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Bloch's Cycle Complex of Zero Cycles with $\Mathbb Z/P^ N $ Coefficients

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Bloch's Cycle Complex of Zero Cycles with $\Mathbb Z/P^ N $ Coefficients BLOCH’S CYCLE COMPLEX OF ZERO CYCLES WITH Z/pn COEFFICIENTS FROM GROTHENDIECK’S COHERENT DUALITY POINT OF VIEW FEI REN Abstract. Let X be a separated scheme of finite type over k with k being a perfect field of positive characteristic p. In this work we define a complex Kn,X,log via Grothendieck’s duality theory of coherent sheaves following [Kat87] and build up a quasi-isomorphism from the Kato-Moser complex of logarithmic de Rham-Witt sheaves νen,X to Kn,X,log for the ´etale topology, and also for the Zariski topology under the extra assumption k = k. Combined with Zhong’s quasi-isomorphism from Bloch’s cycle complex c e ZX to νn,X [Zho14, 2.16], we deduce certain vanishing, ´etale descent properties as well as invariance under rational resolutions for higher Chow groups of 0-cycles with Z/pn-coefficients. Contents Introduction 2 Part 1. The complexes 5 1. Kato’s complex Kn,X,log,t 5 1.1. Definition of Kn,X,log 5 d 1.2. Comparison of WnΩX,log with Kn,X,log 6 1.2.1. Compatibility of C′ with the classical Cartier operator C 7 1.2.2. Proof of Theorem 1.7: C for the top Witt differentials on the affine space 9 1.2.3. Proof of Theorem 1.7: C′ for the top Witt differentials on the affine space 11 1.2.3.1. Trace map of the canonical lift FX of absolute Frobenius FX 11 1.2.3.2. C′ for top Witt differentials e 14 1.2.4. Criterion for surjectivity of C′ − 1 e 18 d 1.2.5. Comparison between WnΩX,log and Kn,X,log 20 1.3. Localization triangle associated to Kn,X,log 20 1.3.1. Definition of TrWnf,log 20 1.3.2. TrWnf,log in the case of a nilpotent immersion 21 1.3.3. Localization triangles associated to Kn,X,log 22 1.4. Functoriality 24 1.5. Etale´ counterpart Kn,X,log,´et 25 c 2. Bloch’s cycle complex ZX,t 28 M 3. Kato’s complex of Milnor K-theory CX,t 28 4. Kato-Moser’s complex of logarithmic de Rham-Witt sheaves νn,X,t 30 arXiv:2104.09662v1 [math.AG] 19 Apr 2021 Part 2. The maps e 30 M 5. Construction of the chain map ζn,X,log,t : CX,t −→ Kn,X,log,t 30 M 5.1. Construction of the chain map ζn,X,t : CX,t −→ Kn,X,t 30 M 5.2. Functoriality of ζn,X,t : CX,t → Kn,X,t 36 5.3. Extend to Kn,X,log,t 37 M n 5.4. ζn,X,log,t : CX,t/p ≃ νn,X,t −→ Kn,X,log,t is a quasi-isomorphism 38 c M M 6. Combine ψX,t : ZX,t → CX,t with ζn,X,log,t : CX,t −→ Kn,X,log,t 40 c M 6.1. The map ψX,t : ZX,t →e CX,t 40 c n ≃ 6.2. ζn,X,log,t ◦ ψX,t : ZX,t/p −→ Kn,X,log,t is a quasi-isomorphism 40 Part 3. Applications 41 7. De Rham-Witt analysis of νn,X,t and Kn,X,log,t 41 8. Higher Chow groups of zero cycles 45 8.1. Vanishing and finitenesse results 45 1 2 FEI REN 8.2. Etale´ descent 46 8.3. Birational geometry and rational singularities 47 8.4. Galois descent 48 Appendix 49 A. Semilinear algebra 49 References 51 Introduction Let X be a separated scheme of finite type over k of dimension d with k being a perfect field of c n positive characteristic p. In this work, we show that Bloch’s cycle complex ZX of zero cycles mod p is quasi-isomorphic to the Cartier operator fixed part of a certain dualizing complex from coherent duality theory. From this we obtain new vanishing results for the higher Chow groups of zero cycles with mod pn coefficients for singular varieties. c As the first candidate for a motivic complex, Bloch introduced his cycle complex ZX in [Blo86] under i the framework of Beilinson-Lichtenbaum. Let i be an integer, and ∆ = Spec k[T0,...,Ti]/( Tj − c 1). Here Z := z0(−, −•) is a complex of sheaves in the Zariski or the ’etale topology. The global X P sections of its degree (−i)-term z0(X,i) is the free abelian group generated by dimension i-cycles in X × ∆i intersecting all faces properly and the differentials are the alternating sums of the cycle-theoretic intersection of the cycle with each face (cf. Section 2). In this article we define a complex Kn,X,log via Grothendieck’s duality theory of coherent sheaves following the idea in [Kat87] and build up a quasi-isomorphism from the Kato-Moser complex of logarithmic de Rham-Witt sheaves νn,X (namely the Gersten complex of logarithmic de Rham-Witt sheaves, which is introduced and