THE MOTIVIC COHOMOLOGY OF CLASSICAL QUOTIENTS OF MGL

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1 THE MOTIVIC COHOMOLOGY OF CLASSICAL QUOTIENTS OF MGL MARC HOYOIS Contents 1. The Hurewicz map for MGL 1 2. Complements on the motivic Steenrod algebra 3 3. Complements on H-modules 6 4. The motivic homology of regular quotients of MGL 7 5. Key lemmas The motivic cohomology of quotients of BP 12 References 13 In this note we compute the mod l motivic cohomology of some quotients of algebraic cobordism over a perfect field k of characteristic not equal to l, which is prime, as modules over the motivic Steenrod algebra. The methods we use are elementary and work equally well to compute the mod l singular cohomology of the analogous quotients of complex cobordism, such as connective Morava K-theory, at least if l is odd (if l = 2 the topological Steenrod algebra has a different structure and some modifications are required). The motivic computation requires a little more care, however, because of the nontrivial action of Steenrod operations on the motivic cohomology of the base field. For example, if k does not have a root of 1 then the Bockstein does not act trivially on H (MGL; Z/2) since βτ = ρ. One application of these computations is the Hopkins Morel isomorphism MGL/(a 1, a 2,... ) HZ over fields of characteristic zero (proved in [Hoy12] assuming the results of this paper). Another is the construction of algebraic Morava K-theory as an MGL-module K(n) = k(n)[vn 1 ], where k(n) is such that H k(n) = A /A Q n. Throughout this paper k is a perfect field (when not an integer... ) and l char k is a prime number. We denote by H the Eilenberg Mac Lane spectrum HZ/l. A smooth scheme is a smooth scheme of finite type over k. 1. The Hurewicz map for MGL We briefly review the computations of [NSØ09]. Let E be an oriented ring spectrum. If BGL k is the infinite Grassmannian of k-planes, we have E BGL k+ = E [[c 1,..., c k ]], where c i is the ith Chern class of the tautological vector bundle. This computation is obtained in the limit from the computation of the cohomology of the finite Grassmannian Gr(k, n), which is a free E -module of rank ( n k). From [Hu05, Theorem A.1], we know that Σ Gr(k, n) + is dualizable in SH(k). If X is the dual, then the canonical map E X E E Y E (X Y ) is a natural transformation between homological functors of Y that preserves direct sums, so it is an isomorphism for any cellular Y. It follows that the duality between Σ Gr(k, n) + and X Date: July 24,

2 2 MARC HOYOIS induces a duality between E Gr(k, n) + and E X = E, Gr(k, n) +. In the limit we obtain canonical isomorphisms E BGL k+ = HomE (E, BGL k+, E ), E BGL k+ = HomE,c(E, BGL k+, E ), where Hom E,c denotes continuous maps for the inverse limit topology on E BGL k+. Taking the limit as k, we get duality isomorphisms for BGL. There are similar results for finite products of such spaces. Now BGL has a multiplication BGL BGL BGL classifying the direct sum of vector bundles, which makes E BGL + into a Hopf algebra over E. From the formula giving the total Chern class of a direct sum of vector bundles we obtain the formula for the comultiplication on E BGL + = E [[c 1, c 2,... ]]: (c n ) = c i c j. i+j=n If β i E BGL + denotes the element which is dual to c i 1 with respect to the monomial basis of E BGL +, then β 0 = 1 and it follows from algebraic considerations that the dual E -algebra E BGL + is a polynomial algebra on the elements β i for i 1 (see for example [MM79, p. 176]). Since the restriction map E BGL + E P + simply kills all higher Chern classes, the β i s span E P + E BGL +. The multiplication MGL r MGL s MGL r+s is compatible with the multiplication BGL r BGL s BGL r+s under the Thom isomorphism, and so the dual Thom isomorphism E BGL + = E MGL is an isomorphism of E -algebras. Thus, E MGL is also a polynomial algebra E MGL = E [b 1, b 2,... ], where b n E 2n,n Σ 2, 1 MGL 1 is dual to the image of c n 1 under the Thom isomorphism E P + = E Σ 2, 1 MGL 1. As in topology, the elements b i are the coefficients of a power series defining a strict isomorphism between the two formal group laws on π (MGL MGL). If we recall that the groupoid of formal group laws and strict isomorphisms is represented by a Hopf algebroid (L, LB) with L = Z[a 1, a 2,... ] and LB = L[b 1, b 2,... ], the strict isomorphism on MGL MGL gives a map of Hopf algebroids (L, LB) (MGL, MGL MGL) sending a i to an element of MGL 2i,i which we also call a i, and sending b i to b i MGL 2i,i MGL. The elements a i MGL become zero in HZ (since HZ carries the additive group law), and the formula for the right unit of (L, LB) ([Rav03, Theorem A2.1.16]) allows us to compute the image of a i under the Hurewicz map h: MGL HZ MGL = HZ [b 1, b 2,... ]. Namely, the elements c(b i ) Z/l[b 1, b 2,... ] are determined by the recursive formula ( ) i+1 c(b i ) b j = 1 (where b 0 = 1), and we have h(a i ) = i 0 j 0 { lc(b i ) if i = l k 1 for some prime l, c(b i ) otherwise. Proposition 1.1 (Divisibility properties of h(a i )). Let n i Z be such that h(a i ) n i b i mod (b 1,..., b i 1 ), and let l be prime. If i = l k 1 for some k, then h(a i ) 0 mod l but n i 0 mod l 2. Otherwise, n i 0 mod l. Proof. Obvious since c(b i ) b i modulo decomposables.

