Quotients of MGL,

Their Slices and Their Geometric Parts

Marc Levine, Girja Shanker Tripathi^{1}

Received: January 10, 2015

Abstract. Let x_{1}, x_{2}, … be a system of homogeneous polynomial generators for the Lazard ring 𝕃^{∗} = MU^{2∗} and let MGL_{S} denote Voevodsky’s algebraic cobordism spectrum in the motivic stable homotopy category over a base-scheme S [Vo98]. Relying on Hopkins-Morel-Hoyois isomorphism [Hoy] of the 0th slice s_{0}MGL_{S} for Voevodsky’s slice tower with MGL_{S}/(x_{1}, x_{2}, …) (after inverting all residue characteristics of S), Spitzweck [S10] computes the remaining slices of MGL_{S} as s_{n}MGL_{S} = Σ_{T}^{n} Hℤ⊗𝕃^{−n} (again, after inverting all residue characteristics of S). We apply Spitzweck’s method to compute the slices of a quotient spectrum MGL_{S}/({x_{i} : i ∈ I}) for I an arbitrary subset of ℕ, as well as the mod p version MGL_{S}/({p, x_{i} : i ∈ I}) and localizations with respect to a system of homogeneous elements in ℤ[{x_{j} : j ∉ I}]. In case S = Spec k, k a field of characteristic zero, we apply this to show that for E a localization of a quotient of MGL as above, there is a natural isomorphism for the theory with support

Ω_{∗}(X) ⊗_{𝕃^{−∗}} E^{−2∗,−∗}(k) → E_{X}^{2m−2∗,m−∗}(M)

for X a closed subscheme of a smooth quasi-projective k-scheme M, m = dim_{k} M.

To Sasha Merkurjev with warmest regards on his 60th birthday

Contents

Introduction  408
1. Quotients and homotopy colimits in a model category  409
2. Slices of effective motivic module spectra  418
3. The slice spectral sequence  424
4. Slices of quotients of MGL  426
5. Modules for oriented theories  430
6. Applications to quotients of MGL  438
References  440

1Both authors wish to thank the Humboldt Foundation for financial support

Documenta Mathematica · Extra Volume Merkurjev (2015) 407–442

Introduction

This paper has a two-fold purpose. We consider Voevodsky’s slice tower on the motivic stable homotopy category SH(S) over a base-scheme^{2} S [Vo00]. For E in SH(S), we have the nth layer s_{n}E in the slice tower for E. Let MGL denote Voevodsky’s algebraic cobordism spectrum in SH(S) [Vo98] and let x_{1}, x_{2}, … be a system of homogeneous polynomial generators for the Lazard ring 𝕃_{∗}. Via the classifying map for the formal group law for MGL, we may consider x_{i} as an element of MGL^{−2i,−i}(S), and thereby as a map x_{i} : Σ^{2i,i}MGL → MGL, giving the quotient MGL/(x_{1}, x_{2}, …). Spitzweck [S10] shows how to build on the Hopkins-Morel-Hoyois isomorphism [Hoy]

MGL/(x_{1}, x_{2}, …) ≅ s_{0}MGL

to compute all the slices s_{n}MGL of MGL. Our first goal here is to extend Spitzweck’s method to handle quotients of MGL by a subset of {x_{1}, x_{2}, …}, as well as localizations with respect to a system of homogeneous elements in the ring generated by the remaining variables; we also consider quotients of such spectra by an integer. Some of these spectra are Landweber exact, and the slices are thus computable by the results of Spitzweck on the slices of Landweber exact spectra [S12], but many of these, such as the truncated Brown-Peterson spectra or Morava K-theory, are not. The second goal is to extend results of [DL14, L09, L15], which consider the “geometric part” X ↦ E^{2∗,∗}(X) of the bi-graded cohomology defined by an oriented weak commutative ring T -spectrum E and raise the question: is the classifying map

E^{∗}(k) ⊗_{𝕃^{∗}} Ω^{∗} → E^{∗}

Quotients of MGL, Their Slices and Geometric Parts 409

an isomorphism of oriented cohomology theories, that is, is the theory E^{∗} a theory of rational type in the sense of Vishik [Vi12]? Starting with the case E = MGL, discussed in [L09], which immediately yields the Landweber exact case, we have answered this affirmatively for “slice effective” algebraic K-theory in [DL14], and extended to the case of slice-effective covers of a Landweber exact theory in [L15]. In this paper, we use our computation of the slices of a quotient of MGL to show that the classifying map is an isomorphism for the quotients and localizations of MGL described above. The paper is organized as follows: in §1 and §2, we abstract Spitzweck’s method from [S10] to a more general setting. In §1 we give a description of quotients in a suitable symmetric monoidal model category in terms of a certain homotopy colimit. In §2 we begin by recalling some basic facts and the slice tower and its construction. We then apply the results of §1 to the category of R-modules for R ∈ SH(S) a commutative ring T -spectrum (with some additional technical assumptions), developing a method for computing the slices of an R-module M, assuming that R and M are effective and that the 0th slice s_{0}M is of the form M/({x_{i} : i ∈ I}) for some collection {[x_{i}] ∈ R^{−2d_{i},−d_{i}}(S), d_{i} < 0} of elements in R-cohomology of the base-scheme S; see theorem 2.3. We also discuss localizations of such R-modules and the mod p case (corollary 2.4 and corollary 2.5). We discuss the associated slice spectral sequence for such M and its convergence properties in §3, and apply these results to our examples of interest: truncated Brown-Peterson spectra, Morava K-theory and connective Morava K-theory, as well as the Landweber exact examples, the Brown-Peterson spectra BP and the Johnson-Wilson spectra E(n), in §4. The remainder of the paper discusses the classifying map from algebraic cobordism Ω_{∗} and proves our results on the rationality of certain theories. This is essentially taken from [L15], but we need to deal with a technical problem, namely, that it is not at present clear if the theories [MGL/({x_{i} : i ∈ I})]^{2∗,∗} have a multiplicative structure. For this reason, we extend the setting used in [L15] to theories that are modules over ring-valued theories. This extension is taken up in §5 and we apply this theory to quotients and localizations of MGL in §6. We are grateful to the referee for suggesting a number of improvements to an earlier version of this paper, especially for pointing out to us how to use works of Spitzweck to extend many of our results to an arbitrary base-scheme.

2In this paper a “scheme” will mean a noetherian separated scheme of finite Krull dimension.

Documenta Mathematica · Extra Volume Merkurjev (2015) 407–442

1. Quotients and homotopy colimits in a model category

In this section we consider certain quotients in a model category and give a description of these quotients as a homotopy colimit (see proposition 1.9). This is an abstraction of the methods developed in [S10] for computing the slices of MGL. In what follows, we use the term “fibrant replacement” of an object x in a model category C to mean a morphism α : x → x^{f} in C, where x^{f} is fibrant and α is a cofibration and a weak equivalence. A cofibrant replacement of x is similarly a morphism β : x^{c} → x in C with x^{c} cofibrant and β a fibration and a weak equivalence. Let (C, ⊗, 1) be a closed symmetric monoidal simplicial pointed model category with cofibrant unit 1. We assume that 1 admits a fibrant replacement α : 1 → 1 such that 1 is a 1-algebra in C, that is, there is an associative multiplication map μ_{1} : 1 ⊗ 1 → 1 such that μ_{1} ∘ (α ⊗ id) and μ_{1} ∘ (id ⊗ α) are the respective multiplication isomorphisms 1⊗1 → 1, 1⊗1 → 1. We assume in addition that the functor K ↦ 1 ⊗ K, giving part of the simplicial structure, is a symmetric monoidal left Quillen functor. For a cofibrant object T in C, the map T ≅ T ⊗1 ⟶^{id⊗α} T ⊗1 is a cofibration and weak equivalence. Indeed, the functor T ⊗ (−) preserves cofibrations, and also maps that are both a cofibration and a weak equivalence, whence the assertion.

Remark 1.1. We will be applying the results of this section to the following situation: M is a cofibrantly generated symmetric monoidal simplicial model category satisfying the monoid axiom [ScSh, definition 3.3]; ; we assume in addition that the functor K ↦ e ∧ K, e the unit in M, giving part of the simplicial structure, is a symmetric monoidal left Quillen functor. We fix in addition a commutative monoid R in M, cofibrant in M, and C is the category of R-modules in M, with model structure as in [ScSh, §4], that is, a map is a fibration or a weak equivalence in C if and only if it is so as a map in M, and cofibrations are determined by the LLP with respect to acyclic fibrations. By [ScSh, theorem 4.1(3)], the category R-Alg of monoids in C has the structure of a cofibrantly generated model category, with fibrations and weak equivalence those maps which become a fibration or weak equivalence in M, and each cofibration in R-Alg is a cofibration in C. The unit 1 in C is just R and we may take α : 1 → 1 to be a fibrant replacement in R-Alg.

Let {x_{i} : T_{i} → 1 | i ∈ I} be a set of maps with cofibrant sources T_{i}. We assign each T_{i} an integer degree d_{i} > 0. Let 1/(x_{i}) be the homotopy cofiber (i.e., mapping cone) of the map x_{i} : 1⊗T_{i} → 1 and let p_{i} : 1 → 1/(x_{i}) be the canonical map. Let A = {i_{1}, …, i_{k}} be a finite subset of I and define 1/({x_{i} : i ∈ A}) as

1/({x_{i} : i ∈ A}) := 1/(x_{i_{1}}) ⊗ … ⊗ 1/(x_{i_{k}}).

Of course, the object 1/({x_{i} : i ∈ A}) depends on a choice of ordering of the elements in A, but only up to a canonical symmetry isomorphism. We could for example fix the particular choice by fixing a total order on A and taking the product in the proper order. The canonical maps p_{i}, i ∈ I composed with the map 1 → 1 give rise to the canonical map

p_{I} : 1 → 1/({x_{i} : i ∈ A})

defined as the composition

1 ⟶^{μ^{−1}} 1^{⊗k} → 1^{⊗k} ⟶^{p_{i_{1}} ⊗...⊗p_{i_{k}}} 1/({x_{i} : i ∈ A}).

For finite subsets A ⊂ B ⊂ I, define the map

ρ_{A⊂B} : 1/({x_{i} : i ∈ A}) → 1/({x_{i} : i ∈ B})

as the composition

1/({x_{i} : i ∈ A}) ⟶^{μ^{−1}} 1/({x_{i} : i ∈ A}) ⊗ 1

⟶^{id⊗p_{B\A}} 1/({x_{i} : i ∈ A}) ⊗ 1/({x_{i} : i ∈ B \ A}) ≅ 1/({x_{i} : i ∈ B}).

where the last isomorphism is again the symmetry isomorphism. Because C is a symmetric monoidal category with unit 1, we have a well-defined functor from the category P_{fin}(I) of finite subsets of I to C:

1/(−) : P_{fin}(I) → C

sending A ⊂ I to 1/({x_{i} : i ∈ A}) and sending each inclusion A ⊂ B to ρ_{A⊂B}.

Definition 1.2. The object 1/({x_{i} : i ∈ I}) of C is defined by

1/({x_{i}}) = hocolim_{A∈P_{fin}(I)} 1/({x_{i} : i ∈ A}).

More generally, for M ∈ C, we define M/({x_{i} : i ∈ I}) as

M/({x_{i} : i ∈ I}) := 1/({x_{i} : i ∈ I}) ⊗ QM,

where QM → M is a cofibrant replacement for M. In case the index set I is understood, we often write these simply as 1/({x_{i}}) or M/({x_{i}}).

Remark 1.3. 1. The object 1/(x_{i}) is cofibrant and hence the objects 1/({x_{i} : i ∈ A}) are cofibrant for all finite sets A. As a pointwise cofibrant diagram has cofibrant homotopy colimit [Hir03, corollary 14.8.1, example 18.3.6, corollary 18.4.3], 1/({x_{i} : i ∈ I}) is cofibrant. Thus M/({x_{i} : i ∈ I}) := 1/({x_{i} : i ∈ I}) ⊗ QM is also cofibrant.

2. We often select a single cofibrant object T and take T_{i} := T^{⊗d_{i}} for certain integers d_{i} > 0. As T is cofibrant, so is T^{⊗d_{i}}. In this case we set deg T = 1, deg T^{⊗d_{i}} = d_{i}.

We let [n] denote the set {0, …, n} with the standard order and Δ the category with objects [n], n = 0, 1, …, and morphisms the order-preserving maps of sets. For a small category A and a functor F : A → C, we let hocolim_{A} F_{∗} denote the standard simplicial object of C whose geometric realization is hocolim_{A} F, that is

hocolim_{A} F_{n} = ⋁_{σ:[n]→A} F (σ(0)).

Lemma 1.4. Let {x_{i} : T_{i} → 1 : i ∈ I_{1}}, {x_{i} : T_{i} → 1 : i ∈ I_{2}} be two sets of maps in C, with cofibrant sources T_{i}, and with I_{1}, I_{2} disjoint index sets. Then there is a canonical isomorphism

1/({x_{i} : i ∈ I_{1} ⨿ I_{2}}) ≅ 1/({x_{i} : i ∈ I_{1}}) ⊗ 1/({x_{i} : i ∈ I_{2}}).

Proof. The category P_{fin}(I_{1} ⨿ I_{2}) is clearly equal to P_{fin}(I_{1}) × P_{fin}(I_{2}). For functors F_{i} : A_{i} → C, i = 1, 2, [hocolim_{A_{1}×A_{2}} F_{1} ⊗ F_{2}]_{∗} is the diagonal simplicial space associated to the bisimplicial space (n, m) ↦ [hocolim_{A_{1}} F_{1}]_{n} ⊗ [hocolim_{A_{2}} F_{2}]_{m}. Thus

hocolim_{A_{1}×A_{2}} F_{1} ⊗ F_{2} ≅ hocolim_{A_{2}}[hocolim_{A_{1}} F_{1}] ⊗ F_{2}.

This gives us the isomorphism

1/({x_{i} : i ∈ I_{1} ⨿ I_{2}})

= hocolim_{(A_{1},A_{2})∈P_{fin}(I_{1})×P_{fin}(I_{2})} 1/({x_{i} : i ∈ A_{1}}) ⊗ 1/({x_{i} : i ∈ A_{2}})

≅ hocolim_{A_{1}∈P_{fin}(I_{1})} 1/({x_{i} : i ∈ A_{1}}) ⊗ hocolim_{A_{2}∈P_{fin}(I_{2})} 1/({x_{i} : i ∈ A_{2}})

= 1/({x_{i} : i ∈ I_{1}}) ⊗ 1/({x_{i} : i ∈ I_{2}}).

