Schemes

Depends on

• A locally ringed space
• Affine schemes and their coordinate rings

Used by

• Euler characteristic in a proper flat family is locally constant Corollary
• Finite morphisms are proper Corollary
• Global functions on geometrically connected and geometrically reduced proper schemes Corollary
• Upper semicontinuity of fibre cohomology dimensions Corollary
• A quotient stack need not be a scheme Counterexample
• Affine open subschemes Definition
• Algebraic spaces over a scheme, defined as fppf sheaves Definition
• Derived schemes and the cotangent complex of a morphism Definition
• Etale neighbourhoods and elementary etale neighbourhoods of a point Definition
• Fppf coverings and the fppf site Definition
• Ideal sheaves Definition
• Integral schemes Definition
• Irreducible components as schemes Definition
• Locally factorial scheme Definition
• Locally Noetherian and Noetherian schemes Definition
• Morphisms of schemes Definition
• Projective scheme of a homogeneous quotient and its standard affine charts Definition
• Quasi-coherent module on a scheme Definition
• Quasi-compact and quasi-separated schemes Definition
• Sheaf total quotient rings Definition
• Split augmented simplicial schemes and matching objects Definition
• Square-zero extensions, small extensions and first-order thickenings Definition
• The reduction of a scheme Definition
• Weil divisor normal noetherian scheme Definition
• The empty morphism is finite, proper and projective Example
• Under AC, effective divisors on normal proper curves give finite subschemes of the same degree Example
• A meromorphic unit has locally finite nonzero order support Lemma
• A quasi-compact image stable under specialization is closed Lemma
• A quasi-compact immersion with nonclosed image has a boundary specialization Lemma
• Affine neighbourhood containing component generic points Lemma
• Affine pushforward algebra localizes Lemma
• Closed gluing of two projective three-spaces is proper Lemma
• Closed immersions are affine quotients and survive base change Lemma
• Field extension preserves the graded pieces and the total length of a zero-dimensional projective quotient Lemma
• Finite morphisms over a projective variety over any field Lemma
• Finite morphisms survive base change and composition Lemma
• Finite projective complex for proper flat coherent cohomology Lemma
• Flat deformations form a Zariski sheaf of groupoids Lemma
• Flat field extension commutes with coherent cohomology Lemma
• Fpqc covers are universally submersive Lemma

…and 36 more results.

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, Schemes, Definition 9.1 (standard reference, not scraped)