3 THE MOTIVIC COHOMOLOGY OF CLASSICAL QUOTIENTS OF MGL 3 We abbreviate a l k 1 to v k, with v 0 = l. The special rôle played by these elements is explained by the following corollary of Proposition 1.1. Corollary 1.2. The kernel of h: L H MGL is (v 0, v 1,... ) and its range is Z/l[c(b i ) i l k 1]. Remark 1.3. The formula for h(a i ) shows that the composition L MGL HZ MGL is injective. In particular, L MGL is injective. 2. Complements on the motivic Steenrod algebra We denote by A the motivic Steenrod algebra at l. This is the subalgebra of all bistable natural transformations H ( ; Z/l) H ( ; Z/l) (as functors on the pointed homotopy category H (k)) generated by H, the power operations P i, and the Bockstein β. For simplicity, we assume that the canonical surjection H H lim n H +2n, +n (K(Z/l(n), 2n); Z/l) is an isomorphism. This is unfortunately only known if the base field k admits resolutions of singularities (see [Hoy12, 4.1]). Under this assumption, A is a subalgebra of H H (it is a subalgebra of the right-hand side in general), and in particular H E is a left module over A for every spectrum E. Remark 2.1. With a little more care we could remove this assumption for the main results of this paper (Theorems 4.1, 4.6, and 6.3). We know that A acts on H MGL in any case, since H MGL is the limit of the cohomology of the spaces MGL n. Our computations will then show that A acts appropriately on H E for any quotient E of MGL that we will consider. Since A contains H, it inherits the left and right H -module structures of H H. Voevodsky has shown that there exists a coproduct on A compatible with the intrinsic structure of H H, that is, there is a commutative diagram H H H (H H) H H? H H A A? A. Here the tensor products? introduce the relations ax y = x ay for a H. This coproduct is coassociative and cocommutative, and although A? A is not a ring it is multiplicative in the sense that (x) (y) is well-defined by the obvious formula and equals (xy), for all x, y A. Let A, denote the dual of A (by dual we will always mean the monoidal dual in the category of graded-weighted left H -modules). The fact that A has a left basis over H such that in any given bidegree only finitely many basis elements can appear with nonzero coefficients (which follows from Theorem 3.3) implies that it is in turn the dual of A,. The structure of A that we discussed so far makes (H, A ) into a Hopf algebroid (without coinverse) in the category of graded-weighted commutative rings (or Z/l-algebras) such that the canonical maps H (H i ) A i preserve this structure (i.e., form a morphism of cosimplicial rings). Conversely, the structure of A is determined by that of A. For the spectra E appearing in this paper it will be the case that H E is dual to H E and therefore is a comodule over A. Remark 2.2. For a general cellular spectrum E, we do not know that H E is a comodule over A because we do not know that H H is flat over H. Define a Hopf algebroid (A, Γ) as follows. Let A = Z/l[ρ, τ], Γ = A[τ 0, τ 1,..., ξ 1, ξ 2,... ]/(τ 2 i τξ i+1 ρτ i+1 ρτ 0 ξ i+1 ).