□

Remark 1.5. Via this lemma, we have the isomorphism for all M ∈ C,

M/({x_{i} : i ∈ I_{1} ⨿ I_{2}}) ≅ (M/({x_{i} : i ∈ I_{1}})/({x_{i} : i ∈ I_{2}}).

Let I be the category of formal monomials in {x_{i}}, that is, the category of maps N : I → ℕ, i ↦ N_{i}, such that N_{i} = 0 for all but finitely many i ∈ I, and with a unique map N → M if N_{i} ≥ M_{i} for all i ∈ I. As usual, the monomial in the x_{i} corresponding to a given N is ∏_{i∈I} x_{i}^{N_{i}}, written x^{N}. The index N = 0, corresponding to x^{0} = 1, is the final object of I. Take an i ∈ I. For m > k ≥ 0 integers, define the map

×x_{i}^{m−k} : 1 ⊗ T_{i}^{⊗m} → 1 ⊗ T_{i}^{⊗k}

as the composition

1⊗T_{i}^{⊗m} = 1⊗T_{i}^{⊗m−k} ⊗T_{i}^{⊗k} ⟶^{id_{1}⊗x_{i}^{⊗m−k}⊗id_{T{⊗k \atop i}}} 1^{⊗m−k+1}⊗T_{i}^{⊗k} ⟶^{μ⊗id} 1⊗T_{i}^{⊗k}.

In case k = 0, we use 1 instead of 1 ⊗ 1 for the target; we define ×x^{0} to be the identity map. The associativity of the maps μ_{1} shows that ×x_{i}^{m−k} ∘ ×x_{i}^{n−m} = ×x_{i}^{n−k}, hence the maps ×x_{i}^{n} all commute with each other. Now suppose we have a monomial in the x_{i}; to simplify the notation, we write the indices occurring in the monomial as {1, …, r} rather than {i_{1}, …, i_{r}}. This gives us the monomial x^{N} := x_{1}^{N_{1}} · . . . · x_{r}^{N_{r}}. Define

T_{∗}^{N} := 1 ⊗ T_{1}^{⊗N_{1}} ⊗ … ⊗ 1 ⊗ T_{r}^{⊗N_{r}} ⊗ 1;

in case N_{i} = 0, we replace …⊗1⊗1⊗1⊗T_{i+1}^{⊗M_{i+1}}⊗… with …⊗1⊗T_{i+1}^{⊗M_{i+1}}⊗…, and we set T_{∗}^{0} := 1. Let N → M be a map in I, that is N_{i} ≥ M_{i} ≥ 0 for all i. We again write the relevant index set as {1, …, r}. Define the map

×x^{N−M} : T_{∗}^{N} → T_{∗}^{M}

as the composition

T_{∗}^{N} \underrightarrow{⨂_{j=1}^{r} ×x_{j}^{N_{j} −M_{j}}} 1 ⊗ T_{1}^{⊗M_{1}} ⊗ … ⊗ 1 ⊗ T_{r}^{⊗M_{r}} ⊗ 1 ⟶^{μ_{M}} T_{∗}^{⊗M};

the map μ_{M} is a composition of ⊗-product of multiplication maps μ_{1} : 1⊗1 → 1, with these occurring in those spots with M_{j} = 0. In case N_{i} = M_{i} = 0, we simply delete the term ×x_{i}^{0} from the expression. The fact that the maps μ_{1} satisfy associativity yields the relation

×x^{M−K} ∘ ×x^{N−M} = ×x^{N−K}

and thus the maps ×x^{N−M} all commute with each other. Defining D_{x}(N) := T_{∗}^{N} and D_{x}(N → M) = ×x^{N−M} gives us the I-diagram

D_{x} : I → C.

We consider the following full subcategories of I. For a monomial M let I_{≥M} denote the subcategory of monomials which are divisible by M, and for a positive integer n, recalling that we have assigned each T_{i} a positive integral degree d_{i}, let I_{deg≥n} denote the subcategory of monomials of degree at least n, where the degree of N := (N_{1}, …, N_{k}) is N_{1}d_{1} + ⋯ + N_{k}d_{k}. One defines similarly the full subcategories I_{>M} and I_{deg>n}.

Let I^{∘} be the full subcategory of I of monomials N ≠ 0 and I_{≤1}^{∘} ⊂ I^{∘} be the full subcategory of monomials N for which N_{i} ≤ 1 for all i. We have the corresponding subdiagrams D_{x} : I^{∘} → C and D_{x} : I_{≤1}^{∘} → C of D_{x}. For J ⊂ I a subset, we have the corresponding full subcategories J ⊂ I, J^{∘} ⊂ I^{∘} and J_{≤1}^{∘} ⊂ I_{≤1}^{∘} and corresponding subdiagrams D_{x}. If the collection of maps x_{i} is understood, we write simply D for D_{x}. Let F : A → C be a functor, a an object in C, c_{a} : A → C the constant functor with value a and φ : F → c_{a} a natural transformation. Then φ induces a canonical map φ̃ : hocolim_{A} F → a in C. As in the proof of [S10, Proposition 4.4], let C(A) be the category A with a final object ∗ adjoined and C(F, φ) : C(A) → C the functor with value a on ∗, with restriction to A being F, and which sends the unique map y → ∗ in C(A), y ∈ A, to φ(y). Let [0, 1] be the category with objects 0, 1 and a unique non-identity morphism 0 → 1, and let C(A)^{Γ} be the full subcategory of C(A) × [0, 1] formed by removing the object ∗ × 1. We extend C(F, φ) to a functor C(F, φ)^{Γ} : C(A)^{Γ} → C by C(F, φ)^{Γ}(y × 1) = pt, where pt is the initial/final object in C.

Lemma 1.6. There is a natural isomorphism in C

hocolim_{C(A)^{Γ}} C(F, φ)^{Γ} ≅ hocofib(φ̃ : hocolim_{A} F → a).

Proof. For a category A we let N (A) denote the simplicial nerve of A. We have an isomorphism of simplicial sets N (C(A)) ≅ Cone(N (A), ∗), where Cone(N (A), ∗) is the cone over N (A) with vertex ∗. Similarly, the full subcategory A × [0, 1] of C(A)^{Γ} has nerve isomorphic to N (A) × Δ[1]. This gives an isomorphism of N (C(A)^{Γ}) with the push-out in the diagram

\xymatrix{N (A) \ar@{^{(}->}[r] \ar@{^{(}->}[d]_{id×δ_{0}} & Cone(N (A), ∗) \\ N (A) × Δ[1].}

This in turn gives an isomorphism of the simplicial object hocolim_{C(A)^{Γ}} C(F, φ)_{∗}^{Γ} with the pushout in the diagram

\xymatrix{hocolim_{A} F \ar@{^{(}->}[r] \ar@{^{(}->}[d] & C(hocolim_{A} F, a) \\ C(hocolim_{A} F, pt).}

This gives the desired isomorphism. □

Lemma 1.7. Let J ⊂ K ⊂ I be finite subsets of I. Then the map

hocolim_{J_{≤1}^{∘}} D_{x} → hocolim_{K_{≤1}^{∘}} D_{x}

induced by the inclusion J ⊂ K is a cofibration in C.

Proof. We give the category of simplicial objects in C, C^{Δ^{op}}, the Reedy model structure, using the standard structure of a Reedy category on Δ^{op}. By [Hir03, theorem 19.7.2(1), definition 19.8.1(1)], it suffices to show that

hocolim_{J_{≤1}^{∘}} D_{∗} → hocolim_{K_{≤1}^{∘}} D_{∗}

is a cofibration in C^{Δ^{op}}, that is, for each n, the map

φ_{n} : hocolim_{J_{≤1}^{∘}} D_{n} ⨿_{L^{n} hocolim_{J{∘ \atop ≤1}} D_{∗}} L^{n} hocolim_{K_{≤1}^{∘}} D_{∗} → hocolim_{K_{≤1}^{∘}} D_{n}

is a cofibration in C, where L^{n} is the nth latching space. We note that

hocolim_{J_{≤1}^{∘}} D_{n} = ⋁_{σ∈N (J_{≤1}^{∘})_{n}} D(σ(0)),

where we view σ ∈ N (J_{≤1}^{∘})_{n} as a functor σ : [n] → J_{≤1}^{∘}; we have a similar description of hocolim_{K_{≤1}^{∘}} D_{n}. The latching space is

L^{n} hocolim_{J_{≤1}^{∘}} D_{∗} = ⋁_{σ∈N (J_{≤1}^{∘})_{n}^{deg}} D(σ(0)),

where N (J_{≤1}^{∘})_{n}^{deg} is the subset of N (J_{≤1}^{∘})_{n} consisting of those σ which contain an identity morphism; L^{n} hocolim_{K_{≤1}^{∘}} D_{∗} has a similar description. The maps

L^{n} hocolim_{J_{≤1}^{∘}} D_{∗} → hocolim_{J_{≤1}^{∘}} D_{n}, L^{n} hocolim_{J_{≤1}^{∘}} D_{∗} → L^{n} hocolim_{K_{≤1}^{∘}} D∗,

L^{n} hocolim_{K_{≤1}^{∘}} D_{∗} → hocolim_{K_{≤1}^{∘}} D_{n}, hocolim_{J_{≤1}^{∘}} D_{n} → hocolim_{K_{≤1}^{∘}} D_{n}

are the unions of identity maps on D(σ(0)) over the respective inclusions of the index sets. As N (K_{≤1}^{∘})_{n}^{deg} ∩ N (J_{≤1}^{∘})_{n} = N (J_{≤1}^{∘})_{n}^{deg}, we have

hocolim_{J_{≤1}^{∘}} D_{n} ⨿_{L^{n} hocolim_{J{∘ \atop ≤1}} D_{∗}} L^{n} hocolim_{K_{≤1}^{∘}} D_{∗} ≅ hocolim_{J_{≤1}^{∘}} D_{n} ⋁ C,

where

C = ⋁_{σ∈N (K_{≤1}^{∘})_{n}^{deg}\N (J_{≤1}^{∘})_{n}^{deg}} D(σ(0)),

and the map to hocolim_{K_{≤1}^{∘}} D_{n} is the evident inclusion. As D(N) is cofibrant for all N, this map is clearly a cofibration, completing the proof. □

We have the n-cube □^{n}, the category associated to the partially ordered set of subsets of {1, …, n}, ordered under inclusion, and the punctured n-cube □_{0}^{n} of proper subsets. We have the two inclusion functors i_{n}^{+}, i_{n}^{−} : □^{n−1} → □^{n}, i_{n}^{+}(I) := I ∪ {n}, i_{n}^{−}(I) = I and the natural transformation ψ_{n} : i_{n}^{−} → i_{n}^{+} given as the collection of inclusions I ⊂ I ∪ {n}. The functor i_{n}^{−} induces the functor i_{n0}^{−} : □^{n−1} → □_{0}^{n}. For a functor F : □^{n} → C, we have the iterated homotopy cofiber, hocofib_{n}F, defined inductively as the homotopy cofiber of hocofib_{n−1}(F (ψ_{n})) : hocofib(F ∘ i_{n}^{−}) → hocofib(F ∘ i_{n}^{+}). Using this inductive construction, it is easy to define a natural isomorphism hocofib_{n}F ≅ hocolim_{□_{0}^{n+1}} F̂, where F̂ ∘ i_{n+10}^{−} = F and

F̂(I) = pt if n ∈ I. The following result, in the setting of modules over a model of MGL as a commutative 𝕊-algebra, is proven in [S10, Lemma 4.3 and Proposition 4.4]. We give here a somewhat different proof in our context, which allows for a wider application.

Lemma 1.8. Assume that I is countable. Then there is a canonical isomorphism in Ho C

1/({x_{i} | i ∈ I}) ≅ hocofib[hocolim_{I^{∘}} D_{x} → hocolim_{I} D_{x}].

Proof. As 1 is the final object in I, the collection of maps ×x^{N} : T_{∗}^{N} → 1 defines a weak equivalence π : hocolim_{I} D_{x} → 1. In addition, for each N ∈ I^{∘}, the comma category N/I_{≤1}^{∘} has initial object the map N → N̄, where N̄_{i} = 1 if N_{i} > 0, and N̄_{i} = 0 otherwise. Thus I_{≤1}^{∘} is homotopy right cofinal in I^{∘} (see e.g. [Hir03, definition 19.6.1]). Since D_{x} is a diagram of cofibrant objects in C, it follows from [Hir03, theorem 19.6.7] that the map hocolim_{I_{≤1}^{∘}} D_{x} → hocolim_{I^{∘}} D_{x} is a weak equivalence. This reduces us to identifying 1/({x_{i}}) with the homotopy cofiber of π_{≤1}^{∘} : hocolim_{I_{≤1}^{∘}} D_{x} → 1, where π_{≤1}^{∘} is the composition of π with the natural map : hocolim_{I_{≤1}^{∘}} D_{x} → hocolim_{I} D_{x}. Next, we reduce to the case of a finite set I. Take I = ℕ. Let P_{fin}(I) be the category of finite subsets of I, ordered by inclusion, consider the full subcategory P_{fin}^{O}(I) of P_{fin}(I) consisting of the subsets I_{n} := {1, …, n}, n = 1, 2 …, and let I_{n,≤1}^{∘} ⊂ I_{≤1}^{∘} be the full subcategory with all indices in I_{n}. As P_{fin}^{O}(I) is cofinal in P_{fin}(I), we have

colim_{n} hocolim_{I_{n,≤1}^{∘}} D_{x} ≅ hocolim_{I_{≤1}^{∘}} D_{x}.

Take n ≤ m. By lemma 1.7 the the map hocolim_{I_{n,≤1}^{∘}} D_{x} → hocolim_{I_{m,≤1}^{∘}} D_{x} is a cofibration in C. Thus, using the Reedy model structure on C^{ℕ} with ℕ considered as a direct category, the ℕ-diagram in C, n ↦ hocolim_{I_{n,≤1}^{∘}} D_{x}, is a cofibrant object in C^{ℕ}. As ℕ is a direct category, the fibrations in C^{ℕ} are the pointwise ones, hence ℕ has pointwise constants [Hir03, definition 15.10.1] and therefore [Hir03, theorem 19.9.1] the canonical map

hocolim_{n∈ℕ} hocolim_{I_{n,≤1}^{∘}} D_{x} → colim_{n∈ℕ} hocolim_{I_{n,≤1}^{∘}} D_{x}

is a weak equivalence in C. This gives us the weak equivalence in C

hocolim_{n} hocolim_{I_{n,≤1}^{∘}} D_{x} → hocolim_{I_{≤1}^{∘}} D_{x}.

