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Cycles, Transfers, and Motivic Homology Theories
... Proof. For the proof, we need use bivariant cycle cohomology developed in Chapter 4 in [10] and we also use the notations in op.cit. For these definitions, we refer to [10]. ...
... For the proof, we need use bivariant cycle cohomology developed in Chapter 4 in [10] and we also use the notations in op.cit. For these definitions, we refer to [10]. Via Proposition 5.8, Theorem 8.3 (the homotopy invariance) in [10] and Theorem 3.5, we have: ...
... For these definitions, we refer to [10]. Via Proposition 5.8, Theorem 8.3 (the homotopy invariance) in [10] and Theorem 3.5, we have: ...
In this paper we describe the category of motives for an elliptic curve in the sense of Voevodsky as a derived category of dg modules over a commutative differential graded algebra in the category of representations over some reductive group.
... It is a little more subtle, but using the material in [SV00b] one can make a version Cor k of SmCor k whose objects are the objects of Sch k which contains SmCor k as a full subcategory in the obvious way. 12 One can find expositions of this in [Ivo07], [CD19], [Kel17], and now also the Stacks Project [Sta18]. ...
... We should instead be talking about fibre and cofibre sequences in some kind of stable ∞-categories, for example pretriangulated dg-categories. 12 In the notation of [SV00b] one takes hom Cor k (X, Y ) = c equi (X × Y /X, 0). 13 Strictly, speaking, L cdh L A 1 is only known to be . So in general one should take the localisation L cdh,A 1 with respect to both classes at once, rather than L cdh L A 1 . ...
... We began the talk with a category of motives, and then ended up with a motivic cohomology. We observed in Remark 21 that correspondences as they are defined in [SV00b], the first step to building DM eff k , can't see nilpotents. There is however a category, not of motives, but of stable motivic homotopy types which can see nilpotents, namely the category of motivic spectra MS S of Annala and Iwasa, [AI22], further developed by them with Hoyois, [AHI24]. ...
These are expanded notes from a talk at the RIMS Workshop, Algebraic Number Theory and Related Topics, December 13th, 2023. We discussed Elmanto-Morrow's motivic complex, the procdh sheafification of the classical motivic complex, and their comparison. The procdh topology and the comparison is joint work with Shuji Saito. The comparison was obtained through joint discussion with Morrow, and its proof relies heavily on the main results of [EM23].
... Constructions of a triangulated category of mixed motives over a perfect base-field were given by Hanamura [57][58][59], Voevodsky [127], and myself [83]. All three categories yield Bloch's higher Chow groups as the categorical motivic cohomology, however, Voevodsky's sheaf-theoretic approach has had the most far-reaching consequences and has been widely adopted as the correct solution. ...
... define isomorphic objects in DM eff .k/, in particular, are isomorphic in the derived category of Nisnevich sheaves on Sm k . The details of these constructions and results are carried out in [127] (with a bit of help from [117]). ...
... Working over a base-field k and for m prime to the characteristic, we have the isomorphism of the étale sheafification Z=m.r/ ét with the étale sheaf ˝r m . The étale sheaves Z=m.r/ ét can be considered as objects in a version of Voevodsky's DM constructed using the étale topology rather than the Nisnevich topology, and their categorical cohomology agrees with the usual étale cohomology [127,Chap. 5 For S D Spec ƒ, with ƒ a mixed characteristic .0; p/ dvr, the complex Z=p n .r/ ...
... where is the connecting map of the long exact sequence in motivic cohomology of the pair (C, C T ) and γ is the Gysin map ( [7] Ch. 5, also studied by Déglise [6]), by ...
... In Sect. 2 below, I review the relevant foundations of the derived category of motives, mainly following the book of Mazza, Voevodsky, and Weibel [11]. The texts by Voevodsky [15] and Friedlander, Suslin, Voevodsky [7] are also used. In Sect. ...
... In this section, we will give some definitions of concepts relating to motives that will be used in the proof of Theorem 1. See [7,11,15] for additional background information. ...
Using Voevodsky’s derived category of motives, we prove a reciprocity law in motivic cohomology of a smooth projective morphism of dimension 1 over a smooth scheme over a perfect field.