4 4 MARC HOYOIS The structure maps η L, η R, ɛ, and are given by the formulas η L : A Γ, η L (ρ) = ρ, η L (τ) = τ, η R : A Γ, η R (ρ) = ρ, η R (τ) = τ + ρτ 0, ɛ: Γ A, ɛ(ρ) = ρ, ɛ(τ) = τ, ɛ(τ k ) = 0, ɛ(ξ k ) = 0, : Γ Γ A Γ, (ρ) = ρ 1, (τ) = τ 1, k 1 (τ k ) = τ k τ k + ξk i li τ i, i=0 k 1 (ξ k ) = ξ k ξ k + ξk i li ξ i. The coinverse map c: Γ Γ is determined by the identities it must satisfy. Namely, we have c(ρ) = ρ, c(τ) = τ + ρτ 0, k 1 c(τ k ) = τ k ξk ic(τ li i ), i=0 k 1 c(ξ k ) = ξ k ξk ic(ξ li i ). We will not use this map. We view H as an A-algebra via the map A H defined as follows: it sends 1 Z/l to 1 H 0,0 ; if l is odd it sends both ρ and τ to 0, while if l = 2 it sends ρ to the class of 1 in and τ to the generator of i=1 H 1, 1 = k /(k ) 2 H 0, 1 = µ 2 (k) = Z/2 (recall that char k 2 if l = 2). We will also denote by ρ, τ H the images of ρ, τ A under this map. So if l 2, ρ = τ = 0 in H ; all the arguments in this paper work regardless of what ρ and τ are, so with this setup we do not have to worry about the parity of l from now on. Voevodsky s computation of the motivic Steenrod algebra ([Voe03]) is summarized in the following theorem. Theorem 2.3. A is isomorphic to Γ A H with i=1 τ k = (2l k 1, l k 1) and ξ k = (2l k 2, l k 1). The map H A dual to the left action of A on H is a left coaction of (A, Γ) on the ring H, and the Hopf algebroid (H, A ) is isomorphic to the twisted tensor product of (A, Γ) with H. This means that A = Γ A H ; η L and ɛ are extended from (A, Γ); η R : H A is the coaction; : A A H A is induced by the diagonal of Γ and η R to the second factor; c: A A is induced by the coinverse of Γ and η R.

5 THE MOTIVIC COHOMOLOGY OF CLASSICAL QUOTIENTS OF MGL 5 [I know of no a priori interpretation of the coinverse of (H, A ), but it is determined by the other structure maps. In the case where A = H H, the coinverse is therefore induced by the twist H H H H.] As usual, for a sequence E = (ɛ 0, ɛ 1,... ) with ɛ i {0, 1}, we set τ(e) = τ ɛ0 0 τ ɛ and for a sequence R = (r 1, r 2,... ) of nonnegative integers we set ξ(r) = ξ r1 1 ξr Then the products τ(e)ξ(r) form a basis of A as a left H -module, and if ρ(e, R) denotes the dual basis of A, we have ρ(e, R) = Q(E)P R = Q ɛ0 0 Qɛ P R, where Q(E) is dual to τ(e), Q i to τ i, and P R to ξ(r). We have P n,0,0,... = P n and for degree reasons P R is a polynomial in the P n s with Z/l coefficients. Finally, for i 1 we set q i = P ei, where e i is the sequence with 1 in the ith position and 0 elsewhere. From the formula for (ξ k ) we see at once that q i = P li P li 1... P l P 1. Two sequences R and S can be added termwise, and we write S R if there exists a sequence T such that S + T = R. We write e 0 or for a sequence of zeros. The following lemma completely describes the coproduct on the basis elements ρ(e, R). It is proved by dualizing the product on A. Explicit formulas for the products of elements ρ(e, R) are more complicated, so we will only derive partial formulas as needed. Lemma 2.4 (Cartan formulas). (P R ) = E=(ɛ 0,ɛ 1,... ) R τ i 0 ɛi 1+R 2=R E Q(E)P R1 Q(E)P R2 ; (Q k ) = Q k Q k + k i=1 E 1+E 2=[k i+1,k 1] ρi Q k i Q(E 1 ) Q k i Q(E 2 ). It is also easy to prove that the subalgebra of A generated by ρ and Q i, i 0, is an exterior algebra in the Q i s over Z/l[ρ] H (but the algebra generated by Q i and H, which is the left H -submodule on the operations Q(E), is not even commutative if ρ 0). A well-known result in topology is that the left and right ideals generated by Q i, i 0, coincide and are generated by Q 0 as two-sided ideals. This can fail altogether in the motivic Steenrod algebra: for example, if ρ 0, Q 0 τ and τq 0 are the unique nonzero elements of degree (1, 1) in the right and left ideals and Q 0 τ τq 0 = ρ, so neither ideal is included in the other and in particular neither ideal is a two-sided ideal. A true statement is that the H -bimodules generated by those various ideals coincide, but this is not useful if we are interested in the one-sided ideals. Lemma 2.5. The left ideal of A generated by {Q i i 0} is the free left H -module on the basis {ρ(e, R) E }. Proof. Define a matrix c by the rule P R Q(E) = E,R c E,R E,R ρ(e, R ). Then c E,R E,R is the coefficient of ξ(r) τ(e) in (τ(e )ξ(r )). The only term in (ξ(r )) that can be a factor of ξ(r) τ(e) is ξ(r ) 1, so we must have R R for c E,R E,R to be nonzero. If R = R, we must further have a term 1 τ(e) in (τ(e )), and it is easy to see that this cannot happen unless also E = E, in which case ξ(r) τ(e) appears with coefficient 1 in (τ(e)ξ(r)). It is also clear that c E,R,R = 0 if E. Combining these three facts, we can write P R Q(E) = ρ(e, R) + c E,R E,R ρ(e, R ). E R R