Since ℕ is contractible, the canonical map hocolim_{ℕ} 1 → 1 is a weak equivalence in C, giving us the weak equivalences

hocofib[hocolim_{I_{≤1}^{∘}} D_{x} → 1]

∼ hocofib[hocolim_{n∈ℕ} hocolim_{I_{n,≤1}^{∘}} D_{x} → hocolim_{n∈ℕ} 1]
∼ hocolim_{n∈ℕ}[hocofib[hocolim_{I_{n,≤1}^{∘}} D_{x} → 1]].

Thus, we need only exhibit isomorphisms in Ho C

ρ_{n} : hocofib[hocolim_{I_{n,≤1}^{∘}} D_{x} → 1] → 1/(x_{1}, …, x_{n}) := 1/(x_{1}) ⊗ … ⊗ 1/(x_{n}),

which are natural in n ∈ ℕ. By lemma 1.6 we have a natural isomorphism in C,

hocofib[hocolim_{I_{n,≤1}^{∘}} D_{x} → 1] ≅ hocolim_{C(I_{n,≤1}^{∘})^{Γ}} C(D_{x}, π)^{Γ}.

However, I_{n,≤1}^{∘} is isomorphic to □_{0}^{n} by sending N = (N_{1}, …, N_{n}) to I(N) := {i | N_{i} = 0}. Similarly, C(I_{n,≤1}^{∘}) is isomorphic to □^{n}, and C(I_{n,≤1}^{∘})^{Γ} is thus isomorphic to □_{0}^{n+1}. From our discussion above, we see that hocolim_{C(I_{n,≤1}^{∘})^{Γ}} C(D_{x}, π)^{Γ} is isomorphic to hocofib_{n}C(D_{x}, π), so we need only exhibit isomorphisms in Ho C

ρ_{n} : hocofib_{n}C(D_{x}, π) → 1/(x_{1}) ⊗ … ⊗ 1/(x_{n})

which are natural in n ∈ ℕ. We do this inductively as follows. To include the index n in the notation, we write C(D_{x}, π)_{n} for the functor C(D_{x}, π) : □^{n} → C. For n = 1, hocofib_{1}C(D_{x}, π)_{1} is the mapping cone of μ_{1}∘(×x_{1}⊗id) : 1⊗T_{1}⊗1 → 1, which is isomorphic in Ho C to the homotopy cofiber of ×x_{1} : 1 ⊗ T_{1} → 1. As this latter homotopy cofiber is equal to 1/(x_{1}), so we take ρ_{1} : hocofib_{1}C(D_{x}, π)_{1} → 1/(x_{1}) to be this isomorphism. We note that C(D_{x}, π)_{n} ∘ i_{n}^{+} = C(D_{x}, π)_{n−1} and C(D_{x}, π)_{n} ∘ i_{n}^{−} = C(D_{x}, π)_{n−1} ⊗ T_{n} ⊗ 1. Define C(D_{x}, π)′_{n} by C(D_{x}, π)′_{n}∘i_{n}^{−} = C(D_{x}, π)_{n−1}⊗1⊗T_{n}⊗1, C(D_{x}, π)′_{n}∘i_{n}^{+} = C(D_{x}, π)_{n−1} ⊗ 1, with the natural transformation C(D_{x}, π)′_{n} ∘ ψ_{n} given as

C(D_{x}, π)_{n−1} ⊗ 1 ⊗ T_{n} ⊗ 1 ⟶^{(id⊗μ)∘(id⊗×x_{n}⊗id_{1})} C(D_{x}, π)_{n−1} ⊗ 1.

The evident multiplication maps give a weak equivalence C(D_{x}, π)′_{n} → C(D_{x}, π)_{n}, giving us the isomorphism in Ho C

ρ_{n} : hocofib_{n}C(D_{x}, π)_{n} → 1/(x_{1}) ⊗ … ⊗ 1/(x_{n})

defined as the composition

hocofib_{n}C(D_{x}, π)_{n} ≅ hocofib_{n}C(D_{x}, π)′_{n}
≅ hocofib(hocofib_{n−1}(C(D_{x}, π)_{n−1} ⊗ 1 ⊗ T_{n})

⟶^{hocofib_{n−1}(id⊗×x_{n})} hocofib_{n−1}(C(D_{x}, π)_{n−1} ⊗ 1))

≅ hocofib(hocofib_{n−1}(C(D_{x}, π)_{n−1}) ⊗ 1 ⊗ T_{n}

⟶^{id⊗×x_{n}} hocofib_{n−1}(C(D_{x}, π)_{n−1}) ⊗ 1)

≅ hocofib_{n−1}(C(D_{x}, π)_{n−1}) ⊗ hocofib(×x_{n} : 1 ⊗ T_{n} → 1)

= hocofib_{n−1}(C(D_{x}, π)_{n−1}) ⊗ 1/(x_{n})

⟶^{ρ_{n−1}⊗id} 1/(x_{1}) ⊗ … ⊗ 1/(x_{n−1}) ⊗ 1/(x_{n}).

Via the definition of hocofib_{n},

hocofib_{n}C(D_{x}, π)_{n} = hocofib[hocofib_{n−1}(C(D_{x}, π)_{n} ∘ i_{n}^{−})

⟶^{hocofib_{n−1}(C(D_{x},π)_{n−1}(ψ_{n}))} hocofib_{n}(C(D_{x}, π)_{n} ∘ i_{n}^{+}]

and the identification C(D_{x}, π)_{n} ∘ i_{n}^{+} = C(D_{x}, π)_{n−1}, we have the canonical map hocofib_{n−1}(C(D_{x}, π)_{n−1}) → hocofib_{n}(C(D_{x}, π)_{n}). One easily sees that the diagram

\xymatrix{hocofib[hocolim_{I_{n−1,≤1}^{∘}} D_{x} → 1] \ar[d]_{∼} \ar[r] & hocofib[hocolim_{I_{n,≤1}^{∘}} D_{x} → 1] \ar[d]^{∼} \\ hocofib_{n−1}(C(D_{x}, π)_{n−1}) \ar[r] \ar[d]_{ρ_{n−1}} & hocofib_{n}(C(D_{x}, π)_{n}) \ar[d]^{ρ_{n}} \\ 1/(x_{1}) ⊗ … ⊗ 1/(x_{n−1}) \ar[r]_{ρ_{\{1,...,n−1\}⊂\{1,...,n\}}} & 1/(x_{1}) ⊗ … ⊗ 1/(x_{n})}

commutes in Ho C, giving the desired naturality in n. □

Now let M be an object in C, let QM → M be a cofibrant replacement and form the I-diagram D_{x} ⊗ QM : I → C, (D_{x} ⊗ QM)(N) = D_{x}(N) ⊗ QM.

Proposition 1.9. Assume that I is countable. Let M be an object in C. Then there is a canonical isomorphism in Ho C

M/({x_{i} | i ∈ I}) ≅ hocofib[hocolim_{I^{∘}} D_{x} ⊗ QM → hocolim_{I} D_{x} ⊗ QM].

Proof. This follows directly from lemma 1.8, noting the definition of M/({x_{i} | i ∈ I}) as [1/({x_{i} | i ∈ I})] ⊗ QM and the canonical isomorphism

hocofib[hocolim_{I^{∘}} D_{x} ⊗ QM → hocolim_{I} D_{x} ⊗ QM]

≅ hocofib[hocolim_{I^{∘}} D_{x} → hocolim_{I} D_{x}] ⊗ QM.

□

Proposition 1.10. Let F : I_{deg≥n} → C be a diagram in a cofibrantly generated model category C. Suppose for every monomial M of degree n the natural map hocolim F|_{I_{>M}} → F(M) is a weak equivalence. Then the natural map

hocolim F|_{deg≥n+1} → hocolim F

is a weak equivalence.

Proof. This is just [S10, lemma 4.5], with the following corrections: the statement of the lemma in loc. cit. has “hocolim F|_{I_{≥M}} → F(M) is a weak equivalence” rather than the correct assumption “hocolim F|_{I_{>M}} → F(M) is a weak equivalence” and in the proof, one should replace the object Q(M) with colim Q|_{I>M} rather than with colim Q|_{I≥M}. □

2. Slices of effective motivic module spectra

In this section we will describe the slices for modules for a commutative and effective ring T -spectrum R, assuming certain additional conditions. We adapt the constructions used in describing slices of MGL in [S10]. Let us first recall from [Vo00] the definition of the slice tower in SH(S). We will use the standard model category Mot := Mot(S) of symmetric T -spectra over S, T := 𝔸^{1}/𝔸^{1} \ {0}, with the motivic model structure as in [J00], for defining the triangulated tensor category SH(S) := Ho Mot(S). For an integer q, let Σ_{T}^{q} SH^{eff}(S) denote the localizing subcategory of SH(S) generated by S_{q} := {Σ_{T}^{q} Σ_{T}^{∞}X_{+} | p ≥ q, X ∈ Sm/S}, that is, Σ_{T}^{q} SH^{eff}(S) is the smallest triangulated subcategory of SH(S) which contains S_{q} and is closed under direct sums and isomorphisms in SH(S). This gives a filtration on SH(S) by full localizing subcategories

⋯ ⊂ Σ_{T}^{q+1}SH^{eff}(S) ⊂ Σ_{T}^{q} SH^{eff}(S) ⊂ Σ_{T}^{q−1}SH^{eff}(S) ⊂ ⋯ ⊂ SH(S).

The set S_{q} is a set of compact generators of Σ_{T}^{q} SH(S) and the set ∪_{q}S_{q} is similarly a set of compact generators for SH(S). By Neeman’s triangulated version of Brown representability theorem [N97], the inclusion i_{q} : Σ_{T}^{q} SH^{eff}(S) → SH(S) has a right adjoint r_{q} : SH(S) → Σ_{T}^{q} SH^{eff}(S). We let f_{q} := i_{q}∘r_{q}. The inclusion Σ_{T}^{q+1}SH^{eff}(S) → Σ_{T}^{q} SH^{eff}(S) induces a canonical natural transformation f_{q+1} → f_{q}. Putting these together forms the slice tower

(2.1)  ⋯ → f_{q+1} → f_{q} → ⋯ → id.

For each q there exists a triangulated functor s_{q} : SH(S) → SH(S) and a canonical and natural distinguished triangle

f_{q+1}(E) → f_{q}(E) → s_{q}(E) → Σf_{q+1}(E)

in SH(S). In particular, s_{q}(E) is in Σ_{T}^{q} SH^{eff}(S) for each E ∈ SH(S). Pelaez has given a lifting of the construction of the functors f_{q} to the model category level. For this, he starts with the model category Mot and forms for each n the right Bousfield localization of Mot with respect to the objects Σ_{T}^{m}F_{n}X_{+} with m − n ≥ q and X ∈ Sm/S. Here F_{n}X_{+} is the shifted T - suspension spectrum, that is, Σ_{T}^{m−n}X_{+} in degree m ≥ n, pt in degree m < n, and with identity bonding maps. Calling this Bousfield localization Mot_{q}, the functor r_{q} is given by taking a functorial cofibrant replacement in Mot_{q}. As the underlying categories are all the same, this gives liftings f̃_{q} of f_{q} to endofunctors on Mot. The technical condition on Mot invoked by Pelaez is that of cellularity and right properness, which ensures that the right Bousfield localization exists; this follows from the work of Hirschhorn [Hir03]. Alternatively, one can use the fact that Mot is a combinatorial right proper model category, following work of J. Smith, detailed for example in [B10]. The combinatorial property passes to module categories, and so this approach will be useful here. The category Mot is a closed symmetric monoidal simplicial model category, with cofibrant unit the sphere (symmetric) spectrum 𝕊_{S} and product ∧. Let R be a commutative monoid in Mot. We have the model category C := R-Mod of R-modules, as constructed in [ScSh]. The fibrations and weak equivalences are the morphisms which are fibrations, resp. weak equivalences, after applying the forgetful functor to Mot; cofibrations are those maps having the left lifting property with respect to trivial fibrations. This makes C into a pointed closed symmetric monoidal simplicial model category; C is in addition cofibrantly generated and combinatorial. Assuming that R is a cofibrant object in Mot, the free R-module functor, E ↦ R ∧ E, gives a left adjoint to the forgetful functor and gives rise to a Quillen adjunction. For details as to these facts and a general construction of this model category structure on module categories, we refer the reader to [ScSh]; another source is [Hov], especially theorem 1.3, proposition 1.9 and proposition 1.10. The model category R-Mod inherits right properness from Mot. We may therefore form the right Bousfield localization C_{q} with respect to the free R-modules R ∧ Σ_{T}^{m}F_{n}X_{+} with m − n ≥ q and X ∈ Sm/S, and define the endofunctor f̃_{q}^{R} on C by taking a functorial cofibrant replacement in C_{q}. By the adjunction, one sees that Ho C_{q} is equivalent to the localizing subcategory of Ho C (compactly) generated by {R ∧ Σ_{T}^{m}F_{n}X_{+} | m − n ≥ q, X ∈ Sm/S}. We denote this localizing subcategory by Σ_{T}^{q} Ho C^{eff}, or Ho C^{eff} for q = 0. We call an object M of C effective if the image of M in Ho C is in Ho C^{eff}, and denote the full subcategory of effective objects of C by C^{eff}. Just as above, Neeman’s results give a right adjoint r_{q}^{R} to the inclusion i_{q}^{R} : Σ_{T}^{q} C^{eff} → C and the composition f_{q}^{R} := i_{q}^{R} ∘ r_{q}^{R} is represented by f̃_{q}^{R}. One recovers the functors f_{q} and f̃_{q} by taking R = 𝕊_{S}.

Lemma 2.1. Let R be a cofibrant commutative monoid in Mot. The functors f_{q}^{R} : Ho C → Ho C and their liftings f̃_{q}^{R} have the following properties.

(1) Each f_{n}^{R} is idempotent, i.e., (f_{n}^{R})^{2} = f_{n}^{R}.

