 00 Induction Step: If V0 = V \ R* 6= 0, then we use R* = R* \ S[V #]p. We need to show first that R*00! D*(R* ). Now 1 e(R00) max(e(R), 1). If it is0strictly0less0than0e(R), we appeal to the induction hypothesis to conclude t* *hat R* = D*(R* ) If the exponent is still e(R), we0obtain0the same isomorphism from the special case conclusion. Thus we obtain R* ! D*(R*). Then we have p *00 * * * R* ! R ! D (R ) ! R , and we must show that D*(R*) = R*. We again appeal to Proposition 2.1. Since U0 = V \ R* = V \ D*(R*) by definition of D*(R*), we have dimFR*00(F D*(R*)) = dimFR*00(F D*(R*)) Thus F D*(R*) = F R* and R* = D*(R*). Remark 3.1. The use of the Jacobson Correspondence in Proposition 2.1 to compute the relative derivations DerFR*00(F R*) is very helpful. If the full L* *ie algebra of derivations Der(F R*) were known, one could seek to explicitly compu* *te DerFR*00(F R*). However, while it is true that Der(F R*) can be spanned by line* *ar combinations of Steenrod operations, more than just the Milnor primitives are required in general. For example, if R* = F2[t1, t22] = F2[x, y] then the standard partial derivative basis for Der(F R*) is given by ~ ~ ~ ~ @=@x x-2Sq1 @=@y = (y4 + x4y2)-1(Sq(0,2)- x4Sq2) 10 C. W. WILKERSON 0in0terms of Steenrod operations. In this case, the constant field is (F R*)* *p, R* = F2[x2, y], and DerFR*00(F R*) = F R*{Sq1} = F R*{@=@x}, by direct computation. However, by Proposition 2.1 one achieves the same result by observing that with V \ R* = {0, x}, any Milnor primitive, since it is non-z* *ero on x will span the relative derivations. Such direct computation has its drawba* *cks for more complicated examples. 4. Proofs of Theorems 2, 3, and 5 We are left to provide proofs for those statements in Theorem 2 involving the automorphism groups. It is not obvious that the automorphisms computed in various possible ambient categories should agree. The essential point is that t* *hese are relative automorphisms that do preserve all structures on a large sub-objec* *t. Again these arguments are similar to those in Wilkerson, [20]. Proposition 4.1. Let R* be a sub-Ap-algebra of S[V #] such that S[V #] is algebraic over R*. Let L* be a graded field in F S[V #]. Then a) If i : F R* ! L is a separable extension of graded fields, then L has a u* *nique Ap action respecting the Cartan formula and such that i is a map of Ap-algebras. Thus any automorphism of L* which is the identity on R respects this Ap-structu* *re. b) If j : F R* ! F S[V #] is purely inseparable, then any graded field autom* *or- phism of F S[V #] which restricts to an Ap-automorphism of F R* is itself alrea* *dy an Ap-automorphism of F S[V #]. Hence, for SR* = D*(R*) as in Section Three c) AutR*(D*(R*)) = AutFR* (F D*(R*)). d) AutD*(R*)(S[V #]) = AutFD*(R*)(F S[V #]) = Id and e) AutR*(S[V #]) = AutFR* (F S[V #]) = AutR*(D*(R*)) and these preserve Steenrod operations. Proposition 4.2. Let U0 U1. . .V be an increasing filtration of V #, and D* the associated diagonal algebra. Then the following groups coincide: a) G1 : the subgroup of GL(V #) which respects the filtration. b) G2 : the gradation respecting, Steenrod action preserving algebra automor- phisms of D*. 3) G3 : the gradation respecting, Steenrod action preserving field automor- phisms of F D*. We remark that for S[V #] itself the automorphism groups in several differ- ent plausible categories coincide. However this property is not inherited for * *the "diagonal" subalgebras D*: Example 4.3. A gradation preserving algebra automorphism of D* need not respect the Steenrod algebra action. Details of 4.3: Let D* = F2[t1, t2, t23], and define _(t1) = t1, _(t2) = t2, but _(t23) = t23+ * *t1t2. Extend this to all of D*. Then _ does not commute with Sq1 on t23. Thus Theorem 4.1.a has non-trivial content. If one observes