Cohomology of smooth curves
Proposition 3.99. Algorithm 3.98 is correct and halts in an effectively bounded number of field operations.
Proof. By construction, the algorithm decides correctly whetherT1,k ∼= T2,k, but this is equivalent toT1,ksep ∼=T2,ksep.
We use this to computeR1f∗.
Algorithm 3.98. Suppose that given as input is a factorial field k, a finite locally free mor- phism X → A1
k (or X → P1k) with X a smooth connected curve over k,G a finite locally
constant sheaf of groups on X´etof degree coprime with the characteristic of k.
Output: R1f∗G as a finiteGal(ksep/k)-set of representatives of isomorphism classes of
Gksep-torsors on Xksep.
• Compute a finite ´etale Galois morphismg:Y→Xwith Galois groupΓand withYconnected, such thatg−1Gis constant, say with fibreG(withΓ-action).
• Compute a finite purely inseparable extension l of ksuch that the normal completionYl ofYlis smooth.
• SetT =Torsaffl,Y
l (orT =Tors
proj
l,Yl in the projective case).
• Using an absolute primary decomposition algorithm, compute a finite Galois extensionl0oflover which the connected components of ObTlsepare defined.
• Compute a finite extensionl00 ofl0, and for every connected component of ObTl0anl00-rational point on it; i.e. aΓ-equivariantG-torsor onYl00.
• Attach to every such torsor its restriction toYl00, and then its quotient byΓ.
• Letk00be the separable closure ofkinl00.
• LetTdenote the finite set ofG-torsors onXk00 obtained this way.
• For every t ∈ T andγ ∈ Gal(k00/k), find using Algorithm 3.96γt ∈ Tby enumeration.
• Outputthe finite Gal(ksep/k)-setT, andhalt.
Proposition 3.99. Algorithm 3.98 is correct and halts in an effectively bounded number of field operations.
Proof. This follows directly from Corollary 3.89 in the affine case, and from Corol-
lary 3.68 in the projective case.
Before considering functoriality in G, we first consider quotients of finite ´etale morphisms by finite locally constant sheaves of groups onX´et.
Algorithm 3.100. Suppose that given as input is a factorial field k, a finite locally free morphism X → A1
k (or X → P1k) with X a smooth connected curve over k, a finite ´etale
scheme Y over X, a finite locally constant sheafGof groups on X´etacting on Y.
Output: the quotient of Y by the action ofG.
3.20 Poincar´e duality
• Compute a finite ´etale Galois morphismg: X0 →Xwith Galois groupΓand withYconnected, such thatg−1Gis constant, say with fibreG(withΓ-action).
• SetY0 =X0×XY.
• OutputΓ\(G\Y0)andhalt.
Hence we can computeR1f∗functorially as follows.
Algorithm 3.101. Suppose that given as input is a factorial field k, a finite locally free morphism X → A1
k (or X → P1k) with X a smooth connected curve over k, ϕ:G → H
a morphism of finite locally constant sheaves of groups on X´et of degree coprime with the
characteristic of k.
Output: theGal(ksep/k)-equivariant map R1ϕ:R1f∗G →R1f∗H.
• Letlbe a finite Galois extension ofksuch thatR1f∗GandR1f∗Hsplit com- pletely overl.
• Outputthe map sending a Gl-torsor T → Xl to aHl-torsor isomorphic to Hl⊗Gl T =Gl\(Hl×Xl T)→Xl (whereG acts byg(h,t) = (hg−1,gt)), and
halt.
Finally, ifGis commutative, thenR1f∗Gis an abelian group, and we can compute its group structure.
Algorithm 3.102. Suppose that given as input is a factorial field k, a finite locally free morphism X → A1
k (or X →P1k) with X a smooth connected curve over k, ϕ:G → Ha
morphism of finite locally constant sheaves of abelian groups on X´etof degree coprime with
the characteristic of k.
Output: the addition map R1f∗G ×kR1f∗G →R1f∗G.
• Letlbe a finite Galois extension ofksuch thatR1f∗Gsplits completely over
l.
• Outputthe map sending a pair(T1,T2)ofGl-torsors to aGl-torsor isomorphic toT1⊗Gl T2=Gl\(T1×Xl T2)(whereGlacts byg(t1,t2) = (t1g−1,gt2)), and
halt.
3.20 Poincar´e duality
Note that we have now computedR0f
!,R0f∗, andR1f∗of a smooth connected curve
f: X→Speck. We compute the rest using Poincar´e duality; we recall its statement first.
Let Λbe a finite ring annihilated by n ∈ Z, let X be a scheme, and let Mbe a finite locally constant sheaf of Λ-modules on X´et. Then we denote thed-th Tate twist M ⊗Z/nZ(µn)⊗d of M byM(d); note that this doesn’t depend the choice
of the annihilator n, and that we can compute this if X is a smooth curve or the spectrum of a field. Write moreoverM∨forHom(M,Λ), which we can compute by Algorithm 3.92.
Theorem 3.103(Poincar´e duality, SGA4.3 [1, Exp. XVIII, Sec. 3.2.6]). LetΛbe a finite ring that is injective as aΛ-module, let f:X →Speck be a smooth curve over a field, and letMbe a finite locally constant sheaf ofΛ-modules on X´et. Then for q = 0, 1, 2we have
Chapter 3 Cohomology of smooth curves
In other words, we have the identities
R1f!M=R1f∗ M∨(1) ∨ R2f!M= f∗ M∨(1) ∨ R2f∗M=f! M∨(1) ∨ .
Therefore we indeed have an algorithm as in Algorithm 2.2, as desired.