Group schemes and motivic spectra

By a theorem of Mandell-May-Schwede-Shipley the stable homotopy theory of classical $S^1$-spectra is recovered from orthogonal spectra. In this paper general linear, special linear, symplectic, orthogonal and special orthogonal motivic spectra are introduced and studied. It is shown that the stable homotopy theory of motivic spectra is recovered from each of these types of spectra. An application is given for the localization functor $C_*\mathcal Fr:SH_{nis}(k)\to SH_{nis}(k)$ in the sense of [15] that converts the Morel-Voevodsky stable motivic homotopy theory $SH(k)$ into the equivalent local theory of framed bispectra [15].


INTRODUCTION
In the 90's several approaches to the stable homotopy theory of S 1 -spectra were suggested.In [21] several comparison theorems relating the different constructions were proven showing that all of the known approaches to highly structured ring and module spectra are essentially equivalent.
Mandell, May, Schwede and Shipley [21] proved that the stable homotopy theory of classical topological S 1 -spectra is recovered from orthogonal spectra.In [24] Østvaer conjectured that the stable homotopy theory of motivic spectra can be recovered from motivic GL-spectra, in which the role of the orthogonal groups as in topology [21] is played by the general linear group schemes GL n -s.In this paper this conjecture is solved in the affirmative.
We follow [21] to develop the formal theory of diagram motivic spectra in Section 2. The framework allows lots of flexibility so that the reader can construct further interesting examples.For our purposes we work with diagram motivic spectra coming from group schemes GL n -s, SL ns, Sp n -s, O n -s and SO n -s (see Section 3).These group schemes act on motivic spheres.We also refer to the associated motivic spectra as general linear, special linear, symplectic, orthogonal and special orthogonal motivic spectra or just GL-, SL-, Sp-, O-, SO-motivic spectra.
One of the tricky concepts in the stable homotopy theory of classical symmetric spectra is that of semistability.Semistable symmetric spectra are important for understanding the difference between stable equivalences and maps inducing π * -isomorphisms, that is, isomorphisms of the classical stable homotopy groups (in contrast with most other categories of spectra, not all stable equivalences of symmetric spectra induce π * -isomorphisms).The same concept of semistability occurs in the stable homotopy theory of motivic spectra.We show in Section 4 that every GL-, SLor Sp-motivic spectrum is semistable ragarded as a symmetric motivic spectrum.This fact is the motivic counterpart of the classical result in topology saying that every orthogonal S 1 -spectrum of topological spaces is semistable.
We then define in Section 5 stable model structures on the categories of diagram motivic spectra.The main result of the paper is proven in Section 6 which compares ordinary/symmetric motivic spectra with GL-, SL-, Sp-, O-and SO-motivic spectra respectively (cf.Mandell-May-Schwede-Shipley [21, 0.1]). 2 -spectra are all Quillen equivalences with respect to the stable model structure:

Theorem (Comparison). Let k be any field. The following natural adjunctions between categories of T -and T
(1) An application of the Comparison Theorem is given in Section 7 for the localizing functor in the sense of [15].Recall that a new approach to the classical Morel-Voevodsky stable homotopy theory SH(k) was suggested in [15] and is based on the functor C * F r.This approach has nothing to do with any kind of motivic equivalences and is briefly defined as follows.We start with the local stable homotopy category of sheaves of S 1 -spectra SH nis S 1 (k).Then stabilizing SH nis S 1 (k) with respect to the endofunctor G ∧1 m ∧ −, we arrive at the triangulated category of bispectra SH nis (k).We then apply an explicit localization functor that first takes a bispectrum E to its naive projective cofibrant resolution E c and then one sets in each bidegree C * F r(E) i, j := C * Fr(E c i, j ).The localization functor C * F r is isomorphic to the big framed motives localization functor M b f r of [14] (see [15] as well).Then SH new (k) is defined as the category of C * F r-local objects in SH nis (k).By [15, Section 2] SH new (k) is canonically equivalent to Morel-Voevodsky's SH(k).
Using the Comparison Theorem above, we define new functors C * F r G ,n on SH nis (k) that depend on n 0 and the choice of the family of groups G = {GL k } k 0 , {SL 2k } k 0 , {Sp 2k } k 0 , {O 2k } k 0 , {SO 2k } k 0 .In Theorem 7.3 we prove that C * F r and C * F r G ,n are naturally isomorphic.As a result, one can incorporate linear algebraic groups into the theory of motivic infinite loop spaces and framed motives developed in [14].
Throughout the paper we denote by S a Noetherian scheme of finite dimension.We write Sm/S for the category of smooth separated schemes of finite type over S. Sm/S comes equipped with the Nisnevich topology [23, p. 95].We denote by (Shv • (Sm/S), ∧, pt + ) the closed symmetric monoidal category of pointed Nisnevich sheaves on Sm/S.The category of pointed motivic spaces M • is, by definition, the category ∆ op Shv • (Sm/k) of pointed simplicial Nisnevich sheaves.Unless otherwise specified, we shall always deal with the flasque local (respectively motivic) model structure on M • in the sense of [19].Both model structures are weakly finitely generated in the sense of [10].
Acknowledgements.The author is very grateful to Alexey Ananyevskiy, Semen Podkorytov and Matthias Wendt for numerous helpful discussions.He also thanks Aravind Asok, Andrei Druzhinin and Sergey Gorchinsky for various comments.

DIAGRAM MOTIVIC SPACES AND DIAGRAM MOTIVIC SPECTRA
We refer the reader to [7] for basic facts of enriched category theory.We mostly adhere to [21] in this section.Suppose C is a small category enriched over the closed symmetric monoidal category of pointed motivic spaces M • .Following [21] If C is a symmetric monoidal M • -category with monoidal product ⋄ and monoidal unit u, then , where S 0 := pt + , is the identity and the following diagram commutes: The following lemma is straightforward.
Then the categories of (right) R-modules and of C -spectra over R are isomorphic.
A theorem of Day [9] also implies the following The right hand side refers to the If R is commutative, then C R is symmetric monoidal with monoidal product ⋄ R on objects being defined as the monoidal product ⋄ in C .Its unit object is the unit object The proof of the following fact literally repeats that of [21, 2.2], which is purely categorical and is not restricted by topological categories only.