6 6 MARC HOYOIS So we can use induction on the -order of R to prove that for all E, ρ(e, R) is an H -linear combination of elements of the form P R Q(E ) with E. In particular, ρ(e, R) is in the left ideal if E, which proves one inclusion. Conversely, let ρ(e, R)Q i be in the left ideal. By what was just proved this is an H -linear combination of elements of the form P R Q(E )Q i ; because the Q i s generate an exterior algebra, such an element is either 0 or ±P R Q(E ) with E. The above formula shows directly that this is in turn an H -linear combination of elements of the desired form. Corollary 2.6. The operations P R form a basis of A /A (Q 0, Q 1,... ) as a left H -module. Proof. Immediate by Lemma 2.5. We will denote by P the left H -submodule of A generated by the elements ξ(r); it is clearly a left A -comodule algebra (but it is not a Hopf algebroid in general, since it may not even be a right H -module). As an H -algebra it is the polynomial ring H [ξ 1, ξ 2,... ]. By Lemma 2.5, the inclusion P A is dual to the projection A A /A (Q 0, Q 1,... ). 3. Complements on H-modules The goal of this section is to formulate an appropriate finiteness condition on the homology of cellular spectra so that their homology and cohomology are dual to one another and satisfy Künneth formulas. We work in the stable -category of H-modules. This is a closed symmetric monoidal category, and we denote by M the dual of an H-module M. Note that if M = H E, then π M is the motivic homology of E and π, M is its motivic cohomology. In this section H can be HR for any commutative ring R, and the next two lemmas are obviously true for an arbitrary E ring spectrum. Lemma 3.1. Let M and N be H-modules. If M is cellular and π N is flat over H, then the canonical map π M H π N π (M H N) is an isomorphism. Proof. Since this is a natural map between homological functors of M preserving sums, it suffices to check that it is an isomorphism when M = Σ p,q H, but this is obvious. An H-module is split if it is equivalent to an H-module of the form Σ pα,qα H. Lemma 3.2. For an H-module M the following conditions are equivalent: (1) M is split; (2) π M is free over π H; (3) M is cellular and π M is free over π H. α Proof. It is clear that (1) implies (2) and (3). Assuming (2) or (3), use a basis of π M to define a morphism of H-modules α Σpα,qα H M. This morphism is then a π -isomorphism (or a π -isomorphism between cellular H-modules) and so it is an equivalence. The following theorem summarizes the standard vanishing results for motivic cohomology (which we use freely elsewhere in this paper). Theorem 3.3. Let A be an abelian group, X a smooth scheme, and p, q Z. Then the group HA p,q X + is zero under any of the following conditions: (1) q < 0; (2) p > q + dim X; (3) p > 2q. As a consequence of Theorem 3.3, if M α Σpα,qα H, the family of bidegrees (p α, q α ) is determined by M (in fact, if M p,q Z HV p,q where V p,q is a Z/l-module, then V p,q = π rig p,qm).

