De Rham cohomology of smooth proper pairs

such that

OX(−Z)(U) =t₁· · ·t_rOX(U),

for some 0≤r≤ndepending onU. We always assume thatrin the above formula is minimal in the sense that notj,j= 1, . . . , r, can be left out. For simplicity, we callU together with thet₁, . . . , t_n apleasant open affine ofX and apleasant neighbourhood of any pointP ofU.

For the remainder of this chapter, let(X, Z)be a smooth proper pair over W(k)whereX is of relative dimension nandZ nontrivial.

3.1. De Rham cohomology of smooth proper pairs

This section is divided into three subsections. In the first we define differentials with logarithmic poles and the de Rham cohomology of (X, Z). The middle subsection contains Theorem B and its proof. The third part specialises to the case of curves.

Differentials with logarithmic poles. — To define the de Rham cohomology of (X, Z) one introduces the sheaf Ω_(X,Z) of differentials on X with logarithmic poles alongZ: LetU,t1, . . . , tn, be a pleasant open affine. SinceU is ´etale overAⁿ_W(k), the differentialsdt1, . . . , dtn form anOX(U)-basis of ΩX(U). Set

dt˜j df=

(dtj/tj if j ≤r dtj otherwise.

The sections of Ω_(X,Z) overU are defined as Ω(X,Z)(U)^df=

n

M

j=1

OX(U) ˜dtj (3.4)

⊆Ω_X(Z)(U).

This definition glues and gives rise to a subsheaf Ω_(X,Z)of Ω_X(Z). We let

∂˜_j: OX(U)→OX(U) (3.5)

be the map sending an elementf to the coefficient of ˜dt_j in df∈Ω_(X,Z)(U).

As usual, we set Ωⁱ_(X,Z)^df=Vi

Ω_(X,Z). The Ωⁱ_(X,Z) together with the universal deriva-tionddefine a complex Ω^•_(X,Z). Thede Rham cohomlogy of (X, Z) is

H^•_dR(X, Z)^df=H^•(Ω^•_(X,Z)).

Main theorem and proof. — To compute zeta functions one is interested in the Frobenius action on the lattices Hⁱ_dR(X, Z)/(tor) by virtue of the isomorphisms of (3.2). In many cases, the only interesting action is the one on Hⁿ_dR(X, Z)/(tor). For geometrically integral curves, this was already remarked in the introduction. Simi-larly, when X is a hypersurface inPⁿ⁺¹_k , one is usually interested in the Frobenius action on Hⁿ⁺¹_dR (Pⁿ⁺¹_W(k), X) [Ger07, AKR11] or Hⁿ_dR(X, Z) [Wal09] withZa smooth hyperplane section.

To find a description of Hⁿ_dR(X, Z)/(tor) akin to (3.1), let us compare the hypercoho-mology of Ω^•_(X,Z)to the hypercohomology of the complex

Ω^•_(X,Z)(E)^df= Ω^•_(X,Z)⊗OX(E), whereE is an effective divisor onX such that

E≤`Z and Ω^•_(X,Z)(E) is acyclic.

We say that a complex of quasi-coherent sheaves is acyclic if every sheaf in this complex is acyclic. In this case

Hⁿ(Ω^•_(X,Z)(E)) = H⁰(Ωⁿ_(X,Z)(E))

dH⁰(Ωⁿ⁻¹_(X,Z)(E)).

For example, if X is a geometrically integral curve of genus g, then Ω^•_(X,Z)(E) is acyclic if and only ifOX(E) is. By Riemann-Roch the latter happens wheneverE is of degree at least 2g−1. Let

Q^•= Coker

Ω^•_(X,Z),→Ω^•_(X,Z)(E) .

Lemma 3.1. — The complexH^•(Q^•) vanishes upon multiplication byp^{blogp`c</sup>. Proof. — There is a spectral sequence E<sub>2</sub><sup>rs</sup> = R<sup>r</sup>Γ(H<sup>s</sup>(·)) ⇒ H<sup>r+s</sup>(·) [GM03, Para. III.7.14] so it suffices to show that the cohomology sheaves H<sup>i</sup>(Q<sup>•</sup>) vanish upon multiplication byp<sup>blogp`c}. Taking cohomology commutes with cokernels (in fact with any direct limit) and so Hⁱ(Q^•) equals the cokernel of Hⁱ(Ω^•_(X,Z))→Hⁱ(Ω^•_(X,Z)(E)).

