Cellular objects in isotropic motivic categories

Our main purpose is to describe the category of isotropic cellular spectra over flexible fields. Guided by [6], we show that it is equivalent, as a stable $\infty$-category equipped with a $t$-structure, to the derived category of left comodules over the dual of the classical topological Steenrod algebra. In order to obtain this result, the category of isotropic cellular modules over the motivic Brown-Peterson spectrum is also studied, and isotropic Adams and Adams-Novikov spectral sequences are developed. As a consequence, we also compute hom sets in the category of isotropic Tate motives between motives of isotropic cellular spectra.


Introduction
Isotropic categories are local versions of motivic categories, obtained by, roughly speaking, killing all anisotropic varieties.Although they often have a handier structure than their global versions, they exhibit some key characteristics of both motivic and classical topological phenomena.In [21], Vishik introduced the isotropic triangulated category of motives and computed the isotropic motivic cohomology of the point, which is strongly related to the Milnor subalgebra.By following this lead, we studied in [19] the isotropic stable motivic homotopy category.In particular, we identified the isotropic motivic homotopy groups of the sphere spectrum with the cohomology of the topological Steenrod algebra, i.e. the E 2page of the classical Adams spectral sequence.These results are quite surprising since they show that topological objects naturally arise from isotropic environments, which could lead to a fruitful exchange between topology and isotropic motivic theory.
Motivic categories, constructed by Morel and Voevodsky (see [16] and [24]) in order to study algebraic varieties by topological means, are extremely rich categories.Even over an algebraically closed field, they are more complex than the respective topological counterparts.For example, while every object in the classical stable homotopy category is cellular, i.e. built up by attaching spheres, not every motivic spectrum is cellular, since many algebro-geometric phenomena come into the picture.In spite of this, it is still interesting to understand the structure of the category of cellular objects in motivic stable homotopy theory.This project was initiated by Dugger and Isaksen in [3] and much attention has been dedicated to it since then.Our work, in particular, is concerned with understanding the structure of the subcategories of cellular objects in isotropic categories, which we believe could shed light on the deep interconnection with topology.
We have already highlighted that motivic categories are particularly challenging to study.For example, one of the difficulties that one does not encounter in classical topology is the presence of an object τ that appears in various incarnations throughout motivic homotopy theory, sometimes as an element of the motivic cohomology of the ground field and sometimes as a map in the 2-complete motivic stable homotopy groups of spheres.Hence, the principal task is to find first some substitutes of the original motivic categories and tools which could help in the process of analysing them.In the case of algebraically closed fields, for example, topological realisation is a very helpful tool, since it allows to study the initial motivic category by looking at its deformation τ = 1, which happens to be just the classical stable homotopy theory (see [4]).However, in this process part of the information is lost and one could try to recover it by studying other deformations, for example τ = 0.This was done by Isaksen in [9], Gheorghe in [5] and Gheorghe-Wang-Xu in [6].More precisely, in [9] the stable motivic homotopy groups of Cτ , i.e. the cofiber of τ , are identified with the E 2 -page of the classical Adams-Novikov spectral sequence, while in [5] the motivic spectrum Cτ is provided with an E ∞ -ring structure inducing an isomorphism of rings with higher products between π * * (Cτ ) and the classical Adams-Novikov E 2 -page.A parallel result for isotropic categories was obtained in [19], where the isotropic sphere spectrum X has been equipped with an E ∞ -ring structure inducing an isomorphism of rings with higher products between π * * (X) and the classical Adams E 2 -page.Moreover, in [6] the category of Cτ -cellular spectra is described, which is proved to be equivalent as a stable ∞-category equipped with a t-structure (see [13]) to the derived category of left BP * BP-comodules concentrated in even degrees, where BP is the Brown-Peterson spectrum and BP * BP its BP-homology.
In this work, we intend to follow a similar path for isotropic categories.Recall that a field k is called flexible if it is a purely transcendental extension of countable infinite degree over some other field.In our situation it is really essential to work over flexible fields since, as highlighted in [21], these are the ground fields over which the isotropic categories behave particularly well.For example, over algebraically closed fields, due to the lack of anisotropic varieties, the isotropic category would be just the same as the original motivic category, so in this case the isotropic localization produces nothing new.We are encouraged by the evident parallel between the computations of π * * (Cτ ) over complex numbers (see [9] and [5]), on the one hand, and of π * * (X) over flexible fields (see [19]), on the other.More precisely, we have been guided by the idea that studying the isotropic stable motivic homotopy category over a flexible field is similar in some sense to studying the stable ∞-category of Cτ -cellular spectra in the motivic stable homotopy category over complex numbers.Indeed, they obviously share some common features which is highlighted by the following theorem that is the main result of this paper.where A * is the classical dual Steenrod algebra and X − Mod b cell,HZ/2 is the stable ∞-category of HZ/2complete X-cellular modules having MBP-homology non-trivial in only finitely many Chow-Novikov degrees (the superscript "b" stands for "bounded", see Definition 8.4).
As a consequence, we obtain that the category of isotropic cellular spectra is completely algebraic, which makes it easier to study.Moreover, it is deeply related to classical topology, as foreseeable from results in [21] and [19].
