A motivic decomposition for push-forwards of constant torsion sheaves 27

2.2 Motivic decompositions of push-forward sheaves in the presence of a section . 25

2.2.2 A motivic decomposition for push-forwards of constant torsion sheaves 27

invertible onS then one gets the constant sheaves Λ^X,p “Λ^X and Λ^S,p “Λ^S back.

From now on, we assumeS to be a connected at most one-dimensional regular scheme such that all closed points have perfect residue fields and such that all remaining points have residue fields of characteristic 0. Although we will need the result below only ford“1we formulate it for generaldas it will make no difference for the proof.

Theorem 2.8. Suppose π :X Ñ S has a section s:S Ñ X. Then the sheaves Λ^S,ppnq and Λ^S,ppn´dqr´2ds split off as direct summands of the complex Rπ_˚Λ^X,ppnq. If d “1 we will write ^pR¹π_˚Λ^X_p pnqr´1s for the remaining summand, i.e. we will have the canonical decomposition

Rπ˚Λ^X,ppnq » Λ^S,ppnq ‘ ^pR¹π˚Λ^X,ppnqr´1s ‘ Λ^S,ppn´1qr´2s. (2.10) Proof. It is enough to prove the claim for the restriction πr1{ps:Xr1{ps ÑSr1{ps to the open part, i.e. to show that Λ^Sr1{pspnq andΛ^Sr1{pspn´dqr´2dssplit off as direct summands of Rπr1{ps_˚Λ^Xr1{pspnq. In fact, an application ofj_p,!¹ will then reproduce the original claim since

j_p,!¹ Rπr1{ps˚Λ^Xr1{pspnq “Rpj_p¹πr1{psq!Λ^Xr1{pspnq “Rπ!jp,!Λ^Xr1{pspnq “Rπ˚Λ^X,ppnq.

Therefore we may assume that pis invertible on S.

We will use the adjunctionsπ^˚ $π_˚ ands^˚ $s_˚$s^! to construct maps Λ^Spnq ÝÑ^ϕ0 Rπ˚Λ^Xpnq ÝÑ^ψ0 Λ^Spnq,

Λ^Spn´dqr´2ds ÝÑ^ϕ2d Rπ˚Λ^Xpnq ÝÑ^ψ2d Λ^Spn´dqr´2ds (2.11) that compose as

˜ ψ₀ ψ_2d

¸

˝

´

ϕ₀ ϕ_2d

¯

“

˜ id 0

0 id

¸ , thereby proving the proposition.

The existence of ϕ0, ψ0 is a formal consequence of functoriality of the push-forwardπ˚s˚“ pπsq_˚ “id and exactness ofπ^˚, s^˚, s_˚. Usings^˚Λ^X “Λ^S andπ^˚Λ^S “Λ^X we define

ϕ₀: Λ^Spnq ÝÑRπ_˚π^˚Λ^Spnq “Rπ_˚Λ^Xpnq, ψ₀:Rπ_˚Λ^Xpnq ÑRπ_˚s_˚s^˚Λ^Xpnq “s^˚Λ^Xpnq “Λ^Spnq.

Consider the shift of the above maps for anpd´nq-twist by 2ddegrees:

Λ^Spd´nqr2ds

ϕ0r2ds

ÝÝÝÝÑ Rπ_˚Λ^Xpd´nqr2ds

ψ0r2ds

ÝÝÝÝÑ Λ^Spd´nqr2ds.

Apply p´q^_“RH

om

Sp´,Qp{Zpq. Corollary 2.7 shows that one obtains Λ^Spn´dqr´2ds

ϕ^_₀r´2ds

ÐÝÝÝÝÝÝÝ Rπ_˚Λ^Xpnq

ψ^_₀r´2ds

ÐÝÝÝÝÝÝÝ Λ^Spn´dqr´2ds.

We let ϕ_2d“ψ^_₀r´2dsandψ_2d“ϕ^_₀r´2ds.

ψ0ϕ0 “id is clear since both maps are adjoints of identity maps. Therefore, also ψ2dϕ2d“ pψ₀ϕ₀q^_r´2ds “ 0. Next, one has ψ_2dϕ₀ “ 0 for degree reasons. More precisely, ϕ₀ is

the inclusion of the lowest degreeΛ^Spnq “R⁰π_˚Λ^Xpnq into Rπ_˚Λ^Xpnq whileψ_2d factors through the top degreeτ^ě2dRπ_˚Λ^Xpnq –R^2dπ_˚Λ^Xpnq. Similarly, we must haveψ0ϕ_2d“0 sinceϕ_2dmapsΛ^Spn´dqr´2dsinto the top degree ofRπ_˚Λ^Xpnqwhileψ₀ is trivial in degrees ą0. This completes the proof.

