Sheafification of a presheaf

Depends on

• The plus construction for a presheaf
• The second plus construction is a sheaf

Used by

• Sheafification is idempotent Corollary
• A global section of the quotient that does not lift, and its nonzero connecting class Counterexample
• The constant presheaf need not be a sheaf on a disconnected open set Counterexample
• Cartier divisor Definition
• Extension by zero for abelian sheaves on an open subspace Definition
• Inverse image presheaf and inverse image sheaf Definition
• Kernel sheaves are objectwise, while cokernels and images are sheafified Definition
• Picard group of a scheme Definition
• Principal parts of an invertible sheaf on a curve Definition
• Sheaf of relative Kähler differentials Definition
• Sheaf total quotient rings Definition
• Tensor product of abelian sheaves and its total complex Definition
• Tensor product of sheaves of modules Definition
• A meromorphic unit has locally finite nonzero order support Lemma
• Abelian sheaves form a Grothendieck category Lemma
• Associator, symmetry and unitors of the abelian sheaf tensor product Lemma
• Cartier divisor local equation equivalence Lemma
• Constant sheaves on irreducible spaces are flasque and acyclic Lemma
• Dual of a line bundle is its tensor inverse 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
• Filtered colimits and sheaf cohomology on Noetherian spaces Lemma
• Filtered colimits of sheaves and sections over compact opens Lemma
• Finite filtration of a generated subsheaf of the constant integer sheaf Lemma
• Higher direct images localize over an affine base Lemma
• Injective abelian sheaves are flasque Lemma
• Local-section formula for derived direct image Lemma
• Morphisms from the constant sheaf are global sections Lemma
• Sections on a compact set extend to a neighbourhood Lemma
• The constant sheaf is the sheaf of locally constant functions Lemma
• The image sheaf is the sheafification of the presheaf image Lemma
• The sheaf attached to a module on an affine scheme Lemma
• Cartier divisors on a normal Noetherian scheme give Weil divisors Theorem
• Grothendieck vanishing on a Noetherian space Theorem
• Kernels and cokernels of quasi-coherent modules Theorem
• Sheafification is left adjoint to the inclusion of sheaves into presheaves Theorem
• Sheafification preserves stalks Theorem

Dependency tree · two levels

7 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, Sheaves on Spaces, Section 17 (standard reference, not scraped)
• Ravi Vakil, Foundations of Algebraic Geometry, Class 3, Section 4.7 (standard reference, not scraped)