0 (hence, Assumption 2.1 holds). Also, let CB denote the category of quasi-elementary sub-Hopf algebras of B and their sub-Hopf algebras. In this section we present an Avrunin-Scott theorem for B-modules, Theorem 3.4. Before proceeding further we need to make the following assumption. This assumption holds for group algebras (in which case quasi-elementary is the same as elementary abelian) and for sub-Hopf algebras of the Steenrod algebra (see Section 5). Assumption 3.1. If Q is a quasi-elementary sub-Hopf algebra of B and Q0 Q, then Q0 is quasi-elementary. In this case, the induced map res*Q;Q0: |Q0|k ! |Q|k is injective. We will now proceed to prove the stratification theorem for modules. The method of proof will follow the presentation of the analogous group theoretic result found in [Ben , Chapter 5]). For the sake of clarity to the reader we have included details of some of the proofs along with the logical progression of the relevant theorems. The first result provides a decomposition of the variety |B|M into a union of closed subvarieties. Proposition 3.2. For any finitely generated B-module M, we have [ |B|M = res*B;Q|Q|M : Q2CB Proof. This follows from Theorem 2.4(b), just as in [Ben , Thm. 5.7.4]. SUPPORT VARIETIES FOR THE STEENROD ALGEBRA 9 Lemma 3.3. (a) Suppose C / B with B==C = k[x]=(x2). Let v 2 Ext 1B(k; k) be the image of a non-zero element in Ext 1B==C(k; k). Fix z 2 Ext *C(k; k); if resB;C(z) 2 JC (M), then some power of z lies in the ideal generated by JB (M) and v. (b) Let p be odd and C/B with B==C = k[x]=(xp). Let v 2 Ext 1B(k; k) be the image of a non-zero element in Ext 1B==C(k; k). Fix z 2 Ext *C(k; k); if resB;C(z) 2 JC (M), then some power of z lies in the ideal generated by JB (M) and fiPe0(v). (c) Let Q0 Q be quasi-elementary sub-Hopf algebras of B and M be a Q-module. Then (identifying |Q0|k with its image under |Q0|k ,! |Q|k) we have |Q0|M = |Q0|k \ |Q|M : Proof. The first two assertions, (a) and (b), can be proved as in [Ben , Lem. 5.7.6] by applying [P , Lemma 3.2]. Part (c) follows by using parts (a)-(b) and [HS , Lemma A.11], just as in [Ben , Prop. 5.7.7]. The next result provides a stratification of the variety |B|M into a disjoint union of locally closed subvarieties. Theorem 3.4. Let M be a finitely generated B-module. (a) The variety |B|M is a disjoint union of subvarieties a (3.1) |B|M = res*B;Q(|Q|+M): Q2CB Furthermore, the map res*B;Qinduces an inseparable isogeny |Q|+M ! res*B;Q(|Q|+M): (b) The natural map lim-!Q2CB|Q|M ! |B|M is an inseparable isogeny. Proof. By Lemma 3.3(c), we see that [ |Q|+M = |Q|M \ res*Q;Q0|Q0|M ; Q0isnot free}; (b) cQ (M) = dim |Q|M = dim (|Q|M ). Proof. Let V (LQ ) denote the restricted enveloping algebra of LQ with the usual coalgebra structure and let L]Q denote the dual of LQ . For any V (LQ )-module M, it was shown in [FP1 , (1.4)] there exists a finite map oeo : S*(L]Q) ! H*(V (LQ ); k) which induces a map oe : |LQ |k ! LQ such that (|LQ |M ) = {x 2 LQ : x[p]= 0; M|is not free}: Here, |LQ |M = Maxspec (RV (LQ)=IV (LQ)(M)). A simple calculation shows that there is an isomorphism ffio : S*(Z]Q) ! H*(E(ZQ ); k), and one can show (for example, by using [Mil , 2.2 and 2.3] or by imi- tating the argument