The stalk of the affine structure sheaf at a prime is A_p

Depends on

• The localization construction extends to the structure sheaf on Spec A
• Sections and restrictions on distinguished opens of an affine scheme
• The stalk of a presheaf at a point
• Localisation at a prime ideal: $R_{\mathfrak p}=(R\setminus\mathfrak p)^{-1}R$
• Universal property of localisation: maps that invert $S$ factor uniquely through $S^{-1}R$

Used by

• Global functions on geometrically connected and geometrically reduced proper schemes Corollary
• Spec A with its structure sheaf is a locally ringed space Corollary
• The degree of a divisor descends to the Picard group of a normal proper curve Corollary
• A non-quasi-coherent module with H1 on an affine scheme Counterexample
• A nonradical ideal need not enlarge every tangent space Counterexample
• A regular point that is not smooth: a purely inseparable thickening Counterexample
• Finite type need not be locally free Counterexample
• Proper cohomology need not be finite for noncoherent sheaves Counterexample
• Stalk and fibre are different Counterexample
• The equation must define the intended scheme Counterexample
• Delta invariant of a curve singularity Definition
• Locally factorial scheme Definition
• Locally free sheaves of finite rank Definition
• Multiplicity of a hypersurface equation at a rational point Definition
• Regular points of locally Noetherian schemes Definition
• Sheaf total quotient rings Definition
• Smoothness over a field by geometric regularity Definition
• The intrinsic Zariski tangent space Definition
• The residue field at a point of an affine scheme Definition
• Weil divisor normal noetherian scheme Definition
• A nondegenerate projective quadric Example
• A projective cone with a smooth conic base Example
• Divisor of a rational function on the projective line Example
• Principal divisors on the projective line have degree zero Example
• Pulling a divisor back along the cusp normalization Example
• The spectrum of a field is a one-point affine scheme Example
• The spectrum of a product ring is a disjoint union Example
• The twists on the projective line have degree n Example
• Under AC, effective divisors on normal proper curves give finite subschemes of the same degree Example
• A dominant map has a surjective differential on a dense source open Lemma
• A regular point lies on one irreducible component Lemma
• A surjective ring map induces a closed immersion of affine spectra Lemma
• A transverse hyperplane slice is smooth at the chosen point Lemma
• Affine-local flatness Lemma
• All initial forms define the tangent cone Lemma
• An associated sheaf restricts to an associated sheaf on an affine open Lemma
• Closed immersions are affine quotients and survive base change Lemma
• Fpqc covers are universally submersive Lemma
• Function field of an integral finite-type scheme Lemma
• Multiplicity one is the smooth hypersurface test Lemma

…and 24 more results.

Dependency tree · two levels

18 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, Lemma 26.5.4 (standard reference, not scraped)