... The key difference from [3] is that we use Geometric Presentation Lemma of Ojanguren and Panin ([8], 10.1) instead of Gabber's. It allows us to use transfer arguments which were motivated by paper [13] and developed in [8], [9] and [14]. The main advantage of this techniques is that it can be extended to the case we are interesting in -when the sheaf of coefficients F is defined not only over the base field k but over some smooth variety over k. ...
... The following definition is inspired by the notion of a good triple used by Voevodsky in [13]. ...
In the present paper we discuss questions concerning the arithmetic resolution for etale cohomology. Namely, consider a smooth quasi-projective variety X over a field k together with the local scheme U at a point x. Let Y be a smooth proper scheme over U. We prove there is the Gersten-type exact sequence for etale cohomology with coefficients in a locally constant etale sheaf F of Z/nZ-modules on Y which has finite stalks and (n,char(k))=1.
... Given a closed immersion of a divisor D inside a smooth scheme X, we show that the E 2 -terms of our spectral sequence can be described as the cdh-hypercohomology of a subcomplex of the complex of equidimensional cycles of Friedlander-Suslin-Voevodsky on a scheme S X . This scheme is obtained from gluing two copies of X along D. If X is projective, these E 2 -terms are shown to coincide with the motivic cohomology with compact support [12] of the complement of D in X. ...
... Definitions and Notation. We will write k for a perfect field of exponential characteristic p (in some cases we will assume that the field k admits resolution of singularities [12,Definition 3.4]). Let Sch k be the category of separated schemes of finite type over k and Sm k be the full subcategory of Sch k consisting of smooth schemes over k. ...
We construct a motivic spectral sequence for the relative homotopy invariant K-theory of a closed immersion of schemes . The -terms of this spectral sequence are the cdh-hypercohomology of a complex of equi-dimensional cycles. Using this spectral sequence, we obtain a cycle class map from the relative motivic cohomology group of 0-cycles to the relative homotopy invariant K-theory. For a smooth scheme X and a divisor , we construct a canonical homomorphism from the Chow groups with modulus \CH^i(X|D) to the relative motivic cohomology groups appearing in the above spectral sequence. This map is shown to be an isomorphism when X is affine and .
... ) provided by results of [8,19,20]. ...
... By , where c = codim X x. Since E = R(F ) is A 1 -invariant, it is strict A 1 -invariant by [9], and by [19,Theorem 4.37] the Cousin complex ...
We prove that a motivic equivalence of objects of the form \begin{equation*} X/(X-x)\simeq X^\prime/(X^\prime-x^\prime) \end{equation*} in or over a scheme B, where x and are closed points of smooth B-schemes X and , implies an isomorphism of residue fields, i.e. For a given , , , and closed points x and that residue fields are simple extensions of the ones of B, we show an isomorphism of groups and prove that it leads to an equivalence of subcategories. Additionally, using the result on perverse homotopy heart by F.~D\'eglise and N.~Feld and F.~Jin and the strict homotopy invariance theorem for presheaves with transfers over fields by the first author, we prove an equivalence of the Rost cycle modules category and the homotopy heart of over a field k with integral coefficients.
... From the start we use Suslin's equidimensional complexes Z p eq (X ; * ) to define higher Chow groups [13]. The advantage of representing a higher Chow class by a cycle ϒ in U × ∆ n which is equidimensional and dominant cycle over the algebraic simplex ∆ n , is that ϒ induces a transform operation ϒ ∨ from suitable classes of currents in the compactification P n of ∆ n to currents in the variety U ; see (3.2). ...
... Proof of Theorem 2. 13. In order to compare ψ m,n# (Θ m+n ) and Θ m × Θ n , we push them forward using the isomorphism φ m ×φ n ∶ ∆ m ×∆ n ≅ → m × n and compare the resulting currents in m × n = m+n . ...
This paper utilizes the properties of transforms of currents under equidimensional cycles, as introduced in [4], to establish the multiplicative nature of the resulting regulator map, in the derived category. The construction relies on a synthetic presentation of the fundamental triples of currents from [4], which exhibits group-like behavior under an extended Eilenberg-Zilber morphism. A key component of the analysis is a character of the permutation Hopf algebra SSym that takes values in the function field of A ∞ Q. CONTENTS
... That is, DM gm B is the compact part and Ind(DM gm B ) = DM B . If k is a field there is a canonical equivalence DM gm B (Spec(k)) = DM gm (k; )ޑ with (the ∞-categorical enhancement of) Voevodsky's category of geometric motives with rational coefficients [48]. ...