Thus the lemma is a theorem by Abott et al. [AKR11, Thm. 2.2.5].

One can now describe lattices that are “close” to Hⁱ_dR(X, Z)/(tor) and “almost”

Frobenius-stable: The short exact sequence

0→Ω^•_(X,Z)→Ω^•_(X,Z)(E)→Q^•→0 induces the long exact sequence

. . .→Hⁱ⁻¹(Q^•)→Hⁱ_dR(X, Z)→Hⁱ(Ω^•_(X,Z)(E))→Hⁱ(Q^•)−→. . . . (3.6) Since p^blogp`cHⁱ(Q^•) vanishes, both kernel and cokernel of the map Hⁱ_dR(X, Z) → Hⁱ(Ω^•_(X,Z)(E)) are torsion. Thus, there is a natural embedding

Hⁱ_dR(X, Z)

(tor),→Hⁱ(Ω^•_(X,Z)(E))

(tor), (3.7)

3.1. DE RHAM COHOMOLOGY OF SMOOTH PROPER PAIRS 35

whose cokernel is killed by p^blogp`c, see also [vdB08, Prop. 3.5.4], [Wal09, Prop. 5.1.11], or [Tui11, Cor. 3.7.6]. By (3.2), the sequence of maps Hⁿ_dR(X, Z)/(tor) ,→Hⁿ(Ω^•_(X,Z)(E))/(tor),→Hⁿ_dR(XK\ZK) is Frobenius-equivariant. Consequently,

p^blogp`cF

Hⁱ(Ω^•_(X,Z)(E)) (tor)

⊆Hⁱ(Ω^•_(X,Z)(E))

(tor). (3.8) Let us now give a description of the Frobenius-stable lattice Hⁿ_dR(X, Z)/(tor) itself.

Since Ωⁿ_(X,Z)(E) andQⁿ are the last nonzero sheaves in their respective complexes, there are maps H⁰(Ωⁿ_(X,Z)(E))→Hⁿ(Ω^•_(X,Z)(E)) and H⁰(Qⁿ)→Hⁿ(Q^•). These sit in the following commutative square in which ϕis defined as the diagonal.

H⁰(Ωⁿ_(X,Z)(E))

//

ϕ '' H⁰(Qⁿ)

Hⁿ(Ω^•_(X,Z)(E)) //Hⁿ(Q^•)

(3.9)

The kernel ofϕis closely related to the de Rham cohomology of (X, Z). To describe this relation, it is useful to consider the stalks of Ωⁿ_(X,Z)(E) at points ofX. In partic-ular, the completion of a stalk along the intersection of the irreducible components of Z passing through the point is of interest. To this end, for each pointP ofX write

i_P ^df= X

p⊇OX(−Z)_P, minimal

p,

where the sum is over the minimal primes of O_X,P that contain OX(−Z)_P. For an OX,P-moduleM, set

Mˆ ^df= lim

←−M/iⁱ

PM.

Lemma 3.2. — Let U,t1, . . . , tn be a pleasant open affine. The minimal primes of OX(U) that contain OX(−Z)(U) are exactly the ideals t_jOZ(U), j = 1, . . . , r. In particular, ifP is a point ofU andsthe number of irreducible components ofZ that pass through P, thens≤r and the tj can be ordered such that

i_P =

s

X

j=1

tjOX,P.

This observation is certainly not original to this thesis, but lacking a reference we give our own proof.

Proof. — If p is a minimal prime of OX(U) containing OX(−Z)(U), then its inverse image q in W(k)[x1, . . . , xn] is a minimal prime containing the ideal x₁· · ·x_rW(k)[x₁, . . . , x_n] [Mil80, Prop. I.3.17]. Hence q equals x_jW(k)[x₁, . . . , x_n] for an appropriate 1≤j≤r. SinceU is unramified overAⁿ_W(k), tj generatesp.

Conversely, by minimality of r (see the definition of a pleasant open affine) each t_jOX(U), j = 1, . . . , r, is a proper ideal. Moreover, it cannot properly contain a minimal prime p over OX(−Z)(U): Otherwise, p would equal some tiOX(U). But thenj must equali sinceZ is reduced. Thus any minimal prime pover tjOX(U) is also minimal overOX(−Z)(U). Thenpequalst_iOX(U) for somei. Again, sinceZ is reduced, we deduce thatj equalsi. HencetjOX(U) is prime.