(2) f_{n}^{R}Σ_{T}^{1} = Σ_{T}^{1} f_{n−1}^{R} for n ∈ ℤ.

(3) Each f̃_{n}^{R} commutes with homotopy colimits.

(4) Suppose that R is in SH^{eff}(S). Then the forgetful functor U : Ho R-Mod → SH(S) induces an isomorphism U ∘ f_{q}^{R} ≅ f_{q} ∘ U as well as an isomorphism U ∘ s_{q}^{R} ≅ s_{q} ∘ U, for all q ∈ ℤ.

Proof. (1) and (2) follow from universal property of triangulated functors f_{n}^{R}. In case R = 𝕊_{S}, (3) is proved in [S10, Cor 4.6]; the proof for general R is the same. For (4), it suffices to prove the result for f_{q} and f_{q}^{R}. Take M ∈ C. We check the universal property of Uf_{q}^{R}M → UM: Since R is in SH^{eff}(S) and the functor − ∧ R is compatible with homotopy cofiber sequences and direct sums, −∧R maps Σ_{T}^{q} SH^{eff}(S) into itself for each q ∈ ℤ. As U(R∧E) = R∧E, it follows that U(Σ_{T}^{q} Ho R-Mod^{eff}) ⊂ Σ_{T}^{q} SH^{eff}(S) for each q. In particular, U(f_{q}^{R}(M)) is in Σ_{T}^{q} SH^{eff}(S). For p ≥ q, X ∈ Sm/S, we have

Hom_{SH(S)}(Σ_{T}^{p} Σ_{T}^{∞}X_{+}, U(f_{q}^{R}(M))) ≅ Hom_{Ho C}(R ∧ Σ_{T}^{p} Σ_{T}^{∞}X_{+}, f_{q}^{R}(M))
≅ Hom_{Ho C}(R ∧ Σ_{T}^{p} Σ_{T}^{∞}X_{+}, M)
≅ Hom_{SH(S)}(Σ_{T}^{p} Σ_{T}^{∞}X_{+}, U(M)),

so the canonical map U(f_{q}^{R}(M)) → f_{q}(U(M)) is therefore an isomorphism. □

From the adjunction Hom_{C}(R, M) ≅ Hom_{Mot}(𝕊_{S}, M) and the fact that 𝕊_{S} is a cofibrant object of Mot, we see that R is a cofibrant object of C. Thus C is a closed symmetric monoidal simplicial model category with cofibrant unit 1 := R and monoidal product ⊗ = ∧_{R}. Similarly, T_{R} := R ∧ T is a cofibrant object of C. Abusing notation, we write Σ_{T} (−) for the endofunctor A ↦ A ⊗ T_{R} of C. The compatibility of the simplicial monoidal structure with monoidal structure of C follows directly from the construction of C. We recall that the category Mot satisfies the monoid axiom of Schwede-Shipley [ScSh, definition 3.3]; the reader can see for example the proof of [Hoy, lemma 4.2]. Following remark 1.1, there is a fibrant replacement R → 1 in C such that 1 is an R-algebra; in particular, R → 1 is a cofibration and a weak equivalence in both C and in Mot, and 1 is fibrant in in both C and in Mot. For each x̄ ∈ R^{−2d,−d}(S), we have the corresponding element x̄ : T_{R}^{⊗d} → R in Ho C, which we may lift to a morphism x : T_{R}^{⊗d} → 1 in C. Thus, for a collection of elements {x̄_{i} ∈ R^{−2d_{i},−d_{i}}(S) | i ∈ I}, we have the associated collection of maps in C, {x_{i} : T_{R}^{⊗d_{i}} → 1 | i ∈ I} and thereby the quotient object 1/({x_{i}}) in C. Similarly, for M an R-module, we have the R-module M/({x_{i}}), which is a cofibrant object in C. We often write R/({x_{i}}) for 1/({x_{i}}).

Lemma 2.2. Suppose that R is in SH^{eff}(S). Then for any set

{x̄_{i} ∈ R^{−2d_{i},−d_{i}}(S) | i ∈ I, d_{i} > 0}

of elements of R-cohomology, the object R/({x_{i}}) is effective. If in addition M is an R-module and is effective, then M/({x_{i}}) is effective.

Proof. This follows from lemma 2.1 since f_{n}^{R} is a triangulated functor and C^{eff} is closed under homotopy colimits. □

Let A be an abelian group and SA the topological sphere spectrum with Acoefficients. For a T -spectrum E let us denote the spectrum E ∧ SA by E ⊗ A. Of course, if A is the free abelian group on a set S, then E ⊗ A = ⊕_{s∈S}E. Let {x̄_{i} ∈ R^{−2d_{i},−d_{i}}(S) | i ∈ I, d_{i} > 0} be a set of elements of R-cohomology, with I countable. Suppose that R is cofibrant as an object in Mot and is in SH^{eff}(S). Let M be in C^{eff} and let QM → M be a cofibrant replacement. By lemma 1.8, we have a homotopy cofiber sequence in C,

hocolim_{I^{∘}} D_{x} ⊗ QM → QM → M/({x_{i}}).

Clearly hocolim_{I^{∘}} D_{x}⊗QM is in Σ_{T}^{1} Ho C^{eff}, hence the above sequence induces an isomorphism in Ho C

s_{0}^{R}M ⟶^{σ_{M}} s_{0}^{R}(M/({x_{i}})).

Composing the canonical map M/({x_{i}}) → s_{0}^{R}(M/({x_{i}})) with σ_{M}^{−1} gives the canonical map

π_{M}^{R} : M/({x_{i}}) → s_{0}^{R}M

in Ho C. Applying the forgetful functor gives the canonical map in SH(S)

π_{M} : U(M/({x_{i}})) → U(s_{0}^{R}M) ≅ s_{0}(UM).

This equal to the canonical map U(M/({x_{i}})) → s_{0}(U(M/({x_{i}}))) composed with the inverse of the isomorphism s_{0}(UM) → s_{0}(U(M/({x_{i}}))).

Theorem 2.3. Let R be a commutative monoid in Mot(S), cofibrant as an object in Mot(S), such that R is in SH^{eff}(S). Let X = {x̄_{i} ∈ R^{−2d_{i},−d_{i}}(S) | i ∈ I, d_{i} > 0} be a countable set of elements of R-cohomology. Let M be an Rmodule in C^{eff} and suppose that the canonical map π_{M} : U(M/({x_{i}})) → s_{0}(UM) is an isomorphism. Then for each n ≥ 0, we have a canonical isomorphism in Ho C,

s_{n}^{R}M ≅ Σ_{T}^{n} s_{0}^{R}M ⊗ ℤ[X]_{n},

where ℤ[X]_{n} is the abelian group of weighted-homogeneous degree n polynomials over ℤ in the variables {x_{i}, i ∈ I}, deg x_{i} = d_{i}. Moreover, for each n, we have a canonical isomorphism in SH(S),

s_{n}UM ≅ Σ_{T}^{n} s_{0}UM ⊗ ℤ[X]_{n}.

Proof. Replacing M with a cofibrant model, we may assume that M is cofibrant in C; as R is cofibrant in Mot, it follows that UM is cofibrant in Mot. Since π_{M} = U(π_{M}^{R}), our assumption on π_{M} is the same as assuming that π_{M}^{R} is an isomorphism in Ho C. By construction, π_{M}^{R} extends to a map of distinguished triangles

[figure: (hocolim_{I^{∘}} D_{x}) ⊗ M; α; M; M/({x_{i}}); π_{M}^{R}; Σ(hocolim_{I^{∘}} D_{x}) ⊗ M; Σα; f_{1}^{R}M; M; s_{0}^{R}M; Σf_{1}^{R}M,]

Documenta Mathematica · Extra Volume Merkurjev (2015) 407–442

and thus the map α is an isomorphism. We note that α is equal to the canonical map given by the universal property of f_{1}^{R}M → M. We will now identify f_{n}^{R}M in terms of the diagram D_{x}|_{I_{deg≥n}} ⊗ M, proving by induction on n ≥ 1 that the canonical map hocolim D_{x} ⊗ M|_{deg≥n} → f_{n}^{R}M in Ho C is an isomorphism. As I^{∘} = I_{deg≥1}, the case n = 1 is settled. Assume the result for n. We claim that the diagram

f̃_{n+1}^{R}[D_{x} ⊗ M|_{deg≥n}] : I_{deg≥n} → C

satisfies the hypotheses of proposition 1.10. That is, we need to verify that for every monomial M of degree n the natural map

hocolim f̃_{n+1}^{R}[D_{>M} ⊗ M] → f̃_{n+1}^{R}[D(M) ⊗ M]

is a weak equivalence in C. This follows by the string of isomorphisms in Ho C

hocolim f̃_{n+1}^{R}[D_{>M} ⊗ M] ≅ hocolim f̃_{n+1}^{R}[Σ_{T}^{n} D_{deg≥1} ⊗ M]
≅ hocolim Σ_{T}^{n} f̃_{1}^{R}[D_{deg≥1} ⊗ M]
≅ Σ_{T}^{n} f_{1}^{R} hocolim[D_{deg≥1} ⊗ M]
≅ Σ_{T}^{n} f_{1}^{R}f_{1}^{R}M
≅ Σ_{T}^{n} f_{1}^{R}M
≅ f_{n+1}^{R}Σ_{T}^{n} M
≅ f_{n+1}^{R}[D(M) ⊗ M].

Applying proposition 1.10 and our induction hypothesis gives us the string of isomorphisms in Ho C

f_{n+1}^{R}M ≅ f_{n+1}^{R}f_{n}^{R}M ≅ f_{n+1}^{R} hocolim[D_{x} ⊗ M|_{deg≥n}]

≅ hocolim f̃_{n+1}^{R}[D_{x} ⊗ M|_{deg≥n}] ≅ hocolim f̃_{n+1}^{R}[D_{x} ⊗ M|_{deg≥n+1}]

≅ hocolim D_{x} ⊗ M|_{deg≥n+1},

the last isomorphism following from the fact that D_{x}(x^{N}) ⊗ M is in Σ_{T}^{|N|}C^{eff}, and hence the canonical map f̃_{n+1}^{R}[D_{x} ⊗ M] → D_{x} ⊗ M is an objectwise weak equivalence on I_{deg≥n+1}. For the slices s_{n} we have

s_{n}^{R}M := hocofib(f̃_{n+1}^{R}M → f̃_{n}^{R}M) ≅ hocofib(f̃_{n+1}^{R}f̃_{n}^{R}M → f̃_{n}^{R}M)
≅ hocofib(hocolim f̃_{n+1}^{R}[D_{deg≥n} ⊗ M] → hocolim D_{deg≥n} ⊗ M)
≅ hocolim hocofib(f̃_{n+1}^{R}[D_{deg≥n} ⊗ M] → D_{deg≥n} ⊗ M).

At a monomial of degree greater than n, the canonical map f̃_{n+1}^{R}[D_{deg≥n} ⊗ M] → D_{deg≥n} ⊗ M is a weak equivalence, and at a monomial M of degree n the homotopy cofiber is given by

hocofib(f̃_{n+1}^{R}[D(M) ⊗ M] → D(M) ⊗ M) = hocofib((f̃_{n+1}^{R}[Σ_{T}^{n} M] → Σ_{T}^{n} M)

≅ hocofib(Σ_{T}^{n} f̃_{1}^{R}M → Σ_{T}^{n} M) ≅ Σ_{T}^{n} s_{0}^{R}M

Let s̃_{0}^{R} be the functor on C^{eff}, N ↦ hocofib(f̃_{1}^{R}N → N), and let F_{n}M : I_{deg≥n} → C^{eff} be the diagram

F_{n}(M) = \begin{cases}pt & for deg M > n \\ Σ_{T}^{n} s̃_{0}^{R}M & for deg M = n. \end{cases}

We thus have a weak equivalence of pointwise cofibrant functors

hocofib(f̃_{n+1}^{R}[D_{deg≥n} ⊗ M] → D_{deg≥n} ⊗ M) → F_{n} : I_{deg≥n} → C,

and therefore a weak equivalence on the homotopy colimits. As we have the evident isomorphism in Ho C

hocolim_{I_{deg≥n}} F_{n} ≅ ⊕_{M,deg M=n}Σ_{T}^{n} s_{0}^{R}M,

this gives us the desired isomorphism s_{n}^{R}M ≅ Σ_{T}^{n} s_{0}^{R}M ⊗ ℤ[X]_{n} in Ho C. Applying the forgetful functor and using lemma 2.1 gives the isomorphism s_{n}UM ≅ Σ_{T}^{n} s_{0}UM ⊗ ℤ[X]_{n} in SH(S). □

Corollary 2.4. Let R, X and M be as in theorem 2.3. Let Z = {z_{j} ∈ ℤ[X]_{e_{j}}} be a collection of homogeneous elements of ℤ[X], and let M[Z^{−1}] ∈ C be the localization of M with respect to the collection of maps ×z_{j} : M → Σ_{T}^{−e_{j}}M. Then there are natural isomorphisms

s_{n}^{R}M[Z^{−1}] ≅ Σ_{T}^{n} s_{0}^{R}M ⊗ ℤ[X][Z^{−1}]_{n},
s_{n}UM[Z^{−1}] ≅ Σ_{T}^{n} s_{0}UM ⊗ ℤ[X][Z^{−1}]_{n}.

Proof. Each map ×z_{j} : M → Σ_{T}^{−e_{j}}M induces the isomorphism ×z_{j} : M[Z^{−1}] → Σ_{T}^{−e_{j}}M[Z^{−1}] in Ho C, with inverse ×z_{j}^{−1} : Σ_{T}^{−e_{j}}M[Z^{−1}] → M[Z^{−1}]. Applying f_{q}^{R} gives us the map in Ho C

×z_{j} : f_{q}^{R}M → f_{q}^{R}Σ_{T}^{−e_{j}}M ≅ Σ_{T}^{−e_{j}}f_{q+e_{j}}^{R}M.

