Locally Noetherian and Noetherian schemes

Depends on

• Schemes
• Affine open subschemes
• Noetherian commutative rings and modules

Used by

• Degree is additive on invertible sheaves over a proper curve Corollary
• Euler characteristic in a proper flat family is locally constant Corollary
• Global functions on geometrically connected and geometrically reduced proper schemes Corollary
• The degree of a divisor descends to the Picard group of a normal proper curve Corollary
• A cusp family defeats the missing source hypothesis Counterexample
• A Weil divisor that is not Cartier at the vertex of the quadric cone Counterexample
• Finite type need not be locally free Counterexample
• Frobenius linear systems have nonreduced general members Counterexample
• The equation must define the intended scheme Counterexample
• The intersection product needs Cartier or complementary-dimension hypotheses Counterexample
• Base points and base-point-free linear systems Definition
• Canonical bundle and canonical divisors Definition
• Complete linear system Definition
• Degree of an invertible sheaf on a proper one-dimensional scheme Definition
• Delta invariant of a curve singularity Definition
• Divisors on a smooth proper curve Definition
• Genus and arithmetic genus of a curve Definition
• Geometric genus of a singular curve Definition
• Grothendieck groups of coherent sheaves and of vector bundles on a scheme Definition
• Intersection multiplicity of closed subschemes at a point Definition
• Intersection numbers of Cartier divisors on a smooth projective surface Definition
• Locally factorial scheme Definition
• Normal scheme modifications and normalized point blowups Definition
• Order of an ideal sheaf at a point Definition
• Principal parts of an invertible sheaf on a curve Definition
• Regular points of locally Noetherian schemes Definition
• S-dense open subschemes and S-rational maps Definition
• Simple normal crossings divisors and simultaneous normal crossings position Definition
• The cotangent complex of a morphism of schemes Definition
• The Riemann-Roch dimension l(D) Definition
• Weil divisor normal noetherian scheme Definition
• A coherent closed-point skyscraper Example
• All twists on the projective line Example
• Pulling a divisor back along the cusp normalization Example
• The twists on the projective line have degree n Example
• A regular point lies on one irreducible component Lemma
• A transverse hyperplane slice is smooth at the chosen point Lemma
• A vector bundle on the projective line has a line subbundle of maximal degree Lemma
• All initial forms define the tangent cone Lemma
• An S-rational map defined after a faithfully flat smooth base change is defined Lemma

…and 66 more results.

Dependency tree · two levels

10 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, Foundations of Algebraic Geometry, Section 6.3 (standard reference, not scraped)