MOTIVIC SPECTRA ASSOCIATED WITH GROUP SCHEMES
After collecting basic facts for C -spectra over a ring object R in [C , M • ], where C is a symmetric monoidal M • -category, in this section we give particular examples we shall work with in this paper.The framework we have fixed above allows a lot of flexibility and we invite the interested reader to construct further examples.A canonical choice for a ring object, which we denote by S or by S C if we want to specify the choice of the diagram M • -category C , is the motivic sphere spectrum S = (S 0 , T, T 2 , . ..), where T n is the Nisnevich sheaf A n S /(A n S − 0).Another natural choice is the motivic sphere T 2spectrum S = (S 0 , T 2 , T 4 , . ..) consisting of the even dimensional spheres T 2n .The latter spectrum is necessary below when working with, say, special linear or symplectic groups.From the homotopy theory viewpoint, stable homotopy categories of motivic T -and T 2 -spectra are Quillen equivalent (see, e.g., [25, 3.2]).Where it is possible we follow the terminology and notation of [21] in order to be consistent with the classical topological examples.
We should stress that in all our examples below the category of diagrams C is defined in terms of group schemes.Our first example is elementary, but most important for our analysis.

Example (Ordinary motivic T -spectra).
Let N be the (unbased) category of non-negative integers Z 0 , with only "identity morphisms motivic spaces" between them.Precisely, The symmetric monoidal structure is given by addition m + n, with 0 as unit.An N -space is a sequence of based motivic spaces.The canonical enriched functor S = S N takes n ∈ Z n 0 to T n .It is a ring object of [N , M • ], but it is not commutative since permutations of motivic spheres T n are not identity maps.This is a typical difficulty in defining the smash product in stable homotopy theory.A motivic T -spectrum is an N -spectrum over S .Let Sp N T (S) denote the category of Nspectra over S .Since T n is the n-fold smash product of T , the category Sp N T (S) is isomorphic to the category of ordinary motivic T -spectra Sp T (S).
The shift desuspension functors to N -spectra are given by (F m A) n = A ∧ T n−m (by definition, T n−m = * if n < m).The smash product of N -spaces (not N -spectra!) is given by The category N S such that an N -spectrum is an N S -space has morphism motivic spaces The category of ordinary motivic T 2 -spectra Sp N T 2 (S) is defined in a similar fashion.As we have noticed above, S N is not commutative, and hence the category of N -spectra Sp N T (S) does not have a smash product that makes it a closed symmetric monoidal category.In all other examples below the ring object S C ∈ [C , M • ] is commutative, and therefore the category of C -spectra over S C is closed symmetric monoidal.The first classical example is that for symmetric spectra (we refer the reader to [20] for further details).

Example (Symmetric motivic T -spectra).
Let Σ be the (unbased) category of finite sets m = {1, . . ., m}.By definition, 0 := / 0. Its morphisms motivic spaces Σ(m, n) are given by symmetric groups canonically regarded as group S-schemes.Precisely, Notice that the underlying category associated with Σ is n 0 Σ n .The symmetric monoidal structure on Σ is given by concatenation of sets m ⊔ n and block sum of permutations, with 0 as unit.Commutativity of the monoidal product is given by the shuffle permutation χ m,n : m ⊔ n ∼ = −→ n ⊔ m from the symmetric group Σ m+n .The category [Σ, M • ] is isomorphic to the category of symmetric sequences of pointed motivic spaces, i.e. the category of non-negatively graded pointed motivic spaces with symmetric groups actions.
The canonical enriched functor S = S Σ takes n to T n (Σ n permutes the n copies of T or, equivalently, the coordinates of The shift desuspension functors to symmetric spectra are given by In turn, the smash product of Σ-spaces is given by The category Σ S such that a Σ-spectrum is a Σ S -space (see Theorem 2.5) has morphism spaces We shall write Sp Σ T (S) to denote the category of symmetric motivic T -spectra.The category of symmetric motivic T 2 -spectra Sp Σ T 2 (S) is defined in a similar fashion.