Similar to the Cohen structure theorem for regular local rings containing a field [Bou06, Prop. VIII.5.6 & Thm. VIII.5.2], there is

Proposition 3.3. — Let P be a point of X. There is a section O_X,P/i_P ,→O_X,P

to the canonical epimorphism OX,P OX,P/i_P. If s is the number of irreducible components ofZ that pass throughP, then there is anO_X,P/i_P-algebra isomorphism

OX,P/i_P

[[T1, . . . , Ts]]−→^∼ OˆX,P

Tj 7→tj,

where thetj are part of thet1, . . . , tn of a pleasant neighbourhood ofP ordered in such a way that t1, . . . , ts generatei_P.

Abott et al. use a similar power series representation for certain rational functions in their proof of the central ingredient to our Lemma 3.1.

Proof. — Write i = i_P. Let us prove the first claim first. By assumption on U, there is a morphism U → Aⁿ_W(k) of finite type (´etale even). The closed immer-sion V ^df= V(t₁, . . . , t_s) ,→ U is induced by the closed immersion A^n−s_W(k) ,→ Aⁿ_W(k) (corresponding toW(k)[x1, . . . , xn]W(k)[xs+1, . . . , xn]) and base change to along U → Aⁿ_W(k). The latter immersion has an obvious retraction and hence, so does V ,→U. In particular, there is a retraction locally at P and this corresponds to a section ofOX,P OX,P/i.

The image of the sectionO_X,P/i,→O_X,P has trivial intersection withi. Thei-adic filtration onO_X,P is exhaustive by definition and separated sinceO_X,P is a Noetherian domain [Bou89a, Prop. III.3.5]. Hence, the proposition is proven if we can show that the images of the t_j, j = 1, . . . , s, in the graded algebra gr ˆOX,P associated to OˆX,P and i form an algebraically free system of generators over OX,P/i [Bou89a, Prop. III.2.11].

To this end, let us adapt a proof of Bourbaki [Bou06, Proof of Thm. VIII.5.1]: The O_X,P/i-modulei/i²is free of ranks[Liu06, Lem. 6.3.6] and so the symmetric algebra

3.1. DE RHAM COHOMOLOGY OF SMOOTH PROPER PAIRS 37

S of i/i² over OX,P/i is the polynomial ring

OX,P/i

[T1, . . . , Ts]. Let grOX,P be the graded algebra associated toOX,P andi. Note that it coincides with gr ˆOX,P. To prove the proposition, let us show that the canonical morphismγ:S →grO_X,P is an isomorphism.

It is certainly epi (so the map in the proposition is epi, too). If it is not iso, then there is a homogeneous polynomial h(T₁, . . . , T_r) of degree dat least 1 that gets mapped to 0 by γ, i.e. h(t1, . . . , ts) ∈ I^d+1. In other words, there is a polynomial relation 0 =h(t1, . . . , ts) +“terms of higher degree in thetj” overOX,P. But thedtj are part of anOX,P-basis of Ω_X,P sinceU is ´etale overAⁿ_W(k), so this cannot happen. Hence γ is iso.

LetP be a point ofX. Then

Ωˆⁿ_(X,Z)(E)_P ∼= Ωⁿ_(X,Z),P ⊗OX(E)_P ⊗OˆX,P.

To describe ˆΩⁿ_(X,Z)(E)_P we still need to determine the stalkOX(E)_P. IfU,t1, . . . , tn

are a pleasant neighbourhood of P as in Lemma 3.2, then OX(E)_P is the ideal t^−v₁ ^P1(E)· · ·t^−vs ^Ps(E)OX,P. Let us show that the numbers vP_j(E) are already deter-mined by data from the special fibre of (X, Z): On the one hand, O_X(E)_P equals t^−v₁ ^P1(E)· · ·t^−v_s ^Ps(E)OX,P. On the other hand, going through the proof of Lemma 3.2 again, this time overkinstead ofW(k), we deduce thatvP_j(E) equalsv_P

j(E), where P_j is the irreducible component ofZ that corresponds tot_j. Hence,

OX(E)_P =t^−v₁ ^P1(E)· · ·t^−vs ^{P s(E)}OX,P. (3.10) Now specialise to the case where P is a generic point of Z. Then the number s of irreducible components ofZ that pass through P equals 1. Hencei_P is generated by t1, andOX,P/i_P andOZ,P =OX,P/t1OX,P coincide. We write

t_{P ,j}^df=t_j and t_P ^df=t_{P ,1}. By the proposition, there is an isomorphism

t^−vP(E)