7 THE MOTIVIC COHOMOLOGY OF CLASSICAL QUOTIENTS OF MGL 7 Definition 3.4. A split H-module will be called psf (short for proper and slicewise finite) if it is equivalent to α Σpα,qα H where the bidegrees (p α, q α ) satisfy the following condition: they are all contained in the cone q 0, p 2q, and for each q there are only finitely many α such that q α = q. As we will see, this condition is satisfied in many interesting cases. For example, if E is any Grassmannian or Thom space of the tautological bundle thereof, or if E = MGL, then E is cellular and the calculus of oriented cohomology theories shows that H E is free over H, with finitely many generators in each bidegree (2n, n). It follows from Lemma 3.2 that H E is psf. If the base field has characteristic zero, it is known that H H is split and that H H = A (see [Hoy12, Lemma 4.5]), so again H H is psf. Proposition 3.5. Let M and N be psf H-modules. Then (1) M H N is psf; (2) M is split; (3) the pairing M H M H is perfect; (4) the canonical map M H N (M H N) is an equivalence; (5) the pairing π M H π M H is perfect; (6) the canonical map π M H π N π (M H N) is an isomorphism; (7) the canonical map π M H π N π (M H N) is an isomorphism. Proof. (1) is clear from the definition. Let M = α Σpα,qα H. Theorem 3.3 and the psf condition imply that the canonical maps Σ pα,qα H Σ pα,qα H and Σ pα, qα H Σ pα, qα H α α α α are equivalences (compare Figure 1 (a) and (b)). This implies (2), (3), and (4). In particular, the two inclusions α π Σ ±pα,±qα H α π Σ ±pα,±qα H are isomorphisms, which shows (5). (6) is a just special case of Lemma 3.1, and (7) follows from (2), (4), and Lemma 3.1. q q p = 2q (a) p (p, q) d (b) p Figure 1. (a) The proper cone. (b) If d = dim X, the shaded area is the potentially nonzero locus of H p+,q+ X + according to Theorem 3.3. If for every (p, q) and every d there are only finitely many bidegrees (p α, q α ) in the shaded area, it follows that the canonical map α Σpα,qα H α Σpα,qα H is a π - isomorphism. 4. The motivic homology of regular quotients of MGL Recall from Corollary 1.2 that h(l) is the subring Z/l[c(b i ) i l k 1] H MGL. In the sequel we will view the ring h(l) as a direct summand of Z/l[b 1, b 2,... ] via some splitting of the inclusion, chosen once and for all. π : Z/l[b 1, b 2,... ] h(l) Theorem 4.1. The coaction : H MGL A Z/l Z/l[b 1, b 2,... ] factors through P Z/l Z/l[b 1, b 2,... ] and the composition 1 π H MGL P Z/l Z/l[b 1, b 2,... ] P Z/l h(l)

8 8 MARC HOYOIS is an isomorphism of left A -comodule algebras. Towards proving this theorem we explicitly compute the coaction of A on H MGL. Since it is an H -algebra map, it suffices to compute (b i ) for i 1. Consider the zero section s: P + MGL 1 as a map in H (k). In cohomology this map is the composition of the Thom isomorphism and multiplication by the top Chern class c 1. In homology, it therefore sends β i to 0 if i = 0 and to b i 1 otherwise. Thus, (1) (b i ) = (s (β i+1 )) = (1 s ) (β i+1 ). The action of A on c n 1 H P + = H [c 1 ] is determined by the Cartan formulas (Lemma 2.4). For degree reasons Q i acts trivially on elements in H (2,1) P +, and we get P R (c n 1 ) = a n,r c n+ R 1, Q i (c n 1 ) = 0, where a n,r is the multinomial coefficient given by ( ) n a n,r = n i 1 r, i, r 1, r 2,... understood to be 0 if i 1 r i > n. Dualizing, we obtain (β n ) = a m,r ξ(r) β m, whence by (1), m+ R =n (2) (b n ) = m+ R =n a m+1,r ξ(r) b m. Lemma 4.2. The H -algebra map f : H MGL P defined by { ξ k if n = l k 1, f(b n ) = 0 otherwise. is a map of left A -comodules. Proof. If m is of the form l k 1, then the coefficient a m+1,r vanishes mod l unless R = l k e i for some i 0, in which case a m+1,r = 1 and n = l k+i 1. Comparing (2) with the formula for (ξ k ) shows that f is a comodule map. Proof of Theorem 4.1. The formula (2) shows that factors through P Z/l Z/l[b 1, b 2,... ]. Let g be the map to be proved an isomorphism. Note that it is a comodule algebra map since is and π is a ring map. Formula (2) shows further that g is extended from a map By Lemma 4.2, g : Z/l[b 1, b 2,... ] Z/l[ξ 1, ξ 2,... ] Z/l h(l). g(b lk 1c(b i )) ξ k c(b i ) modulo decomposables, so g is surjective. Now g is a map between Z/l-modules of the same finite dimension in each bidegree, so g and hence g are isomorphisms. The isomorphism g of Theorem 4.1 is certainly not a map of h(l)-modules, but it is easy to modify it so that it preserves the h(l)-module structure as well. To do this consider the ring map f : P H MGL, f(ξk ) = g 1 (ξ k 1), which is clearly an A -comodule map. Corollary 4.3. The map f and the inclusion of h(l) induce an isomorphism P Z/l h(l) = H MGL of left A -comodules algebras and of h(l)-modules.

