Furthermore, they accept the validity of the Artin-Verdier duality prediction, i.e. the existence of a perfect pairing. This will later explain the appearance of the Bloch-Kato conductorApXq in the special value formulas.
The Artin-Verdier ètale topos and compact support cohomology
One formulation of Artin-Verdier duality uses compact support cohomology which we will briefly review here. For any abelian sheaf on Uét Milne's cohomology with compact support, the cohomology of the complex RΓpcpUét,Fq is given via the respective triangle.
The Duality Statement
HpcipXét,Zpnq{mq ˆH2d`1´ipXét,Zpd´nq{mq ÝÑ Hpc2d`1pXét,Zpdq{mq Ñ Q{Z is a perfect pairing of finite abelian groups for all integers, i.e. Zp1qX{prbLZp1qX{pr ÝÑ Zp2qX{pr (1.3), so that for all prime powers pr the induced pairing of cohomology groups.
Lichtenbaum’s product map and Spiess’ complex
Moreover, for every prime power pr the complex Zp2qX{pr is cohomologically concentrated in degrees 0,1,2. Since the left side is cohomologically concentrated at degree 3, it shows that HipZp2qX{pq “0 for alliă0.
A further construction of the p-torsion product map for arithmetic surfaces a
The Gersten complex of logarithmic deRham-Witt slices of a variety X (of which the right-hand side of (1.11) is an example) is known to be concentrated to one degree only if X is normal crossing. So it is sufficient to find a rational function f PkpNq "kpZq which has a simple zero at exactly one Pi0 and is non-zero at all remaining Pi.
Saito’s duality result and duality on the closed part
SiOη is a DVR and the canonical map Brkpηq ÑBrKη injects, we get Zr´1s ÝÑ RΓηpX,Gmq ÝÑ BrKη. Note that SE can be viewed as the collection of all finite v sites of the residual field kpηq.
Duality on the open part
We will use the structure map f :U Ñ U to reduce the above statement to the usual Artin-Verdier duality for number fields. For (V) we used the Verdier duality addition Rf˚ $ Rf. S) follows from smooth base change applied to :U ÑU.
Proof of the global duality statement
Consider the long exact order on compact support cohomology of the triangle (1.25) as well as the long exact order of (1.26). The Hasse-Weil ζ function, on the other hand, is an object associated with the generic fiber X "XF and is therefore independent of the integral model X of X.
Motivic decompositions of push-forward sheaves in the presence of a section . 25
A motivic decomposition for push-forwards of constant torsion sheaves 27
ÝÑ2d R2dπ˚ΛXpnq ÝÑTrπ ΛSpn´dqr´2ds, (2.12) where Trπ is the trace map of Poincaré duality for etalé cohomology and ϕ12d is obtained by applying Rπ˚to the adjointness of a cohomology Rπ˚to the adjointness of a cohomology Rπ˚ isr´d´2. » ΛXpnq. Finally, for n“0 the Remark shows 2.9Rπ˚Z»Z‘τě1Rπ˚Z. Furthermore, the long exact order for the derivative functor Rπ˚ associated with it.
Motivic decompositions for complex manifolds
The motivic picture and notation
This should provide something analogous to the perverse t-structure on the derived category of l-adic sheaves on varieties over finite fields. Let APDpXq be a complex in the derived category of discs on any fixed topology of X.
Deligne Cohomology
A similar theory is expected to hold for real regular Using (2.22) we reduce to simple cohomology and by considering real and complex localities separately we obtain the ranking (A.12).
Étale motivic cohomology
- Completed motivic cohomology
- The case n “ 1
- Compact support cohomology and the perfect pairing conjecture
- Torsion of motivic cohomology
- Weil-étale cohomology
Proposition A.14 shows that the rightmost terms of the above sequences are the same, so we actually have. In this section, we establish an analogous vanishing result for the rotation parts of the motivational cohomology h1 part of X. A direct comparison of the cohomology groups of the two parts (or, alternatively, Remark 2.9) shows that pR1πr1{ps˚ μbnp ‚ r´1sis centered on positive degrees.