... From the start we use Suslin's equidimensional complexes Z p eq (X ; * ) to define higher Chow groups [18]. The advantage of representing a higher Chow class by a cycle ϒ in U × n which is equidimensional and dominant cycle over the algebraic simplex n , is that ϒ induces a transform operation ϒ ∨ from suitable classes of currents in the compactification P n of n to currents in the variety U ; see (4.2). ...
This paper utilizes the properties of transforms of currents under equidimensional cycles, as introduced in (dos Santos et al. in J Lond Math Soc (2) 106(3): 2511–2561, 2022), to establish the multiplicative nature of the resulting regulator map, in the derived category. The construction relies on a synthetic presentation of the fundamental triples of currents from [6], which exhibits group-like behavior under an extended Eilenberg–Zilber morphism. A key component of the analysis is a character of the permutation Hopf algebra that takes values in the function field of .
... Sometimes, we work in a larger category DM ef f − (F ) of effective motives of Voevodsky over F . The construction of this category can be found e.g. in [FSV00]. ...
In the present article we investigate ordinary and equivariant Rost motives. We provide an equivariant motivic decomposition of the variety X of full flags of a split semisimple algebraic group over a smooth base scheme, study torsion subgroup of the Chow group of twisted forms of X, define some equivariant Rost motives over a field and ordinary Rost motives over a general base scheme, and relate equivariant Rost motives with classifying spaces of some algebraic groups.
... An example of the latter, not covered by the separable extensions of [Bal16a], can be obtained by 'modding out' coefficients in motivic categories, see [VSF00,Chap. 5]. For instance, if K = DM gm (X; Z) F −→ DM gm (X; Z/p) = L then we have im(Spc(F )) = supp(Z/p). ...
We prove a few results about the map Spc(F) induced on tensor-triangular spectra by a tensor-triangulated functor F. First, F is conservative if and only if Spc(F) is surjective on closed points. Second, if F detects tensor-nilpotence of morphisms then Spc(F) is surjective on the whole spectrum. In fact, surjectivity of Spc(F) is equivalent to F detecting the nilpotence of some class of morphisms, namely those morphisms which are nilpotent on their cone.
... Triangulated structure is not enough: The goal is to construct a rigid tensor category of motivic noncommutative spaces which allows basic operations like pull-back, push-forward and finite correspondences (as morphisms). In the classical setting, we have a construction of D M as a triangulated category due to Voevodsky (see e.g., [26]). However, one would like to extract the right category of motives inside it (possibly as an abelian rigid tensor category). ...
In this survey article we discuss a framework of noncommutative geometry with differential graded categories as models for spaces. We outline a construction of the category of noncommutative spaces and also include a discussion on noncommutative motives. We propose a motivic measure with values in a motivic ring. This enables us to introduce certain zeta functions of noncommutative spaces.
... The concept of a nice triple was introduced in [PSV,Defn. 3.1] and is very similar to that of a standard triple introduced by Voevodsky [Voe,Defn. 4.1], and was in fact inspired by the latter notion. Let k be a field, let X be a k-smooth irreducible affine k-variety, and let x 1 , x 2 , . . . ...
In a series of papers [Pan0], [Pan1], [Pan2], [Pan3] we give a detailed and better structured proof of the Grothendieck--Serre's conjecture for semi-local regular rings containing a finite field. The outline of the proof is the same as in [P1],[P2],[P3]. If the semi-local regular ring contains an infinite field, then the conjecture is proved in [FP]. Thus the conjecture is true for regular local rings containing a field. The present paper is the one [Pan1] in that new series. Theorem 1.1 is one of the main result of the paper. It is also one of the key steps in the proof of the Grothendieck--Serre's conjecture for semi-local regular rings containing a field (see [Pan3]). The proof of Theorem 1.1 is completely geometric.