In order to achieve our main results, we need several tools.In particular, it is necessary to develop and study isotropic versions of both the Adams spectral sequence and the Adams-Novikov spectral sequence.This requires a focus on the motivic Brown-Peterson spectrum MBP (see [20]) from an isotropic point of view.In particular, we note that the isotropic Brown-Peterson spectrum is an E ∞ -ring spectrum, in contrast to the topological picture where BP has shown not to admit an E ∞ -ring structure by Lawson in [11].Then, we use techniques developed by Gheorghe-Wang-Xu in [6], based on Lurie's results (see [13]), to, first, describe in algebraic terms the category of isotropic MBP-cellular modules and, then, the category of all isotropic cellular spectra.At the end, we are also able to provide some results about the cellular subcategory of the isotropic triangulated category of motives, i.e. the category of isotropic Tate motives.
Outline.We now briefly present the contents of each section of this paper.In Section 2, we provide the main notations that are followed throughout this work.Then, we move on to Section 3 by recalling isotropic categories and their main properties, mostly referring to results in [21] and [19].Since we are mainly interested in cellular objects, we recall in Section 4 definitions and some of the main results from [3], which are useful in the rest of the paper.Section 5 is devoted to a deep analysis of the isotropic motivic Adams spectral sequence, which was already initiated in [19].These results are used in Section 6 to study the motivic Brown-Peterson spectrum from an isotropic perspective.In particular, we compute its isotropic stable homotopy groups.Sections 7 and 8 are modeled on Sections 3, 4 and 5 of [6].More precisely, in Section 7 we endow the isotropic motivic Brown-Peterson spectrum with an E ∞ -ring structure and, then, identify as a triangulated category the category of isotropic MBP-cellular spectra with the category of bigraded F 2 -vector spaces.In Section 8, after developing an isotropic Adams-Novikov spectral sequence, we describe the category of isotropic cellular spectra in algebraic terms as the derived category of comodules over the dual of the Steenrod algebra equipped with a t-structure.Finally, in Section 9, we provide an algebraic description of the hom-sets in the category of isotropic motives between motives of isotropic cellular spectra, which is a step forward in the understanding of the category of isotropic Tate motives.

Notation
Let us start by fixing some notations we use throughout this paper.bigraded mod 2 topological Steenrod algebra and its dual i.e.G 2q,q = A q , G p,q = 0 for p = 2q and similarly for the dual We denote hom-sets in SH(k) by [−, −] and the suspension S p,q ∧X of a motivic spectrum X by Σ p,q X.Moreover, if E is a motivic E ∞ -ring spectrum, the stable ∞-category of E-modules (see [13]) is denoted by E − Mod, its smash product by − ∧ E − and hom-sets in its homotopy category by [−, −] E .
If R is an algebra and C a coalgebra, then we denote by R − Mod and C − Comod the categories of left R-modules and left C-comodules respectively.Hom-sets in these categories are both denoted by Hom R (−, −) and Hom C (−, −) and it will be clear from the context if they are meant to be hom of modules or comodules.For a bigraded object M * * (respectively M * * ) we also denote by Σ p,q M * * (respectively Σ p,q M * * ) its suspension, i.e. the bigraded object defined by Σ p,q M a,b = M a−p,b−q (respectively Σ p,q M a,b = M a+p,b+q ).The convention for bigraded homomorphisms between bigraded objects is the following: Hom p,q (M * * , N * * ) = Hom 0,0 (Σ p,q M * * , N * * ) and Hom p,q (M * * , N * * ) = Hom 0,0 (Σ p,q M * * , N * * ) where Hom 0,0 (−, −) denotes the bidegree preserving homomorphisms.Moreover, the bounded derived categories of R − Mod and C − Comod are denoted by D b (R − Mod) and D b (C − Comod) respectively.

Isotropic motivic categories
In this section we want to introduce the main categories we consider throughout this paper, namely isotropic motivic categories.These categories are built from the respective motivic ones by, roughly speaking, killing all anisotropic varieties.We refer to [21, Section 2] and [19,Section 2] for more details on the construction and properties of isotropic categories.
Let us recall first the definition of flexible field from [21].
Definition 3.1.A field k is called flexible if it is a purely transcendental extension of countable infinite degree, i.e. k = k 0 (t 1 , t 2 , . . . ) for some other field k 0 .
Once and for all we consider a flexible base field k of characteristic different from 2. We proceed by recalling the definition of a fundamental object in SH(k) for the construction of the isotropic stable motivic homotopy category SH(k/k).Definition 3.2.Denote by Q the disjoint union of all connected anisotropic (mod 2) varieties over k, i.e. varieties which do not have closed points of odd degree, and by Č(Q) its Čech simplicial scheme, i.e.Č(Q) n = Q n+1 with face and degeneracy maps given respectively by partial projections and partial diagonals.We define the isotropic sphere spectrum We recall from [19, Section 2] that X is an idempotent monoid, i.e. there is an equivalence X ∧ X ∼ = X induced by the map S → X, and so an E ∞ -ring spectrum (see [19,Proposition 6.1]).Definition 3.3.The full triangulated subcategory X ∧ SH(k) of SH(k) will be called the isotropic stable motivic homotopy category, and denoted by SH(k/k).
This triangulated category has very nice properties, in particular it is both localising and colocalising (see [19,Section 2]).The very same construction was done first for DM(k) by Vishik in [21], by tensoring the triangulated category of motives with the idempotent M(X), where M : SH(k) → DM(k) is the motivic functor.Definition 3.4.The full triangulated subcategory M(X) ⊗ DM(k) of DM(k) will be called the isotropic category of motives, and denoted by DM(k/k).