9 THE MOTIVIC COHOMOLOGY OF CLASSICAL QUOTIENTS OF MGL 9 Proof. We have g 1 (ξ k 1) b l k 1 modulo decomposables. It follows that the map is surjective and hence, as in the proof of Theorem 4.1, an isomorphism. Remark 4.4. Even though P is not a Hopf algebroid, the proof of the comodule algebra structure theorem [Rav03, Theorem A1.1.17] shows that H MGL = P Z/l C as left A -comodules and C-modules, where C is the algebra of primitives in Z/l[b 1, b 2,... ]. Since h(l) C for formal reasons, it follows from dimension counting that C = h(l). Let x L be a homogeneous element such that h(x) H MGL is nonzero. Then multiplication by h(x) is injective and so there is a short exact sequence 0 H Σ x MGL H MGL H (MGL/x) 0. It follows that H (MGL/x) = H [b 1, b 2,... ]/h(x) and comparing with the isomorphism of Corollary 4.3 we deduce that the map P Z/l h(l)/h(x) H (MGL/x) induced by f and the inclusion is an isomorphism of left A -comodules and of h(l)-modules. By induction we obtain the following result. Lemma 4.5. Let x = (x 1, x 2,... ) be a sequence of homogeneous elements in L which is regular modulo the kernel of h: L H MGL. Then the maps f and h(l)/h(x) H (MGL/x) induce an isomorphism P Z/l h(l)/h(x) = H (MGL/x) of left A -comodules and h(l)-modules. We can now undo the modification of Corollary 4.3: Theorem 4.6. Let x = (x 1, x 2,... ) be a sequence of homogeneous elements in L which is regular modulo the kernel of h: L H MGL. Then the coaction of A on H (MGL/x) induces an isomorphism H (MGL/x) = P Z/l h(l)/h(x) of left A -comodules. Proof. Let g be the isomorphism of Lemma 4.5 and let g be the map to be proved an isomorphism. Then g g(1 b) = 1 b and g g(ξ k 1) ξ k 1 modulo decomposables, so g g and hence g are isomorphisms by the usual argument. We will denote by BP the MGL-module MGL/(a i i l k 1). By Theorem 4.6, H BP = P where 1 P corresponds to the image of 1 H MGL in the left-hand side. Remark 4.7. The motivic spectrum BP defined in various ways elsewhere coincides with the l-localization of the spectrum that we abusively call BP, but the distinction is invisible in mod l cohomology. As we noted in 3, H MGL is a psf H-module. Dualizing Theorem 4.1, we therefore deduce that the map A /A (Q 0, Q 1,... ) Z/l h(l) H MGL, [φ] m φ(m), is an isomorphism of left A -modules. Here h(l) is the subcoalgebra of Z/l[x 1, x 2,... ] H MGL dual to π. If x is a regular sequence modulo the kernel of h, the computation of H (MGL/x) shows, with Lemma 3.2, that H MGL/x is a split direct summand of H MGL, so it is also psf. By dualizing Theorem 4.6, we obtain a computation of H (MGL/x). For example, in the maximal ideal case, we obtain that the map A /A (Q 0, Q 1,... ) H BP, [φ] φ(τ), where τ : BP H is the lift of the Thom class, is an isomorphism of left A -modules.

10 10 MARC HOYOIS 5. Key lemmas We will denote by τ : MGL H the universal Thom class. Lemma 5.1. Let R = (r 1, r 2,... ). Then P R (τ) H MGL is dual to i 1 bri l i 1 H MGL. Proof. τ is dual to 1, so we must look for monomials m Z/l[b 1, b 2,... ] such that (m) has a term of the form ξ(r) 1. By Lemma 4.2, such a term can only appear if m is a monomial in b li 1, and this monomial must be i 1 bri l i 1. Define Φ k by the homotopy fiber sequence Since Q 0 = β, we have Φ 0 = HZ/l 2. Σ 2lk 2,l k 1 Q k H Φ k H Σ 2lk 1,l k 1 H. Lemma 5.2 (Voevodsky). Let k 1 and n = l k 1. For any lift