that D*(R*) is also a Hopf-algebra, then it is true that Hopf-algebra automorphisms of D*(R*) respect the Stennrod algebra action. INSEPARABLE FORMS OF RINGS OF INVARIANTS 11 Proof. Proof of Theorem 2 We first observe that if R* is replaced by its maximal separable extension in S[V #], S*R*, then Section II proves Theorem 2 in this special case, with W (S** *R*) = {Id} . However, it is clear that the filtrations and diagonal subalgebras defi* *ned by R* and S*R*agree. It remains to check that the action of W (R*) preserves the filtration on V . But if ypi is separable over R*, then any Galois conjugate of* * it is also separable over R*. The identification of the various possible definitions * *of the automorphism groups is done in 4.1. Proof. Proof of Theorem 3 Let R* = Fp[y1, . .,.yn] be the polynomial algebra, and D*(R*) its maximal separable extension in S[V #]. Any derivation of R* or its fraction field F R* * *into F D*(R*) extends uniquely to a derivation on F D*(R*), since the extension is s* *ep- arable. More explicitly, if ffi 2 Der(F R*) and ff 2 F D*(R*) has minimal separ* *able equation over F R* of f(ff) = ffN + b1ffN-1 + . .+.b0 then X ffiff = -( ffN-iffibi)=f0(ff) Since f is separable, there are no repeated roots and f(ff) 6= 0 . In the case * *at hand of polynomial algebras, the partial derivatives in {yi} and {zj} form bases res* *pec- tively for Der(F R*) and Der(F D*(R*)). But from the above {@=@yi} is also a basis for Der(F D*(R*)) over F D*(R*). The Jacobian |@yi=@zj| is the determinant of the change of basis matrix for these two bases of Der(F D*(R*)) over F D*(R** *), and hence must be non-zero. Proof. Proof of Prop.4.2 Certainly, G1 G2 G3 We know that D* = Un(F R*). Since G3 induces an automorphism of D*, evidently G3 G2. But the induced automorphism on D* extends uniquely to S[V #], by the rule that if vpN 2 D*, then _(v) = (_vpN)1=pN. Since p-th roots are unique if they exist, th* *is extension is well defined, since the question of whether an element is a p-th p* *ower is detectable by the Steenrod action. Hence G1 G2. Since Un(F D*) = D*, one has G2 = G3 immediately. Proof. Proof of Theorem V Statement 1) is directly from Lam, [9] For 2) consider the p-th root closure of (R*)00in S[V #]. That is, y 2 (R*)0* *0if and only if there exists some N 0 with ypN 2 R*. Then by definition, (S[V #])* *p\ (R*)00 ((R*)00)p and 1) applies. That is, c0 2 (R*)00, and there exists N 0 * *with N cp0 2 R*. This argument holds for any non-zero ideal of R* which is closed under the Steenrod algebra_action_also. ___ To prove 3), let R* = Un(F R*) . If y 2 R* consider the conductor ideal C(y, R*) R* consisting of those elements r 2 R* for which ry 2 R*. Let S(y, R* **) be the radical of this ideal, that is, the set of all elements which have a pow* *er in C(y, R*). Finally, define ___ * " * S(R* , R ) = y2__R*S(y, R ) ___ Then we claim that S(R* , R*) is a non-zero ideal in R* which is closed under t* *he action of the Steenrod algebra. If so, then by part 2) of this theorem, some po* *wer 12 C. W. WILKERSON ___ of c0 falls in S(R* , R*). That is, ___ -1 * S-1zR*= Sz R ___ It remains to check that S(R* , R*) is closed under the Steenrod algebra act* *ion. It suffices to check this for each S(y, R*). (this need not be true for each C(* *y, R*)). Let |y| = 2M, and r 2 C(y, R*). Now use the Cartan formula to compute MpM ipMpM X jpM pM (i-j)pM i pM P ipr y = (P r )y + P r P y = (P r) y j>0 The sparseness of the first expansion is due to the treatment of p-th powers by the Steenrod algebra. But moreover, since y is unstable of dimension 2M, each term on the right hand side except the first vanishes, since (i - j)pM > M. Th* *is demonstrates that if r 2 C(y, R*), then P ir 2 S(y, R*), and by replacing r by a suitable p-th power, that P_iS(y,_R*) S(y, R*).