Flach and Morin use the pairing perfection (2.25) to construct a morphism αX,n:RHompRΓpX,Qp2´nqq,Qr´6sq ÝÑ RΓpX,Zpnqq.
Betti cohomology and Weil-étale cohomology with compact support
De Rham and derived de Rham cohomology
Therefore – asF is torsion-free – mapF ÑG must be injective and furthermore LX{S »H0pLX{Sq. 2.37). The last quasi-isomorphism follows by direct inspection or alternatively from the short exact sequence (2.1) in [3]. We now assume that all special fibers of X are reduced instead of having an embedding of X in P2Z. The subscheme Z ØX of non-smooth points is then 0-dimensional.
Let 1tpnqddRpXq also be defined by Eq. tpnqddRpXq:“tpnqddRpSq ¨1tpnqddRpXq´1¨tpn´1qddR pSq. i) The symbol pR1ΓddRpX{Zq{Fn) itself is undefined.
Completions of L- and ζ-functions
We will explain these conjectures using the decompositions into motivic degrees of the various cohomology groups elaborated in the last chapter. In particular, for n “ 1 the above will appear to be equivalent to the Birch and Swinnerton-Dyer conjecture. In particular, VO(X,1) is equivalent to the vanishing-order part of the Birch and Swinnerton-Dyer conjecture.
For the second part, note that ords“nLpH1pXq, sq “0 for ně2 since the infinite product expression for LpH1pXq, sq converges for Repsq ±3{2.
Integral Structures
C2´n:“ tcvj2´n|v|8,1ďjďgu is a basis for the integral structure of HWi,npXqR determined byHWi,npXq, since the duality (2.34) is a duality of integral lattices. Similarly, let BdRp2q be a basis for the image of H0pS, ωFq´bÝÑR1HdR2 pX8{Rq. Most of the work towards the above identity is hidden in the definitions of B1 " p2πiqB´YB" and BdR "B10dRYB01dR as self-dual bases.
Specializing to i“2 gives the commutative square. 2.23) simplifies for i “ 2 to the restriction of the Poincaré duality pairing of algebraic topology.
The Regulator RpX q
Since Poincaré's duality also holds integrally, h2,nD does not contribute to the determinant of the upper right-hand decomposition h2,nD ˝ρ2:H2,npXqRÑHD1,2´npXq˚ of the square. We conclude that the set of classesDp :“ trCjps P pΛpqRu1ďjădppq is the basis of the image Λp inside pΛpqR. Raynaud's; Bosch, Liu). This should serve as an intuition as to why #KerImαβ does not depend on the specific fiber Xp other than the value of njppq.
Due to the splitting provided by Theorem 3.8(iii), the above sequence yields an orthogonal decomposition.
The Fundamental Line
The Trivialization Factor for n “ 1
The diagram below shows the relevant parts of the long exact sequences induced by the h1 part of (3.27). It also shows nulls, such as asp1TXi,nqRif, if they carry information about the integral structures involved and therefore give rise to the numerical value of 1Λ8pX,nq. The quotient of the alternating product of torsion cardinalities associated with the two sequences passing through HW,c‚,npXq is equal.
The integral networks of 1HW,c2,1pXqR and 1HW,c3,1pXqR characterized by vertical maps are generated by PYB`;1 and P˚, respectively.
Trivialization Factors for n ě 2
Trivialization Factors for n ď 0
The Correction Factor
Thus, each term in (3.30) decomposes and we can define iΛppX, nq as the trivialization factor of the hi component of (3.30). We retain the notation of the proof and provide a more geometric version of it for X with good reduction. The motific degree-1 part of the H1 groups of the long exact sequence belonging to Dpp1q is given by.
The exponential map in (3.39) is the base change of the inverse of the above logarithm in Qp.