P O_Z,P[[t_P]] ˜dt_P −→^∼ Ωˆⁿ_(X,Z),P(E), with ˜dt_P = ∧_jdt˜_{P ,j}. For an element ω_P of ˆΩⁿ

(X,Z),P(E), let ω⁽ⁱ⁾

P ∈ O_Z,P be the coefficient oftⁱ

P

dt˜_P. Finally, let

∂˜_{P ,j}:O_X,P →O_X,P be the germ of the map ˜∂j:OX(U)→OX(U) of (3.5).

Theorem B (Generalised van den Bogaart’s Lemma) Let ϕbe as in(3.9). There is a natural isomorphism

Kerϕ

dH⁰(Ωⁿ⁻¹_(X,Z)(E))−→^∼ Hⁿ_dR(X, Z)

(tor). (3.11)

Furthermore with the above notation, Kerϕ=

ω∈H⁰(Ωⁿ_X(Z+E))|For all generic pointsP ofZ and i=−v_P(E), . . . ,−1, there existf⁽ⁱ⁾

P ,1, . . . , f⁽ⁱ⁾

P ,n∈OZ,P

such that ω⁽ⁱ⁾

P =if⁽ⁱ⁾

P ,1+

n

X

j=2

∂˜_{P ,j}f⁽ⁱ⁾

P ,j .

IfX is a geometrically integral curve andE=`Z, then tensoring(3.11)withK yields the left-hand isomorphism in(3.1).

One can also describe Hⁿ_dR(X) by comparing the hypercohomologies of the complexes Ω^•_X and Ω^•_(X,Z)(E). ForX a geometrically integral curve of genusg,Z a closed point of degree 1, andE= (2g−1)Z, this was done by van den Bogaart in his thesis [vdB08, Sec. 3.3]. He defines a map that is exactly the map ϕ from above and shows that there exists a natural isomorphism Kerϕ/dH⁰(OX((2g−1)Z))−→^∼ H¹_dR(X) [vdB08, Lem. 3.3.10]. His proof was the inspiration for this section and gives Theorem B its name.

While in theory it might suffice to describe the cohomology of X to determine the zeta function of X, in actual computations it can prove advantageous to use the cohomology of a pair (X, Z). In the case of hypersurfaces in Pⁿ⁺¹_k , as noted before, one usually considers (Pⁿ⁺¹_W(k), X) or (X, Z) withZ a smooth hyperplane section. For curves there are further examples where it is particularly easy to describe a basis for the cohomology of (X, Z) (Remark 3.8).

Proof. — On the one hand, the hypercohomology of the cokernel complex is torsion (Lemma 3.1) and thus the long exact sequence (3.6) induces 0→Hⁿ_dR(X, Z)/(tor)→ Hⁿ(Ω^•_(X,Z)(E))→Hⁿ(Q^•). On the other hand, Ω^•_(X,Z)(E) is acyclic and so its hy-percohomology is given by the cohomology of the complex of global sections. In par-ticular, Hⁿ(Ω^•_(X,Z)(E)) is the quotient of H⁰(Ωⁿ_(X,Z)(E)) bydH⁰(Ωⁿ⁻¹_(X,Z)(E)). Hence

3.1. DE RHAM COHOMOLOGY OF SMOOTH PROPER PAIRS 39

there is the following diagram whose row and column are exact.

H⁰(Ωⁿ⁻¹_(X,Z)(E))

d

H⁰(Ωⁿ_(X,Z)(E))

ϕ

&&

0 //Hⁿ_dR(X, Z)

(tor) //Hⁿ(Ω^•_(X,Z)(E))

//Hⁿ(Q)

0

This gives (3.11). Since tensoring withK is flat, this isomorphism induces the left-hand isomorphism in (3.1) ifX is a geometrically integral curve andE=`Z. Turning to the kernel of ϕ, fix an open affine covering U = {Ui}i∈I of X. Let us writeC^•(F)=^dfC^•(U,F) for the ˇCech complex of a sheafF after applying the global sections functor, i.e. C^r(F) = ⊕#J=r+1F(UJ) where J ⊆ I and UJ = ∩j∈JUj. The hypercohomology of a complexF^• of quasi-coherent sheaves onX is thus given by the cohomology of the total complexSC^•(F^•) associated to the double complex C^•(F^•) [GM03, Sec. III.7]. Recall that SCⁿ(F^•) = ⊕r+s=nC^r(F^s). With this identification, the mapϕ(thought of as concatenation of the top and the right-hand map in (3.9)) becomes the concatenation of the maps in the following diagram.