in [Ben , Thm. 5.8.3]) that the map on varieties ffi : |E(ZQ )| ! ZQ has the property that ffi(|E(ZQ )|M ) = {x 2 ZQ : x2 = 0; M| is not free}: Since Q ~=Q0E0 as Hopf algebras with Q0 = V (LQ ) and E0 = E(ZQ ) as algebras we can use oeo and ffio to get a finite map (4.1) o : S*((LQ ZQ )]) ! H*(Q0; k) H*(E0; k) ~=H*(Q; k): This induces the map = (oe; ffi) : |Q|k ! LQ ZQ . The first assertion is now proven by using the fact that |Q|M ~=|Q0|M x|E0|M . The second assertion follows by Theorem 2.2 and the fact that is finite. Proposition 4.3. Suppose Q0 Q, with associated Lie algebras LQ0 LQ and vector spaces ZQ0 ZQ . Let M and N be finitely generated Q-modules. Then (a) (|Q0|M ) = (|Q|M ) \ (LQ0 ZQ0). (b) (|Q|MN ) = (|Q|M ) [ (|Q|N ). (c) (|Q|MN ) = (|Q|M ) \ (|Q|N ), if : |Q|k ! LQ ZQ is an isomorphism. 12 DANIEL K. NAKANO AND JOHN H. PALMIERI Proof. The first part (a) follows immediately from Theorem 4.2. The second assertion (b) is true because |Q|MN ~=|Q|M [ |Q|N . Part (c) holds because of Theorem 3.6 and the fact that |Q|M is a subvariety of |Q|k. We will show in Lemma 5.3 that for any quasi-elementary sub-Hopf algebra Q of the Steenrod algebra, : |Q|k ! LQ ZQ is an isomor- phism. Now we show that any closed homogeneous subvariety of |B|k is realizable as |B|M for some finitely generated module M. Theorem 4.4. Let B be a finite-dimensional graded cocommutative Hopf algebra satisfying Assumptions 3.1 and 4.1, and suppose that for each Q 2 CB , the map : |Q|k ! LQ ZQ is an isomorphism. (a) Any closed homogeneous variety of |B|k is equal to |B|M for some finite-dimensional B-module M. (b) If M is a finite-dimensional indecomposable B-module then the corresponding projective variety Proj(|B|M ) is connected. Proof. First we describe the support varieties for particular modules, the Li's; from this, we will be able to prove part (a) of the theorem. For i 2 Hn (B; k) = Hom B(n(k); k), let Li be the kernel of the map i : n(k) ! k. Here, 1(k) is the kernel of the map P0 ! k and n(k) is the kernel of the map Pn-1 ! Pn-2 for n 2 in (2.1) for N = k. Also, let |B|k be the set of all maximal ideals in H*(B; k) containing i. We claim that |B|Li =. In order to prove this, it suffices to assume that B = Q is a quasi- elementary Hopf algebra_see the remarks preceding [Ben , Prop. 5.9.1] and Theorem 3.4. We have o : S*((LQ ZQ )]) ! H*(Q; k) (4.1). Let i be such that there is an F 2 S*((LQ ZQ )]) with o(F ) = i. By extending the results in [FP2 , (4.2)] it follows that (|Q|Li) is Z(F ) where Z(F ) is the zero set of F . Now observe that () Z(F ) by definition of . If m 2 |Q|Li, then F 2 (o)-1(m) because (|Q|Li) = Z(F ); therefore, i 2 m and |Q|Li Z(F ). Applying it follows that Z(F ) = (|Q|Li) (), thus () = Z(F ) and (|Q|Li) = (). Since is an isomorphism, we have |Q|Li =. Now assume W is a closed homogeneous subvariety of |B|k and let I(W ) = (i1; i2; : :i:s) H*(B; k) be the corresponding ideal (o is finite so we may choose ij such that there exists Fj which maps to ij under o). Let M = Li1 Li2 : : :Lis. Then by Theorem 3.6 |B|M = \si=1|B|Li = \si=1= W: The second assertion (b) follows by the same argument given in [Ben , Thm. 5.12.1] with (a) and Theorem 3.6. SUPPORT VARIETIES FOR THE STEENROD