The Functional Equation
In particular, CpS, nq “pn´1q!´m also holds for ně1 when assuming TC(S,n) instead of the conjecture CEPpQppnqq from Hodge ngap-adic Theory. Therefore – when we assume TC(X,n) for all integers n – the functional equation FE(X) holds for all integers s“n if and only if for all. The second quasi-isomorphism of (3.45) is due to the Poincare Duality which holds integrally; the third and fourth follow directly from the deterministic formalism.
Therefore, the trivialization factors x28pX, nq and x28pS, nq arise entirely from a comparison of the integral structures of the complexes in (3.44) (and are analogous for S).
Summary of special value results
In this section, we will review the construction of Bloch motivic cyclic complexes Zpnq “ ZpnqX for arithmetic schemes X. We define the Bloch cyclic complex Zpnq “ ZpnqX as a chain complex of abelian sheaves at the etalon site X, derived from the Dold-Kahn correspondence applied to the simplex ZXn p´, ‚q after re-indexing via‚ Ø 2n' ‚. ZpnqX is cohomologically concentrated in degrees ď n`d. If d ą n, it is even concentrated in degrees ď2n. iii) Fornă0 one defines Zpnq:“À. pjp,!pµbnp8qr´1where jp :Xr1{ps ÑX denotes a canonical open embedding.
So we will often call Zpnq Bloch cycle complexes. and Levine [18] use different indexing to obtain a cycle complex – here denoted Zpnq˜ – that relates to Bloch's complex via Zpnq˜ X "Zpd´nqXr2ds. i) (Beilinson-Soulé) ZpnqX is cohomologically concentrated in non-negative degrees. ii).
G R -equivariant cohomology
We fix a real embedding σ and adopt the notations of the remarks preceding the proof. commute sinceFpXqGR “ pπ˚FqpX{GRqfor any GR equivariant gerfF onX. RΓpGR, X, ZpnqXq »RΓpX{GR, Rπ˚ZpnqXq. We analyze the restrictions of Rπ˚Zpnq to the closed and the open part separately. Abandoning the first quarter yields the complex. which is quasi-isomorphic to Zn‘À. Summing up, we get i˚Rπ˚ZpnqX “Γ˚XG.
SpecC“SpecCis just a point, the analogue of (A.1), collapses to the identity of a one-point space.
Comparison between motivic and completed motivic cohomology
So it suffices to show that Z{2SpRq and Z{2SpRqr´1 separate as direct sums of RπR,˚Z{2SpRq. Using the derived function formalism as in the proof of Theorem 2.8, the identity πRsR“id immediately shows that RΓpSpRq,Z{2q divides as the direct summation of RΓpXpRq,Z{2q (see also Remark 2.9). Then the complexes ZpnqS and Zpn´1qSr´2 are divided as direct summaries of Rπ˚ZpnqX in the derived category of sheaves in the site Artin-Verdier étaleX “Xét e X.
We will conclude with a summary of all torsion information for motivic and completed motivic cohomology.
Supplementary material for derived de Rham cohomology
Now fix a scheme X and let ShpXZarq and ShRingspXZarq denote the topoi of sets of sets and rings on the Zariski site XZar of X. Applying the resulting functors to the π´1OS module OX yields the Shπ´1OS-AlgpXZarq-simplex X{S “Pπ´1OSpOXq. SinceΩ‚X,tors are concentrated on the non-smooth ends of X, the conductorApXq depends only on the bad fibers of X.
Write ip:ZpãÑX for the closed embedding of the subscheme of singular pointsZp of the special fiber Xp intoX.
Overview of computed cohomology groups
Rank tables
A general formula for ApXq - involving Swan characters of Galois representations given by the l-adic étal cohomology of X .
Motivic and Weil-étale motivic cohomology tables
Diagram of cohomology groups
Classical algebraic k-theory and connections to arithmetic (Proc. Conf., Seattle, Wash., Battelle Memorial Inst., 1972), pages 349–446.