Example (GL-motivic T -spectra).
Let GL be the (unbased) category whose objects are the non-negative integers Z 0 .Its morphisms motivic spaces GL(m, n) are given by the following group S-schemes: The symmetric monoidal structure on GL is given by addition of integers and standard concatenation GL m × GL n → GL m+n by block matrices.Commutativity of the monoidal product is given by the shuffle permutation matrix χ m,n ∈ GL m+n .The canonical enriched functor S = S GL takes n to T n (GL n acts on Note that there is a canonical M • -functor ι : Σ → GL mapping n to n and mapping permutations to their permutation matrices such that S Σ = S GL • ι. The shift desuspension functors to GL-spectra are given by the induced motivic spaces (we refer the reader to [16] for basic facts on equivariant homotopy theory) In turn, the smash product of GL-spaces is given by The category GL S such that a GL-spectrum is a GL S -space (see Theorem 2.5) has morphism spaces A typical example of a GL-spectrum is the algebraic cobordism T -spectrum MGL (this follows from [25,Section 4]).We shall write Sp GL T (S) to denote the category of GL-motivic T -spectra.3.4.Example (SL-motivic T 2 -spectra).In contrast to general linear groups, special linear groups contain only even permutations as their permutation matrices.We can equally define the "SLcategory" as in Example 3.3 whose objects are all non-negative integers.The problem with such a M • -category of diagrams is that it is not symmetric monoidal (unless characteristic is 2), and hence a problem with defining corresponding ring objects.To fix the problem, we work with even non-negative integers 2Z 0 .We define morphisms motivic spaces SL(2m, 2n) by the following group S-schemes: With these embeddings of symmetric groups into even-dimensional special linear groups the diagram category SL becomes a symmetric monoidal M • -category.The symmetric monoidal structure on SL is given by addition of integers and standard concatenation SL 2m × SL 2n → SL 2m+2n by block matrices.Commutativity of the monoidal product is given by the shuffle permutation matrix χ 2m,2n = i n (χ m,n ) ∈ SL 2m+2n .The canonical enriched functor S = S SL takes 2n to T 2n (SL 2n acts on T 2n = A 2n S /(A 2n S − 0) in a canonical way).It is a commutative ring object of [SL, M • ] because each SL 2n contains Σ n as permutation matrices defined above.An SL-motivic T 2 -spectrum is an SL-spectrum over S .Note that there is a canonical M • -functor ι : Σ → SL mapping n to 2n and σ ∈ Σ n to i n (σ ) such that the symmetric sphere T 2 -spectrum (S 0 , T 2 , T 4 , . ..) equals S SL • ι.If there is no likelihood of confusion we shall also denote the symmetric sphere T 2 -spectrum (S 0 , T 2 , T 4 , . ..) by S Σ whenever we work with T 2 -spectra.Notice that this T 2 -spectrum S Σ is a commutative ring object of [Σ, M • ] and the category of right modules over S Σ is isomorphic to the category of symmetric T 2 -spectra Sp Σ T 2 (S).
The shift desuspension functors to SL-spectra are given by the induced motivic spaces In turn, the smash product of SL-spaces is given by The category SL S such that an SL-spectrum is an SL S -space (see Theorem 2.5) has morphism spaces SL S (2m, 2n) = (SL 2n ) + ∧ SL 2n−2m T 2n−2m .A typical example of an SL-spectrum is the algebraic special linear cobordism T 2 -spectrum MSL in the sense of Panin-Walter [25, Section 4].We shall write Sp SL T 2 (S) to denote the category of SL-motivic T 2 -spectra.

3.5.
Example (Symplectic motivic T 2 -spectra).Following [25,Section 6] we write the standard symplectic form on the trivial vector bundle of rank 2n as The canonical symplectic isometry (O S , ω 2n ) ∼ = (O S , ω 2 ) ⊕n gives rise to a natural action of Σ n .It permutes the n orthogonal direct summands, and hence one gets an embedding i n : Σ n ֒→ Sp 2n , which sends permutations to the same permutation matrices as in Example 3.4.Let Sp have objects 2Z 0 and let morphisms motivic spaces Sp(2m, 2n) be defined by the following group S-schemes: With embeddings of symmetric groups into symplectic groups above the diagram category Sp becomes a symmetric monoidal M • -category.The symmetric monoidal structure on Sp is given by addition of integers and standard concatenation Sp 2m × Sp 2n → Sp 2m+2n by block matrices.Commutativity of the monoidal product is given by the shuffle permutation matrix χ 2m,2n ∈ Sp 2m+2n .The canonical enriched functor S = S Sp takes 2n to T 2n (Sp 2n acts on T 2n = A 2n S /(A 2n S − 0) in a canonical way).It is a commutative ring object of [Sp, M • ] because each Sp 2n contains Σ n as permutation matrices defined above.A symplectic motivic T 2 -spectrum is an Sp-spectrum over S .Note that there is a canonical M • -functor ι : Σ → Sp mapping n to 2n and σ ∈ Σ n to i n (σ ) such that the symmetric sphere T 2 -spectrum S Σ = (S 0 , T 2 , T 4 , . ..) equals S Sp • ι.
The shift desuspension functors to symplectic spectra are given by the induced motivic spaces In turn, the smash product of Sp-spaces is given by The category Sp S such that an Sp-spectrum is an Sp S -space (see Theorem 2.5) has morphism spaces Sp S (2m, 2n) = (Sp 2n ) + ∧ Sp 2n−2m T 2n−2m .A typical example of a symplectic spectrum is the algebraic symplectic cobordism T 2 -spectrum MSp in the sense of Panin-Walter [25, Section 6].We shall write Sp Sp T 2 (S) to denote the category of symplectic motivic T 2 -spectra.
In the next two examples we suppose 1  2 ∈ S and follow the terminology and notation of [8].Denote by q 2m the standard split quadratic form We define O 2m := O(q 2m ) and SO 2m := SO(q 2m ).
3.6.Example (Orthogonal motivic T 2 -spectra).Let O have objects 2Z 0 and let morphisms motivic spaces O(2m, 2n) be defined by the following group S-schemes: The corresponding embeddings of symmetric groups into orthogonal groups are the same with those of Example 3. ] because each O 2n contains Σ n as permutation matrices defined above.An orthogonal motivic T 2 -spectrum is an O-spectrum over S .Note that there is a canonical M • -functor ι : Σ → O mapping n to 2n and σ ∈ Σ n to i n (σ ) such that the symmetric sphere T 2 -spectrum S Σ = (S 0 , T 2 , T 4 , . . . equals S O • ι.
The shift desuspension functors to orthogonal spectra are given by the induced motivic spaces In turn, the smash product of O-spaces is given by The category O S such that an O-spectrum is an O S -space (see Theorem 2.5) has morphism spaces O S (2m, 2n) We shall write Sp O T 2 (S) to denote the category of orthogonal motivic T 2 -spectra.3.7.Example (SO-motivic T 2 -spectra).The definition of this type of motivic T 2 -spectra literally repeats Example 3.6 if we replace O 2n with SO 2n in all relevant places.The shift desuspension functors to SO-spectra are given by the induced motivic spaces The category SO S such that an SO-spectrum is an SO S -space (see Theorem 2.5) has morphism spaces SO S (2m, 2n) We shall write Sp SO T 2 (S) to denote the category of SO-motivic T 2 -spectra.