As f_{q+e_{j}}^{R}M is in Σ_{T}^{q+e_{j}}Ho C^{eff}, both Σ_{T}^{−e_{j}}f_{q+e_{j}}^{R}M and f_{q}^{R}M are in Σ_{T}^{q} Ho C^{eff}. The composition

Σ_{T}^{−e_{j}}f_{q+e_{j}}^{R}M → Σ_{T}^{−e_{j}}M ⟶^{×z_{j}^{−1}} M[Z^{−1}]

gives via the universal property of f_{q}^{R} the map Σ_{T}^{−e_{j}}f_{q+e_{j}}^{R}M → f_{q}^{R}M[Z^{−1}]. Setting |N| = ∑_{j} N_{j}e_{j}, this extends to give a map of the system of monomial multiplications

×z^{N−M} : Σ_{T}^{−|N|}f_{q+|N|}^{R}M → Σ_{T}^{−|M|}f_{q+|M|}^{R}M

to f_{q}^{R}M[Z^{−1}]; the universal property of the truncation functors f_{n} and of localization shows that this system induces an isomorphism

hocolim_{N∈I^{op}} Σ_{T}^{−|N|}f_{q+|N|}^{R}M ≅ f_{q}^{R}M[Z^{−1}]

in Ho C. As the slice functors s_{q} are exact and commute with hocolim, we have a similar collection of isomorphisms

hocolim_{N∈I^{op}} Σ_{T}^{−|N|}s_{q+|N|}^{R}M ≅ s_{q}(M[Z^{−1}]).

Theorem 2.3 gives us the natural isomorphisms

Σ_{T}^{−|N|}s_{q+|N|}^{R}M ≅ Σ_{T}^{q} s_{0}^{R}M ⊗ ℤ[X]_{q+|N|};

via this isomorphism, the map ×z_{j} goes over to id_{Σ_{T}^{q} s_{0}^{R}M} ⊗ ×z_{j}, which yields the result. □

Corollary 2.5. Let R, X and M be as in theorem 2.3. Let Z = {z_{j} ∈ ℤ[X]_{e_{j}}} be a collection of homogeneous elements of ℤ[X], and let M[Z^{−1}] ∈ C be the localization of M with respect to the collection of maps ×z_{j} : M → Σ_{T}^{−e_{j}}M. Let m ≥ 2 be an integer. We let M[Z^{−1}]/m := hocofib × m : M[Z^{−1}] → M[Z^{−1}]. Then there are natural isomorphisms

s_{n}^{R}M[Z^{−1}]/m ≅ Σ_{T}^{n} s_{0}^{R}M/m ⊗ ℤ[X][Z^{−1}]_{n},
s_{n}UM[Z^{−1}]/m ≅ Σ_{T}^{n} s_{0}UM/m ⊗ ℤ[X][Z^{−1}]_{n}.

This follows directly from corollary 2.4, noting that s_{n}^{R} and s_{n} are exact functors.

Remark 2.6. Let P be a multiplicatively closed subset of ℤ. We may replace Mot with its localization Mot[P^{−1}] with respect to P in theorem 2.3, corollary 2.4 and corollary 2.5, and obtain a corresponding description of s_{n}^{R}M and s_{n}UM for a commutative monoid R in Mot[P^{−1}] and an effective R-module M. For P = ℤ\{p^{n}, n = 1, 2, …}, we write Mot⊗ℤ_{(p)} for Mot[P^{−1}] and SH(S)⊗ ℤ_{(p)} for Ho Mot ⊗ ℤ_{(p)}.

3. The slice spectral sequence

The slice tower in SH(S) gives us the slice spectral sequence, for E ∈ SH(S), X ∈ Sm/S, n ∈ ℤ,

(3.1)  E_{2}^{p,q}(n) := (s_{−q}(E))^{p+q,n}(X) ⟹ E^{p+q,n}(X).

This spectral sequence is not always convergent, however, we do have a convergence criterion:

Lemma 3.1 ([L15, lemma 2.1]). Suppose that S = Spec k, k a perfect field. Take E ∈ SH(S). Suppose that there is a non-decreasing function f : ℤ → ℤ with lim_{n→∞} f(n) = ∞, such that π_{a+b,b}E = 0 for a ≤ f(b). Then the for all Y, and all n ∈ ℤ, the spectral sequence (3.1) is strongly convergent.^{3}

This yields our first convergence result. For E ∈ SH(S), Y ∈ Sm/S, p, q, n ∈ ℤ, define

H^{p−q}(Y, π_{−q}^{μ}(E)(n − q)) := Hom_{SH(S)}(Σ_{T}^{∞}Y_{+}, Σ^{p+q,n}s_{−q}(E)).

Here Σ^{a,b} is suspension with respect to the sphere S^{a,b} ≅ S^{a−b} ∧ 𝔾_{m}^{∧b}. This notation is justified by the case S = Spec k, k a field of characteristic zero. In this case, there is for each q a canonically defined object π_{q}^{μ}(E) of Voevodsky’s “big” triangulated category of motives DM(k), and a canonical isomorphism

EM_{𝔸^{1}}(π_{q}^{μ}(E)) ≅ Σ_{T}^{q} s_{q}(E),

where EM_{𝔸^{1}} : DM(k) → SH(k) is the motivic Eilenberg-MacLane functor. The adjoint property of EM_{𝔸^{1}} yields the isomorphism

H^{p−q}(Y, π_{−q}^{μ}(E)(n − q)) := Hom_{DM(k)}(M(Y), π_{−q}^{μ}(E)(n − q)[p − q])

≅ Hom_{SH(S)}(Σ_{T}^{∞}Y_{+}, Σ^{p+q,n}s_{−q}(E)).

We refer the reader to [P11, RO08, Vo04] for details.

Proposition 3.2. Let R be a commutative monoid in Mot(S), cofibrant as an object in Mot(S), with R in SH^{eff}(S). Let X := {x̄_{i} ∈ R^{−2d_{i},−d_{i}}(S)} be a countable set of elements of R-cohomology, with d_{i} > 0. Let P be a multiplicatively closed subset of ℤ and let M be an R[P^{−1}]-module, with UM ∈ SH(S)^{eff}[P^{−1}]. Suppose that the canonical map

U(M/({x_{i}})) → s_{0}UM

is an isomorphism in SH(S)[P^{−1}]. Then

1. The slice spectral sequence for M^{∗∗}(Y) has the following form:

E_{2}^{p,q}(n) := H^{p−q}(Y, π_{0}^{μ}(M)(n − q)) ⊗_{ℤ} ℤ[X]_{−q} ⟹ M^{p+q,n}(Y).

2. Suppose that S = Spec k, k a perfect field. Suppose further that there is an integer a such that M^{2r+s,r}(Y) = 0 for all Y ∈ Sm/S, all r ∈ ℤ and all s ≥ a. Then the slice spectral sequence converges strongly for all Y ∈ Sm/S, n ∈ ℤ.

Proof. The form of the slice spectral sequence follows directly from theorem 2.3, extended via remark 2.6 to the P -localized situation. The convergence statement follows directly from lemma 3.1, where one uses the function f(r) = r − a. □

We may extend the slice spectral sequence to the localizations M[Z^{−1}] as in corollary 2.4.

3As spectral sequence {E_{r}^{pq}} ⇒ G^{p+q} converges strongly to G^{∗} if for each n, the spectral sequence filtration F^{∗}G^{n} on G^{n} is finite and exhaustive, there is an r(n) such that for all p and all r ≥ r(n), all differentials entering and leaving E_{r}^{p,n−p} are zero and the resulting maps E_{r}^{p,n−p} → E_{∞}^{p,n−p} = Gr_{F}^{p} G^{n} are all isomorphisms.

Documenta Mathematica · Extra Volume Merkurjev (2015) 407–442

Proposition 3.3. Let R, X, P and M be as in proposition 3.3 and assume that all the hypotheses for (1) in that proposition hold. Let Z = {z_{j} ∈ ℤ[X]_{e_{j}}} be a collection of homogeneous elements of ℤ[X], and let M[Z^{−1}] ∈ C be the localization of M with respect to the collection of maps ×z_{j} : M → Σ_{T}^{−e_{j}}M. Then the slice spectral sequence for M[Z^{−1}]^{∗∗}(Y) has the following form:

E_{2}^{p,q}(n) := H^{p−q}(Y, π_{0}^{μ}(M)(n − q)) ⊗_{ℤ} ℤ[X][Z^{−1}]_{−q} ⟹ M[Z^{−1}]^{p+q,n}(Y).

Suppose further that S = Spec k, k a perfect field, and there is an integer a such that M^{2r+s,r}(Y) = 0 for all Y ∈ Sm/S all r ∈ ℤ and all s ≥ a. Then the slice spectral sequence converges strongly for all Y ∈ Sm/S, n ∈ ℤ.

The proof is same as for proposition 3.2, using corollary 2.4 to compute the slices of M[Z^{−1}].

Remark 3.4. Let R be a commutative monoid in Mot, with R ∈ SH^{eff}(S). Suppose that there are elements a_{i} ∈ R^{2f_{i},f_{i}}(S), i = 1, 2, …, f_{i} ≤ 0, so that M is the quotient module R/({a_{i}}). Suppose in addition that there is a constant c such that R^{2r+s,r}(Y) = 0 for all Y ∈ Sm/S, r ∈ ℤ, s ≥ c. Then M^{2r+s,r}(Y) = 0 for all Y ∈ Sm/S, r ∈ ℤ, s ≥ c. Indeed

M := hocolim_{n} R/(a_{1}, a_{2}, …, a_{n}),

so it suffices to handle the case M = R/(a_{1}, a_{2}, …, a_{n}), for which we may use induction in n. Assuming the result for N := R/(a_{1}, a_{2}, …, a_{n−1}), we have the long exact sequence (f = f_{n})

… → N^{p+2f,q+f}(Y) ⟶^{×a_{n}} N^{p,q}(Y) → M^{p,q}(Y) → N^{p+2f+1,q+f}(Y) → ….

Thus the assumption for N implies the result for M and the induction goes through.

4. Slices of quotients of MGL

The slices of a Landweber exact spectrum have been described by Spitzweck in [S12, S10], but a quotient of MGL or a localization of such is often not Landweber exact. We will apply the results of the previous section to describe the slices of the motivic truncated Brown-Peterson spectra BP ⟨n⟩, effective motivic Morava K-theory k(n) and motivic Morava K-theory K(n), as well as recovering the known computations for the Landweber examples [S12], such as the Brown-Peterson spectra BP and the Johnson-Wilson spectra E(n). Let MGL_{p} be the commutative monoid in Mot ⊗ ℤ_{(p)} representing p-local algebraic cobordism, as constructed in [PPR, §2.1]^{4}. As noted in loc. cit., MGL_{p} is a cofibrant object of Mot ⊗ ℤ_{(p)}. The motivic BP was first constructed by Vezzosi in [Ve01] as a direct summand of MGL_{p} by using Quillen’s idempotent theorem. Here we construct BP and BP ⟨n⟩ as quotients of MGL_{p}; the effective Morava K-theory k(n) is similarly a quotient of MGL_{p}/p. Our explicit description of the slices allows us to describe the E_{2}-terms of slice spectral sequences for BP and BP ⟨n⟩. The bigraded coefficient ring π_{∗,∗}MGL_{p}(S) contains π_{2∗}MU ≃ 𝕃_{∗}, localized at p, as a graded subring of the bi-degree (2∗, ∗) part, via the classifying map for the formal group law of MGL; see for example [Hoy, remark 6.3]. The ring 𝕃_{∗p} := 𝕃_{∗} ⊗_{ℤ} ℤ_{(p)} is isomorphic to polynomial ring ℤ_{(p)}[x_{1}, x_{2}, ⋯] [A95, Part II, theorem 7.1], where the element x_{i} has degree 2i in π_{∗}MU, degree (2i, i) in π_{∗,∗}MGL_{p} and degree i in 𝕃_{∗}. The following result of Hopkins-Morel-Hoyois [Hoy] is crucial for the application of the general results of the previous sections to quotients of MGL and MGL_{p}.

4This gives MGL as a symmetric spectrum, we take the image in the p-localized model structure to define MGL_{p}.

Documenta Mathematica · Extra Volume Merkurjev (2015) 407–442

Theorem 4.1 ([Hoy, theorem 7.12]). Let p be a prime integer, S an essentially smooth scheme over a field of characteristic prime to p. Then the canonical maps MGL_{p}/({x_{i} : i = 1, 2, …}) → s_{0}MGL_{p} → Hℤ_{(p)} are isomorphisms in SH(S). In case S = Spec k, k a perfect field of characteristic prime to p, the inclusion 𝕃_{∗p} ⊂ π_{2∗,∗}MGL_{p}(S) is an equality.

This has been extended by Spitzweck. He has constructed [S13] a motivic Eilenberg-MacLane spectrum Hℤ in Spt_{ℙ^{1}}(X) with a highly structured multiplication, for an arbitrary base-scheme X. For X smooth and of finite type over a Dedekind domain, Hℤ represents motivic cohomology defined as Bloch’s higher Chow groups [Vo02]; this theory agrees with Voevodsky’s motivic cohomology for smooth schemes of finite type over a perfect field. In addition, Spitzweck has extended theorem 4.1 to an arbitrary base-scheme.

Theorem 4.2 ([S13, theorem 11.3], [S14, corollary 6.6]). Let p be a prime integer and let S be a scheme whose positive residue characteristics are all prime to p. Then the canonical maps MGL_{p}/({x_{i} : i = 1, 2, …}) → s_{0}MGL_{p} → Hℤ_{(p)} are isomorphisms in SH(S). In case S = Spec A, A a Dedekind domain with all residue characteristics prime to p and with trivial class group, the inclusion 𝕃_{∗p} ⊂ π_{2∗,∗}MGL_{p}(S) is an equality.

We define a series of subsets of the set of generators {x_{i} | i = 1, 2 …},

B_{p}^{c} = {x_{i} : i ≠ p^{k} − 1, k ≥ 1},
B_{p} = {x_{i} : i = p^{k} − 1, k ≥ 1},
B⟨n⟩_{p}^{c} = {x_{i} : i ≠ p^{k} − 1, 1 ≤ k ≤ n},
B⟨n⟩_{p} = {x_{i} : i = p^{k} − 1, 1 ≤ k ≤ n},
k⟨n⟩_{p} = {x_{p^{n}−1}}.

We also define

k⟨n⟩_{p}^{c} = {x_{i} : i ≠ p^{n} − 1, and x_{0} = p} ⊂ {p, x_{i} | i = 1, 2 …}.

Definition 4.3 (BP, BP ⟨n⟩ and E(n)). The Brown-Peterson spectrum BP is defined as

BP := MGL_{p}/({x_{i} | i ∈ B_{p}^{c}}),

the truncated Brown-Peterson spectrum BP ⟨n⟩ is defined as

BP ⟨n⟩ := MGL_{p}/({x_{i} | i ∈ B⟨n⟩_{p}^{c}})

and the Johnson-Wilson spectrum E(n) is the localization

E(n) := BP ⟨n⟩[x_{p^{n}−1}^{−1}].