studied in [Kat86a, §1][Mos99, (1.3)-(1.5)]) to Kn,X,log for the ´etale topology and also for the Zariski topologye under the c extra assumption k = k. Combined with Zhong’s quasi-isomorphism from Bloch’s cycle complex ZX to νn,X [Zho14, 2.16], we deduce certain vanishing and finiteness properties as well as invariance under rational resolutions for higher Chow groups of 0-cycles with Z/pn-coefficients. The proofs in this article aree self-contained in respect to Kato’s work [Kat87]. Let us briefly recall Kato’s work in [Kat87] and introduce our main object of studies: Kn,X,log. Let π : X → Spec k be the structure morphism of X. Let WnX := (|X|, WnOX ), where |X| is the underlying topological space of X, and WnOX is the sheaf of length n truncated Witt vectors. Let Wnπ : WnX → Spec Wnk be the morphism induced from π via functoriality. According to Grothendieck’s duality theory, ! there exists an explicit Zariski complex Kn,X of quasi-coherent sheaves representing (Wnπ) Wnk (such a complex Kn,X is called a residual complex, cf. [Har66, VI.3.1]). There is a natural Cartier operator ′ d d C : Kn,X → Kn,X , which is compatible with the classical Cartier operator C : WnΩX → WnΩX in d the smooth case via Ekedahl’s quasi-isomorphism (see Theorem 1.7). Here WnΩX denotes the degree d := dim X part of the de Rham-Witt complex. We define the complex Kn,X,log to be the mapping cone of C′−1. What Kato did in [Kat87] is the FRP counterpart, where FRP is the ”flat and relatively perfect” topology (this is a topology with ´etale coverings and with the underlying category lying in between the small and the big ´etale site). Kato then showed that Kn,X,log in the topology FRP acts as a dualizing complex in a rather big triangulated subcategory of the derived category of Z/pn-sheaves, containing all coherent sheaves and sheaves like logarithmic de Rham-Witt sheaves [Kat87, 0.1]. Kato also showed that in the smooth setting, Kn,X,log is concentrated in one degree and this only nonzero cohomology sheaf is the top degree logarithmic de Rham-Witt sheaf [Kat87, 3.4]. For the latter, an analogy on the small ´etale site naturally holds. R¨ulling later observed that with a trick from p−1-linear algebra, [Kat87, 3.4] can be done on the Zariski site as well, as long as one assumes k = k (cf. Proposition 1.15). Comparing this with the Kato-Moser complex νn,X , which is precisely the Gersten resolution of the logarithmic de Rham-Witt sheaf in the smooth setting, one gets an identification in the smooth setting νn,X ≃ Kn,X,log on the Zariski topology. Similar ase in [Kat87, 4.2] (cf. Proposition 1.21), R¨ulling also built up the localization sequence for Kn,X,log on the Zariski site in his unpublished notes (cf. Propositione 1.22). c Compared with the localization sequence for ZX [Blo94, 1.1] and for νn,X (which trivially holds in the Zariski topology), it is reasonable to expect a chain map relating these objects in general. ≃ The aim of this article is to build a quasi-isomorphism ζlog : νn,X −→e Kn,X,log in the singular setting, c such that when pre-composed with Zhong’s quasi-isomorphism ψ : ZX → νn,X [Zho14, 2.16], it gives e e CYCLE COMPLEX VIA COHERENT DUALITY 3 another perspective of Bloch’s cycle complex with Z/pn-coefficients in terms of Grothendieck’s coherent duality theory. More precisely, we prove the following result. Theorem 0.1 (Theorem 5.10, Theorem 6.1). Let X be a separated scheme of finite type over k with k being a perfect field of positive characteristic p. Then there exists a chain map ≃ ζlog,´et : νn,X,´et −→ Kn,X,log,´et, which is a quasi-isomorphism. If moreover k = k, the chain map e ≃ ζlog,Zar : νn,X,Zar −→ Kn,X,log,Zar is also a quasi-isomorphism. Composed with Zhong’s quasi-isomorphisme ψ, we have the following composition of chain maps c n ≃ ζlog,´et ◦ ψ´et : ZX,´et/p −→ Kn,X,log,´et which is a quasi-isomorphism. If moreover k = k, the composition c n ≃ ζlog,Zar ◦ ψZar : ZX,Zar/p −→ Kn,X,log,Zar is also a quasi-isomorphism. We explain more on the motivation behind the definition of Kn,X,log. In the smooth setting, the d logarithmic de Rham-Witt sheaves can be defined in two ways: either as the subsheaves of WnΩX generated by log forms, or as the invariant part under the Cartier operator C.
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