SEMISTABLE MOTIVIC SPECTRA
One of the tricky concepts in the stable homotopy theory of classical symmetric spectra is that of semistability.The same concept of semistability occurs in the stable homotopy theory of motivic symmetric T -or T 2 -spectra.
Namely, following Röndigs, Spitzweck and Østvaer [26], a motivic symmetric T -spectrum (likewise T 2 -spectrum) E is said to be semistable if the natural map is a stable weak equivalence of underlying (non-symmetric) motivic spectra.In level n it is defined as the composite map of the twist isomorphism, the nth structure map of the spectrum E and the cyclic permutation χ n,1 = (1, 2, . . . ,n + 1).
Similarly to the classical symmetric S 1 -spectra (see, e.g., [28, I.3.16]) a motivic symmetric Tor T 2 -spectrum X is semistable if for every n and every even permutation σ ∈ Σ n the action of σ on X n coincides with the identity in the pointed motivic unstable homotopy category [26, 3.2].
It follows from Examples 3.3-3.7 that every G-spectrum, where G ∈ {GL, SL, Sp}, is a symmetric T -or T 2 -spectrum.It follows from [27, 3.2] that every orthogonal S 1 -spectrum of topological spaces is semistable.The following theorem is a motivic counterpart of that fact.Proof.GL-, SL-or Sp-motivic spectra have the property that the action of the symmetric group Σ n on the motivic spaces of GL-, SL-or Sp-motivic spectra factors through the action of GL n , SL 2n and Sp 2n respectively.Therefore, even permutations are A 1 -homotopic to identity (see [11,Section 2]).
In more detail, this means that if E is a G-spectrum and σ ∈ Σ n is an even permutation, then there is an A 1 -homotopy E n → Hom(A 1 , E n ) between the action of σ and the identity map.It follows that the action of σ on E n coincides with the identity in the pointed motivic unstable homotopy category, and hence E is semistable by [26, 3.2].
As a consequence of the preceding theorem, we get rid of the semistability phenomenon for GL-, SL-or Sp-motivic spectra.Typical examples of such motivic spectra are MGL, MSL and MSp.It will follow from Theorem 6.1 that symmetric motivic spectra are Quillen equivalent to GL-, SL-or Sp-motivic spectra.Therefore we can make symmetric motivic spectra GL-, SL-or Sp-motivic spectra by extending the group action and then compute the latter spectra within GL-, SL-or Sp-motivic spectra for which the phenomenon of semistability is irrelevant.

MODEL STRUCTURES FOR C -SPECTRA
Throughout this section C is a small category of diagrams enriched over M • .Recall that M • is equipped with the flasque motivic model structure in the sense of [19].This model structure is simplicial, monoidal, proper, cellular and weakly finitely generated in the sense of [10].It follows from [23, 3.2.13]that the smash product preserves motivic weak equivalences.Furthermore, M • satisfies the monoid axiom in the sense of [29].In the flasque model structure every sheaf of the form X /U is cofibrant, where U ֒→ X is a monomorphism in Sm/S.In particular, the sheaf T n , n 0, is flasque cofibrant.
gory, and the monoid axiom in the sense of [29] holds.
( Recall that ordinary and symmetric motivic spectra have Quillen equivalent stable model structures (see, e.g., [20, 4.31]).We want to extend the stable model structure further to diagram spectra of Examples 3.3-3.7.To define it, we fix a symmetric monoidal diagram M • -category C together with a faithful strong symmetric monoidal functor of M • -categories ι : Σ → C and a sphere ring spectrum S = S C such that S Σ = S C • ι.We shall always assume that S = (S 0 , K, K ∧2 , . ..) with K = T or K = T 2 .By Theorem 2.5 we identify the corresponding categories of spectra with categories [Σ S , M • ] and [C S , M • ].As above, one has a natural adjunction where L is the enriched left Kan extension and U is the forgetful functor, is a Quillen pair with respect to the stable model structure.
Since ordinary T -or T 2 -spectra are Quillen equivalent to symmetric spectra (see [20, 4.31]), the preceding proposition implies the following 5.6.Corollary.The canonical adjunction where L is the enriched left Kan extension and U is the forgetful functor, is a Quillen pair with respect to the stable model structure.
The main goal of the paper is to show that the adjunction of the previous proposition is a Quillen equivalence for C being GL, SL, Sp, O and SO if we make a further assumption that the base scheme S is the spectrum Spec k of a field k.This is treated in the next section.

THE COMPARISON THEOREM
Throughout this section k is any field.We shall freely operate with various equivalent models for SH(k) like T -/P 1 -spectra or (S 1 , G ∧1 m )-bispectra.It will always be clear which of the models is used.
The natural Quillen equivalences k) between ordinary and symmetric motivic T -or T 2 -spectra are well-known (see, e.g., [20, 4.31]).The purpose of this section is to establish Quillen equivalences between spectra having a further structure given by various families of group schemes.Namely, we are now in a position to formulate the main result of the paper which compares ordinary/symmetric motivic spectra with GL-, SL-, Sp-, O-and SO-motivic spectra respectively (cf.Mandell-May-Schwede-Shipley [21, 0.1]).