Ker

C⁰(Ωⁿ_(X,Z)(E))→C¹(Ωⁿ_(X,Z)(E))

//Ker C⁰(Qⁿ)→C¹(Qⁿ)

 _

Ker SCⁿ(Q^•)→SCⁿ⁺¹(Q^•)

Ker SCⁿ(Q^•)→SCⁿ⁺¹(Q^•)

/Im SCⁿ⁻¹(Q^•)→SCⁿ(Q^•) An element (ωU)U∈Uof Ker

C⁰(Ωⁿ_(X,Z)(E))→C¹(Ωⁿ_(X,Z)(E))

is in the kernel ofϕ if and only ifωU + Ωⁿ_(X,Z)(U) lies in the image ofQⁿ⁻¹(U)→Qⁿ(U) for allU ∈U.

SinceQ^•is supported onZ, it suffices to check this at the stalks of the closed points of Z. In fact, it suffices to check this at the stalks of the generic points ofZ (Lemma 3.4) of which there are only finitely many. LetP be such a point. We set

dt˜^(j)

P =∧k6=jdt˜_{P ,k}.

Then by (3.4) and Proposition 3.3, Q_Pⁿ⁻¹∼= Ωⁿ⁻¹_(X,Z),P(E)/Ωⁿ⁻¹

(X,Z),P

∼= ˆΩⁿ⁻¹

(X,Z),P(E)/Ωˆⁿ⁻¹

(X,Z),P

∼=

n

M

j=1

−1

M

i=−v_P(E)

OZ,Ptⁱ_Pdt˜^(j)

P .

and similarly,

Qⁿ_P ∼=

−1

M

i=−v_P(E)

O_Z,Ptⁱ_Pdt˜_P.

With respect to the bases ˜dt^(j)

P , j= 1, . . . , n, and ˜dt_P, the mapQ_Pⁿ⁻¹→Qⁿ_P is X

i

f⁽ⁱ⁾

P ,1tⁱ_P,X

i

f⁽ⁱ⁾

P ,2tⁱ_P, . . . ,X

i

f⁽ⁱ⁾

P ,ntⁱ_P

! 7→X

i



if⁽ⁱ⁾

P ,1+

n

X

j=2

∂˜_{P ,j}f⁽ⁱ⁾

P ,j



tⁱ_P.

The calculation is straightforward: d(f⁽ⁱ⁾

P ,jtⁱ

P

dt˜^(j)

P ) = ∂˜_{P ,j}f⁽ⁱ⁾

P ,jtⁱ

P

dt˜_P whenever j is at least 2. Only the case j equals 1 is interesting: Here, d(f⁽ⁱ⁾

P ,1tⁱ

P

dt˜⁽¹⁾

P )

=

if⁽ⁱ⁾

P ,1+ ˜∂_{P ,1}f⁽ⁱ⁾

P ,1

tⁱ

P

dt˜_P. But f⁽ⁱ⁾

P ,1 belongs to OZ,P = OX,P/t_{P ,1}OX,P, so

∂˜_{P ,1}f⁽ⁱ⁾

P ,1 vanishes. Noting that H⁰(Ωⁿ_(X,Z)(E)) equals H⁰(Ωⁿ_X(Z+E)) concludes the proof.

The following lemma was used in the proof.

Lemma 3.4. — With the notation of Theorem B and its proof, let P be a generic point of Z and Qbe a closed point in the closure of P. If ω_P+ Ωⁿ

(X,Z),P lies in the image ofQ_Pⁿ⁻¹→Qⁿ_P, thenω_Q+ Ωⁿ

(X,Z),Q lies in the image of Qⁿ⁻¹_Q →Q_Qⁿ. Proof. — Without loss of generality, we assume that the pleasant neighbourhoodU, t1, . . . , tn ofP that is used in Theorem B is also a pleasant neighbourhood ofQ, and thatP_j=V(t_j),j= 1, . . . , s, are the irreducible components ofZ that pass through Q. AsP is fixed, we drop the indexP for thetjand ˜∂j in this proof. As before, write P =P1 andt=t1. By Proposition 3.3 and (3.10),