Definition 4.4 (Morava K-theories k(n) and K(n)). Effective Morava Ktheory k(n) is defined as

k(n) := MGL_{p}/({x_{i} | i ∈ k⟨n⟩_{p}^{c}}) ≅ BP ⟨n⟩/(x_{p−1}, …, x_{p^{n−1}−1}, p).

Define Morava K-theory K(n) to be the localization

K(n) := k(n)[x_{p^{n}−1}^{−1}].

The spectra BP , BP ⟨n⟩, E(n), k(n) and K(n) are MGL_{p}-modules. BP and E(n) are Landweber exact. We let C denote the category of MGL_{p}-modules.

Lemma 4.5. The MGL_{p}-module spectra BP , BP ⟨n⟩ and k(n) are effective. BP and E(n) have the structure of oriented weak commutative ring T -spectra in SH(S).

Proof. The effectivity of these theories follows from lemma 2.2 and the fact that homotopy colimits of effective spectra are effective. The ring structure for BP and E(n) follows from the Landweber exactness (see [NSO09]). □

We first discuss the effective theories BP , BP ⟨n⟩ and k(n).

Proposition 4.6. Let p be a prime and S a scheme with all residue characteristics prime to p. Then in SH(S):

1. The zeroth slices of both BP and BP ⟨n⟩ are isomorphic to p-local motivic Eilenberg-MacLane spectrum Hℤ_{(p)}, and the zeroth slice of k(n) is isomorphic to Hℤ/p.

2. The quotient maps from MGL_{p} induce isomorphisms

s_{0}BP ≃ (s_{0}MGL)_{p} ≃ s_{0}BP ⟨n⟩,
s_{0}k(n) ≃ (s_{0}MGL)_{p}/p.

3. The respective quotient maps from BP, BP ⟨n⟩ and k(n) induce isomorphisms

BP /({x_{i} : x_{i} ∈ B_{p}}) ≃ s_{0}BP ,
BP ⟨n⟩/({x_{i} : x_{i} ∈ B⟨n⟩_{p}}) ≃ s_{0}BP ⟨n⟩,
k(n)/(x_{p^{n}−1}) ≃ s_{0}k(n).

Proof. By theorem 4.1 (in case S is essentially smooth over a field) or theorem 4.2 (for general S), the classifying map MGL → Hℤ for motivic cohomology induces isomorphisms

MGL_{p}/({x_{i} : i = 1, 2, …}) ≅ s_{0}MGL_{p} ≅ Hℤ_{(p)}

in SH(S) ⊗ ℤ_{(p)}.

Now let S ⊂ ℕ be a subset and S^{c} its complement. By remark 1.5, we have an isomorphism

(MGL_{p}/({x_{i} : i ∈ S^{c}}))/({x_{i} : i ∈ S}) ≅ MGL_{p}/({x_{i} : i ∈ ℕ}).

Also, as x_{i} is a map Σ^{2i,i}MGL_{p} → MGL_{p}, i > 0, the quotient map MGL_{p} → MGL_{p}/({x_{i} : i ∈ S^{c}}) induces an isomorphism

s_{0}MGL_{p} → s_{0}[MGL_{p}/({x_{i} : i ∈ S^{c}})].

This gives us isomorphisms

(MGL_{p}/({x_{i} : i ∈ S^{c}}))/({x_{i} : i ∈ S}) ≅
≅ s_{0}[MGL_{p}/({x_{i} : i ∈ S^{c}})] ≅ s_{0}MGL_{p},

with the first isomorphism induced by the quotient map

MGL_{p}/({x_{i} : i ∈ S^{c}} → (MGL_{p}/({x_{i} : i ∈ S^{c}}))/({x_{i} : i ∈ S}).

Taking S = B_{p}, B⟨n⟩_{p}, {x_{p^{n}−1}} proves the result for BP, BP ⟨n⟩ and k(n), respectively. □

For motivic spectra E = BP , BP ⟨n⟩, k(n), E(n) and K(n) defined in 4.3 and 4.4 let us denote the corresponding topological spectra by E^{top}. The graded coefficient rings E_{∗}^{top} of these topological spectra are

E_{∗}^{top} ≃ \{ \begin{matrix}ℤ_{p}[v_{1}, v_{2}, ⋯] & E = BP \\ ℤ_{p}[v_{1}, v_{2}, ⋯, v_{n}] & E = BP ⟨n⟩ \\ ℤ_{p}[v_{1}, v_{2}, ⋯, v_{n}, v_{n}^{−1}] & E = E(n) \\ ℤ/p[v_{n}] & E = k(n) \\ ℤ/p[v_{n}, v_{n}^{−1}] & E = K(n) \end{matrix} \}

where deg v_{n} = 2(p^{n} − 1). The element v_{n} corresponds to the element x̄_{n} ∈ MGL^{2n,n}(k).

Corollary 4.7. Let p be a prime integer and let S be a scheme whose positive residue characteristics are all prime to p. Then in SH(S), the slices of Brown-Peterson, Johnson-Wilson and Morava theories are given by

s_{i}E ≃ \{ \begin{matrix}Σ_{T}^{i} H_{ℤ_{p}} ⊗ E_{2i}^{top} & E = BP , BP ⟨n⟩ and E(n) \\ Σ_{T}^{i} H_{ℤ/p} ⊗ E_{2i}^{top} & E = k(n) and K(n) \end{matrix} \}

where E_{2i}^{top} is degree 2i homogeneous component of coefficient ring of the corresponding topological theory.

Proof. The statement for BP and BP ⟨n⟩ follows from theorem 2.3, and remark 2.6. The case of E(n) follows from corollary 2.4 and the cases of k(n) and K(n) follow from corollary 2.5. □

Theorem 4.8. Let p be a prime integer and let S be a scheme whose positive residue characteristics are all prime to p. The slice spectral sequence for any of the spectra E = BP , BP ⟨n⟩, k(n), E(n) and K(n) in SH(S) has the form

E_{2}^{p,q}(X, m) = H^{p−q}(X, Z(m − q)) ⊗_{ℤ} E_{−2q}^{top} ⇒ E^{p+q,m}(X),

where Z = ℤ_{p} for E = BP , BP ⟨n⟩ and E(n), and Z = ℤ/p for E = k(n) and K(n). In case S = Spec k and k is perfect, these spectral sequences are all strongly convergent.

Proof. The form of the slice spectral sequence for E follows from corollary 4.7. The fact that the slice spectral sequences strongly converge for S = Spec k, k perfect, follows from remark 3.4 and the fact that MGL^{2r+s,r}(Y) = 0 for all Y ∈ Sm/S, r ∈ ℤ and s ≥ 1. This in turn follows from the Hopkins-Morel-Hoyois spectral sequence

E_{2}^{p,q}(n) := H^{p−q}(Y, ℤ(n − q)) ⊗ 𝕃_{−q} ⟹ MGL^{p+q,n}(Y),

which is strongly convergent by [Hoy, theorem 8.12]. □

5. Modules for oriented theories

We will use the slice spectral sequence to compute the “geometric part” E^{2∗,∗} of a quotient spectrum E = MGL_{p}/({x_{i_{j}}}) in terms of algebraic cobordism, when working over a base field k of characteristic zero. As the quotient spectra are naturally MGL_{p}-modules but may not have a ring structure, we will need to extend the existing theory of oriented Borel-Moore homology and related structures to allow for modules over ring-based theories.

5.1. Oriented Borel-Moore homology. We first discuss the extension of oriented Borel-Moore homology. We use the notation and terminology of [LM09, §5]. Let Sch/k be the category of quasi-projective schemes over a field k and let Sch/k′ denote the subcategory of projective morphisms in Sch/k. Let Ab_{∗} denote the category of graded abelian groups, Ab_{∗∗} the category of bi-graded abelian groups.

Definition 5.1. Let A be an oriented Borel-Moore homology theory on Sch/k [LM09, definition 5.1.3]. An oriented A-module B is given by

(MD1) An additive functor B_{∗} : Sch/k′ → Ab_{∗}, X ↦ B_{∗}(X). (MD2) For each l. c. i. morphism f : Y → X in Sch/k of relative dimension d, a homomorphism of graded groups f^{∗} : B_{∗}(X) → B_{∗+d}(Y). (MD3) For each pair (X, Y) of objects in Sch/k a bilinear graded pairing

A_{∗}(X) ⊗ B_{∗}(Y) → B_{∗}(X ×_{k} Y)
u ⊗ v ↦ u × v

which is associative and unital with respect to the external products in the theory A. These satisfy the conditions (BM1), (BM2), (PB) and (EH) of [LM09, definition 5.1.3]. In addition, these satisfy the following modification of (BM3).

(MBM3) Let f : X′ → X and g : Y′ → Y be morphisms in Sch/k. If f and g are projective, then for u′ ∈ A_{∗}(X′), v′ ∈ B_{∗}(Y′), one has

(f × g)_{∗}(u′ × v′) = f_{∗}(u′) × g_{∗}(v′).

If f and g are l. c. i. morphisms, then for u ∈ A_{∗}(X), v ∈ B_{∗}(Y), one has

(f × g)^{∗}(u × v) = f_{∗}(u) × g_{∗}(v).

Let f : A → A′ be a morphism of Borel-Moore homology theories, let B be an oriented A-module, B′ an oriented A′-module. A morphism g : B → B′ over f is a collection of homomorphisms of graded abelian groups g_{X} : B_{∗}(X) → B′_{∗}(X), X ∈ Sch/k such that the g_{X} are compatible with projective pushforward, l. c. i. pull-back and external products.

We do not require the analog of the axiom (CD) of [LM09, definition 5.1.3]; this axiom plays a role only in the proof of universality of Ω_{∗}, whereas the universality of Ω for A-modules follows formally from the universality for Ω among oriented Borel-Moore homology theories (see proposition 5.3 below).

Example 5.2. Let N_{∗} be a graded module for the Lazard ring 𝕃_{∗} and let A_{∗} be an oriented Borel-Moore homology theory. Define A_{∗}^{N}(X) := A_{∗}(X) ⊗_{𝕃_{∗}} N_{∗}. Then with push-forward f_{∗}^{N} := f_{∗}^{A} ⊗ id_{N_{∗}}, pull-back f_{N}^{∗} := f_{A}^{∗} ⊗ id_{N_{∗}}, and product u × (v ⊗ n) := (u × v) ⊗ n, for u ∈ A_{∗}(X), v ∈ A_{∗}(Y), n ∈ N_{∗}, A_{∗}^{N} becomes an oriented A-module. Sending N_{∗} to A_{∗}^{N} gives a functor from graded 𝕃_{∗}-modules to oriented A-modules. In case k has characteristic zero, we note that, for A_{∗} = Ω_{∗}, we have a canonical isomorphism θ_{N_{∗}} : Ω_{∗}^{N_{∗}}(k) ≅ N_{∗}, as the classifying map 𝕃_{∗} → Ω_{∗}(k) is an isomorphism [LM09, theorem 1.2.7].

Just as for a Borel-Moore homology theory, one can define operations of A_{∗}(Y) on B_{∗}(Z) via a morphism f : Z → Y, assuming that Y is in Sm/k: for a ∈ A_{∗}(Y), b ∈ B_{∗}(Z), define a ∩_{f} b ∈ B_{∗}(Z) by

a ∩_{f} b := (f, id_{Z})^{∗}(a × b),

where (f, id_{Z}) : Z → Y ×_{k} Z is the (transpose of) the graph embedding. As Y is smooth over k, (f, id_{Z}) is an l. c. i. morphism, so the pullback (f, id_{Z})^{∗} is defined. Similarly, B_{∗}(Y) is an A_{∗}(Y)-module via

a ∪_{Y} b := δ_{Y}^{∗} (a × b).

These products satisfy the analog of the properties listed in [LM09, §5.1.4, proposition 5.2.1].

Proposition 5.3. Let A be an oriented Borel-Moore homology theory on Sch/k and let B be an oriented A-module. Let ϑ_{A} : Ω_{∗} → A_{∗} be the classifying map. There is a unique morphism θ_{A/B} : Ω_{∗}^{B_{∗}(k)} → B_{∗} over ϑ_{A} such that θ_{A/B}(k) : Ω_{∗}^{B_{∗}(k)}(k) → B_{∗}(k) is the canonical isomorphism θ_{B_{∗}(k)}.

Proof. For X ∈ Sch/k, b ∈ B_{∗}(k) and u ∈ Ω_{∗}(X), we define θ_{A/B}(u ⊗ b) := ϑ_{A}(u) × b ∈ B_{∗}(X ×_{k} k) = B_{∗}(X). It is easy to check that this defines a morphism over ϑ_{A}. Uniqueness follows easily from the fact that the product structure in A and Ω is unital. □

5.2. Oriented duality theories. Next, we discuss a theory of modules for an oriented duality theory (H, A). We use the notation and definitions from [L08]. In particular, we have the category SP of smooth pairs over k, with objects (M, X), M ∈ Sm/k, X ⊂ M a closed subset, and where a morphism f : (M, X) → (N, Y) is a morphism f : M → N in Sm/k such that f^{−1}(Y) ⊂ X.

Definition 5.4. Let A be a bi-graded oriented ring cohomology theory, in the sense of [L08, definition 1.5, remark 1.6]. An oriented A-module B is a bi-graded cohomology theory on SP, satisfying the analog of [L08, definition 1.5], that is: for each pair of smooth pairs (M, X), (N, Y) there is a bi-graded homomorphism

× : A_{X}^{∗∗}(M) ⊗ B_{Y}^{∗∗}(N) → B_{X×Y}^{∗∗} (M ×_{k} N)

satisfying

(1) associativity: (a × b) × c = a × (b × c) for a ∈ A_{X}^{∗∗}(M), b ∈ A_{Y}^{∗∗}(N), c ∈ B_{Z}^{∗∗}(P).

(2) unit: 1 × a = a.

(3) Leibniz rule: Given smooth pairs (M, X), (M, X′), (N, Y) with X ⊂ X′ we have

∂_{M×N,X′×N,X×N}(a × b) = ∂_{M,X′,X}(a) × b

for a ∈ A_{X′\X}^{∗∗}(M \ X), b ∈ B_{Y}^{∗∗}(N). For a triple (N, Y′, Y) with Y ⊂ Y′ ⊂ N, a ∈ A_{X′}^{m,∗}(M), b ∈ B_{Y′\Y}^{∗∗} (N \ Y) we have

∂_{M×N,M×Y′,M×Y} (a × b) = (−1)^{m}a × ∂_{N,Y′,Y} (b).