Theorem (Comparison). The following natural adjunctions between categories of T -and T 2spectra are all Quillen equivalences with respect to the stable model structure of Definition 5.4:
(1) We postpone its proof but first verify several statements which are of independent interest.Recall that a motivic space A ∈ M • is an A1 -n-connected if the Nisnevich sheaves π A 1 i (A) ∼ = * for i n.For any B ∈ SH(k), denote by π A 1 i,n (B) the sheaf associated to the presheaf B is said to be connected if π A 1 i,n (B) = 0 for i < n.We also set SH(k) ℓ := Σ ℓ S 1 SH(k) 0 and refer to the objects of SH(k) ℓ as (ℓ − 1)-connected.We define the category of (ℓ − 1)connected S 1 -spectra SH S 1 (k) ℓ in a similar fashion.We say that a motivic space A ∈ M • is stably (ℓ − 1)-connected, ℓ 0, if its suspension S 1 -spectrum is in SH S 1 (k) ℓ (i.e.all its negative sheaves of stable homotopy groups are zero below ℓ).Finally, a motivic space A ∈ M • is (ℓ − 1)biconnected, ℓ 0, if its suspension bispectrum (or its P 1 -/T -spectrum) is in SH(k) ℓ .6.2.Remark.In the language of framed motives [14] if A ∈ M • is (ℓ − 1)-biconnected and the base field is (infinite) perfect then the framed motive M f r (A c ) (respectively the motivic space C * Fr(A c ) gp with 'gp' standing for group completion of the sectionwise H-space C * Fr(A c )), where A c is a cofibrant resolution of A in the projective model structure of spaces, is locally (ℓ − 1)connected as an S 1 -spectrum (respectively as a motivic space).
It is well-known that the suspension bispectrum of a space is connected.The following statement is a further extension of this fact. 1  6.3.Proposition.Let n > 0 and let A ∈ M • be an A 1 -(n − 1)-connected or stably (n − 1)connected pointed motivic space.Then A is (n − 1)-biconnected.
Proof.Let A f be a motivically fibrant replacement of A. First observe that the suspension S 1spectrum Σ ∞ S 1 A f is locally (n − 1)-connected.Indeed, the zeroth space of the spectrum is locally (n− 1)-connected by assumption, and hence each mth space A ∧ S m of the spectrum is locally (m + n − 1)-connected.Morel's stable A 1 -connectivity theorem [22] Since A f is locally (n − 1)-connected by assumption, it follows that each Then each weight S 1 -spectrum B(q) is motivically fibrant and locally (n − 1)-connected.
Since B is a levelwise motivically fibrant bispectrum, then its stabilization in the The proof for stably (n−1)-connected motivic spaces is similar to that for A 1 -(n−1)-connected spaces.
The proof of the preceding proposition also implies the following 6.4.Corollary.Under the assumptions of Proposition 6.3 the space A ∧C is (n − 1)-biconnected for any C ∈ M • .
The next result is crucial for proving Theorem 6.1.

Theorem. Given a pointed motivic space C ∈ M • , the following natural maps are all stable motivic equivalences of ordinary motivic T -and T 2 -spectra:
(1) λ n : F n+1 (C ∧ T ) → F n C, where shift desuspension functors are defined by (1) in Example 3.3 for GL-spectra; , where the shift desuspension functors are defined by (3) in Example 3.5 for symplectic spectra; (4) λ n : F 2n+2 (C ∧ T 2 ) → F 2n C, where the shift desuspension functors are defined by (4) in Example 3.6 for orthogonal spectra provided that char k = 2; (5) λ n : F 2n+2 (C ∧ T 2 ) → F 2n C, where the shift desuspension functors are defined by (5)  Proof of Theorem 6.5.(1).This is the case of GL-motivic spectra.By definition (see ( 1)), For q n + 1, λ n (q) is the canonical quotient map Since T n ∧ − reflects stable motivic equivalences of ordinary T -spectra by [20, 3.18], our statement reduces to showing that T n ∧ λ n is a stable motivic equivalence in Sp N T (k).
The map T n ∧ λ n takes the form Since GL q acts on T q , it follows from [16, 1.2] that the latter map is isomorphic to the map λ ′ q : (GL q /GL q−n−1 ) + ∧C ∧ T q → (GL q /GL q−n ) + ∧C ∧ T q .Here GL q /GL q−n−1 , GL q /GL q−n are smooth schemes of Remark 6.6.Set, and The structure of T -spectra on F ′ n+1 (C ∧ T ) and F ′ n (C) are obvious.It is induced by the action of T on the right.
Consider a commutative diagram of ordinary motivic T -spectra where , α, β are induced by the the following injective maps in M • : They send the basepoint of S 0 to + and the unbasepoint to GL q−n−1 and GL q−n respectively.Note that the left vertical arrow is a stable motivic equivalence in Sp N T (k).Observe that α and β are isomorphic to counit andjunction maps To show that T n ∧ λ n is a stable motivic equivalence it is enough to show that α and β are stable motivic equivalences.