ψ : MGL Φ k of τ, ψ(a n ) π 2n,n Φ k = Z/l is nonzero. Proof. If E is the cofiber of a n : Σ 2n,n 1 MGL, we must show that ψ(a n ) does not lift to E. Since H,n = 0, there exists a unique lift ˆτ : E H of the Thom class: a n Σ 2n,n 1 MGL E Σ 2n+1,n 1 0 H τ ˆτ and a lift of ψ(a n ) to E would be a lift of ˆτ to Φ k, so it suffices to show that Q kˆτ 0. Recall that Q k = q k β βq k. Again, βˆτ is the unique lift of βτ = 0, so βˆτ = 0. We must therefore show that βq kˆτ 0, i.e., that q kˆτ does not lift to a class in H 2n,n (E; Z/l 2 ). Consider the diagram of exact sequences H 2n,n (E; Z/l 2 ) H 2n,n E H 2n+1,n E β H 2n,n (MGL; Z/l 2 ) H 2n,n MGL H 2n+1,n MGL β a n (HZ/l 2 ) 0,0. By Lemma 5.1, q k τ is dual to b n. Thus, an arbitrary lift of q k τ to H (MGL; Z/l 2 ) has the form x + ly where x is dual to b n and y is any cohomology class. It suffices to show that no such class lifts to H (E; Z/l 2 ). By Proposition 1.1, we have y(a n ) 0 mod l and x(a n ) 0 mod l 2, so (x + ly)(a n ) 0 mod l 2. Theorem 5.3. Let k 0 and n = l k 1. Let MGL E be a map of MGL-modules which is an isomorphism on H 0,0, H 2n, n, and H 2n 1, n. Denote by τ : E H and ˆτ : E/a n H the unique lifts of the universal Thom class. If Q k τ = 0, then the square E/a n ˆτ δ Σ 2n+1,n E Σ 2n+1,n τ H Q k Σ 2n+1,n H commutes up to multiplication by an element of Z/l. Proof. Consider the diagram

11 THE MOTIVIC COHOMOLOGY OF CLASSICAL QUOTIENTS OF MGL 11 a n Σ 2n,n E E E/a n Σ 2n+1,n E δ Σ 2n,n τ ψ τ Σ 2n+1,n τ Σ 2n,n H Φ k H Σ 2n+1,n H. Since Q k τ = 0, we can choose a lift ψ of τ as indicated. Suppose that the left square commutes up to Z/l. Then there exists a map τ : E/a n H making the middle square commute and making the right square commute up to Z/l ; in particular, τ lifts τ, so that τ = ˆτ. Thus, it suffices to prove that the left square commutes up to Z/l. If k = 0, since Φ 0 = HZ/l 2, it is clear that the left square actually commutes, so suppose that k 1. By Lemma 5.2, a n(ψ) Φ 0,0 k Σ2n,n E is nonzero since it fits into the commutative diagram Σ 2n,n 1 Q k η a n Σ 2n,n MGL a n MGL Σ 2n,n E a n (ψ) a n E ψ Φ k. By our assumptions, H 2n,n Σ 2n,n E = Z/l and the map H 2n,n Σ 2n,n E Φ 0,0 k Σ2n,n E is an isomorphism since H 0,0 Σ 2n,n E = 0 and H 1,0 Σ 2n,n E = 0. So it sends Σ 2n,n τ to a Z/l -multiple of a n(ψ), as desired. Lemma 5.4. Let k 0. Denote by ˆτ : MGL/v k H the unique lift of the universal Thom class. If i k, then Q iˆτ = 0. Proof. Suppose first that k 1. Then βˆτ = 0 for degree reasons, so it suffices to show that βq iˆτ = 0 for i 1. Let n = l i 1 and consider the diagram of exact sequences H 2n,n Σ Q k MGL δ β H 2n+1,n Σ Q k MGL H 2n,n (MGL/v k ; Z/l 2 ) H 2n,n (MGL/v k ) H 2n+1,n (MGL/v k ) β H 2n,n (MGL; Z/l 2 ) H 2n,n MGL H 2n+1,n MGL p β v k H 2n,n (Σ v k MGL; Z/l 2 ). By Lemma 5.1, q i τ H MGL is dual to b n. Let x H 2n,n (MGL; Z/l 2 ) be dual to b n, and let m Z/l 2 [b 1, b 2,... ] be any monomial. Then v k(x), m = x, (v k ) (m) = x, h(v k )m, which is zero for any m since b n is the only monomial with which x pairs nontrivially. Thus, v k (x) = 0 and x lifts to an element ˆx H2n,n (MGL/v k ; Z/l 2 ). Let y be the image of ˆx in H 2n,n (MGL/v k ). Then p y = q i τ = p q iˆτ, so y q iˆτ = δ (z) for some z. For degree reasons, z = λz for some λ H 1,1 and z H (2,1) MGL, so β(z) = β(λ)z + λβ(z ) = 0 and therefore βq iˆτ = β(y δ z) = βy = 0.