Remark 2.9. The construction ofϕ₀, ψ₀ did not requireΛ^X to be torsion orpto be invertible on S. So, the above proof more generally shows thatA^S splits off as a direct summand of Rπ˚A^X for the constant sheafA^X associated to any abelian groupA. Since bothX andS are connected one has π_˚A^X “A^S and consequently

Rπ_˚A^X »A^S‘τ^ě1Rπ_˚A^X.

Remark 2.10. If π : X Ñ S is smooth and proper, and S the spectrum of a field of characteristic unequal topthe above decomposition is well-known and the motivic components

pRⁱπ_˚Λ^Xpnqcoincide with the cohomological components Rⁱπ_˚Λ^Xpnq for degreesi“0,2d.

In particular, if d“1, they are identical for all degrees, i.e. one then has Rπ_˚Λ^Xpnq » R⁰π_˚Λ^Xpnq ‘ R¹π_˚Λ^Xpnqr´1s ‘ R²π_˚Λ^Xpnqr´2s.

This also follows from direct computations when observing that (2.11) may be rewritten as Λ^Spn´dqr´2ds ^ϕ

1

ÝÑ2d R^2dπ_˚Λ^Xpnq ÝÑ^Trπ Λ^Spn´dqr´2ds, (2.12) where Tr_π is the trace map from Poincaré duality for etalé cohomology and ϕ¹_2d is obtained from applyingRπ˚to the adjoint of a cohomological purity isomorphismRs^!Λ^Spn´dqr´2ds » Λ^Xpnq. We omit the details.

For a general proper regular arithmetic surface π : X Ñ S the maps in (2.12) are not necessarily isomorphisms. This suggests that the motivic decomposition of Rπ˚Λ^Xpnq arises from the (standard) cohomological degree componentsRⁱπ_˚Λ^Xpnq as follows: Split R²π_˚Λ^Xpnq into a component dual to R⁰π_˚Λ^Xpnq and into another component describing the obstruction ofX from being smooth and then regroup the latter to the motivic degree 1 part. This pattern is familiar from (2.3) and the contained definition ^pLpH¹pXq, sq :“

ΠpX, sqLpH¹pXq, sq showing that thestandard Hasse-Weil function ζHWpX, sq differs from themotivicfunctionζpX, sqonly by additional terms in motivic degree1that are characterized entirely by the bad fibers of X.

2.2.3 A motivic decomposition for Rπ_˚Zpnq

The fundamental insight for all following computations of motivic cohomology will be that Theorem 2.8 has an analogue for Bloch’s cycle complexes. We will need a technical preparation.

For any mapf :X ÑY between arithmetic schemes and any étale neighborhoodV ÑY define

Z_YⁿpV, jq^f :“ tZ PZ_YⁿpV, jq | pfˆid_∆jq^´1pZq intersects all faces properlyu.

For a sections:S ÑX of π and nPZwe introduce the technical assumption

FPB(s,n)

The inclusion of simplicial structures Z_Xⁿ p´,2n´ ‚q^s ãÑZ_Xⁿ p´,2n´ ‚q gives rise to a quasi-isomorphism of associated derived complexes. In other words,

DKpZ_Xⁿ p´,2n´ ‚q^sq »Zpnq^X.

FPB(s,n) ensures the existence of a functorial pull-back morphisms^˚Zpnq^X ÑZpnq^S whose adjointZpnq^X Ñs_˚Zpnq^S is given over an étale neighborhoodU ÑX by the map

ZⁿpU, jq^s ÝÑ Zⁿps^´1U, jq

Z ÞÑ

$

’’

&

’’

%

psˆid_∆jq^´1Z

if for all closed pointsxP∆j:

ZX pX ˆ txuqhas codim0inX ˆ txu, or is a finite union ofverticaldivisors ofX ˆ txu

0 otherwise

(2.13)

Note that for nď1 one has a morphism s^˚Zpnq^X ÑZpnq^S that is functorial in suncondi- tionally since this is well-known for the sheavesQp{Zppnq,Z, andGm.

The analogue of FPB(s,n) for morphisms between smooth varieties over fields are known (cf. [18] property 4 following Thm 1.1).

Theorem 2.11. Suppose π : X Ñ S is of relative dimension d “ 1 and has a section s : S Ñ X. If n ě 2 assume that s satisfies the condition FPB(s,n). Then, for any integern, the complexesZpnq^S andZpn´1q^Sr´2ssplit off as direct summands ofRπ_˚Zpnq^X. When writing ^pR¹π˚Zpnq^Xr´1s for the remaining summand we arrive at the canonical

decomposition

Rπ_˚Zpnq^X » Zpnq^S ‘ ^pR¹π_˚Zpnq^Xr´1s ‘ Zpn´1q^Sr´2s. (2.14) Proof. Letně1. As before, we will prove the theorem by exhibiting maps

Zpnq^S ÝÑ^ϕ0 Rπ˚Zpnq^X ÝÑ^ψ0 Zpnq^S,

Zpn´1q^Sr´2s ÝÑ^ϕ2 Rπ˚Zpnq^X ÝÑ^ψ2 Zpn´1q^Sr´2s that compose as

˜ ψ₀ ψ2

¸

˝

´

ϕ0 ϕ2

¯

“

˜ id 0 0 id

¸ .