ALGEBRA 13 5 Applications to the Steenrod algebra In this section we apply the results in the previous three sections to finite sub-Hopf algebras of the mod p Steenrod algebra (for all p) and the mod p reduced powers (for p odd). Recall that we defined Hopf algebras P and A in Section 1. The sub-Hopf algebras of P and A have been classified; for example, any sub-Hopf algebra B of A has the form i n1 n2 * j B = k[1; 2; 3; : :]: E[o0; o1; o2; : :]:=(p1 ; p2 ; : :;:o0e0; o1e1; : :): for exponents ni 2 {0; 1; 2; : :}:[ {1} and ei 2 {0; 1} satisfying cer- tain conditions (see [AD ] and [AM2 ]). The sequence of exponents (n1; n2; : :;:e0; e1; : :):is called the profile function for B and we write B = A(n1; n2; : :;:e0; e1; : :):. The analogous results hold for sub-Hopf algebras of P , and we use similar notation. We define sub-Hopf algebras E(m); Q(m) P and R(m) A: E(m) = P (0;_0;_:_:;:0;-0z_____"; m + 1; m + 1; m + 1; : :):; m 0; m Q(m) = P (0;_0;_:_:;:0-z____"; 1; m; m; m; : :):; m 2; m-2 R(0) = A(0; 0; 0; : :;:0; 0; 0; : :):; R(1) = A(1; 1; 1; : :;:1; 0; 0; : :):; R(m) = A(0;_0;_:_:;:0-z____"; 1; m; m; m; : :;:1;_1;_:_:;:1-z____"; 0; 0; 0; :* * :):; m 2: m-2 m We will show that a sub-Hopf algebra Q of P is quasi-elementary if and only if Q E(m) for some m when p = 2, or Q Q(m) for some m when p is odd. Similarly, Q A is quasi-elementary if and only if Q R(m) for some m 0. First we compute the cohomology of these Hopf algebras. Lemma 5.1. (a) For p = 2, any Q E(m) is an exterior algebra on the Pts's that it contains; e.g., E(m) = E(Pts: t m + 1; s m): Hence Ext *Q(k; k) is a polynomial algebra, with one generator hts for each Pts2 Q. (b) For p odd, any Q Q(m) is isomorphic as an algebra to the restricted universal enveloping algebra on the Pts's that it con- tains. Furthermore, Ext *Q(k; k) is of the form P T , where P is a polynomial algebra with one generator bts for each Pts2 Q, 14 DANIEL K. NAKANO AND JOHN H. PALMIERI and T consists of nilpotent elements (T is a subquotient of the exterior algebra E(hts: Pts2 Q)). (c) For p odd, any R R(m) is isomorphic as a Hopf algebra to the tensor product of an exterior algebra on the Qn's in R with the algebra generated by the Pts's in R. The latter, considered as a sub-Hopf algebra of P , is a sub-Hopf algebra of Q(m). Hence Ext *R(k; k) is of the form P 0 P T , where P 0is a polynomial algebra with one generator vn for each Qn 2 R, and P and T are as in part (b). Proof. For part (a), it is well-known that E(m) (and hence any sub- Hopf algebra Q E(m)) is exterior; see [Mar ], for example. The cohomology computation is standard. Part (c) follows from part (b) and an easy computation with Milnor basis elements. We prove part (b) for Q = Q(m); the reader can easily do the more general case. Note that the collection of Pts's under consideration is S = {Pm0-1} [ {Pvu: v m; m - 1 u 0}: These all satisfy (Pts)p = 0; also, the only non-zero commutator rela- tions between them are the following: for all v m, we have [Pm0-1; Pvm-1] = Pm0+v-1: Let L be the vector space spanned by S. Then L is a graded restricted Lie algebra contained in Q(m), and L generates Q(m) as an algebra. Therefore, there is an induced epimorphism of algebras V (L)!!Q(m): By