The map α fits in a level cofiber sequence of T -spectra where GL n+1 is pointed at the identity matrix and GL n+i /GL i−1 is pointed at GL i−1 .
We claim that F ′′ n+1 (C ∧ T ) is isomorphic to zero in SH(k).This is equivalent to saying that is isomorphic to zero in SH(k) (we use here [20, 3.18]).Every T -spectrum E = (E 0 , E 1 , . ..) has the layer filtration . By the proof of [4, 2.1.3]the "projection onto the first column map" GL n /GL n−1 → A n \ 0 is a motivic equivalence of spaces.It follows from [4, 2.1.4]that A n \ 0 is A 1 -(n − 2)-connected for n 2, and hence so is GL n /GL n−1 .If we consider a fibre sequence of motivic spaces . This is only possible when F ′′ (C ∧ T ) ∼ = 0 in SH(k), and our claim follows.Thus α is a stable equivalence, because its cofiber Using the same arguments, β is a stable equivalence as well, and hence so is λ n as stated.
We shall need the following useful fact.Proof.We prove the statement for GL-motivic T -spectra, because the proof for the other cases is similar.Denote by where cyl refers to the ordinary mapping cylinder map, P is the family of Definition 5.4 corresponding to ordinary T -spectra.Similarly, set where P is the family of Definition 5.4 corresponding to GL-spectra.Then P N (respectively with respect to the stable model structure.Also, the left Kan extension functor of Corollary 5.2 The proof of Theorem 6.5 shows that the commutative square in Sp N T (k) / / F GL n C with vertical maps being the counit maps consists of stable motivic equivalences.Since the cylinder maps are preserved by the forgetful functor , it follows that U ( P GL ) is a family of injective stable motivic equivalences.
Let J be a family of generating trivial flasque cofibrations [19, 3.2(b)] for M • .By [19, 3.10] domains and codomains of the maps in J are finitely presentable.Recall that the set of maps in is a family of generating trivial cofibrations for the pointwise model structure of Proposition 5.1 (see, e.g., the proof of [10, 4.2]).By construction, L(P N J ) = P GL J .We set An augmented family of P GL -horns is the following family of trivial cofibrations: Λ(P GL ) = P GL J ∪ P GL .Observe that domains and codomains of the maps in Λ(P GL ) are finitely presentable.It can be proven similarly to [18, 4.2] that a map f : A → B is a fibration in the stable model structure with fibrant codomain if and only if it has the right lifting property with respect to Λ(P GL ).
By [20, 2.12] a map f : X → Y in Sp N T (k) is a stable motivic equivalence if and only if it induces a weak equivalence f * : Map * (Y,W ) → Map * (X ,W ) of Kan complexes for all stably fibrant injective T -spectra W .It follows that a pushout of an injective stable motivic equivalence is an injective stable motivic equivalence.Since all colimits in Sp GL T (k) are computed in Sp N T (k), it follows that a pushout of a coproduct of maps from Λ(P GL ) computed in Sp GL T (k) is a stable motivic equivalence in Sp N T (k), because every map of Λ(P GL ) is an injective stable motivic equivalence in Sp N T (k).In particular, U sends Λ(P GL )-cell complexes to stable motivic equivalence in Sp N T (k).We now apply the small object argument to the family Λ(P GL ) in order to fit f : X → Y of the proposition into a commutative diagram with X → L P X , Y → L P Y being Λ(P GL )-cell complexes and L P X , L P Y stably fibrant GLspectra (hence stably fibrant ordinary spectra by Corollary 5.6).Notice that L P f is a level motivic equivalence.Our statement now follows.
Proof of Theorem 6.1.We only prove that : U is a Quillen equivalence with respect to the stable model structure, because the other cases are proved in a similar fashion.
The proof of Theorem 6.5 shows that the counit map β n : with γ n the counit map.Since β n , ϕ n ,U L(ϕ n ) are stable motivic equivalences, then so is γ n .It follows from [10, 3.5] that γ := colim n γ n : ) is a stable motivic equivalence, because Sp N T (k) is a weakly finitely generated model category.Let δ : L(E) → RL(E) be a fibrant resolution of L(E) in Sp GL T (k).Then U (δ ) is a stable motivic equivalence in Sp N T (k) by Proposition 6.7.We see that the composition ) is a stable motivic equivalence for any cofibrant E ∈ Sp N T (k).Since U plainly reflects stable equivalences between fibrant GL-spectra, (L,U ) is a Quillen equivalence by [17, 1.3.16].This completes the proof of the theorem.
We discuss an application of Theorem 6.1 in the next section concerning the localization functor C * F r of [15].

