The genus of a proper curve under extension of the base field #
Let X be a proper scheme over a field k, and let X_K = X ×_k K be its base change to a field
extension K / k. Then X_K has the same genus as X: dim_K H¹(X_K, 𝒪) = dim_k H¹(X, 𝒪)
(TauCeti.AlgebraicGeometry.genus_eq_of_isPullback_of_isProper). This is flat base change of
cohomology (TauCeti.AlgebraicGeometry.finrank_cohomology_trivial_eq_of_isPullback), and needs no
smoothness, reducedness, irreducibility or dimension hypothesis, which matters because X_K need
not be reduced or irreducible when K / k is inseparable. In degree zero, flat base change
(TauCeti.AlgebraicGeometry.finrank_cohomology_zero_trivial_eq_of_isPullback) shows that
H⁰(X, 𝒪_X) = k implies H⁰(X_K, 𝒪) = K. The hypothesis H⁰(X, 𝒪_X) = k cannot be dropped
there: for X = ℙ¹_L viewed as a k-scheme with L / k finite of degree > 1, the ring
H⁰(X_L, 𝒪) = L ⊗_k L has dimension [L : k] > 1 over L.
For a smooth proper connected curve the finite-dimensionality of H¹(X, 𝒪_X) is automatic, so
there the invariance holds without finiteness hypotheses
(TauCeti.AlgebraicGeometry.genus_eq_of_isPullback_of_smooth); the dimension of H¹ is also
computed on a cover by two affine opens (Scheme.exists_two_affine_opens_of_smooth). It is the
invariance of the genus that comparisons of curves and of their Jacobians across field extensions
rely on.
References #
- R. Hartshorne, Algebraic Geometry, Proposition III.9.3 (flat base change).
- The Stacks Project, Lemma 0BY9.
The genus of a smooth proper connected curve is invariant under extension of the base
field: dim_K H¹(X_K, 𝒪) = dim_k H¹(X, 𝒪).
For a smooth proper connected curve, H¹(X, 𝒪_X) stays finite-dimensional under extension of
the base field.
The genus of a proper scheme over a field is invariant under extension of the base field:
dim_K H¹(X_K, 𝒪) = dim_k H¹(X, 𝒪). The scheme need not be smooth, reduced, irreducible or of
dimension one. The finite-dimensionality of H¹ after base change, needed to form the genus of
Y, follows from that before by
finiteDimensional_cohomology_trivial_of_isPullback_of_isProper.
The genus of a smooth proper connected curve is invariant under extension of the base
field, stated with Scheme.genus. The finite-dimensionality of H¹ before and after base
change, needed to form both genera, follows from the geometric hypotheses.