The following result tells us that the isotropic stable motivic homotopy category is nothing else but the stable ∞-category of X-modules.
Proposition 3.5.The isotropic stable motivic homotopy category SH(k/k) is equivalent to the stable ∞-category X − Mod of modules over the motivic E ∞ -ring spectrum X.
Recall that motivic cohomology with Z /2-coefficients is represented by the motivic Eilenberg-MacLane spectrum HZ/2.Then, we define isotropic motivic cohomology as the cohomology theory represented by the motivic E ∞ -ring spectrum X ∧ HZ/2.Definition 3.8.For any X in SH(k), we define the isotropic motivic cohomology of X as and the isotropic motivic homology of X as The isotropic motivic cohomology of the point was computed by Vishik in [21].We report the result in the next theorem.Theorem 3.9.Let k be a flexible field.Then, for any i ≥ 0 there exists a unique cohomology class r i of bidegree and Q j r i = δ ij , where Q j are the Milnor operations.At this point, we want to introduce the isotropic motivic Steenrod algebra A * * (k/k) and its dual A * * (k/k).They are defined respectively as the isotropic motivic cohomology and homology of the motivic Eilenberg-MacLane spectrum.
Definition 3.10.The isotropic motivic Steenrod algebra is defined by and its dual by The structure of A * * (k/k) was studied in [19, Section 3].We summarise the main results in the next proposition.
By projecting the motivic Cartan formulas (see [23,Propositions 9.7 and 13.4]) to the isotropic category one gets a coproduct on A * * (k/k) given by: This coproduct structures A * * (k/k) as a coalgebra whose dual is described as an H * * (k/k)-algebra by where τ i is the dual of the Milnor operation Q i and ξ j is the dual of the motivic cohomology operation Sq ) is given by (see [23,Lemma 12.11]): Remark 3.12.By Proposition 3.11, the projection from A * * (k/k) to its quotient by the left ideal generated by Milnor operations provides a homomorphism

Cellular motivic spectra
In this work, we are mostly interested in cellular objects of isotropic motivic categories.We recall from [3,Remark 7.4] that the category of cellular motivic spectra, which we denote by SH(k) cell , is the localising subcategory of SH(k) generated by the spheres Σ p,q S. Similarly, the category of Tate motives, which we denote by DM(k) T ate , is the localising subcategory of DM(k) generated by the Tate motives T (q)[p].If E is a motivic E ∞ -ring spectrum, then we denote by E − Mod cell the stable ∞-category of E-cellular modules, i.e. the localising subcategory of E − Mod generated by Σ p,q E. Definition 4.1.The category of X-cellular modules will be called the category of isotropic cellular motivic spectra, and denoted by SH(k/k) cell .In the same way, the full localising subcategory of DM(k/k) generated by the objects M(X)(q)[p] will be called the category of isotropic Tate motives, and denoted by DM(k/k) T ate .
A fundamental property of the category of cellular objects is that isomorphisms can be detected by motivic homotopy groups, as reported in the following result.Proposition 4.2.Let E be a motivic E ∞ -ring spectrum and X → Y be a map of E-cellular motivic spectra that induces isomorphisms on π p,q for all p and q in Z.Then, the map is a weak equivalence.
Another essential advantage of dealing with cellular objects is that they allow the construction of very useful convergent spectral sequences.
If M is a left E-cellular motivic spectrum, then there is a conditionally convergent spectral sequence Proof.See [3, Propositions 7.7 and 7.10].

The isotropic motivic Adams spectral sequence
In this section we recall the construction of the isotropic motivic Adams spectral sequence (see [19,Section 4]).Moreover, we study the circumstances under which the E 2 -page is expressible in terms of Ext-groups over the isotropic motivic Steenrod algebra.
Definition 5.1.Let Y be an isotropic motivic spectrum, i.e. an object in X − Mod.Then, the standard isotropic motivic Adams resolution of Y consists of the Postnikov system where X ∧ HZ/2 is defined by the following exact triangle in SH(k): By applying motivic homotopy groups functors π * * to the previous Postnikov system we get an unrolled exact couple, which induces in turn a spectral sequence with E 1 -page described by and first differential In general, differentials on the E r -page have tri-degrees given by We call this spectral sequence isotropic motivic Adams spectral sequence.
The isotropic Adams spectral sequence converges to the homotopy groups of a motivic spectrum closely related to Y , namely its X ∧ HZ/2-nilpotent completion that we denote by Y ∧ X ∧ HZ/2 .Before proceeding, let us recall from [2, Section 5] how to construct the E-nilpotent completion of a spectrum Y for a homotopy ring spectrum E. Definition 5.2.Let E be a homotopy ring spectrum and Y a motivic spectrum in SH(k).First, define E by the following distinguished triangle in SH(k): This way one gets an inverse system and the E-nilpotent completion of Y is the motivic spectrum defined by Note that, by [ , the isotropic motivic Adams spectral sequence strongly converges to π * * (Y ∧ X ∧ HZ/2 ).It only remains to notice that, since Y is a X-module, its HZ/2-nilpotent and X ∧ HZ/2-nilpotent completions coincide.In fact, after smashing with X the morphism of distinguished triangles since X is an idempotent in SH(k).It follows that X ∧HZ/2 ∼ = X ∧X ∧ HZ/2, and so X ∧HZ/2 n ∼ = X ∧(X ∧ HZ/2) n for any n.Therefore, since Y ∼ = X ∧Y one obtains that which is what we wanted to show.