ON THE LOCALIZATION FUNCTOR C * F r
Throughout this section k is an (infinite) perfect field.As usual, we assume char k = 2 whenever we deal with orthogonal or special orthogonal motivic spectra.Recall that SH nis (k) is the triangulated category obtained from the local stable homotopy category of sheaves of S 1 -spectra SH nis S 1 (k) by stabilizing SH nis S 1 (k) with respect to the endofunctor G ∧1 m ∧ −.Let T be a triangulated category.Following [1], we define a localization in T as a triangulated endofunctor L : T → T together with a natural transformation η : id → L such that Lη X = η LX for any X in T and η induces an isomorphism LX ∼ = LLX .We refer to L as a localization functor in T .Such a localization functor determines a full subcategory Ker L whose objects are those X such that LX = 0.An object X ∈ T is said to be L-local if η X : X → LX is an isomorphism.
The computation of localization functors and their full subcategories of local objects is enormously hard in practice.In particular, if T = SH nis (k) and S is the full subcategory of SH nis (k) compactly generated by the shifted cones of the arrows pr X : then the Bousfield localization theory in compactly generated triangulated categories says that there exists a localisation functor By definition, the Morel-Voevodsky stable motivic homotopy category SH(k) is the quotient category SH nis (k)/S .
A new approach to the classical stable homotopy theory SH(k) of Morel-Voevodsky [23] was suggested in [15].This approach has nothing to do with any kind of motivic equivalences and is briefly defined as follows.There exists an explicit localization functor that first takes a bispectrum E to its naive projective cofibrant resolution E c and then one sets in each bidegree C * F r(E) i, j := C * Fr(E c i, j ) (we refer the reader to [14] for the definition of C * Fr(X ), X ∈ M • ).We should note that the localization functor C * F r is isomorphic to the big framed motives localization functor M b f r of [14] (see [15] as well).We then define SH new (k) as the category of C * F r-local objects in SH nis (k).By [15, Section 2] SH new (k) is canonically equivalent to Morel-Voevodsky's SH(k).
The localization functor C * F r is also of great utility when dealing with another model for SH(k), constructed in [15].This model recovers all motivic bispectra as certain covariant functors on Fr 0 (k) taking values in A 1 -local framed S 1 -spectra.In particular, this model of SH(k) implies that π A 1 i, j (E)-s have more information than just the naive bigraded sheaves.Namely, they are recovered from certain covariant functors π f r i (E) on Fr 0 (k) taking values in strictly A 1 -invariant framed sheaves.Thus the functors π f r i (E) have one index only corresponding to the S 1 -direction (in this way we get rid of the second index).These are reminiscent of the classical stable homotopy groups of ordinary S 1 -spectra.It is therefore useful to think of the π A 1 i, j (E) as the richer information "π f r i (E)".Theorems 6.1 and 6.5 give rise to an equivalent model for the localization functor C * F r (see below).It involves smooth algebraic varieties of the form G n+k /G n , where G n , n 0, is GL n , SL 2n , Sp 2n , O 2n or SO 2n .Below we shall write G to denote the family In other words, if we consider the P 1 -spectrum then Fr G ,n (X ) equals the 0th space of the spectrum Θ ∞ P 1 (Y ).Notice that G n+k /G n -s incorporated into the definition are all smooth algebraic varieties.In turn, if G is {SL 2k } k 0 , {Sp 2k } k 0 , {O 2k } k 0 or {SO 2k } k 0 and n 0 is even, then Fr G ,n (X ) is defined as above if we take the colimit over even q-s.
Using the terminology of [14], we define the (G , n)-framed motive M G ,n f r (X ) of X as the Segal S 1 -spectrum associated with the (sectionwise) Γ-space m ∈ Γ op → C * Fr G ,n (X ∧ m + ), where C * stands for the Suslin complex.
If we want to specify the choice of groups, we write below C * Fr GL,n (X ), C * Fr SL,2n (X ), C * Fr Sp,2n (X ), C * Fr O,2n (X ), and C * Fr SO,2n (X ) (respectively, we write M GL,n f r (X ), M SL,2n f r (X ), Let ∆ op Fr 0 (k) be the category of simplicial objects in Fr 0 (k).There is an obvious fully faithful functor spc : Fr 0 (k) → • (Sm/k) sending an object X ∈ Fr 0 (k) to the Nisnevich sheaf X + .It induces a fully faithful functor Denote the image of this functor by T .Also, we shall write → T to denote the motivic spaces which are filtered colimits of objects in T coming from filtered diagrams in ∆ op Fr 0 (k) under the functor spc.7.2.Theorem.Suppose X ∈ → T .Under the notation of Definition 7.1 there is a natural stable local equivalence of S 1 -spectra µ : M f r (X ) → M GL,n f r (X ), where n 0. If n is even and G ∈ {SL, Sp, O, SO} then there is also a natural stable local equivalence of S 1 -spectra µ : M f r (X ) → M G ,n f r (X ).
Proof.We shall prove the theorem for the case G = {GL n } n 0 .The proof for the other choices of G is similar.Without loss of generality we may assume for simplicity X = X + , where X ∈ Sm/k.By the proof of Theorem 6.5 there is a natural stable motivic equivalence of T -spectra where sh −n (Σ ∞ T (X ∧ T n )) = ( * , n−1 . . ., * , X ∧ T n , X ∧ T n+1 , . ..) is the (−n)th shift of Σ ∞ T (X ∧ T n ) and Y as in Definition 7.1.Observe that both spectra are Thom spectra with the bounding constant d 1 in the sense of [11].
Since the map C * Θ ∞ P 1 (sh −n (Σ ∞ T (X ∧T n ))) → C * Θ ∞ P 1 (Y ) is a stable motivic equivalence, it follows from [11, 5.2] that the map of spaces ν : C * Fr(X ∧ T ) → C * Fr GL,n (X ∧ T ) is a local equivalence.By [12, A.1] and the proof of [11, 9.9] both spaces are locally connected.It follows from [14, 6.4] that these are the underlying spaces of (locally) very special Γ-spaces, and so the map of S 1 -spectra Consider a commutative diagram Here f refers to the stable local fibrant replacement of S 1 -spectra and the upper arrow is induced by β .It follows from [14, 7.1] that all spectra are motivically fibrant.Then the map ξ * is a level weak equivalence of motivically fibrant spectra.The proof of [14, 4.1(2)] shows that the vertical arrows are level weak equivalences (we also use [11,Section 9]), and hence so is the upper arrow.It follows that the map M f r (X ) → M GL,n f r (X ) is a stable local equivalence, as was to be shown.Thus if E is a bispectrum then the natural map of bispectra C * F r(E) → C * F r G ,n (E) is a level stable local equivalence.The fact that C * F r G ,n is an endofunctor on SH nis (k) is obvious as well as that both functors are isomorphic on SH nis (k).This completes the proof.
a motivic C -space or just a C -space is an enriched functor X : C → M • .The category of motivic C -spaces and M • -natural transformations between them is denoted by [C , M • ].In the language of enriched category theory [C , M • ] is the category of enriched functors from the M • -category C to the M • -category M • .When C is enriched over unbased motivic spaces, we implicitly adjoin a base object * ; in other words, we then understand C (a, b) to mean the union of the unbased motivic space of maps from a to b in C and a disjoint basepoint.2.1.Definition.For an object a ∈ C , define the evaluation functor Ev

