Endomorphisms and automorphisms of an invertible sheaf #
Let M be an invertible sheaf on a scheme X. Every endomorphism of M is multiplication by a
unique global function: the ring homomorphism Ξ(X, πͺ_X) β+* End M sending a global function to
the scalar multiplication it induces on sections
(AlgebraicGeometry.Scheme.Modules.globalSectionsAction) is bijective. Consequently the
automorphism group of M is the unit group Ξ(X, πͺ_X)Λ£ of the global functions.
The proof is local. Over an open V carrying a rank-one trivialization t of M, an
endomorphism Ο acts on the basis section e_t by a regular function r_t, its trivialization
scalar, and then acts as multiplication by the restriction of r_t on the sections of M over
every open subset of V. The scalars attached to two trivializations agree on the intersection of
their domains, so they glue to a global function r with Ο = r β’ π. Uniqueness is the fact
that a global function is determined by its restrictions to a cover and is read off from its
action on a basis section.
The bijection is the statement that a line bundle has no automorphisms other than the global units.
It is the input for rigidifying line bundles along a section xβ : S β X of f : X β S: a
trivialization along xβ fixes the pullback to S of every automorphism, so no nontrivial
automorphism survives once the restriction Ξ(X, πͺ_X)Λ£ β Ξ(S, πͺ_S)Λ£ along xβ is injective, for
instance when f_* πͺ_X = πͺ_S. Without that injectivity the units in the kernel of the restriction
still act on the rigidified bundle.
Main declarations #
Scheme.Modules.Hom.trivializationScalar: the regular function by which an endomorphism acts on the basis section of a rank-one trivialization, withScheme.Modules.Hom.app_eq_trivializationScalar_smuldescribing the action on all sections over open subsets of the trivializing open, andScheme.Modules.Hom.map_trivializationScalar_eqcomparing two trivializations on their common domain;Scheme.Modules.globalSectionsAction_bijective: for an invertible sheaf, the action of global functions on the sheaf is a bijection onto its endomorphisms;Scheme.Modules.globalSectionsActionRingEquiv: the ring isomorphismΞ(X, πͺ_X) β+* End M;Scheme.Modules.unitsGlobalSectionsMulEquivAut: the group isomorphismΞ(X, πͺ_X)Λ£ β* Aut M.
References #
- R. Hartshorne, Algebraic Geometry, Chapter II, Section 5 and Exercise II.5.1(d)
(
πom(β, β) β πͺ_Xfor an invertible sheaf). - S. Bosch, W. LΓΌtkebohmert, M. Raynaud, NΓ©ron Models, Section 8.1.
The regular function on V by which an endomorphism Ο of M acts on the basis section of
a rank-one trivialization t of M over V: the coordinate of Ο applied to that basis
section.
Equations
- One or more equations did not get rendered due to their size.
Instances For
An endomorphism of M multiplies the basis section of a rank-one trivialization by its
trivialization scalar.
On every open subset W of the domain V of a rank-one trivialization, an endomorphism of
M acts on sections as multiplication by the restriction of its trivialization scalar.
The trivialization scalars of an endomorphism for two rank-one trivializations agree on any common open subset of their domains.
The trivialization scalar of multiplication by a global function is the restriction of that function to the trivializing open.
Distinct global functions act differently on an invertible sheaf.
Every endomorphism of an invertible sheaf is multiplication by a global function.
For an invertible sheaf, the action of global functions is a bijection onto the endomorphisms of the sheaf.
Every endomorphism of an invertible sheaf is multiplication by a unique global function.
Endomorphisms of a line bundle are global functions. For an invertible sheaf M on a
scheme X, the action of global functions is a ring isomorphism Ξ(X, πͺ_X) β+* End M.
Equations
Instances For
Automorphisms of a line bundle are global units. For an invertible sheaf M on a scheme
X, the action of global functions is a group isomorphism Ξ(X, πͺ_X)Λ£ β* Aut M.
Equations
Instances For
An automorphism of an invertible sheaf is multiplication by the global unit corresponding to it.