Remark 5.4.By [14, Section 5.2 and Theorem 1.0.3], the HZ/2-completion of a connective motivic spectrum coincides with its (2, η)-completion.Since all isotropic motivic spectra are 2-power torsion (see Remark 3.6), and so 2-complete, the previous result establishes the convergence of the isotropic Adams spectral sequence for a connective isotropic spectrum to the motivic stable homotopy groups of its η-completion.
Definition 5.5.A spectral sequence {E s,t,u r } is called Mittag-Leffler if for each s, t, u there exists r 0 such that E s,t,u r ∼ = E s,t,u ∞ whenever r > r 0 .
Note that every Mittag-Leffler spectral sequence satisfies the condition lim ← − 1 r E s,t,u r = 0 for any s, t, u (see [2, after Proposition 6.3]).We will see that in many important cases the isotropic Adams spectral sequence is Mittag-Leffler, which guarantees strong convergence.Now, we would like to understand what conditions we need to impose on Y in order to be able to express the E 2 -page of the isotropic Adams spectral sequence in terms of Ext-groups over the isotropic motivic Steenrod algebra.First, we need the following lemmas.
Lemma 5.6.Let k be a flexible field and Y an object in X − Mod.Then, there exists an isomorphism of left Since by [8,Theorem 5.10] HZ/2 ∧ HZ/2 is a split HZ/2-module, i.e. it is equivalent to a wedge sum of the form α∈A Σ pα,qα HZ/2, we have that Remark 5.7.Note that, by the previous lemma, the map Y → X ∧ HZ/2 ∧Y induces in isotropic motivic homology a coaction In the next result, we show that, if the homology of an isotropic cellular spectrum Y is free over H * * (k/k), then the motivic spectrum X ∧ HZ/2 ∧Y is a split X ∧ HZ/2-module.
Lemma 5.8.Let k be a flexible field and Y an object in X − Mod cell such that H iso * * (Y ) is a free left H * * (k/k)-module generated by a set of elements {x α } α∈A , where x α has bidegree (q α )[p α ].Then, there exists an isomorphism of spectra α∈A Σ pα,qα (X ∧ HZ/2) we can represent each generator x α as a map Σ pα,qα S → X ∧ HZ/2 ∧Y , where (q α )[p α ] is the bidegree of x α .For all α ∈ A, this map corresponds bijectively to a map Σ pα,qα (X ∧ HZ/2) → X ∧ HZ/2 ∧Y of X ∧ HZ/2-cellular modules.Hence, we get a map α∈A Σ pα,qα (X ∧ HZ/2) → X ∧ HZ/2 ∧Y of X ∧ HZ/2-cellular modules.In order to check that it is an isomorphism, by Proposition 4.2 it is enough to look at the induced morphisms on homotopy groups.Indeed, we have that on the one hand which sends 1 ∈ Σ pα,qα H * * (k/k) to x α for any α ∈ A, so it is an isomorphism, as we wanted to show.
The next lemma provides us with a condition under which the isotropic cohomology of a spectrum is dual to its isotropic homology.Lemma 5.9.Let k be a flexible field and Y an object in X − Mod such that there is an isomorphism X ∧ HZ/2 ∧Y ∼ = α∈A Σ pα,qα (X ∧ HZ/2) for some set A. Then, for any bidegree (q)[p] there is an isomorphism Proof.Since X ∧ HZ/2 ∧Y ∼ = α∈A Σ pα,qα (X ∧ HZ/2) by hypothesis, we have that On the other hand, we have the following chain of isomorphisms: that concludes the proof.Now, we need to define a certain concept of finiteness which suits the isotropic environment.
Definition 5.10.We say that a set of bidegrees {(q α )[p α ]} α∈A is isotropically finite type if for any bidegree (q)[p] there are only finitely many α ∈ A such that p − p α ≥ 2(q − q α ) ≥ 0.Moreover, we say that a set of bigraded elements {x α } α∈A is isotropically finite type if the corresponding set of bidegrees is so.
Then, for any bidegree (q)[p], the obvious map is an isomorphism.
Proof.First, note that, for any bidegree (q)[p], one has the following commutative diagram π p,q (X ∧ α∈A Σ pα,qα HZ/2) The left vertical arrow is the isomorphism described by the following chain of equivalences where the identification α∈A is due to the fact that the set {(q α )[p α ]} α∈A is isotropically finite type, so for any bidegree (q)[p] only for a finite number of α ∈ A the group H pα−p,qα−q (k/k) is non-zero by Theorem 3.9.The bottom horizontal map is obviously an isomorphism since A * * (k/k) is an F 2 -vector space.The right vertical map is an isomorphism since where the last isomorphism comes from the fact that the set of bidegrees {(q α )[p α ]} α∈A is isotropically finite type, so for any bidegree (q)[p] only for finitely many α ∈ A the group ) is non-trivial by Theorem 3.9.This completes the proof.
At this point, we are ready to present the structure of the E 2 -page of the isotropic Adams spectral sequence, which behaves as in the classical case.
