Extension by zero for abelian sheaves on an open subspace

Depends on

• Restriction of a sheaf to an open subspace
• Sheafification of a presheaf

Used by

• A non-quasi-coherent module with H1 on an affine scheme Counterexample
• Extension by zero can be strictly smaller than direct image on a punctured interval Counterexample
• Derived operations on analytic field sheaves Definition
• Dualizing and support shifts on the complex line Example
• Incidence cochains of a closed interval Example
• A polyhedral dimension bound for every sheaf Lemma
• Abelian sheaves form a Grothendieck category Lemma
• Actual analytic constructible sheaves admit coherent six operations Lemma
• Bounded fibrewise-soft resolutions from a uniform all-sheaf dimension bound Lemma
• Compact-support acyclicity of c-soft sheaves Lemma
• Compact-support calculus along open and closed embeddings Lemma
• Complex algebraic constructible complexes have finite conical cochain models Lemma
• Derived tensor–Hom and inverse/direct image for field sheaves Lemma
• Enough injective sheaves of modules Lemma
• Extension by zero and the closed complement: a short exact sequence Lemma
• Extension-by-zero generators detect sheaf-cohomology vanishing Lemma
• Finite filtration of a generated subsheaf of the constant integer sheaf Lemma
• Finite simplicial constructible sheaves have an actual derived incidence model Lemma
• Flatness criteria and canonical epimorphisms from flat abelian sheaves Lemma
• Global sections on module sheaves have ordinary sheaf cohomology as derived functors Lemma
• Injective abelian sheaves are flasque Lemma
• Injective modules are flasque and Ext from the structure sheaf is cohomology Lemma
• Koszul sheaf Ext of a smooth regular immersion is concentrated in codimension Lemma
• Local-to-global Ext collapse for a regular immersion Lemma
• Locally contractible metric spaces have singular constant sheaf cohomology Lemma
• Shifted smooth complex pullback is exact for the middle perverse t structure Lemma
• Support localization, base change, projection and trace calculus Lemma
• Vanishing compactly supported first cohomology on opens characterises c-softness 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
• Exceptional inverse image as the adjoint of derived proper-support image Theorem
• Extension by zero is left adjoint to restriction and is exact on abelian sheaves Theorem
• Grothendieck vanishing on a Noetherian space Theorem
• Middle perverse t structure glues and has simple intersection complexes Theorem
• Ordinary middle perverse t structure, recollement and intersection complexes 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

• Ravi Vakil, The Rising Sea, Section 2.7 and nearby discussion (standard reference, not scraped)
• The Stacks Project, Section 6.31 (standard reference, not scraped)