Let Φ₀ :π^˚Zpnq^S ÑZpnq^X be the flat pull-back morphism, i.e. the adjoint of the canonical morphism Zpnq^S Ñπ_˚Zpnq^X which, on the level of complexes over an étale neighborhood U ÑS, is given by

ZⁿpU, jq ÝÑZⁿpπ^´1U, jq, Z ÞÑ pπˆid_∆jq^´1Z.

Similarly, letΨ0 :s^˚Zpnq^X Ñ Zpnq^S denote the pull-back morphism (2.13). By virtue of the usual adjunctions, we may now define

ϕ0 :Zpnq^S ÝÑRπ_˚π^˚Zpnq^S

Rπ˚Φ0

ÝÝÝÝÑ Rπ_˚Zpnq^X, ψ₀ :Rπ_˚Zpnq^X ÝÑRπ_˚s_˚s^˚Zpnq^X “s^˚Zpnq^X ÝÑ^Ψ0 Zpnq^S.

One sees directly that ψ0ϕ0 “id, i.e. Zpnq^S splits off as a direct summand ofRπ˚Zpnq^X. Next, we letΦ2:s˚Zpn´1q^Sr´2s ÑZpnq^X to be the morphism which acts on complexes as

Z^n´1ps^´1U, jq ÑZⁿpU, jq, Z ÞÑ psˆid_∆jqpZq.

Cor. 3.2 in [12] shows that Φ2 is well-defined. Note that the adjoint of Φ2 is a quasi- isomorphism Zpn´1q^Sr´2s » Rs^!Zpnq^X showing cohomological purity for Bloch’s cycle complexes (cf. [12] Cor. 7.2(a), Cor. 3.3(a)).

ϕ₂:Zpn´1q^Sr´2s “Rπ_˚s_˚Zpn´1q^Sr´2s Ý^RπÝÝÝ^˚ΦѲ Rπ_˚Zpnq^X.

Finally, letΨ2 :π˚Zpnq^X ÑZpn´1q^Sr´2sbe the proper push-forward map which is given on cycles by

Zⁿpπ^´1U, jq ÑZ^n´1pU, jq, Z ÞÑ

$

&

%

pπˆid_∆jqpZq

if for all closed pointsxP∆_j:

ZXpXˆtxuqhas codimension2inXˆtxu, or

is a finite union ofhorizontaldivisors ofXˆtxu

0 otherwise

(2.15) The conditions on S guarantee that [12] Cor. 3.2 and Cor. 7.2(b) are applicable, showing thatΨ2 extends to a morphism

ψ2:Rπ˚Zpnq^X ÝÑ Zpn´1q^Sr´2s.

Again it is clear that ψ₂ϕ₂ “id. Also, from the explicit descriptions (2.13) and (2.15) we see immediately that the compositions ψ2ϕ0 andψ0ϕ2 must be trivial. Consequently,Zpnq^S and Zpn´1q^Sr´2ssplit off as distinct direct summands of Rπ_˚Zpnq^X.

Let us now considerZp´nq. Recall that Zp´nq^Xr1s “à

p

j_p,!Qp{Zpp´nq.

It suffices to show that

Rπ_˚à

p

j_p,!“à

p

j¹_p,!Rπr1{ps_˚ (2.16) as the claim then follows from applying Theorem 2.8 to j_p,!¹ Qp{Zpp´nq. However, (2.16) is immediate from diagram (2.8) sinceπ˚ “π! and πr1{ps˚ “πr1{ps!.

Finally, for n“0 the Remark 2.9 showsRπ_˚Z»Z‘τ^ě1Rπ_˚Z.Moreover the long exact sequence for the derived functor Rπ˚ associated to

0 ÝÑ Z^X ÝÑ Q^X ÝÑ Q{Z^X ÝÑ 0

provesR¹π_˚Z“0 andτ^ě2Rπ_˚Z“τ^ě1Rπ_˚Q{Zr´1s. Indeed, one hasR^rπ_˚Q“0 for rą0 as can be seen by passing to stalks and recalling that Galois cohomology with rational coefficients vanishes. We may thus write

Rπ˚Z^X “Z^S‘à

p

τ^ě1Rπ˚Qp{Z^X_p r´1s. (2.17) For each pthe Open-Closed-Decomposition (2.8) gives rise to the short exact sequence

0 ÝÑ pQp{Zpq^X,p ÝÑ Qp{Z^X_p ÝÑ ip,˚Qp{Z^Xp ^p ÝÑ 0.