... One of the starting points of this paper was: How can one formulate a Gersten complex for Hochschild homology? There is a general device producing Gersten complexes for A 1 -invariant Zariski sheaves with transfers [Voe00,Thm. 4.37] as well as a coniveau spectral sequence [MVW06,Remark 24.12]. ...
We develop the Hochschild analogue of the coniveau spectral sequence and the Gersten complex. Since Hochschild homology does not have devissage or A^1-invariance, this is a little different from the K-theory story. In fact, the rows of our spectral sequence look a lot like the Cousin complexes in Hartshorne's 1966 "Residues & Duality". Note that these are for coherent cohomology. We prove that they agree by an 'HKR isomorphism with supports'. Using the close ties of Hochschild homology to Lie algebra homology, this gives residue maps in Lie homology, which we show to agree with those \`a la Tate-Beilinson.
... One of the fundamental problems in the study of the theory of motives, as envisioned by Grothendieck, is to construct a universal cohomology theory of varieties and describe it in terms of algebraic cycles. When X is a smooth quasi-projective scheme over a base field k, such a motivic cohomology theory of X is known to exist (see, for example, [60], [39]). Moreover, Bloch [12] showed that this cohomology theory has an explicit description in terms of groups of algebraic cycles, called the higher Chow groups. ...
We show, for a smooth projective variety X over an algebraically closed field k with an effective Cartier divisor D, that the torsion subgroup \CH_0(X|D)\{l\} can be described in terms of a relative {\'e}tale cohomology for any prime . This extends a classical result of Bloch, on the torsion in the ordinary Chow group, to the modulus setting. We prove the Roitman torsion theorem (including p-torsion) for \CH_0(X|D) when D is reduced. We deduce applications to the problem of invariance of the prime-to-p torsion in \CH_0(X|D) under an infinitesimal extension of D.
... The concept of a nice triple was introduced in [PSV,Defn. 3.1] and is very similar to that of a standard triple introduced by Voevodsky [Voe,Defn. 4.1], and was in fact inspired by the latter notion. Let k be a field, let X be a k-smooth irreducible affine k-variety, and let x 1 , x 2 , . . . ...
It is proved that for any cohomology theory A in the sense of [PS] and any essentially k-smooth semi-local X the Cousin complex is exact. As a consequence we prove that for any integer n the Nisnevich sheaf A^n_Nis, associated with the presheaf U |--> A^n(U), is strictly homotopy invariant. Particularly, for any presheaf of S^1-spectra E on the category of k-smooth schemes its Nisnevich sheves of stable A1-homotopy groups are strictly homotopy invariant. The ground field k is arbitrary. We do not use Gabber's presentation lemma. Instead, we use the machinery of nice triples as invented in [PSV] and developed further in [P3]. This recovers a known inaccuracy in Morel's arguments in [M]. The machinery of nice triples is inspired by the Voevodsky machinery of standard triples.
... In his work on motives, Voevodsky introduced homotopy invariant pretheories as contravariant functors on smooth schemes over a field enjoying certain transfer maps [Voe00a, Definition 3.1]. While algebraic K-theory admits transfer maps for relative smooth curves, it is not an example of a pretheory [Voe00a,§3.4]. However, it is the motivating example of a pseudo pretheory in the sense of Friedlander-Suslin [FS02, Section 10]. ...
We prove versions of the Suslin and Gabber rigidity theorems in the setting of equivariant pseudo pretheories of smooth schemes over a field with an action of a finite group. Examples include equivariant algebraic K-theory, presheaves with equivariant transfers, equivariant Suslin homology, and Bredon motivic cohomology.
... Mixed motives and conservativity. We refer to [VSF00] for Voevodsky's theory of mixed motives. Since we work with Q-coefficients, the categories of mixed motives in the Nisnevich andétale topologies are equivalent, with or without transfers; see [Ayo14]. ...
Using recent developments in the theory of mixed motives, we prove that the log Bloch conjecture holds for an open smooth complex surface if the Bloch conjecture holds for its compactification. This verifies the log Bloch conjecture for all -homology planes and for open smooth surfaces which are not of log general type.
