Alphabeta MathHow statement and proof provenance work
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The internal Hom sheaf of two module sheaves
Definition
Let be a ringed space, and let be -modules. The internal Hom sheaf
is the sheaf on defined by for each open set .
Restriction to a smaller open set sends a morphism to its restriction . Multiplication by a local section acts on a morphism by either pre- or post-composition with multiplication by , giving the structure of an -module.
Depends on
Used by
- Twisting the exact sequence of an effective Cartier divisor Corollary
- Derived operations on analytic field sheaves Definition
- Embedded deformations of a closed subscheme Definition
- Internal Hom of module sheaves Definition
- Picard group of a scheme Definition
- Sheaf Ext of coherent modules Definition
- The residue pairing of a line bundle with the dual canonical twist Definition
- Twists on the two-affine projective line Example
- Actual analytic constructible sheaves admit coherent six operations Lemma
- Addition of Cartier divisors is tensor product of their sheaves Lemma
- Adjunction for a smooth closed subvariety Lemma
- Cartesian simplicial sheaves model actual equivariant derived Hom Lemma
- Derived tensor–Hom and inverse/direct image for field sheaves Lemma
- Dual and base change for finite locally free sheaves Lemma
- Dual of a line bundle is its tensor inverse Lemma
- Ext of a locally free cotangent sheaf via sheaf cohomology Lemma
- Finite simplicial constructible sheaves have an actual derived incidence model Lemma
- Functoriality of the residue pairing under line-bundle maps and connecting homomorphisms Lemma
- Koszul sheaf Ext of a smooth regular immersion is concentrated in codimension Lemma
- Local-to-global Ext collapse for a regular immersion Lemma
- Shifted smooth complex pullback is exact for the middle perverse t structure Lemma
- Support localization, base change, projection and trace calculus Lemma
- The twist index on the projective line is an isomorphism invariant Lemma
- Coherent six operations on analytic constructible field sheaves Theorem
- Constructible six functors, proper base change and normalized equivariant duality Theorem
- Constructible Verdier duality and complex orientation Theorem
- Equivariant constructible Ext is degreewise finite and compatible with characteristic zero scalars Theorem
- Exceptional inverse image as the adjoint of derived proper-support image Theorem
- Serre duality for coherent sheaves on a smooth proper curve, Ext form Theorem
- Serre duality for coherent sheaves on projective space Theorem
- Serre duality for finite locally free sheaves on a smooth proper curve Theorem
- Serre duality for line bundles on a smooth proper curve, and the residue realization Theorem
Dependency tree · two levels
5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- The Stacks Project, Section 17.22 (standard reference, not scraped)
- Ravi Vakil, The Rising Sea, Section 2.3.C (standard reference, not scraped)