__ Finally, to show that S(R* , R*) 6= 0, since_R* is notherian, we can choose a finite set {yk} of algebra generators for R*. Then for each k, there exist wk, * *xk 2 R* such that yk = xk=wk. Then the element Y = wk ___ is in S(R* , R*). 5. Discussion and Examples In this section, we make some comments on the algebraic classification of un- stable domains and give some illustrations of the filtration structure pointed * *out by Theorem 2. The special case of polynomial algebras has great historical interest. The a* *p- plication of Theorem 2 to this case is a success, since Theorem IV from [D-M-W] gives very strong restrictions on such algebras to be topologically realizable.* * In particular, the filtration produced in Theorem 2 must degenerate to U0(R*) = V for odd primes p. However from a strictly algebraic viewpoint, there is still a nagging questi* *on as to whether the property of the W -action having a polynomial algebra as its * *ring of invariants is inherited by the action on the diagonal algebras D*(R*), and v* *ice versa. More precisely, Conjecture 5.1. Let W be a subgroup of GL(V #) such that W restricts to an action on D*, a diagonal subalgebra of S[V #]. Then (D*)W is a polynomial algebra if and only if S[V #]W is a polynomial algebra. This conjecture is easily verified if the order of W is prime to p, since in* * that case the ring of invariants is a polynomial algebra if and only if W is generat* *ed by generalized reflections. Without attempting to restrict to just polynomial algebras, Theorem 2 pro- vides give a construction and classification of all Ap - integral domains which* * are noetherian and integrally closed. One can view this in two steps: 1) Choose a particular subgroup W GL(V ). This generates the separable example as S[V #]W 2) The inseparable forms of this algebra W are are then parameterized by W - invariant filtrations of V and choices of exponents, or equivalently, a "diagon* *al" INSEPARABLE FORMS OF RINGS OF INVARIANTS 13 algebra D* such that the action of W on S[V #] restricts to D*. So the insepara* *ble form is R* = (D*)W . Notice that if the representation of W has no properNinvariant subspaces, th* *en there are always only the "standard" copies (S[V #]p )W associated with W. The* *re- fore, non-trivial examples of Theorem 2 must use non-trivial W - filtrations on* * V . Example 5.2. If one restricts the search to classical rationally irreducible* * Weyl groups, the examples are rather restricted. Various forms of G=C0, where C0 is a finite central subgroups, give non-trivial W filtrations. There are two interes* *ting examples for which the filtrations do not involve a trivial submodule in the as* *soci- ated graded. These occur for W (G2) at the prime 3 and W (F4) at the prime 2. We treat the G2 example in detail. For p = 3, this Weyl group provides an example * *of a proper invariant subspace which has a non-trivial W -action. The Weyl group of G2 is the dihedral group D12 of order 12. The mod 3 cohomology of BG2 is a ring of invariants which provides a good illustration of two possible viewpoints of * *the Steenrod algebra action. We can compare the internal view provided by the expli* *cit Steenrod algebra action with the external view provided by the Adams-Wilkerson embedding and its associated Weyl group action. The filtration is easiest to se* *e in terms of the W -action, but it is also visible indirectly in the formulas detai* *ling the Steenrod algebra action. We take the repesentation of D12 to be determined by the two reflections ~ ~ ~ ~ ff = -10 11 , fi = 10 -01 in GL2(Z). So over F3, there are generators for the invariants x4 = t21, and < y12= (t2(t1 + t2)(t1 - t2))2 > . There is only one nontrivial stable subspace for D12, {0, t1, -t1}, and it h* *as no D12 complement. This filtration corresponds