Theorem 5.12.Let k be a flexible field and Y an object in X − Mod cell such that H iso * * (Y ) is a free left H * * (k/k)-module generated by an isotropically finite type set of elements {x α } α∈A .Then, the E 2 -page of the isotropic motivic Adams spectral sequence is described by Proof.First, we want to prove by induction that H iso * * ((X ∧ HZ/2) ∧s ∧ Y ) is a free left H * * (k/k)-module generated by an isotropically finite type set of elements {x α } α∈As for any s ≥ 0. The induction basis is guaranteed by hypothesis after setting A 0 = A. Suppose the statement is true at the s − 1 stage, i.e.H iso * * ((X ∧ HZ/2) Then, by Lemma 5.6, the map (X ∧ HZ/2) ∧s−1 ∧ Y → X ∧ HZ/2 ∧(X ∧ HZ/2) ∧s−1 ∧ Y induces in isotropic motivic homology the monomorphism Hence, the standard Adams resolution induces for any p and q a short exact sequence 0 → H iso p,q ((X ∧ HZ/2) ∧s−1 ∧Y ) → H iso p,q (X ∧ HZ/2 ∧(X ∧ HZ/2) ∧s−1 ∧Y ) → H iso p−1,q ((X ∧ HZ/2) ∧s ∧Y ) → 0. Now, note that, by the very structure of the dual of the isotropic motivic Steenrod algebra, A * * (k/k) is freely generated over H * * (k/k) by a set of generators {1, y β } β∈B which is finite in each bidegree and such that p β ≥ 2q β ≥ 0 for any β ∈ B, where (q β )[p β ] is the bidegree of y β .Hence, the set {y β x α } β∈B,α∈A s−1 is isotropically finite type and freely generates H iso * * ((X ∧ HZ/2) ∧s ∧ Y ) over H * * (k/k), i.e.
Therefore, Lemma 5.8 implies that all X ∧ HZ/2 ∧(X ∧ HZ/2) ∧s ∧ Y are wedges of appropriately shifted X ∧ HZ/2.More precisely, for any s ≥ 0, there exists an isomorphism where A s = B ×A s−1 , from which we deduce, using Lemma 5.11, that the E 1 -page of the isotropic Adams spectral sequence can be described by Moreover, note that . Thus, for any s, t, u we have an isomorphism as we aimed to prove.
By using the isotropic motivic Adams spectral sequence, in [19] we computed the isotropic motivic homotopy groups of the sphere spectrum which can be identified with the E 2 -page of the classical Adams spectral sequence.Theorem 5.13.Let k be a flexible field.Then, the stable motivic homotopy groups of the HZ/2-completed isotropic sphere spectrum are completely described by Proof.See [19,Theorem 5.7].

The motivic Brown-Peterson spectrum
In this section, we recall from [20] the construction of the motivic Brown-Peterson spectrum.Moreover, we compute its isotropic homology and homotopy, which will be useful later on for the construction of the isotropic motivic Adams-Novikov spectral sequence and so for the proofs of our main results.Definition 6.1.Let MGL (2) be the motivic algebraic cobordism spectrum (see [22,Section 6.3]) localised at 2. Then, following [20, Section 5] one defines the motivic Brown-Peterson spectrum at the prime 2 as the colimit of the diagram in SH(k) where e (2) is the motivic Quillen idempotent.
Note, in particular, that MBP is a homotopy commutative ring spectrum and a direct summand of MGL (2) .Proposition 6.2.Let k be a flexible field.Then, there is an isomorphism of H * * (k/k)-modules and an isomorphism of H * * (k/k)-algebras where c i is the ith Chern class in H 2i,i iso (BGL) and b i ∈ H iso 2i,i (BGL) is the dual of c i 1 with respect to the monomial basis, for any i.
Proof.First, note that the maps P 1 → P ∞ and HZ/2 → X ∧ HZ/2 induce a commutative square where the left vertical morphism is the projection ) and c is the only non zero class in H 2,1 (P ∞ ) ∼ = H 2,1 (P 1 ) ∼ = Z /2.If we also denote by c the images of c under the horizontal maps in isotropic motivic cohomology, then the right vertical homomorphism is given by the projection Hence, X ∧ HZ/2 is an oriented motivic spectrum (see [ . .] → h(L) a retraction of the inclusion.In the next proposition, we give a description of isotropic homology and cohomology of the algebraic cobordism spectrum MGL.Proposition 6.3.Let k be a flexible field.Then, the coaction ∆ : . .] and the composition is an isomorphism of left A * * (k/k)-comodule algebras.Dually, the map is an isomorphism of left A * * (k/k)-module coalgebras.
Proof.Since HZ/2 ∧ MGL is a split HZ/2-module (see the remark after [8, Definition 5.4]), from [8, Lemma 5.2] we deduce that as an H * * (k/k)-algebra.From [8, Theorem 6.5] we know that the coaction ∆ : . .] and the composition The next result provides us with the structure of isotropic homology and cohomology of the motivic Brown-Peterson spectrum MBP.Proposition 6.4.Let k be a flexible field.Then, the isotropic motivic homology of MBP is described as a left A * * (k/k)-comodule by Dually, the isotropic motivic cohomology of MBP is described as a left A * * (k/k)-module by Proof.From [8,Remark 6.20], one knows that MBP is equivalent to MGL (2) /x where x is any maximal h-regular sequence, i.e. a sequence of homogeneous elements in L such that h(x) is a regular sequence in h(L) which generates the maximal ideal.Therefore, [8,Theorem 6.11] implies that there exists an isomorphism of A * * (k)-comodules H * * (MBP) ∼ = P * * .