... Inspired by an argument originally due to Bloch and discussed in [11,Appendix A], the result of Saito and Sato was revisited and generalized by Esnault, Kerz and Wittenberg in [10]. Under the assumption that the reduced special fiber X is a simple normal crossing divisor in X , it was observed in [10] that it is possible to replace the classical Chow group (see [15]) CH 0 (X) of the special fiber X with the Friedlander-Voevodsky [12] motivic cohomology H 2d (X, Z(d)), where d = dim k (X), and still prove the existence of an isomorphism (1.1) ρ : CH 1 (X ) ⊗ Z Z/mZ → H 2d (X, Z/mZ(d)), ...
We present a relation between the classical Chow group of relative 0-cycles on a regular scheme , projective and flat over an excellent Henselian discrete valuation ring, and the Levine-Weibel Chow group of 0-cycles on the special fiber. We show that these two Chow groups are isomorphic with finite coefficients under extra assumptions. This generalizes a result of Esnault, Kerz and Wittenberg.
... In the case when k =k then for a smooth projective variety X and n ≥ dim(X) by the Suslin rigidity theorem, the morphism Z X (n) → Rρ * Z X (n) ét is a quasi-isomorphism. For this, see [35,Section 6,Theo. 4.2] and [17,Section 2], and for a proof we refer to [28,Lemma 2.2.2]. ...
... Let k be a field of characteristic zero. As shown by Orgogozo in [Org04] following Voevodsky [VSF00], the category M 0 (k) of Artin motives over k, also known as Artin representations with rational coefficients of the absolute Galois group Gal(k/k) of k form a semi-simple abelian category which fits naturally as the heart of a motivic t-structure on a full subcategory DM 0 c (k) of the triangulated category of Voevodsky motives DM gm (k) over a field. Moreover it is not hard to see that that M 0 (k) embeds fully faithfully in the abelian category M(k) of Nori motives, as shown by Huber and Müller-Stachs in [HMS17,Section 9.4] or [ABV15, Section 4]. ...
Over a scheme of finite type over a field of characteristic zero, we prove that Nori an Voevodsky categories of relative Artin motives, that is the full subcategories generated by the motives of \'etale morphisms in relative Nori and Voevodsky motives, are canonically equivalent. As an application, we show that over a normal base of characteristic zero an Artin motive is dualisable if and only if it lies in the thick category spanned by the motives of finite \'etale schemes. We finish with an application to motivic Galois groups and obtain an analogue of the classical exact sequence of \'etale fundamental groups relating a variety over a field and its base change to the algebraic closure.
... which has implications in the geometrical motives applied to a bundle of geometrical stacks in mathematical physics, as has been studied and showed in [8,11,12]. ...
The tensor structures of triangulated categories in derived categories of ètale sheaves with transfers are considered, taking the tensor product of categories X ⊗ Y = X × Y , in the category Cor k (finite correspondences category) being this the product of the underlying schemes on k. Likewise, is constructed from a total tensor product on the category PSL(k), the generalizations on derived categories using pre-sheaves and contravariant/covariant functors on additive categories to define the exactness of infinite sequences and resolution of spectral sequences of modules in triangu-lated derived categories of objects in ∆ n × A 1 , for morphism of A 1-homotopy. Then through a motives algebra which inherits the generalized tensor product is defined a triangulated category whose motivic cohomology is a hypercoho-mology from the category Sm k ,which has implications in the geometrical motives applied to a bundle of geometrical stacks to field theory. Then we can consider the motives in the hypercohomology to the category DQFT. The mean result will be the creation of theorem that incorporates a 2-simplicial decomposition of ∆ 3 × A 1 , in four triangular diagrams of derived categories from Sm k , which come from a derived category of geometrical motives of DQFT. MSC: 14A15 • 13D09 • 14F08 • 14F20 • 14F42
... Using Suslin's generic equidimensionality results [Sus00] and a simple argument with spectral sequences one can also prove the following result. ...
Using determinantal schemes, we construct explicit cycles in the higher Chow complex Z p (B•GL; *) that represent the universal Chern classes in the higher Chow groups CH p (B•GL; 0). As an application, we use these cycles, along with a canonical stable moving lemma for Karoubi-Villamayor K-theory, to give a direct construction of the Chern class homomorphisms cp,r : KVr(R) → CH p (Spec (R) ; r) for regular algebras over a field k.