Qⁿ_Q=

(−1,...,−1)

M

α=(−v_P

1(E),...,−v_{P s}(E))

(O_X,Q/i_Q)t^α₁¹· · ·t^α_s^sdt.˜

Thus,

ω_Q+ Ωⁿ_(X,Z),Q=X

α

f^(α)

Q t^α₁¹· · ·t^α_s^sdt˜

3.1. DE RHAM COHOMOLOGY OF SMOOTH PROPER PAIRS 41

for appropriatef^(α)

Q ∈O_X,Q/i_Q. Sinceω_P+ Ωⁿ

(X,Z),P is in the image ofQ_Pⁿ⁻¹→Qⁿ_P, there existf⁽ⁱ⁾

P ,j inOX,P/i_P such that ω_P+ Ωⁿ_(X,Z),P =

−1

X

i=−v_P(E)



if⁽ⁱ⁾

P ,1+

n

X

j=2

∂˜jf⁽ⁱ⁾

P ,j



tⁱ₁dt˜

=

n

X

j=1

∂˜_jf⁽ⁱ⁾

P ,j

dt˜ + Ωⁿ_(X,Z),P,

withf_{P ,j}=^dfP

if⁽ⁱ⁾

P ,jtⁱ₁. The mapU →Aⁿ_W(k)is ´etale and so the ideal ofP inOX(U) is generated by{p, t1}. In particular,

O_X,P = O_X,Q

(p,t1)

.

Therefore, each f_{P ,j} lies inOX(F)_Q for some suitable effective divisorF supported onZ. By Proposition 3.3, there is an expansion

f_{P ,j} = X

α≥(−v_P

1(F),...,−v_{P s}(F))

f^(α)

P ,jt^α₁¹· · ·t^α_s^s

for suitablef^(α)

P ,j ∈O_X,Q/i_Q. Hence there is an equality f^(α)

Q =

s

X

j=1

α_jf^(α)

P ,j +

n

X

j=s+1

∂˜_jf^(α)

P ,j ∈O_X,Q/i_Q for all (−v_P

1(F), . . . ,−v_P

s(F)) ≤ α ≤ (−1, . . . ,−1). The element f^(β)

Q vanishes wheneverβi is less than−v_P

i(E) for somei, so the same holds for the right hand side of the equation. We may therefore assume that the summands vanish individually so that







(−1,...,−1)

X

α=(−v_P

1(E),...,−v_{P s}(E))

f^(α)

P ,1t^α₁¹· · ·t^α_s^sdt˜⁽¹⁾, . . . ,

(−1,...,−1)

X

α=(−v_P

1(E),...,−v_{P s}(E))

f^(α)

P ,nt^α₁¹· · ·t^α_s^sdt˜⁽ⁿ⁾







defines the desired preimage ofω_Q+ Ωⁿ_(X,Z),Qin

Qⁿ⁻¹_Q =

n

M

j=1

(−1,...,−1)

M

α=(−v_P

1(E),...,−v_{P s}(E))

(OX,Q/i_Q)t^α₁¹· · ·t^α_s^sdt˜^(j).

The case of curves. — In this subsection, X is a geometrically integral curve of genusg. For such anX, Theorem B takes a particularly nice form: The generic points P of Z coincide with the closed points of Z. Let P be such a point. Its local ring OZ,P is an ´etale extension ofW(k) of degree [k(P) :k] whose completion (with respect to the maximal ideal) is canonically isomorphic toW(k(P)) [Gro64, Rem. 0.19.8.7].

Moreover,nequals 1, so to check whether a differentialωlies in the kernel, it suffices to check that the O(degE) many coefficients ω⁽ⁱ⁾

P , i=−v_P(E), . . . ,−1 lie iniOZ,P. In particular, for anyZ andE there is a natural isomorphism

Kerϕ

dH⁰(OX(E))−→^∼ H⁰(ΩX(Z))⊕M, (3.12) whereMis a freeW(k)-module of rankg. (A quick argument for the rank is as follows:

The rank of Kerϕ

dH⁰(OX(E)) equals the dimension of HdR(XK\ZK). The latter space is the colimit of the spaces H⁰(Ω_XK((r+ 1)Z_K))/dH⁰(OX(rZ_K)) ordered by inclusion. Riemann-Roch says that the dimension of the latter spaces stabilises forr large enough and that it equalsg+ degZK−1. This gives the desired formula since degZK equals degZ.)