comparing dimensions, we see that this map must be an isomor- phism. To compute the cohomology of Q(m), we let D = P (0;_:_:;:0-z__"; m - m-1 1; m-1; m-1; : :):, and we use the Cartan-Eilenberg spectral sequence associated to the central extension D ! Q(m) ! Q(m)==D: Both D and Q(m)==D are polynomial algebras truncated at height p, so we can compute their cohomology to the get the E2-term: E2 ~=k[bts: Pts2 Q(m)] E(hts: Pts2 Q(m)): Just as in the computation in [W , 6.3], d2 is determined by the family of differentials (for all t m) d2(ht0) = hm-1;0ht-m+1;m-1 ; and the spectral sequence collapses at E3. SUPPORT VARIETIES FOR THE STEENROD ALGEBRA 15 We can now show that the sub-Hopf algebras of E(m), Q(m), and R(m) are precisely the quasi-elementary sub-Hopf algebras of P for p = 2, P for p odd, and of A, respectively. Proposition 5.2. (a) For p = 2, a sub-Hopf algebra Q of P is quasi-elementary if and only if Q E(m) for some m. (b) For p odd, a sub-Hopf algebra Q of P is quasi-elementary if and only if Q Q(m) for some m. (c) For p odd, a sub-Hopf algebra Q of A is quasi-elementary if and only if Q R(m) for some m. Proof. Part (a) follows from [W , 6.4], so we assume that p is odd. For part (b), if Q Q(m), then by the computation in Lemma 5.1, Ext1Q(k; k) is spanned as a vector space by a subset of {hts: Pts2 Q}. For each hts2 Ext *Q(k; k), we have fiPe0(hts) = bts, and no product of bts's is nilpotent. Hence Q is quasi-elementary. Conversely, one can apply the arguments in [W , 6.4] to show that if Q is not a proper sub-Hopf algebra of any Q(m), then Q is not quasi-elementary. For part (c), if Q R(m), then Q is quasi-elementary by the previous part and Lemma 5.1. Conversely, assume that Q is quasi-elementary. Suppose n is the smallest integer so that Qn 2 Q. Then Pts2 Q only if s < n_otherwise, if t is the smallest integer so that Ptn2 Q, thennin Q we will have Qn+t = [Qn; Ptn]. In Ext *Q(k; k), then, htn = [pt] and vn = [on] are permanent cycles, and we have the relation htnvn = 0. n+t epn+t p Applying fiPp (not fiP ) gives btnvn = 0, so Q could not be quasi- elementary. Hence as Hopf algebras, Q splits as a tensor product as in the state- ment of Lemma 5.1(c): Q = E Q, with E exterior on a collection of Qn's, and Q0 P . Now, Q A is quasi-elementary if and only if Q0 P is, so apply part (b). Recall that Assumption 2.1 holds for all finite-dimensional graded connected cocommutative Hopf algebras. For the finite-dimensional quasi-elementary sub-Hopf algebras of P and A, Assumption 3.1 follows from Proposition 5.2 and our cohomology calculations. Furthermore, Assumption 4.1 is the content of Lemma 5.1. We record the following lemma, for later use. Lemma 5.3. If Q is a finite-dimensional quasi-elementary sub-Hopf algebra of P or of A, then the map of Theorem 4.2 gives an isomor- phism between |Q|k and LQ , for Q P , or between |Q|k and LQ ZQ , for Q A. 16 DANIEL K. NAKANO AND JOHN H. PALMIERI Proof. By our cohomology calculations, the variety |Q|k is affine of di- mension #D(Q). According to Theorem 4.2, dim |Q|k = dim (|Q|k) = dim kLQ ; hence, (|Q|k) ~=LQ and the result follows. We now present the proofs of the non-trivial parts of the results in Section 1. Proof of Theorem 