... Similarly SH S 1 (k) and Spt P 1 (k) have model structures, which we call the A 1 -model structures, and SH S 1 (k), SH(k) are the respective homotopy categories. For details on the category DM ef f (k), we refer the reader to [3,5]. We will be passing from the unstable motivic (pointed) homotopy category over k, H • (k), to the motivic homotopy category of S 1 -spectra over k, SH S 1 (k), via the infinite (simplicial) suspension functor ...
... Proof. This is a particular case of [3,Theorem 7.1]. ...
Given a smooth projective variety X over a field, consider the -vector space of 0-cycles (i.e. formal finite -linear combinations of the closed points of X) as a module over the algebra of finite correspondences. Then the socle of is absolutely simple, essential and consists of all rationally trivial 0-cycles.
Using a quadratic version of the Bott residue theorem, we give a quadratic refinement of the count of twisted cubic curves on hypersurfaces and complete intersections in a projective space.
In this paper, we study the algebraic cobordism spectrum MSL in the motivic stable homotopy category of Voevodsky over an arbitrary perfect field k. Using the motivic Adams spectral sequence, we compute the geometric part of the -completion of MSL (modulo the maximal subgroup that is l-divisble for all primes ). As an application, we study the Krichever's elliptic genus with integral coefficients, restricted to MSL. We determine its image, and identify its kernel as the ideal generated by differences of SL-flops. This was proved by B. Totaro in the complex analytic setting. In the appendix, we prove some convergence properties of the motivic Adams spectral sequence.
We define Deligne-Beilinson cycle maps for Lichtenbaum cohomology and that with compact supports of an arbitrary complex algebraic variety X. When (m,n)=(2,1), the homological part of our cycle map with compact supports gives a generalization of the Abel-Jacobi theorem and its projection to the Betti cohomology yields that of the Lefschetz theorem on (1,1)-cycles for arbitrary complex algebraic varieties. In general degrees (m,n), we show that the Deligne-Beilinson cycle maps are always surjective on torsion and have torsion-free cokernels. If the version with compact supports induces an isomorphism on torsion, and so does the one without compact supports if We also characterize the algebraic part of Griffiths's intermediate Jacobians with a universal property.
We prove equivariant versions of the Beilinson-Lichtenbaum conjecture for Bredon motivic cohomology of smooth complex and real varieties with an action of the group of order two. This identifies equivariant motivic and topological invariants in a large range of degrees.
Let R be a regular local ring, containing a finite field. Let G be a reductive group scheme over R. We prove that a principal G-bundle over R is trivial, if it is trivial over the fraction field of R. If the regular local ring R contains an infinite field this result is proved in [FP]. Thus the conjecture is true for regular local rings containing a field.
Let k be an algebraically closed field and X a smooth projective k-variety. A famous theorem of A. A. Roitman states that the canonical map from the degree zero part of the Chow group of zero cycles on X to the group of k-points of its Albanese variety induces an isomorphism on torsion prime to the characteristic of k. In the present paper we prove a generalisation to quasi-projective varieties admitting a smooth compactification. As was first observed by Ramachandran, for such a generalisation one should replace the Chow group of zero cycles by Suslin's 0-th algebraic singular homology group and the Albanese variety by the generalised Albanese of Serre. The method of proof is new even in the projective case and makes the motivic nature of the Albanese transparent. We also prove that the generalised Albanese map is an isomorphism if k is the algebraic closure of a finite field.
We express the Mahler measure of an exact polynomial in arbitrarily many variables in terms of Deligne-Beilinson cohomology using the Goncharov polylogarithmic complexes. Applying this to the case of four-variable exact polynomials, we study the relationship between the Mahler measure and the special value of L-functions of K3 surfaces at . This result is an extension of the three-variable case in \cite{Tri23}. In particular, we study the motivic cohomology and Deligne-Beilinson cohomology of singular K3 surfaces. Finally, we prove, under Beilinson's conjecture, that the Mahler measure of the four-variable exact polynomial is expressed in terms of the Riemann zeta function and the L-function of the modular form of weight 3, level 7.