Proposition 3.5. — Let P be a point ofZ of degree1,P be the irreducible compo-nent of Z that reduces to P, and choose E = `P in Theorem B where ` is at least the maximal gap number of P (as point on X). For each gap number r of P, let ωr∈Kerϕbe such that its germ atP lies in

−r−1

X

i=−v_P(E)

ω⁽ⁱ⁾_r tⁱ_Pdt˜_P +ω^(r)_r t^−r

P dt˜_P+ Ω_(X,Z)((r−1)Z)_P

(t_P as in Theorem B) with v(ωr⁽ⁱ⁾) > −i for i = −v_P(E), . . . ,−r − 1 and v(ωr^(r)) = v(r). Then the classes of the ω_r form a basis of the quotient of Kerϕ

dH⁰(OX(E))byH⁰(ΩX(Z)).

Remark 3.6. — First note thatOX(E) is acyclic so that it makes sense to useEin Theorem B: Indeed,O_X(E) is acyclic because` is at least the maximal gap number of P. Since h¹(OX_K(EK)) is at most h¹(O_X(E)) [Liu06, Thm. 5.3.20], the claim follows. Second observe that the basis in the above proposition is determined using the gap numbers ofP as point onX. No knowledge of the gap numbers of the point P_K onX_K (which may differ from those ofP [Sch39,§5]) is required.

Proof. — For each pole numberrofP between 1 andv_P(E), choose an elementf of H⁰(OX(E)) such that its reductionf modulophas pole orderr. This is possible since OX(E) is acyclic and so H⁰(OX(E))⊗k→H⁰(O_X(E)) is surjective. Setωr=df. We are going to show that theωr generate Kerϕmodulo H⁰(ΩX(Z)). A rank argument then shows that the classes of theωrgiven in the proposition form a basis as desired.

3.1. DE RHAM COHOMOLOGY OF SMOOTH PROPER PAIRS 43

Let ω_r = P−1

i=−v_P(E)ωr⁽ⁱ⁾tⁱ

P

dt˜_P be the (beginning of the) expansion of ω_r with re-spect to t_Pdt˜_P according to Proposition 3.3. Given the beginning of the expansion P−1

i=−v_P(E)ϑ⁽ⁱ⁾tⁱ

P

dt˜_P of some element ϑ ∈Kerϕ, we must show that there exists a solution λto the linear system











ω⁽⁻¹⁾₋₁ . . . ω⁽⁻¹⁾

−v_P(E)

... ...

ω^(−v₋₁^P(E)) . . . ω^(−vP(E))

−v_P(E)









 λ=





 ϑ⁽⁻¹⁾

... ϑ^(−vP(E))





.

Since theω_randϑbelong to Kerϕ, thei-th row of the system is divisible byi. After carrying out these divisions, the elements on the diagonal of the matrix become units.

If a column has nonzero entries below the diagonal, then those elements must be a multiple of p (even after the aforementioned divisions). Hence the matrix can be brought into an upper-triangular form with units on the diagonal. A solution λ to the linear system can be found using reverse substitution.

The following lemma is not essential to the remainder of the present thesis. It says that the cohomology of (X, Z) (withX a curve) does not have any torsion. So from now on we will not have to kill any torsion of Hⁱ_dR(X, Z) or Hⁱ_cr(X, Z).

Lemma 3.7. — The moduleHⁱ_dR(X, Z) is a lattice inHⁱ_dR(XK\ZK)for alli∈N.

Lacking a reference, we give a full proof that mimics the proof of [vdB08, Prop. 3.3.2].

Proof. — By (3.2), Hⁱ_dR(X, Z) ⊗K is isomorphic to Hⁱ_dR(XK \ ZK), so it re-mains to show that Hⁱ_dR(X, Z) is torsion-free. To this end, consider the long exact sequence in hypercohomology that is associated to the short exact sequence 0→Ω_(X,Z)[1]^•→Ω^•_(X,Z)→OX[0]^•→0.⁽³⁾ The part in degreeireads

Hⁱ⁻¹(Ω_(X,Z))→Hⁱ(Ω^•_(X,Z))→Hⁱ(OX).

