Geometrical symmetric powers in the motivic homotopy category

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Abstract

Symmetric powers of quasi-projective schemes can be extended via Kan extensions to geometrical symmetric powers of motivic spaces. In this work, we study geometrical symmetric powers and compare them with various symmetric powers in the unstable and stable A1-homotopy category of schemes over a field.

Introduction

Motivic spaces depend on two coordinates, one simplicial and the other geometric, where the last lies on the category of schemes. This viewpoint suggests the possibility of constructing a kind of symmetric power of motivic spaces coming from the geometric coordinate. In [1], Voevodsky established a motivic version of the well known Dold-Thom theorem, in which he considered this construction of what we shall call geometrical symmetric power.

An admissible category1 is a subcategory of schemes over a base field that is closed under taking quotients of schemes by finite group actions and contains the affine line as an interval. Geometrical symmetric powers are left Kan extensions of symmetric powers of schemes in an admissible category. The notion of λ-structure on a model category, or in its homotopy category, was introduced in [2]. It is a categorification of the λ-structure on the K0-ring of an idempotent-complete Q-linear tensor category, studied first by Heinloth, and since by Biglari, Bondarko, Gulekskiĭ, Mazza-Weibel et al. The nth categorical symmetric power of an object in a symmetric monoidal category is the quotient of the nth fold product of the same object by the symmetric group of n letters, which acts by permuting factors. As endofunctors, categorical symmetric powers preserve A1-weak equivalences of motivic spaces, and their left derived functors provide a λ-structure in cofibre sequences on the motivic homotopy category of schemes [2].

The aim of the present work is to develop a systematic study of geometrical symmetric powers in the unstable and stable homotopy category of an admissible category over a field. Our first result shows that geometrical symmetric powers also provide a λ-structure in the unstable motivic setting (Theorem 15):

  • The left derived geometrical symmetric powers provide a λ-structure in cofibre sequences on the pointed unstable motivic homotopy category of an admissible subcategory of quasi-projective schemes over a field.

Moreover, we construct a canonical morphism of λ-structures from the categorical to the geometrical symmetric powers in this setting. It turns out that this morphism is not an isomorphism, cf. Remark 20. A natural question to ask would be whether there is a similar result in the stable setting, and whether categorical and geometrical symmetric powers coincide in this case. Since this comparison fails in the integral case, we deal with this question in the rational stable setting.

The homotopy symmetric powers on a closed symmetric monoidal simplicial model category are defined in terms of homotopy quotients by symmetric groups. There is a canonical morphism of λ-structures from the homotopy to the categorical symmetric powers which is in fact an isomorphism in the stable A1-homotopy category, see [2]. On the other hand, as the rational stable homotopy category of schemes is pseudo-abelian, there is another symmetric power, denoted by Symprn, for a natural number n, constructed by means of the image of a projector induced by the symmetric group of n letters. Let us denote by EQ the canonical functor from the category of separated schemes of finite type over a field to the rational stable A1-homotopy category. We write LSymTn for the left derived functor of the nth categorical symmetric power defined on this stable category, where T stands for the projective line pointed at the infinity. Our second result is the following (Theorem 25):

  • Let k be a nonformally real field and n a natural number. Then, for every quasi-projective k-scheme X, we have the following isomorphismsLSymTnEQ(X)EQ(SymnX)SymprnEQ(X).

Here, EQ(SymnX) may be regarded as the nth geometrical symmetric power of EQ(X).

The paper is organized as follows. In Section 2, we give a survey of admissible categories, simplicial Nisnevich sheaves and categorical λ-structures. Section 3 is devoted to the study of geometrical symmetric powers and their interplay with A1-localization and Künneth towers. In section 4, we prove the main result in the unstable setting (Theorem 15) and construct a canonical morphism of λ-structures from categorical to geometrical symmetric powers. Section 5 is devoted to the formalism of transfers. Finally, in section 6, we prove our main result in the stable setting (Theorem 25).

Section snippets

Preliminaries

From now on, k stands for a base field. Let Sch/k be the category of separated k-schemes of finite type. We write A1 for the affine line over Spec(k). For two k-schemes X and Y, we denote by X×Y the product X×kY, and by X⨿Y the coproduct of X and Y. The identity morphism of Spec(k) is the terminal object of Sch/k, whereas the empty scheme ∅ is its initial object. We say that a full subcategory C of Sch/k is admissible, if it satisfies the following four axioms: (i) Spec(k) and A1 are objects of

Geometrical symmetric powers

In this section, we briefly introduce the notion of geometrical symmetric powers and show that they admit Künneth towers, see Proposition 14.

As in Section 2, let C be a small and admissible subcategory of the category of quasi-projective schemes over a field k, and S the category of Nisnevich sheaves on C. Here and subsequently, Symn stands for the nth categorical symmetric power endofunctor on C. We recall that C+ is the category whose objects have the form U+=U⨿Spec(k), where UC, and whose

Unstable setting

In this section, we show our main result in the unstable setting (Theorem 15) and construct a canonical morphism of λ-structures from categorical to geometrical symmetric powers (Theorem 18).

For each nN, we have a left derived endofunctor LSymgn on the category H(CNis,A1), see Corollary 7.

Theorem 15

The endofunctors LSymgn, for nN, provide a λ-structure on H(CNis,A1).

Proof

LSymg0 is the constant functor with value the unit object and LSymg1 is the identity functor. Let XYZ be a cofibre sequence in H(CNis,

Norm and transfer

Suppose that (D,) and (E,) are two symmetric monoidal categories, and E is also an additive category. Let E:(D,)(E,) denote a monoidal functor. Let G be a finite group, X a G-object in D and ρX:GAut(X) a representation of G on X. The norm of E(X) isNmE(X)=gGE(ρX(g)). Let π:XX/G be the canonical morphism. A transfer of E(π) is a morphismtrE(π):E(X/G)E(X) such that E(π)trE(π)=nidE(X/G) and trE(π)E(π)=NmE(X).

We now assume that E is a Q-linear category, and both categories D,E are

Stable setting

Let T the projective line P1 over a field k, pointed at the infinity. We denote by SHT(k) the stable A1-homotopy category of schemes over k, as defined in [14]. This category is equivalent to the homotopy category of motivic symmetric T-spectra on the smooth Nisnevich site over k with the stable closed model structure [15, Theorem 4.31], and the latter admits a structure of a symmetric monoidal category induced by the smash product of symmetric T-spectra, see [15, Section 4.3]. We write SHT(k)Q

Conclusions

The present results might be generalized for suitable base schemes. In [7], the author studied geometrical symmetric powers and their λ-structure in both unstable and stable motivic setting over a base field. However, the question whether geometrical symmetric powers do or do not preserve stable A1-weak equivalences of motivic symmetric spectra remains open.

Acknowledgements

The present work was written in the framework of the EPSRC grant EP/I034017/1 “Lambda-structures in stable categories”. The author is grateful to Vladimir Guletskiĭ for inspiring explanations and valuable suggestions during his PhD studies at the University of Liverpool. He would like to thank Emmanuel Dror Farjoun, Paul Arne Østvær and Charles Weibel for helpful suggestions. He is also grateful to the referees for the significant improvements of the paper.

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