The residue field at a point of an affine scheme

Depends on

• Spec A with its structure sheaf is a locally ringed space
• The stalk of the affine structure sheaf at a prime is A_p
• $R_{\mathfrak p}/\mathfrak pR_{\mathfrak p}\cong\operatorname{Frac}(R/\mathfrak p)$ is the residue field at $\mathfrak p$

Used by

• Algebraic Bezout formula as a sum of local scheme lengths Corollary
• Euler characteristic in a proper flat family is locally constant Corollary
• Maps from proper integral schemes to the affine line have closed-point image Corollary
• Nontrivial degree-zero line bundles have no sections Corollary
• Upper semicontinuity of fibre cohomology dimensions Corollary
• A genus-zero curve need not be the projective line Counterexample
• Finite type need not be locally free Counterexample
• Stalk and fibre are different Counterexample
• The generic point of the affine line has no relative k-valued coordinate Counterexample
• Cohomology and base-change map Definition
• Degree divisor proper curve Definition
• Etale neighbourhoods and elementary etale neighbourhoods of a point Definition
• Fibre of a module sheaf at a point Definition
• Relative cotangent and tangent spaces Definition
• The intrinsic cotangent space Definition
• The Lie algebra of a group scheme Definition
• Total length of a zero-dimensional projective scheme Definition
• A coherent closed-point skyscraper Example
• An upper jump of h0 in a flat projective family Example
• Compensating h0 and h1 jumps with constant Euler characteristic Example
• Dual numbers give a one-point nonreduced affine scheme Example
• Fitting ideals of a diagonal two-by-two presentation Example
• Incidence projection has closed determinantal image Example
• Principal divisors on the projective line have degree zero Example
• The jump l(D+p) - l(D) ranges from zero to the residue degree Example
• The rank-zero bundle Example
• The spectrum of a field is a one-point affine scheme Example
• The twists on the projective line have degree n Example
• Divisors on the projective line are classified by degree Lemma
• Euler characteristic of a closed point, and invariance under an invertible twist Lemma
• Field extension preserves the graded pieces and the total length of a zero-dimensional projective quotient Lemma
• Field-valued points and local-ring points Lemma
• Geometric Nakayama for finite-type sheaves Lemma
• Global intersection length is the sum of the local multiplicities Lemma
• Global length of a plane complete intersection equals the degree product Lemma
• Monotonicity of L(D) in the divisor Lemma
• Projective-line curve and divisor basics Lemma
• Proper normal curve rational function map Lemma
• The eventual Hilbert function of a zero-dimensional projective quotient equals its total length Lemma
• The exact sequence for adding one point to a divisor Lemma

…and 11 more results.

Dependency tree · two levels

13 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, Section 26.5 (standard reference, not scraped)