12 12 MARC HOYOIS If k = 0, then β = δ p is a morphism of H H-modules and so Q iˆτ = βq iˆτ βq iˆτ = The motivic cohomology of quotients of BP Lemma 6.1. Let E be an MGL-module and let x L be a homogeneous element such that h(x) = 0. If H E is psf, then H E/x is psf. Proof. Since h(x) = 0 in H MGL, we have a short exact sequence 0 H E H (E/x) H Σ 1,0 Σ x E 0. This sequence splits in H -modules since the quotient is free, so we deduce from Lemma 3.2 that H E/x (H E) (H Σ 1,0 Σ x E). Since x is of the form (2n, n), it is clear that H E/x is psf. Lemma 6.2. Let E be an MGL-module and let x L be a homogeneous element such that h(x) = 0. If H E is projective over H MGL, so is H (E/x). Proof. The short exact sequence splits in H MGL-modules. 0 H E H (E/x) H Σ 1,0 Σ x E 0 Theorem 6.3. Let I be a set of nonnegative integers. Then there is an isomorphism of left A -modules A /A (Q i i / I) = H (BP/(v i i I)), given by [φ] φ(τ), where τ : BP/(v i i I) H is the lift of the universal Thom class. Proof. It suffices to prove the theorem when I is finite, and we proceed by induction on the size of I. If I is empty, the theorem is true by Theorem 4.6. Suppose it is true for I and let k / I. Let E = BP/(v i i I). Since MGL/v k is a cellular MGL-module and H E is flat over H MGL by Lemma 6.2, the canonical map H E H MGL H (MGL/v k ) H (E MGL MGL/v k ) = H (E/v k ) is an isomorphism. By Lemma 6.1, all the H-modules H MGL, H E, H MGL/v k, and H E/v k are psf. If we dualize this isomorphism, we therefore get an isomorphism H (E/v k ) = H E H MGL H (MGL/v k ), where τ on the left-hand side corresponds to τ τ on the right-hand side and the A -module structure on the right-hand side is given by φ (x y) = (φ)(x y). By the Cartan formula, i (Q i )(τ τ) = Q i τ τ + τ Q i τ + ψ j (Q i j τ Q i j τ) for some ψ j A? A. By Lemma 5.4, Q i τ H (MGL/v k ) is zero when i k. Using the induction hypothesis, it follows that (Q k )(τ τ) = τ Q k τ, and if i / I {k}, (Q i )(τ τ) = 0. The latter shows that the map [φ] (φ)(τ τ) is well-defined. Then we have a diagram of short exact sequences of left A -modules Q k A /A (Q i i / I) A /A (Q i i / I {k}) A /A (Q i i / I) j=1 = = 1 δ H E H MGL H E H (MGL/v k ) H E H MGL. The right square commutes because the image of 1 A /A (Q i i / I {k}) in the bottom right corner is τ τ either way. For the left square, the two images of 1 are τ δ τ and (Q k )(τ τ) = τ Q k τ. Theorem 5.3 then shows that the left square commutes up to multiplication by an element of Z/l, so the map between extensions is an isomorphism by the five lemma.

13 THE MOTIVIC COHOMOLOGY OF CLASSICAL QUOTIENTS OF MGL 13 Corollary 6.4. Let E = MGL/(l, a 1, a 2,... ) and let τ H 0,0 E be the lift of the Thom class. Then the map A H E, φ φ(τ) is an isomorphism of left A -modules. Remark 6.5. The topological version of Corollary 6.4 shows that the endomorphism algebra of the mod l Eilenberg Mac Lane spectrum is generated by the power operations and the Bockstein. In our setting, this corollary gives a converse to the main theorem of [Hoy12]: if the Hopkins Morel isomorphism holds modulo l, then the motivic Steenrod algebra is actually the full endomorphism algebra. Corollary 6.6. If k has characteristic zero, then H τ : H MGL/(l, a 1, a 2,... ) H H and H τ : H MGL/(a 1, a 2,... ) H HZ are equivalences of H-modules. Proof. By [Hoy12, Lemma 4.5 and Theorem 4.7], these maps are π -isomorphisms between split H-modules, whence equivalences. Example 6.7. As a counterexample to a conjecture one might make regarding the cohomology of spectra of the general form MGL/(a i1, a i2,... ), we observe that if l = 2 and ˆx H (MGL/v 2 ) is a lift of the dual x to b 2, then Q 1ˆx is nonzero. By (2), we have (b 3 ) = 1 b 3 + ξ 2 1 b 1 + ξ ξ 1 b 2. By an analysis similar to those done in section 5, we deduce that for any lift y of q 1 x to Z/4-cohomology, a 3(y) is nonzero, and hence that Q 1ˆx is nonzero. This is only the first example of a general phenomenon which is that in H (MGL/v k ), if ξ(r) b i is a term in (b lk 1) that does not correspond to a term in (ξ k ) and if ˆx is a lift of the dual to b i, then βp R (ˆx) is nonzero. References [Hoy12] M. Hoyois, On the relation between algebraic cobordism and motivic cohomology, 2012, [Hu05] P. Hu, On the Picard group of the A 1 -stable homotopy category, Topology 44 (2005), pp , preprint K-theory:0395 [MM79] I. H. Madsen and R. J. Milgram, The classifying spaces for surgery and cobordism of manifolds, Princeton University Press, 1979 [NSØ09] N. Naumann, M. Spitzweck, and P. A. Østvær, Motivic Landweber Exactness, Doc. Math. 14 (2009), pp , preprint arxiv: v3 [math.ag] [Rav03] D. C. Ravenel, Complex cobordism and stable homotopy groups of spheres, AMS Chelsea Publishing, 2003, [Voe03] V. Voevodsky, Reduced power operations in motivic cohomology, Publ. Math. 98 (2003), pp. 1 57, preprint K-theory:0487