1.1. Part (a) is a combination of Assumption 3.1 and Corollary 3.5. Part (b) follows from Theorem 4.2 and Lemma 5.3. Proof of Theorem 1.2(b). This follows by Proposition 4.3 since is an isomorphism. Proof of Theorem 1.4. The irreducible components of the variety |B|k are the varieties |Q|k, one for each Q 2 ^CB, so the depth of Ext *B(k; k) is bounded above by min Q2C^Bdim |Q|k. Proof of Corollary 1.5.This follows from Theorem 1.1(c) and Theo- rem 1.2(b). Proof of Corollary 1.6.Let T = Gm be the one-dimensional algebraic group over k. If t = (a), a 2 k - {0}, then T acts as a group of automorphisms on B by letting t:b = adeg bb. In this setting the category of graded B-modules is equivalent to the category of B-modules which admit an additional T -structure whose module map B M ! M is T - equivariant. Furthermore, (|Q|M ) is a T -stable variety for any graded B-module (see [FP1 ], [FP2 ]). For part (a) if M is a free B-module, then (|Q|M ) = 0 for all Q 2 CB ; hence by Theorem 1.1(c), H(M; x) = 0 for all x 2 D(B). Conversely, suppose M is not free as a B-module; then (|Q|M ) 6= {0} for some Q 2 CB . Since M is graded the variety (|Q|M ) is T - stable. Since the differentials in B have distinct degrees, it follows that there exists a differential x 2 (|Q|M ) (see [FP1 , Prop. 3.4]), and thus H(M; x) 6= 0. Part (b) can be proved by using the same argument given in [N , Thm. 2.2] (for G1T -modules). Proof of Theorem 1.7. We know that for each quasi-elementary Q A there is a bijection Q : |Q|M ! {x 2 V : x 2 Q; x2 = 0; H(M; x) 6= 0}: Furthermore, these maps are natural with respect to inclusions Q0 Q of quasi-elementary Hopf algebras. Hence we want to show that x 2 V satisfies x2 = 0 if and only if x 2 V \ Q for some quasi-elementary Q A. SUPPORT VARIETIES FOR THE STEENROD ALGEBRA 17 P s 2 s2 Let x = Pst2A;s t1 and u2 t1, so [Pts11; Pvu22] 6= 0, and is in a smaller degree. We conclude with some examples. Let Pn be the sub-Hopf algebra of P with profile function (n+1; n; n-1; : :;:2; 1; 0; 0; 0; : :):. We say that a map ae : H1 ! H2 of commutative k-algebras is an F -isomorphism if H2 is finitely generated as a module over H1, the kernel of ae consistsnof nilpotent elements, and for every y 2 H2, for some n we have yp 2 im ae. Example 5.4. (a) For all p, P0 = k[P10]=((P10)p), so |P0|k is affine 1-space, spanned by P10. (b) For all p, |P1|k is affine 2-space, spanned by P10and P20. This means that H*(P1; k) is F -isomorphic to a rank 2 polynomial algebra. For instance, when p = 2, H*(P1; k) has a subalgebra k[h10; h420] over which H*(P1; k) is finitely generated. (c) When p = 2, then |P2|k is Span{P10; P20; P30} [ Span {P20; P30; P21} Span {P10; P20; P30; P21}: 18 DANIEL K. NAKANO AND JOHN H. 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Quillen, The spectrum of an equivariant cohomology ring I and II, An* *n. of Math. 94 (1971), 549-602. [W] C. Wilkerson, The cohomology algebras of finite dimensional Hopf algebras, Trans. Amer. Math. Soc. 264 (1981), 137-150. Department of Mathematics, Northwestern University, Evanston, IL. 60201 U.S.A. E-mail address: nakano@math.nwu.edu Department of Mathematics, M.I.T., Cambridge, MA 02139 U.S.A E-mail address: palmieri@math.mit.edu