to the family of A3-polynomial algebras parametrized by N, M 0 N 3M (F3[x4, y312]) in F3[x4, y12] = H*(BG2, F3). Here the A3 -actions can be specified as N 3N 3N 3N 3N+13N 3N 3N+1 3N P 1x = -x2, P 3 y = x y , P y = y (x - y ). and analogues for M > 0.MOn the other hand one sees from these formulasMthat one can not introduce a x3 instead of x without also substituting y3 for y. That * *is, M 3N F3[x3 , y ] is not closed under the A3 -action if N < M. This is reflected on the vector sp* *ace level by the fact that there is only one proper invariant subspace. In particu* *lar, there is no W -splitting of the filtration in this case. For G2 for p > 3, any proper invariant subspace would also be a direct sum- mand, but in fact, there are no invariant 1 dimensional subspaces for p > 3. He* *nce 14 C. W. WILKERSON for p > 3 the only inseparable forms of Fp[x4, y12] have the form Fp[x4, y12]pN* *, for N non-negative. The matrices of the W (G2) action at the prime 3 can be pictured as ~ ~ GL1(F3 X 0 GL1(F3 ) in which 0 and X are (1 x 1) F3-matrices. The related W (F4) example at p = 2 has a similar description as ~ ~ GL2(F2 ) X 0 GL2(F2 ) in which 0 and X are (2 x 2) F2-matrices. The analysis is quite similar to the * *G2 example. One could extend these examples to other primes p and n, but these two are natural because of the Lie connection. Example 5.3. We now sketch the classification of noetherian integrally closed unstable domains of rank 2 in F2[t1, t2]. We define some useful classes in this ambient polynomial ring in order to ease the notation later. w1 = t1 + t2, w2 = t1t2 are the Steifel-Whitney classes, and the Dickson invariants are c0 = t1t2(t1 + t2), c1 = t21+ t22+ t1t2 Finally u = t31+ t32+ t21t2 . We now provide the classification by listing the Weyl groups, filtrations, a* *nd exponents that are possible for a given group. 1) W = (id). All subspaces are invariant. The algebras contructed by choices of filtrations and exponents are isomorphic to F2[t2i1, t2j2]. 2) W = Z=2Z . Up to conjugation, we can use the representation ~ ~ ø = 01 10 There is one non-trivial subspace, spanned by w1 . Hence the examples corre- sponding to 0 V0 V and a choice of exponents N and M give rise to algebras isomorphic to N 2M F2[w21, w2 ] F2[w1, w2] with M N. 3) W = Z=3Z. Up to conjugation, we can use the representation ~ ~ æ = 01 11 There are no non-trivial W -invariant subspaces. In this case the invariants ar* *e not polynomial. In fact, R* = S[V #]W = F2[c0, c1, u]=(u2 + c0u + c20+ c31). Any inseparable examples have the form (R*)2N. INSEPARABLE FORMS OF RINGS OF INVARIANTS 15 4) W = GL(V ) and we can use as generators for the representation the matric* *es for æ and ø in 2) and 3) above. Again, there are no proper invariant subspaces,* * so the only inseparable forms are F2[c0, c1]2N. From the inseparability viewpoint, only 1) and 2) are interesting. The reader might wish to attempt this classification of rank 2 unstable polynomial algebras over F2 by direct computation instead of this contruction suggested by Theorem * *2. One special case of 1) and 2) that arises in the classification of the equiv* *ariant cohomology rings associated to involutions on cohomology projective planes is Proposition 5.4. If the polynomial algebra F2[x1, ym ] with |x1| = 1, |ym | * *= m has an unstable A2-action then m = 2N and either a) N F2[x1, y2N] F2[t1, (t2)2 ], |tj| = 1 or b) N-1 * F2[x1, y2N] F2[w1, (w2)2 ] H (BO(2)) In case a) Sqky = 0, 0 < k < 2N In case b) Sqky = 0, 0 < k < 2N-1 and Sq2N-1y = x2N-1y Of course, the formulas for the Steenrod operations in case b) can be pertur* *bed slightly if one replaces the generator y with y + xm . Historical Notes: This paper was originally accepted by J. F. Adams for the Proceedings of the Cambridge Philosophical Society. In the aftermath