2. 4 .
Lemma.Let C be a symmetric monoidal M • -category and R a commutative ring object in [C , M • ].Then the category of R-modules [C , M • ] R has a smash product ∧ R and internal Homfunctor Hom R under which it is a closed symmetric monoidal category with unit R. Let C be a symmetric monoidal M • -category and R a (not necessarily commutative) ring object in [C , M • ].Mandell, May, Schwede and Shipley [21, Section 2] suggested another description of the category of C -spaces over R. Namely, [C , M • ] R can be identified with the category of C Rspaces, where C R (a, b)

Following [ 10 , 2 )
Section 4] [C , M • ] is equipped with the pointwise model structure, where a map f in [C , M • ] is a pointwise motivic weak equivalence (respectively a pointwise fibration) if f (c) is a motivic weak equivalence (respectively fibration) in M • for all c ∈ Ob C .Cofibrations are defined as maps satisfying the left lifting property with respect to all pointwise acyclic fibrations.5.1.Proposition.The following statements are true: (1) [C , M • ] together with pointwise fibrations, pointwise motivic equivalences and cofibrations defined above is a simplicial cellular weakly finitely generated M • -model category.(The pointwise model structure on [

5 . 2 .
).This follows from[10, 4.4].Corollary.Let C be contained in a bigger M • -category of diagrams D. Then the canonical adjunctionL : [C , M • ] ⇄ [D, M • ] :U, where L is the enriched left Kan extension and U is the forgetful functor, is a Quillen pair with respect to the pointwise model structure.5.3.Corollary.The categories of motivic T -and T2 -spectra Sp N T (S), Sp N T 2 (S), Sp Σ T (S), Sp Σ T 2 (S), Sp GL T (S), Sp SL T 2 (S), Sp O T 2 (S) and Sp SO T 2 (S) of Examples 3.1-3.7 are cellular weakly finitely generated proper M • -model categories.Moreover, Sp Σ T (S), Sp Σ T 2 (S), Sp GL T (S), Sp SL T 2 (S), Sp O T 2 (S) and Sp SO T 2 (S) are monoidal M • -model categories, and the monoid axiom holds for them.Proof.This follows from Proposition 5.1 and Theorem 2.5.

5. 4 .
Definition.Following Hovey[18, 8.7], define the stable model structure on [C S , M • ] to be the Bousfield localization with respect to P of the pointwise model model structure on [C S , M • ], where P = {λ n : F n+1 (C ∧ K) → F n C} as C runs through the domains and codomains of the generating cofibrations of M • , and n 0. The weak equivalences of the model category [C S , M • ] will be called stable weak equivalences.Note that if C = Σ then the stable model structure is nothing but the (flasque) stable model structure of symmetric spectra.The preceding definition together with Corollary 5.2 and [18, 2.2] imply the following 5.5.Proposition.The canonical adjunction L

6. 7 .
Proposition.Let G ∈ {GL,SL,Sp,O,SO}.A map f : X → Y of G-spectra in the sense of Examples 3.3-3.7 is a stable equivalence in the sense of Definition 5.4 if and only if it is a stable motivic equivalence of ordinary motivic spectra.
If X → X c is the cofibrant replacement functor in the projective motivic model structure in M • , then X c belongs to → T (see[14,  Section 10]).7.3.Theorem.Under the assumptions of Theorem 7.2 let C * F r G ,n be the functor on bispectra taking an(S 1 , G ∧1 m )-bispectrum E to the bispectrum C * F r G ,n (E) which is defined in each bidegree as C * F r G ,n (E) i, j := C * Fr G ,n (E c i, j ), where E c is a projective cofibrant resolution of E. Then C * F r G ,n is an endofunctor on SH nis (k) and is naturally isomorphic to the localizing functorC * F r : SH nis (k) → SH nis (k) if G = {GL k } k 0 andn is any non-negative integer, or if G ∈ {SL, Sp, O, SO} and n is even nonnegative.In particular, one has a localizing functorC * F r G ,n : SH nis (k) → SH nis (k) such that the category of C * F r G ,n -local objects is SH new (k).Proof.By the Additivity Theorem of[14] C * F r(−,Y ) and C * F r G ,n (−,Y ) are special Γ-spaces for Y a filtered colimit of simplicial schemes from ∆ op Fr 0 (k).Let F be an S 1 -spectrum such that every entry F j of F is a filtered colimit of k-smooth simplicial schemes from ∆ op Fr 0 (k).F has a natural filtration F = colim m L m (F), where L m (F) is the spectrum(F 0 , F 1 , . . ., F m , F m ∧ S 1 , F m ∧ S 2 , . ..).Then C * Fr(F) = C * Fr(colim m L m (F)) = colim m C * Fr(L m (F)), where C * Fr(L m (F)) is the spectrum (C * Fr(F 0 ),C * Fr(F 1 ), . . .,C * Fr(F m ),C * Fr(F m ⊗ S 1 ),C * Fr(F m ⊗ S 2 ), . ..).Similarly, one has C * Fr G ,n (F) = C * Fr G ,n (colim m L m (F)) = colim m C * Fr G ,n (L m (F)), where C * Fr G ,n (L m (F)) is the spectrum (C * Fr G ,n (F 0 ),C * Fr G ,n (F 1 ), . . .,C * Fr G ,n (F m ),C * Fr G ,n (F m ⊗ S 1 ),C * Fr G ,n (F m ⊗ S 2 ), . ..).Observe that sh n C * Fr(L m (F)) = M f r (F m ) and sh n C * Fr G ,n (L m (F)) = M G ,n f r (F m ).By Theorem 7.2 the natural map M f r (F m ) → M G ,n f r (F m) is a stable local equivalence, and hence so is C * Fr(L m (F)) → C * Fr G ,n (L m (F)).Thus the natural map C * Fr(F) → C * Fr G ,n (F) is a stable local equivalence of spectra.
in Example 3.7 for SO-spectra provided that char k = 2. 6.6.Remark.If G is a linear algebraic group over a field k, and H is a closed subgroup, then by G/H we mean the unpointed Nisnevich sheaf associated with the presheaf U → G(U )/H(U ).If G and H are smooth and all H-torsors are Zariski locally trivial, then the sheaf G/H is represented by a scheme (see[3, p. 275]).If there is no likelihood of confusion, we shall denote the scheme by the same symbol G/H.By[3, p. 275] this happens, for example, if H = GL n , SL n or Sp 2n .In turn, if char k = 2 then it is proved similarly to[5, 3.1.9]that the torsors O is a stable motivic equivalence of cofibrant objects.Denote byL GL n (E) := L(L N n (E)).Then L(E) = colim n L GL n (E),because L preserves colimits.By Corollary 5.6 L is a left Quillen functor, and hence L(ϕ n ) : F GL n (E n ) → L GL n (E) is a stable equivalence in Sp GL T (k) by [17, 1.1.12].By Proposition 6.7 U L(ϕ n ) is a stable motivic equivalence in Sp N T (k).Consider a commutative square