Since HZ/2 ∧ MBP is a split HZ/2-module, we deduce from [8, Lemma 5.2] that which proves the first part.The second part follows again from dualization, since G * * is isotropically finite type, by Lemmas 5.8 and 5.9.
Later on, we will also need the isotropic homology and cohomology of MBP ∧ MBP, which is reported in the following proposition.Proposition 6.5.Let k be a flexible field.Then, the isotropic motivic homology of MBP ∧ MBP is described as a left A * * (k/k)-comodule by Dually, the isotropic motivic cohomology of MBP ∧ MBP is described as a left A * * (k/k)-module by Proof.Since HZ/2 ∧ MBP is a split HZ/2-module, by [8, Lemma 5.2] and Proposition 6.4 we obtain that The description of the isotropic cohomology follows again by dualizing the homology isomorphism.Now, we compute the isotropic stable homotopy groups of MBP by using the isotropic Adams spectral sequence developed in the previous section.Theorem 6.6.Let k be a flexible field.Then, the isotropic motivic homotopy groups of MBP are described by Proof.First, note that, by Proposition 6.4, H iso * * (MBP) is freely generated over H * * (k/k) by G * * which is isotropically finite type.Hence, Theorem 5.12 implies that the E 2 -page of the isotropic motivic Adams spectral sequence for X ∧ MBP is given by Now, we deduce from Proposition 6.4 and [19, Theorem 5.4] that Ext s,t,u Therefore, the E 2 -page of the isotropic Adams spectral sequence for X ∧ MBP is concentrated just in the tridegree (0, 0, 0), from which it follows that all differentials from the second on are trivial.Thus, the Mittag-Leffler condition is clearly satisfied, and so strong convergence holds by Proposition 5.3.Then, it immediately follows from Remark 5.4 and the fact that MBP is η-complete that which completes the proof.
In the following sections, it will be also useful to know the isotropic homotopy groups of MBP ∧ MBP that we compute in the next result.Theorem 6.7.Let k be a flexible field.Then, the isotropic motivic homotopy groups of MBP ∧ MBP are described by Proof.The proof of this theorem goes along the lines of the previous one.Since by Proposition 6.5 and G * * ⊗ F 2 G * * is isotropically finite type, by Theorem 5.12 we have that the E 2 -page of the isotropic Adams spectral sequence for X ∧ MBP ∧ MBP is provided by Again, we note that by [19,Theorem 5.4] In particular, since G * * is concentrated on the slope 2 line, we have that all differentials from the second on are trivial by degree reasons.Hence, Mittag-Leffler condition is met, which implies that the spectral sequence is strongly convergent.From all this, it follows as above that that is what we aimed to show.

The category of isotropic cellular MBP-modules
In this section we start by providing X ∧ MBP with an E ∞ -ring structure.This allows us to talk about the stable ∞-category of X ∧ MBP-modules, i.e.X ∧ MBP − Mod, and its cellular part, i.e.X ∧ MBP − Mod cell .Our aim is to focus on isotropic cellular MBP-modules, which is the same as cellular X ∧ MBP-modules.In particular, we completely describe the category X ∧ MBP − Mod cell in algebraic terms.This section is structured along the lines of [6, Section 3].Therefore, before each result we indicate the one from [6] it corresponds to.We hope this could clearly shed light on the deep parallelism between [6] and this work.
Proposition 7.1.The homotopy commutative ring structure on X ∧ MBP extends to an E ∞ -ring structure.
Proof.It follows from [13,Proposition 1.4.4.11] that there exists a t-structure on X − Mod with nonnegative part generated by X 2n,n for any n ∈ Z.By [1, Theorem A.1], X ∧ MGL belongs to the nonnegative parte of this t-structure, and so also X ∧ MBP does.On the other hand, one deduces from Theorem 6.6 and [1, Lemma 2.4] that X ∧ MBP belongs to the non-positive part too.Hence, X ∧ MBP is a homotopy commutative ring spectrum in the heart of the above mentioned t-structure, which means that it is an E ∞ -ring spectrum. 2  Once we know that X ∧ MBP is a motivic E ∞ -ring spectrum, we can consider the stable ∞-category of X ∧ MBP-modules and its homotopy category which is tensor triangulated.In particular, we focus on its cellular part.Proposition 7.2.Let k be a flexible field and Y an object in X ∧ MBP − Mod cell such that π * * (Y ) is isomorphic to the F 2 -vector space α∈A Σ pα,qα F 2 .Then, there exists an isomorphism of spectra α∈A Σ pα,qα (X ∧ MBP) Proof.We follow the lines of the proof of Lemma 5.8.Each generator of π * * (Y ) represents a map Σ pα,qα S → Y .For all α ∈ A, this map corresponds bijectively to a map Σ pα,qα (X ∧ MBP) → Y of X ∧ MBP-cellular modules.Hence, we get a map α∈A Σ pα,qα (X ∧ MBP) → Y of X ∧ MBP-cellular modules that induces an isomorphism on homotopy groups since π * * (X ∧ MBP) ∼ = F 2 by Theorem 6.6.Therefore, it follows from Proposition 4.2 that the above map is an isomorphism of spectra, which completes the proof.