The modules Hⁱ(OX) are free since OX is cohomologically flat. The mod-ules Hⁱ(Ω_(X,Z)) vanish unless i equals 0 (and then the module is free) since Ω_(X,Z)= ΩX(Z) is acyclic. Hence, apart from degree 1, the module Hⁱ(Ω^•_(X,Z)) is a submodule of a free module and thus free itself.

Regarding degree 1, note that the boundary map from H⁰(OX) to H⁰(Ω(X,Z)) is 0.

Indeed, this is equivalent to surjectivity of H⁰(Ω^•_(X,Z)) ,→ H⁰(OX). So let h be an element of H⁰(OX). Since H⁰(Ω^•_(X,Z)) and H⁰(OX) have the same rank (their tensor products withKhave dimension 1 overKsinceX is geometrically integral), a

(3)As usual, when given a sheaf F we denote byF[i]^• the complex withF in degree i and 0 everywhere else.

multipleλhis in the image. Thus the image ofλhin H⁰(Ω(X,Z)) is 0, but this module is free, so the image ofhitself is 0. This meanshis in the image of the inclusion.

So the degree 1 part of the long exact sequence is a short exact sequence. Assume h∈H¹(Ω^•_(X,Z)) is such thatλhequals 0. Since H¹(OX) is free, the image ofhin that module is 0, i.e. his in the image of the map H⁰(Ω_(X,Z))→H¹(Ω^•_(X,Z)). Let h⁰ be a preimage. Then λh⁰ lies in the kernel and thus, since the coboundary map is 0,λh⁰ is 0. Since H⁰(Ω_(X,Z)) is free,h⁰ itself and thereforehmust be 0.

Remark 3.8 (Regarding the choice of E). — Different choices of E influence how easy it is to compute H⁰(Ω¹_X(Z +E)). This in turn affects how easy it is to computeH¹(Ω^•_(X,Z)(E))/(tor) and H¹_dR(X, Z)/(tor). There are at least three factors to consider.

(1) If E is of small degree, then H⁰(Ω¹_(X,Z)(E)) is a module of small rank and thus (hopefully) fast to compute. A lower bound is given by Riemann-Roch: The sheaf OX(E) is acyclic whenever the degree of E is at least 2g−1, and it cannot be acyclic if the degree is less than g. However, degree g is often possible: IfX is hyperelliptic orpat least 2g−1, then (up to a constant field extension) there exists a strict relative normal crossing divisor Z such that Z is of degree 1 and has gap sequence 1, . . . , g [Sch39, S¨atze 6 & 8]. ThusOX(gZ) is acyclic and

H¹(Ω^•_(X,Z)(gZ))

(tor) = H⁰(Ω_X((g+ 1)Z)).

In the higher-dimensional case when X is a hypersurface of degree din P^r_W(k) and Z a smooth hyperplane section, Walker shows that Ω^•_(X,Z)(`Z) is acyclic whenever

` is at least max{(r−2)d+ 1,(r−1)d−r} [Wal09, Thm. 5.1.14]. Specialising to smooth plane curves of genus g, this bound becomesd−2. Therefore, it says that the degree of`Z should be at least 2g−1 (observe that the degree ofZ isdand that the genus of the curve is (d−1)(d−2)/2). This coincides with the bound obtained from Riemann-Roch.

(2) If E is bounded by `Z where ` is small compared to p, then already H¹(Ω_(X,Z)(E))/(tor) is Frobenius-stable, see (3.8). For example, if the degree ofZ is at least 2g−1, then the choiceE=Z yields

H¹_dR(X, Z)∼=H¹(Ω^•_(X,Z)(Z))

(tor) = H⁰(ΩX(2Z))

dH⁰(OX(Z)).

(3) If E is of “special form”, then a basis for H¹_dR(X, Z)/(tor) or a generating set of H⁰(Ω_(X,Z)(E)) may be especially easy to compute. An example of the for-mer is Proposition 4.2. For an example of the latter, let X be nondegenerate. This means that there exists a torus T²_W(k) such that X ∩T²_W(k) can be defined by a Laurent polynomial f(x, y) that is nondegenerate with respect to its Newton poly-tope ∆ [CV09, Def. 1.9]. In his thesis, Tuitman considers the pair (X, Z) where Z =X\(X∩T²_W(k)) [Tui11]. Depending on the Newton polytope ∆, one defines a

Im Dokument Frobenius-stable lattices in rigid cohomology of curves (Seite 43-55)