We write a ∪ b ∈ B_{X∩Y} (M) for δ_{M}^{∗}(a × b), a ∈ A_{X}^{∗∗}(M), b ∈ B_{Y}^{∗∗}(M). In addition, we assume that the “Thom classes theory” [P09, lemma 3.7.2] arising from the orientation on A induces an orientation on B in the following sense: Let (M, X) be a smooth pair and let p : E → M be a rank r vector bundle on M. Then the cup product with the Thom class th(E) ∈ A_{M}^{2r,r}(E)

B_{X}^{∗∗}(M) ⟶^{p^{∗}} B_{p^{−1}(X)}^{∗∗}(E) ⟶^{th(E)∪(−)} B_{X}^{2r+∗,r+∗}(E)

is an isomorphism.

We call an orientation on A that induces an orientation on B as above an orientation on (A, B), or just an orientation on B. Given an orientation ω on A, one has 1st Chern classes in A for line bundles, where for L → M a line bundle over M ∈ Sm/k with zero section s : M → L, one defines c_{1}(L) ∈ A^{2,1}(X) as s^{∗}(th(L)). Let SP′ be the category with the same objects (M, X) as in SP, where a morphism f : (M, X) → (N, Y) is a projective morphism f : M → N such that f(X) ⊂ Y. One proceeds just as in [L08] to show that the orientation on B gives rise to an integration on B. To describe this more precisely, we first need to extend the notion of an integration with support [L08, definition 1.8] to the setting of bi-graded A-modules.

The discussion in [L08] is carried out in the setting of an ungraded cohomology theory; we modify this by introducing a bi-grading on the cohomology theory A as well as on the A-module B as above. An integration with supports for the pair (A, B) is defined by modifying the axioms of [L08, definition 1.8] as follows. We first discuss the modifications for A. The bi-grading is incorporated in that the pushforward map F_{∗} associated to a morphisms F : (M, X) → (N, Y) in SP′ has the form F_{∗} : A_{X}^{∗∗}(M) → A_{Y}^{∗−2d,∗−d}(N) , where d = dim_{k} M − dim_{k} N. With this refinement, the remaining parts of definition 1.8 for A remain the same. For the module B, one requires as above that one has for each morphism F : (M, X) → (N, Y) in SP′ a pushforward map F_{∗} : B_{X}^{∗∗}(M) → B_{Y}^{∗−2d,∗−d}(N). In addition, one modifies the multiplicative structure f^{∗}(−)∪ and ∪ for A in definition 1.8(2) of loc. cit. to bi-graded products

f^{∗}(−)∪ : A_{Z}^{∗∗}(M) ⊗ B_{Y}^{∗∗}(N) → B_{Y ∩f^{−1}(Z)}^{∗∗}(N)

and

∪ : A_{Z}^{∗∗}(M) ⊗ B_{X}^{∗∗}(M) → B_{X∩Z}(M),

and, with these changes, we require that B satisfies the conditions of definition 1.8(2) of loc. cit. We call such a structure an integration with supports on (A, B). Given an integration with supports on (A, B) and an orientation ω on (A, B) we say (as in [L08, definition 1.11]) that the integration with supports is subjected to ω if for each smooth pair (M, X) and each line bundle p : L → M with zero section s : M → L, the compositions

A_{X}^{∗∗}(M) ⟶^{s_{∗}} A_{p^{−1}(X)}^{∗−2,∗−1}(L) ⟶^{s^{∗}} A_{X}^{∗−2,∗−1}(M),

B_{X}^{∗∗}(M) ⟶^{s_{∗}} B_{p^{−1}(X)}^{∗−2,∗−1}(L) ⟶^{s^{∗}} B_{X}^{∗−2,∗−1}(M)

are given by respective cup product with c_{1}(L). We have the analog of [L08, theorem 1.12] in the setting of oriented modules.

Theorem 5.5. Let A be a bi-graded ring cohomology theory with orientation ω and let B be an oriented A-module with orientation induced by ω. Then there is a unique integration with supports on (A, B) subjected to the orientation ω.

The proof is exactly the same way as the proof of theorem 1.12 of loc. cit. We now extend the notion of an oriented duality theory to the setting of modules.

Definition 5.6. Let (H, A) be an oriented duality theory, in the sense of [L08, definition 3.1]. An oriented (H, A)-module is a pair (J, B), where (D1) J : Sch/k′ → Ab_{∗∗} is a functor. (D2) B is an oriented A-module. (D3) For each open immersion j : U → X there is a pullback map j^{∗} : J_{∗∗}(X) → J_{∗∗}(U). (D4) i. For each smooth pair (M, X) and each morphism f : Y → M in Sch/k, there is a bi-graded cap product map

f^{∗}(−)∩ : A_{X}(M) ⊗ H(Y) → H(f^{−1}(X)).

ii. For X, Y ∈ Sch/k, there is a bi-graded external product

× : H_{∗∗}(X) ⊗ J_{∗∗}(Y) → J_{∗∗}(X × Y).

(D5) For each smooth pair (M, X), there is a graded isomorphism

β_{M,X} : J_{∗∗}(X) → B_{X}^{2d−∗,d−∗}(M); d = dim_{k} M.

(D6) For each X ∈ Sch/k and each closed subset Y ⊂ X, there is a map

∂_{X,Y} : J_{∗+1,∗}(X \ Y) → J_{∗∗}(Y).

These satisfy the evident analogs of properties (A1)-(A4) of [L08, definition 3.1], where we make the following changes: Let d = dim_{k} M, e = dim_{k} N. One replaces H with J_{∗∗} throughout (except in (A3)(ii)), and

• in (A1) one replaces A_{Y} (N), A_{X}(M) with B_{Y}^{2d−∗,d−∗}(N),

B_{X}^{2d−∗,d−∗}(M),

• in (A2) one replaces A_{Y} (N), A_{X}(M) with B_{Y}^{2e−∗,e−∗}(N),

B_{X}^{2d−∗,d−∗}(M),

• in (A3)(i) one replaces A_{Y} (M) with B_{Y}^{2d−∗,d−∗}(M) and A_{Y ∩f^{−1}(X)}(N)

with B_{Y ∩f^{−1}(X)}^{2e−∗,e−∗}(N),

• in (A3)(ii) one replaces A_{Y} (M) with B_{Y}^{2e−∗,e−∗}(N) and A_{X×Y} (M ×N) with B_{X×Y}^{2(d+e)−∗,d+e−∗}(M ×N), H(X) with H_{∗∗}(X), H(Y) with J_{∗∗}(Y)

and H(X × Y) with J_{∗∗}(X × Y),
• in (A4) one replaces A_{X\Y} (M \ Y) with B_{X\Y}^{2d−∗,d−∗}(M \ Y).

Remark 5.7. Let (H, A) be an oriented duality theory on Sch/k, for k a field admitting resolution of singularities. By [L08, proposition 4.2] there is a unique natural transformation

ϑ_{H} : Ω_{∗} → H_{2∗,∗}

of functors Sch/k′ → Ab_{∗} compatible with all the structures available for H_{2∗,∗} and, after restriction to Sm/k is just the classifying map Ω^{∗} → A^{2∗,∗} for the oriented cohomology theory X ↦ A^{2∗,∗}(X). We refer the reader to [L08, §4] for a complete description of the properties satisfied by ϑ_{H}. Via ϑ_{H} and the ring homomorphism ρ_{Ω} : 𝕃_{∗} → Ω_{∗}(k) classifying the formal group law for Ω_{∗}, we have the ring homomorphism ρ_{H} : 𝕃_{∗} → H_{2∗,∗}(k). If (J, B) is an oriented (H, A)-module, then via the H_{2∗,∗}(k)-module structure on J_{2∗,∗}(k), ρ_{H} makes J_{2∗,∗}(k) a 𝕃_{∗}-module. We write J_{∗} for the 𝕃_{∗}-module J_{2∗,∗}(k).

Proposition 5.8. Let k be a field admitting resolution of singularities. Let (H, A) be an oriented duality theory and (J, B) an oriented (H, A)-module. There is a unique natural transformation ϑ_{H/J} : Ω_{∗}^{J_{∗}} → J_{2∗,∗} from Sch/k′ → Ab_{∗} satisfying

(1) ϑ_{H/J} is compatible with pullback maps j^{∗} for j : U → X an open immersion in Sch/k.

(2) ϑ_{H/J} is compatible with fundamental classes.

(3) ϑ_{H/J} is compatible with external products.

(4) ϑ_{H/J} is compatible with the action of 1st Chern class operators.

(5) Identifying Ω_{∗}^{J_{∗}}(k) with J_{2∗,∗}(k) via the product map Ω_{∗}(k) ⊗_{𝕃_{∗}} J_{2∗,∗}(k) → J_{2∗,∗}(k), ϑ_{H/J}(k) : Ω_{∗}^{J_{∗}}(k) → J_{2∗,∗} is the identity map.

Proof. For X ∈ Sch/k, we define ϑ_{H/J}(X) by

ϑ_{H/J}(u ⊗ j) = ϑ_{H}(u) × j ∈ J_{2∗,∗}(X ×_{k} Spec k) = J_{2∗,∗}(X),

for u⊗j ∈ Ω_{∗}^{J_{∗}}(X) := Ω_{∗}(X)⊗_{𝕃_{∗}} J_{2∗,∗}(k). The properties (1)-(5) follow directly from the construction. As Ω_{∗}(X) is generated by push-forwards of fundamental classes, the properties (2), (3) and (5) determine ϑ_{H/J} uniquely. □

Remark 5.9. Let k, (H, A) and (J, B) be as in proposition 5.8. Suppose that J_{∗} := J_{2∗,∗} has external products ×_{J} and there is a unit element 1_{J} ∈ J_{0}(k) for these external products. Suppose further that these are compatible with the external products H_{∗}(X) ⊗ J_{∗}(Y) → J_{∗}(X ×_{k} Y), in the sense that

(h × 1_{J}) ×_{J} b = h × b ∈ J_{∗}(X ×_{k} Y)

for h ∈ H_{∗}(X), b ∈ J_{∗}(Y), and that 1_{H} × 1_{J} = 1_{J}. Then ϑ_{H/J} is compatible with external products and is unital. This follows directly from our assumptions and the identity

ϑ_{H/J}((u ⊗ h) × (u′ ⊗ j′)) = ϑ_{H}(u) × ϑ_{H/J}(u′ ⊗ (h × j)).

5.3. Modules for oriented ring spectra. We now discuss the oriented duality theory and oriented Borel-Moore homology associated to a module spectrum for an oriented weak commutative ring T -spectrum. Let ph be the two-sided ideal of phantom maps in SH(S), where a phantom map is a map f : E → F such that f ∘ g = 0 for each compact object A in SH(S) and each morphism g : A → E. Let E be a weak commutative ring T -spectrum, that is, there are maps μ : E ∧ E → E, η : 𝕊_{S} → E in SH(S) that satisfy the axioms for a monoid in SH(S)/ph. An E-module is similarly an object N ∈ SH(S) together with a multiplication map ρ : E ∧ N → E in SH(S) that makes N into a unital E-module in SH(S)/ph (see for example [NSO09, §8], where a weak commutative ring T -spectrum is referred to as a T -spectrum E with a quasi-multiplication μ : E ∧ E → E). Suppose that (E, c) is an oriented weak commutative ring T -spectrum in SH(k), k a field admitting resolution of singularities. We have constructed in [L08, theorem 3.4] a bi-graded oriented duality theory (E′_{∗∗}, E^{∗∗}) by defining E′_{a,b}(X) := E_{X}^{2m−a,m−b}(M), where M ∈ Sm/k is a chosen smooth quasiprojective scheme containing X as a closed subscheme and m = dim_{k} M. Let N be an E-module. For E → M a rank r vector bundle on M ∈ Sm/k and X ⊂ M a closed subscheme, the Thom classes for E give rise to a Thom isomorphism N_{X}^{∗∗}(M) → N_{X}^{2r+∗,r+∗}(E). Using these Thom isomorphisms, the arguments used to construct the oriented duality theory (E′_{∗∗}, E^{∗∗}) go through without change to give N^{∗∗} the structure of an oriented E^{∗∗}-module, and to define an oriented (E′_{∗∗}, E^{∗∗})- module (N′_{∗∗}, N^{∗∗}), with canonical isomorphisms N′_{a,b}(X) ≅ N_{X}^{2m−a,m−b}(M), m = dim_{k} M, and where the cap products are induced by the E-modules structure on N.

5.4. Geometrically Landweber exact modules.

Definition 5.10. Let (E, c) be a weak oriented ring T -spectrum and let N be an E-module. The geometric part of E^{∗∗} is the (2∗, ∗)-part E^{∗} := E^{2∗,∗} of E^{∗∗}, the geometric part of N is the E^{∗}-module N^{2∗,∗}, and the geometric part of N′ is similarly given by X ↦ N′_{∗}(X) := N′_{2∗,∗}(X). This gives us the ℤ-graded oriented duality theory (E′_{∗}, E^{∗}) and the oriented (E′_{∗}, E^{∗})-module (N′_{∗}, N^{∗}).

Let (E, c) be a weak oriented ring T -spectrum and let N be an E-module. By proposition 5.8, we have a canonical natural transformation

ϑ_{E′/N′} : Ω_{∗}^{N′_{∗}(k)} → N′_{∗}

satisfying the compatibilities listed in that proposition. We extend the definition of a geometrically Landweber exact weak commutative ring T -spectrum (see [L15, definition 3.7]) to the case of an E-module:

Definition 5.11. Let (E, c) be a weak oriented ring T -spectrum and let N be an E-module. We say that N is geometrically Landweber exact if for each point η ∈ X ∈ Sm/k

i. The structure map p_{η} : η → Spec k induces an isomorphism p_{η}^{∗} : N^{2∗,∗}(k) → N^{2∗,∗}(η). ii. The product map ∪_{η} : E^{1,1}(η) ⊗ N^{2∗,∗}(η) → N^{2∗+1,∗+1}(η) induces a surjection k(η)^{×} ⊗ N^{2∗,∗}(η) → N^{2∗+1,∗+1}(η).