For each configuration of rational points on the affine line, we define an operation on the group of unstable A1 motivic homotopy classes of endomorphisms of the projective line. We also derive an algebraic formula for the image of such an operation under Cazanave and Morel's unstable degree map, which is valued in an extension of the Grothendieck--Witt group. In contrast to the topological setting, these operations depend on the choice of configuration of points via a discriminant. We prove this by first showing a local-to-global formula for the global unstable degree as a modified sum of local terms. We then use an anabelian argument to generalize from the case of local degrees of a global rational function to the case of an arbitrary collection of endomorphisms of the projective line.
We perform Hochschild homology calculations in the algebro-geometric setting of motives over algebraically closed fields. The homotopy ring of motivic Hochschild homology contains torsion classes that arise from the mod- p p motivic Steenrod algebra and generating functions defined on the natural numbers with finite non-empty support. Under Betti realization, we recover Bökstedt’s calculation of the topological Hochschild homology of finite prime fields.
We compare two cycle class maps on torsion cycles and show that they agree up to a minus sign. The first one goes back to Bloch [6], with recent generalizations to non-closed fields. The second is the étale motivic cycle class map restricted to torsion cycles.
For a large class of cohomology theories, we prove that refined unramified cohomology is canonically isomorphic to the hypercohomology of a natural truncated complex of Zariski sheaves. This generalizes a classical result of Bloch and Ogus and solves a conjecture of Kok and Zhou.
In this preprint our results obtained in arXiv.2304.09465v1 (Main Results) are extended to arbitrary semi-local regular base (with no restrictions on its residue fields).
By using the triangulated category of \'etale motives over a field k, for a smooth projective variety X over k, we define the group as an \'etale analogue of 0-cycles. We study the properties of , giving a description about the birational invariance of such group. We define and present the \'etale degree map by using Gysin morphisms in \'etale motivic cohomology and the \'etale index as an analogue to the classical case. We give examples of smooth projective varieties over a field k without zero cycles of degree one but with \'etale zero cycles of degree one, however, this property is not always true as we present examples where the \'etale degree map is not surjective.
We prove a relative version of a theorem on torsors on the projective line due to Philippe Gille. As a consequence we obtain a ``weak homotopy invariance'' result for torsors under reductive group schemes defined over arbitrary semi-local regular domains. Specifically, only regular semi-local domains with infinite residue fields are regarded in this preprint. However, all results of the present preprint are true (after minor modifications) for arbitrary semi-local regular domains. This will be the topic of our next preprint.
In this paper we show that Bloch's higher cycle class map with finite coefficients of a quasi-projective variety over an algebraically closed field fits naturally in a long exact sequence involving Schreieder's refined unramified homology. We also show that the refined unramified homology satisfies the localization sequence. Using this we conjecture in the end that refined unramified homology is a sheaf cohomology and explain how this is related to known and aforementioned results.
We introduce a notion of a weak elementary fibration and prove that it does exist in certain interesting cases. Our notion is a modification of the M. Artin's notion of an elementary fibration.
The Grothendieck--Serre conjecture predicts that every generically trivial torsor under a reductive group scheme G over a regular local ring R is trivial. The mixed characteristic case of the conjecture is widely open. We consider the following setup. Let A be a mixed characteristic DVR, G a reductive group scheme over A, X an irreducible smooth projective A-scheme, a principal G-bundle over X. Suppose is generically trivial. We prove that in this case is Zariski locally trivial. This result confirms the conjecture.
This paper introduces the trivial fiber topology on schemes. For one-dimensional base schemes, we use it to describe fibrant replacements in the stable motivic homotopy category and motivic infinite loop spaces. We also extend the Garkusha-Panin and Voevodsky strict A 1-invariance theorems to one-dimensional base schemes. The trivial fiber topology plays a central role in the proof of refined localization results for motivic homotopy categories. Moreover, we extend Morel's A 1-connectivity theorem on Nisnevich sheaves of stable motivic homotopy groups. These results open new vistas for computations of motivic invariants over deeper base schemes of arithmetic interest.
We develop the intersection theory of non-archimedean analytic spaces and prove the projection formula and the GAGA principle. As an application, we naturally define the category of finite correspondences of analytic spaces.