Applying Rπ_˚ yields the distinguished triangle

Rπ_˚pQp{Zpq^X_p ÝÑ Rπ_˚Qp{Z^X_p ÝÑ i¹_p,˚Rπ_p,˚Qp{Z^Xp^p ÝÑ. (2.18) Rπ_p,˚Qp{Z^Xp^p is concentrated in degrees 0,1 and an analysis of the long exact sequence associated to (2.18) shows that applying τ^ě1 preserves exactness. This yields

τ^ě1Rπ˚Qp{Z^X_p »Cone

´

i¹_p,˚R¹πp,˚Qp{Z^Xp ^pr´2s ÝÑ τ^ě1Rπ˚pQp{Zpq^X,p

¯ .

We apply Theorem 2.8 to τ^ě1Rπ_˚pQp{Zpq^X,p and verify on stalks that there are no non- trivial morphisms of sheaves i¹_p,˚R¹πp,˚Qp{Z^Xp ^p Ñ pQp{Zpq^S,pp´1q. Thus, we may rewrite the above as

τ^ě1Rπ_˚Qp{Z^Xp »Cone´

i¹_p,˚R¹π_p,˚Qp{Z^Xp^pr´2s ÝÑ ^pR¹π_˚pQp{Zpq^X,pr´1s¯

‘ pQp{Zpq^S,pp´1qr´2s.

Combining this with (2.17) and making the identification

pR¹π_˚Z^X :“à

p

Cone

´

i¹_p,˚R¹π_p,˚Qp{Z^Xp^pr´2s ÝÑ ^pR¹π_˚pQp{Zpq^X,pr´1s

¯

allows us to write

Rπ_˚Z^X » Z^S ‘ ^pR¹π_˚Z^Xr´1s ‘ Zp´1q^Sr´2s as desired.

Remark 2.12. The proof of Theorem 2.11 uses the regularity assumption as follows. For n ď 0 it is implicit in the use of Theorem 2.8 where in turn it is needed for the version of Verdier Duality given in Corollary 2.7. For ně1 it is implicit in the use ofFPB(s,n) since this conjecture is formulated only for regularX. We believe FPB(s,n) to hold only for regularX.

The most important instance of Theorem 2.11 is for the structure map π : X Ñ S of our arithmetic surfaceX as it will allow us to decompose motivic and Weil-étale motivic cohomology into degree 0,1,2 components. However, it is also applicable to localizations π_Zp :X_Zp ÑS_Zp as well as to structure maps of smooth proper curves over fields.

Note that for an elliptic surfaceπ:EÑS one always has a sections:SÑE. In fact, E_F has a rational point andEpOq “E_FpFqsinceE is proper (cf. [31] Cor. IV.4.4(a)). In the case of a general arithmetic surfaceπ :X ÑS one still has an exact triangle forn“1 without assuming the existence of a section.

Let PX{S and P_X⁰_{S denote the étale sheafifications of the functors U{S ÑPicpX ˆSUq and U{SÑPic⁰pX ˆSUq on S respectively.

Proposition 2.13. One has the distinguished triangle

Zp1q^S ÝÑ Rπ_˚Zp1q^X ÝÑ PX{Sr´2s ÝÑ.

Proof. Clearly, τ^ď1Rπ_˚Zp1q^X “ pτ^ď0Rπ_˚G^Xmqr´1s “ π_˚G^Xmr´1s “ G^Smr´1s giving us the truncation triangle

Zp1q^S ÝÑ Rπ˚Zp1q^X ÝÑ pτ^ě1Rπ˚G^X_mqr´1s ÝÑ. Moreover, it is well-known that

R¹π_˚G^Xm–PX{S. (2.19)

It remains to showRⁱπ_˚G^Xm“0 foriě2. Let xãÑX be a geometric point over p. Write S_p“SpecO_p^ur and Xpxq “X ˆSO_X^sh_,x and letπx be the base change ofπ to S_p. Then

`Rⁱπ_˚G^Xm

˘

x “HⁱpXpxq,Gmq “0 for iě2

by Grothendieck’s result [14] Cor. 3.2 (p.98) applied to the surface Xpxqas it is proper and flat over the spectrum S_p of a regular local ring.

Corollary 2.14. If π :X ÑS has a section then ^pR¹π_˚Zp1q –P_X⁰_{Sr´1sand one has the motivic decomposition

Rπ_˚Zp1q^X » Zp1q^S ‘ PX⁰{Sr´1s ‘ Z^Sr´2s.