of his untimely death, the pa* *per slipped through the cracks at the author's end and the final copy was never del* *ivered to the journal. On the other hand, it has been available as a preprint since 1984 and in ele* *c- tronic form on the Hopf Archive since 1991. The paper has been referenced, e.g. (Mitchell-Stong, Aguade, Dwyer-Miller-Wilkerson), and extensively excerpted in the book of Kane, [8]. Related material in a more general context has been deve* *l- oped by M. Neusel [13, 14]. The localization approach is related to the Lannes T - functor, e.g. [4]. One might therefore ask if there are alternative T -theoretic proofs of these theor* *ems. To the best of the author's knowledge, there are not easy T -replacements for t* *he totality of these. However, certainly, there are nice restatements and interpr* *eta- tions of most of Theorem 2. The author has resisted making substantive changes to the paper of record on Hopf. One concession was to change the original notation for H*(BT n, Fp) to match that of the recent paper of Dwyer and Wilkerson, [5], S[V #]. A second has been to supply more comments in this last section. References 1.J. F. Adams and C. W. Wilkerson, Finite H-spaces and algebras over the Steen* *rod algebra, Annals of Mathematics 111 (1980), no. 1, 95-143. 2.W. G. Dwyer, H. R. Miller, and C. W. Wilkerson, The homotopical uniqueness o* *f classifying spaces, J. of Topology 31 (1992), no. 1, 29-45. 16 C. W. WILKERSON 3.W. G. Dwyer and C. W. Wilkerson, Smith theory revisited, Annals of Mathemati* *cs 127 (1988), no. 1, 191-198. 4._____, Smith theory and the functor T, Comment. Math. Helv. 88 (1991), no. 1* *, 1-17. 5._____, Kahler differentials, the T-functor, and a theorem of Steinberg, Tran* *s. Amer. Math. Soc. 350 (1998), no. 12, 4919-4930. 6.N. Jacobson, Lie Algebras, Wiley (Interscience), New York, 1962. 7._____, Lectures in Abstract Algebra, Vol. III., Van Nostrand, Princeton, 196* *4. 8.Richard M. Kane, The Homology of Hopf Spaces, North-Holland Mathematical Lib* *rary, vol. 40, North-Holland Publishing Co., Amsterdam, 1988. 9.S. P. Lam, Unstable algebras over the Steenrod algebras, Ph.D. thesis, Trini* *ty College, Cam- bridge, 1982. 10.S. Lang, Algebra, Addison-Wesley, Reading, 1965. 11.J. Lannes, Sur les espaces fonctionnels dont la source est le classifiant d'* *un p-groupe abelien elementaire., Inst. Hautes Etudes Sci. Publ. Math. (1992), 135-244. 12.S. Mitchell and R. Stong, An adjoint representation for polynomial algebras,* * Proc. Amer. Math. Soc. 101 (1987), no. 1, 161-167. 13.M. Neusel, Inverse Invariant Theory and Steenrod Operations, vol. 146, Memoi* *rs of the AMS, no. 692, American Mathematical Society, Providence, 2000. 14.M. Neusel, Localizations over the Steenrod Algebra. The Lost Chapter, Math. * *Z. 235 (2000), 353-378. 15.D. Quillen, The spectrum of an equivalent cohomology ring I, Annals of Mathe* *matics 94 (1971), 549-572. 16._____, The spectrum of an equivalent cohomology ring II, Annals of Mathemati* *cs 94 (1971), 573-602. 17.D. L. Rector, Noetherian cohomology rings and finite loop spaces with torsio* *n, Journal of Pure and Applied Algebra (1984), 191-217. 18.J.-P. Serre, Sur la dimension cohomologique des groupes profinis, J. of Topo* *logy 3 (1965), 413-420. 19.C. W. Wilkerson, Classifying spaces, Steenrod operations and algebraic closu* *re, J. of Topology 16 (1977), 227-237. 20._____, Integral closure for algebras over the Steenrod algebra, J. of Pure a* *nd Applied Algebra 18 (1979), 49-56. 21.D. Winter, The Structure of Fields, Grad. Texts in Math., vol. 16, Springer * *Verlag, New York, 1974. Department of Mathematics, Wayne State University, Detroit, Michigan Current address: Department of Mathematics, Purdue University, W. Lafayette,* * Indiana 47907-1395 E-mail address: wilker@math.purdue.edu