The previous result implies the following corollary that corresponds to [6,Corollary 3.3].
Corollary 7.3.Let k be a flexible field and X and Y be objects in Proof.It follows from Proposition 7.2 that X ∼ = α∈A Σ pα,qα (X ∧ MBP) and Y ∼ = β∈B Σ p β ,q β (X ∧ MBP) 2 I am grateful to Tom Bachmann for this argument.
for some sets A and B.Then, we have that which concludes the proof.
The next theorem, which corresponds to [6, Theorem 3.8], identifies X ∧ MBP − Mod cell with the category of bigraded F 2 -vector spaces that we denote by F 2 − Mod * * .
Theorem 7.4.Let k be a flexible field.Then, the functor is an equivalence of categories.
Proof.It follows immediately from Proposition 7.2 and Corollary 7.3.Remark 7.5.We want to highlight that the equivalence provided by the previous theorem is actually an equivalence of triangulated categories, where F 2 − Mod * * is structured as a triangulated category in the obvious way.More precisely, the translation functor is the suspension Σ 1,0 and distinguished triangles are of the form where f is a morphism of bigraded F 2 -vector spaces.

The category of isotropic cellular spectra
This section is devoted to the understanding of the structure of the category X − Mod cell that is, as we have already noticed, the category of cellular isotropic spectra SH(k/k) cell .We give a nice algebraic description of this category based on the dual of the topological Steenrod algebra.The results here are the isotropic versions of the ones in [6, Sections 4 and 5].Therefore, the proofs we provide are isotropic adaptations of the respective ones in [6].In the next lemma, which corresponds to [6, Lemma 5.1], we compute the MBP-homology of isotropic MBP-cellular spectra.Proof.Since the motivic spectrum I is by hypothesis in X ∧ MBP − Mod cell , we deduce from Theorem 7.4 that I ∼ = α∈A Σ pα,qα (X ∧ MBP) for some set A. Therefore, by Theorem 6.7 one has that where V ∼ = α∈A Σ pα,qα F 2 .Now, note that by Theorem 6.6 that is what we aimed to show.
following lemma, which corresponds to [6,Lemma 5.3], describes algebraically the hom-sets from isotropic cellular spectra to isotropic MBP-cellular spectra.and Proof.First, note that by arguments similar to the ones in Lemma 5.6 we have an isomorphism for any isotropic spectrum Y and any s ≥ 0. So, we only need to prove the first part of the statement.We achieve this by an induction argument, after noting that obviously the statement holds for s = 0. Now, suppose the statement holds for s − 1, i.e.
Then, the distinguished triangle in SH(k) which completes the proof.
We are now ready to construct the isotropic Adams-Novikov spectral sequence, which corresponds to [6,Theorem 5.6].Before proceeding, we would like to fix some notations.Definition 8.4.Let X be an isotropic spectrum.The Chow-Novikov degree of MBP p,q (X) is the integer p−2q.We denote by X − Mod b cell the category of bounded isotropic cellular spectra, i.e. isotropic cellular spectra whose MBP-homology is non-trivial only for a finite number of Chow-Novikov degrees.
Theorem 8.5.Let k be a flexible field and X and Y objects in X − Mod b cell .Then, there is a strongly convergent spectral sequence where X ∧ MBP is defined by the following distinguished triangle in SH(k) If we apply the functor [Σ * * X, −] we get an unrolled exact couple that induces a spectral sequence with E 1 -page given by and first differential d s,t,u This is what we call the isotropic Adams-Novikov spectral sequence.Note that by Lemmas 8.2 and 8.3 the E 1 -page has a nice description provided by We want to point out that, since X ∧ HZ/2 is a X ∧ MBP-module and X ∧ MBP is X ∧ HZ/2-complete, the subcategories of HZ/2-complete and MBP-complete isotropic spectra coincide.
The next corollary, which corresponds to [6,Corollary 4.7], computes hom-sets from X − Mod b,≥0 cell,HZ/2 to X − Mod b,≤0 cell,HZ/2 in algebraic terms.Proof.As we have already pointed out, the E 1 -page of the isotropic Adams-Novikov spectral sequence is given by E s,t,u Since we are interested in the group [X, Y ], the part of the E 1 -page that is involved consists of the groups in tridegrees (t, t, 0).By hypothesis, X is in X − Mod b,≥0 cell,HZ/2 while Y is in X − Mod b,≤0 cell,HZ/2 , which implies that, among these groups, only E 0,0,0 1 is non-trivial.Since in this tridegree all differentials from the second on are trivial by degree reasons, we have that which completes the proof.
By using the isotropic Adams-Novikov spectral sequence we also get the following corollary that corresponds to [6,  Proof.This follows immediately by noticing that the differentials d s,t,u r : E s,t,u r → E s+r,t+r−1,u r of the isotropic Adams-Novikov spectral sequence are trivial for r ≥ 2 since E s,t,u 2 is trivial for t = 2u.Hence, the spectral sequence is strongly convergent and collapses at the second page, from which we get that that is what we wanted to show.
Before proceeding, we also need the following lemma which essentially corresponds to [6, Lemma 4.10].Lemma 8.9.Let k be a flexible field and M a G * * -comodule concentrated in Chow-Novikov degree 0 which is finitely generated as an F 2 -vector space.Then, there exists an object X in X − Mod ♥ cell,HZ/2 such that M ∼ = MBP * * (X).