Here we use the canonical natural transformation t_{E} : 𝔾_{m} → E^{1,1}(−) defined in [L15, remark 1.5] to define the map k(η)^{×} → E^{1,1}(η) needed in (ii).

The following result generalizes [L15, theorem 6.2] from oriented weak commutative ring T -spectra to modules:

Theorem 5.12. Let k be a field of characteristic zero, N an MGL-module in SH(k), (N′_{∗∗}, N^{∗∗}) the associated oriented (MGL′_{∗∗}, MGL^{∗∗})-module, and N′_{∗} the geometric part of N′. Suppose that N is geometrically Landweber exact. Then the classifying map

ϑ_{MGL′_{∗}/N′_{∗}} : Ω_{∗}^{N′_{∗}(k)} → N′_{∗}

is an isomorphism.

Remark 5.13. Let k be a field of characteristic zero, let (E, c) be an oriented weak commutative ring T -spectrum in SH(S), and let N be an E-module. Via the classifying map φ_{E,c} : MGL → E, N becomes an MGL-module. In addition, the classifying map ϑ_{E′} : Ω_{∗} → E′_{∗} is induced from φ_{E,c} and the classifying map ϑ_{MGL′_{∗}/N′_{∗}} factors through the classifying map ϑ_{E′_{∗}/N′_{∗}} : E_{∗}^{′N′_{∗}(k)} → N′_{∗} as

ϑ_{MGL′_{∗}/N′_{∗}} = ϑ_{E′_{∗}/N′_{∗}} ∘ (φ_{E,c} ⊗ id_{N′_{∗}(k)}).

Thus, theorem 5.12 applies to E-modules for arbitrary (E, c). Moreover, if (E, c) is geometrically Landweber exact in the sense of [L15, definition 3.7], the map ϑ̄_{E′_{∗}} : Ω_{∗}^{E′_{∗}(k)} → E′_{∗} is an isomorphism ([L15, theorem 6.2]) hence the map ϑ_{E′_{∗}/N′_{∗}} is an isomorphism as well.

Proof of theorem 5.12. The proof of theorem 5.12 is essentially the same as the proof of [L15, theorem 6.2]. Indeed, just as in loc. cit., one constructs a commutative diagram (see [L09, (6.4)])

(5.1)

\xymatrix{⊕_{η∈X_{(d)}}k(η)^{×} ⊗ N′_{∗−d+1} \ar[r]^{^{Div^{N}}} & Ω_{∗}^{N′_{∗}(1)}(X) \ar[d]_{ϑ̄^{(1)}} \ar[r]^{i_{∗}} & Ω_{∗}^{N′_{∗}}(X) \ar[d]_{ϑ̄(X)} \ar[r]^{j^{∗}} & ⊕_{η∈X_{(d)}}Ω_{∗}^{N′_{∗}}(η) \ar[r] \ar[d]_{ϑ̄} & 0 \\ ⊕_{η∈X_{(d)}}k(η)^{×} ⊗ N′_{∗−d+1} \ar[r]_{div_{N}} & N_{2∗,∗}^{′(1)}(X) \ar[r]_{i_{∗}} & N′_{2∗,∗}(X) \ar[r]_{j^{∗}} & ⊕_{η∈X_{(d)}}N′_{2∗,∗}(η) \ar[r] & 0}

where we write N′_{∗} for N′_{∗}(k), d is the maximum of dim_{k} X_{i} as X_{i} runs over the irreducible components of X, and N_{2∗,∗}^{′(1)}(X) is the colimit of N′_{2∗,∗}(W), as W runs over closed subschemes of X containing no dimension d generic point of X. A similarly defined colimit of the Ω_{∗}^{N′_{∗}}(W) gives us Ω_{∗}^{N′_{∗}(1)}(X). The maps ϑ̄^{(1)}, ϑ̄(X) and ϑ̄ are all induced by the classifying map ϑ_{MGL′_{∗}/N′_{∗}}. The top row is a complex and the bottom row is exact; this latter fact follows from the surjectivity assumption in definition 5.11(ii). The map ϑ̄ is an isomorphism by part (i) of definition 5.11 and ϑ̄^{(1)} is an isomorphism by induction on d. To show that ϑ̄(X) is an isomorphism, it suffices to show that the identity map on ⊕_{η}N′_{∗−d+1} ⊗ k(η)^{×} extends diagram (5.1) to a commutative diagram. To see this, we note that the map div_{N} is defined by composing the boundary map

∂ : ⊕_{η∈X_{(d)}}N′_{2∗+1,∗}(η) → N_{2∗,∗}^{′(1)}(X)

with the sum of the product maps MGL′_{2d−1,d−1}(η)⊗N′_{∗−d+1}(k) → N′_{2∗+1,∗}(η) and the canonical map t_{MGL}(η) : k(η)^{×} → MGL^{1,1}(η) = MGL′_{2d−1,d−1}(η) (see [L09, remark 1.5]). For MGL′, we have the similarly defined map

div_{MGL} : ⊕_{η∈X_{(d)}}k(η)^{×} ⊗ 𝕃_{∗−d+1} → MGL_{2∗,∗}^{′(1)}(X),

after replacing MGL′_{∗−d+1}(k) with 𝕃_{∗−d+1} via the classifying map 𝕃_{∗} → MGL′_{∗}(k). We have as well the commutative diagram (see [L09, (5.4)])

[figure: ⊕_{η∈X_{(d)}}k(η)^{×} ⊗ 𝕃_{∗−d+1}; Div; Ω_{∗}^{(1)}(X); ϑ_{MGL}^{(1)}; ⊕_{η∈X_{(d)}}k(η)^{×} ⊗ 𝕃_{∗−d+1}; div_{MGL}; MGL_{2∗,∗}^{′(1)}(X),]

which after applying − ⊗_{𝕃_{∗}} N′_{∗} gives us the commutative diagram

(5.2)  [figure: ⊕_{η∈X_{(d)}}k(η)^{×} ⊗ N_{∗−d+1}; ^{Div^{N}}; Ω_{∗}^{N′_{∗}(1)}(X); ϑ_{MGL}^{(1)}⊗id; ⊕_{η∈X_{(d)}}k(η)^{×} ⊗ N_{∗−d+1}; _{\overline{div}_{MGL}}; MGL_{2∗,∗}^{′(1)}(X) ⊗_{𝕃_{∗}} N′_{∗}.]

The Leibniz rule for ∂ gives us the commutative diagram

(5.3)  [figure: ⊕_{η∈X_{(d)}}k(η)^{×} ⊗ N_{∗−d+1}; ^{\overline{div}^{MGL}}; MGL_{2∗,∗}^{′(1)}(X) ⊗_{𝕃_{∗}} N′_{∗}; ∪; ⊕_{η∈X_{(d)}}k(η)^{×} ⊗ N′_{∗−d+1}; div_{N}; N_{2∗,∗}^{′(1)}(X);]

combining diagrams (5.2) and (5.3) yields the desired commutativity. □

6. Applications to quotients of MGL

We return to our discussion of quotients of MGL_{p} and their localizations. We select a system of polynomial generators for the Lazard ring, 𝕃_{∗} ≅ ℤ[x_{1}, x_{2}, …], deg x_{i} = i. Let S ⊂ ℕ, S^{c} its complement and let ℤ[S^{c}] denote the graded polynomial ring on the x_{i}, i ∈ S^{c}, deg x_{i} = i. Let S_{0} ⊂ ℤ[S^{c}] be a collection of homogeneous elements, S_{0} = {z_{j} ∈ ℤ[S^{c}]_{e_{j}}}, and let ℤ[S^{c}][S_{0}^{−1}] denote the localization of ℤ[S^{c}] with respect to S_{0}. We consider a quotient spectrum MGL_{p}/(S) := MGL_{p}/({x_{i} | i ∈ S}) or an integral version MGL/(S) := MGL/({x_{i} | i ∈ S}). We consider as well the localizations

MGL_{p}/(S)[S_{0}^{−1}] := MGL_{p}/(S)[{z_{j}^{−1} | z_{j} ∈ S_{0}}],
MGL/(S)[S_{0}^{−1}] := MGL/(S)[{z_{j}^{−1} | z_{j} ∈ S_{0}}].

and the mod p version

MGL/(S, p)[S_{0}^{−1}] := MGL_{p}/(S)[S_{0}^{−1}]/p.

Proposition 6.1. Let p be a prime, and let S = Spec k, k a perfect field with exponential characteristic prime to p. Let S be a subset of ℕ and S_{0} a set of homogeneous elements of ℤ[S^{c}]. Then the spectra MGL_{p}/(S)[S_{0}^{−1}] and MGL_{p}/(S, p)[S_{0}^{−1}] are geometrically Landweber exact. In case char k = 0, MGL/(S)[S_{0}^{−1}] is geometrically Landweber exact.

Proof. We discuss the cases MGL_{p}/(S)[S_{0}^{−1}] and MGL_{p}/(S, p)[S_{0}^{−1}]; the case of MGL/(S)[S_{0}^{−1}] is exactly the same. Let A be a finitely generated abelian group and let η be a point in some X ∈ Sm/k. Then the motivic cohomology H^{∗}(η, A(∗)) satisfies

H^{2r}(η, A(r)) = H^{2r+1}(η, A(r + 1)) = 0

for r ≠ 0,

H^{0}(η, A(0)) = A, H^{1}(η, A(1)) = k(η)^{×} ⊗_{ℤ} A.

We consider the slice spectral sequences

E_{2}^{p,q}(n) := H^{p−q}(η, ℤ(n − q)) ⊗ ℤ[S^{c}][S_{0}^{−1}]_{−q} ⇒ (MGL_{p}/(S)[S_{0}^{−1}])^{p+q,n}(η)

and

E_{2}^{p,q}(n) := H^{p−q}(η, ℤ/p(n−q))⊗ℤ[S^{c}][S_{0}^{−1}]_{−q} ⇒ (MGL_{p}/(S, p)[S_{0}^{−1}])^{p+q,n}(η)

given by proposition 3.3. As in the proof of theorem 4.8, MGL_{p}^{2n+a,n}(η) = 0 for a > 0 and n ∈ ℤ, and thus by remark 3.4, the convergence hypotheses in proposition 3.3 are satisfied. Thus, these spectral sequences are strongly convergent. As discussed in the proof of [L15, proposition 3.8], the only non-zero E_{2} term contributing to (MGL_{p}/(S)[S_{0}^{−1}])^{2n,n}(η) or to (MGL_{p}/(S, p)[S_{0}^{−1}])^{2n,n}(η) is E_{2}^{n,n}(n), the only non-zero E_{2} term contributing to (MGL_{p}/(S)[S_{0}^{−1}])^{2n−1,n}(η) or contributing to (MGL_{p}/(S, p)[S_{0}^{−1}])^{2n−1,n}(η) is E_{2}^{n,n−1}(n), and all differentials entering or leaving these terms are zero. This gives us isomorphisms

(MGL_{p}/(S)[S_{0}^{−1}])^{2n,n}(η) ≅ ℤ_{(p)}[S^{c}][S_{0}^{−1}]_{−n}
(MGL_{p}/(S, p)[S_{0}^{−1}])^{2n,n}(η) ≅ ℤ/(p)[S^{c}][S_{0}^{−1}]_{−n}
(MGL_{p}/(S)[S_{0}^{−1}])^{2n−1,n}(η) ≅ ℤ_{(p)}[S^{c}][S_{0}^{−1}]_{1−n} ⊗ k(η)^{×}
(MGL_{p}/(S, p)[S_{0}^{−1}])^{2n−1,n}(η) ≅ ℤ/(p)[S^{c}][S_{0}^{−1}]_{1−n} ⊗ k(η)^{×}

from which it easily follows that MGL_{p}/(S)[S_{0}^{−1}] and MGL_{p}/(S, p)[S_{0}^{−1}] are geometrically Landweber exact. □

Corollary 6.2. Let S = Spec k, k a field of characteristic zero. Fix a prime p and let N = MGL/(S)[S_{0}^{−1}], MGL_{p}/(S)[S_{0}^{−1}] or MGL_{p}/(S, p)[S_{0}^{−1}], let (N′, N) be the associated (MGL′, MGL)-module and N′_{∗} the geometric part of N′_{∗∗}. Then the classifying map

ϑ_{N′_{∗}(k)} : Ω_{∗}^{N′_{∗}(k)} → N′_{∗}

is an isomorphism of Ω_{∗}-modules.

This follows directly from theorem 5.12 and proposition 6.1. As an immediate consequence, we have

Corollary 6.3. Let S = Spec k, k a field of characteristic zero. Fix a prime p and let N = BP, BP ⟨n⟩, E(n), k(n) or K(n), let (N′, N) be the associated (MGL′, MGL)-module and N′_{∗} the geometric part of N′_{∗∗}. Then the classifying map

ϑ_{N′_{∗}(k)} : Ω_{∗}^{N′_{∗}(k)} → N′_{∗}

is an isomorphism of Ω_{∗}-modules. In case N = BP or E(n), ϑ_{N′_{∗}(k)} is compatible with external products.

Remark 6.4. Suppose that the theory with supports N^{2∗,∗} has products and a unit, compatible with its MGL^{2∗,∗}-module structure. Then by remark 5.9, the classifying map ϑ_{N′_{∗}(k)} is also compatible with products. In the case of a quotient E of MGL or MGL_{p} by a subset {x_{i} : i ∈ I} of the set of polynomial generators, the vanishing of MGL^{2r+s,r}(k) for s > 0 shows that E^{2∗,∗}(k) = MGL^{2∗,∗}(k)/({x_{i} : i ∈ I}), which has the evident ring structure induced by the natural MGL^{2∗,∗}(k)-module structure. Thus, the rational theory Ω_{∗}^{E_{∗}(k)} has a canonical structure of an oriented Borel-Moore homology theory on Sch/k; the same holds for E a localization of this type of quotient. The fact that the classifying homomorphism ϑ_{E} : Ω_{∗}^{E′_{∗}(k)} → E′_{∗} is an isomorphism induces on E′_{∗} the structure of an oriented Borel-Moore homology theory on Sch/k; it appears to be unknown if this arises from a multiplicative structure on the spectrum level.

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Marc Levine Universität Duisburg-Essen Fakultät Mathematik 45127 Essen Germany marc.levine@uni-due.de Girja Shanker Tripathi Universität Osnabrück Institut für Mathematik 49069 Osnabrück Germany tripathigirja@gmail.com