Proof.Since by hypothesis M is a finite-dimensional F 2 -vector space, by [10,Theorem 3.3] one has a finite filtration of subcomodules such that, for any i, the quotient M i /M i−1 is stably isomorphic to F 2 , i.e.M i /M i−1 ∼ = Σ 2q i ,q i F 2 for some integer q i .We want to prove the statement by induction on i.First, note that by Theorem 6.6 the comodule Σ 2q i ,q i F 2 is the MBP-homology of the isotropic spectrum Σ 2q i ,q i X ∧ HZ/2 for any i.Now, suppose that there exists an object X i−1 in X − Mod ♥ cell,HZ/2 such that M i−1 ∼ = MBP * * (X i−1 ).Then, the short exact sequence 0 ] by Corollary 8.8.Let us define X i as Cone(f i ).Then, we have a long exact sequence in MBP-homology

Note that the connecting homomorphism
described as the Yoneda product with the element g i of Ext 1,0,0 induced by f i in isotropic homotopy groups (see [18,Theorem 2.3.4]).By Corollary 8.8 the isotropic Adams-Novikov spectral sequence collapses at the second page, so g i * = f i * .It follows that the extensions g i and f i coincide which implies that MBP * * (X i ) ∼ = M i , as we wanted to prove.
The next result is the isotropic equivalent of [6, Lemma 4.2].
We are now ready to identify X − Mod Proof.First, note that Corollary 8.7 guarantees that the functor MBP * * is fully faithful.We just need to show that it is essentially surjective.Recall from [7, Propositions 1.4.10,1.4.4 and 1.4.1] that any left G * * -comodule M is a filtered colimit of comodules M α which are finitely generated as F 2 -vector spaces.By Lemma 8.9 all M α are expressible as MBP * * (X α ) for some X α in X − Mod ♥ cell,HZ/2 .Therefore, we have that M ∼ = MBP * * (X) where X = colim X α , which is what we aimed to show.Remark 8.12.Note that G * * − Comod 0 * * is equivalent to the category of left A * -comodules, where A * is the dual of the topological Steenrod algebra.Hence, the previous result can be rephrased by saying that X − Mod ♥ cell,HZ/2 is equivalent to the abelian category of left A * -comodules.
The next proposition that corresponds to [6, Proposition 4.12] provides X − Mod b cell,HZ/2 with a t-structure.cell,HZ/2 , then MBP(X) is concentrated in non-negative Chow-Novikov degrees.Consider the projection MBP(X) → MBP(X) 0 that kills all the elements in positive Chow-Novikov degrees, and note that there exists an object X 0 in X − Mod ♥ cell,HZ/2 such that MBP(X 0 ) ∼ = MBP(X) 0 .Now, by Corollary 8.7 this morphism comes from a map f : X → X 0 such that Σ −1,0 Cone(f ) belongs to X − Mod b,≥1 cell,HZ/2 .Therefore, by [6, Proposition 3.6], the pair (X − Mod b,≥0 cell,HZ/2 , X − Mod b,≤0 cell,HZ/2 ) defines a bounded t-structure on X − Mod b cell,HZ/2 that is what we aimed to prove.
We are now ready to prove the main result of this section that corresponds to [6,Theorem 4.13].In this theorem we identify X − Mod b cell,HZ/2 with the derived category of left G * * -comodules concentrated in Chow-Novikov degree 0.

The category of isotropic Tate motives
We finish in this section by applying previous results in order to obtain information on the category of isotropic Tate motives DM(k/k) T ate .In particular, we get an easy algebraic description for the hom-sets in DM(k/k) T ate between motives of isotropic cellular spectra.First, we prove the following lemma which tells us that the isotropic motivic homology of an isotropic spectrum is always a free H * * (k/k)-module.Lemma 9.1.Let k be a flexible field and X an object in X − Mod.Then, there exists an isomorphism of left H * * (k/k)-modules H iso * * (X) ∼ = H * * (k/k) ⊗ F 2 MBP * * (X).Proof.The Hopkins-Morel equivalence (see [8,Theorem 7.12]) implies in particular that HZ/2 is a quotient spectrum of MBP.It follows that HZ/2 can be obtained from MBP by applying cones and homotopy colimits and, so, it is an MBP-cellular module from which we get by Theorem 7.

1
I am grateful to Tom Bachmann for this argument.
is an isomorphism of left A * * (k)-comodule algebras, where P * * is the subalgebra of A * * (k) defined by H * * (k)[ξ 1 , ξ 2 , . . .].By tensoring the previous composition with H * * (k/k) over H * * (k) we get the desired isomorphism, which completes the first part.The second part follows easily, since G * * ⊗ F 2 h(L) is isotropically finite type, from Lemmas 5.8 and 5.9 by dualizing the homology isomorphism.
then the isotropic motivic Adams spectral sequence for Y is strongly convergent to the stable motivic homotopy groups of the HZ/2-nilpotent completion of Y .
19, Proposition 2.3], if Y is an isotropic motivic spectrum then also Y ∧ E is so.Proposition 5.3.Let Y be an isotropic motivic spectrum.If lim ← − 1 r E s,t,u r = 0 for any s, t, u,