Finite morphism

In algebraic geometry, a morphism f: XY of schemes is a finite morphism if Y has an open cover by affine schemes

V_i = \mbox{Spec} \; B_i

such that for each i,

f^{-1}(V_i) = U_i

is an open affine subscheme Spec Ai, and the restriction of f to Ui, which induces a ring homomorphism

B_i \rightarrow A_i,

makes Ai a finitely generated module over Bi.[1] One also says that X is finite over Y.

In fact, f is finite if and only if for every open affine open subscheme V = Spec B in Y, the inverse image of V in X is affine, of the form Spec A, with A a finitely generated B-module.[2]

For example, for any field k, the morphism from the affine line A1 over k to itself given by xx2 is finite. (Indeed, the polynomial ring k[x] is finitely generated as a module over k[y] by yx2, with generators 1 and x.) By contrast, the inclusion of A1 − 0 into A1 is not finite. (Indeed, the Laurent polynomial ring k[y, y−1] is not finitely generated as a module over k[y].)

Properties of finite morphisms

Morphisms of finite type

For a homomorphism AB of commutative rings, B is called an A-algebra of finite type if B is a finitely generated as an A-algebra. It is much stronger for B to be a finite A-algebra, which means that B is finitely generated as an A-module. For example, for any commutative ring A and natural number n, the polynomial ring A[x1, ..., xn] is an A-algebra of finite type, but it is not a finite A-algebra unless A = 0 or n = 0.

The analogous notion in terms of schemes is: a morphism f: XY of schemes is of finite type if Y has a covering by affine open subschemes Vi = Spec Ai such that f−1(Vi) has a finite covering by affine open subschemes Uij = Spec Bij with Bij an Ai-algebra of finite type. One also says that X is of finite type over Y.

For example, for any natural number n and field k, affine n-space and projective n-space over k are of finite type over k (that is, over Spec k), while they are not finite over k unless n = 0. More generally, any quasi-projective scheme over k is of finite type over k.

The Noether normalization lemma says, in geometric terms, that every affine scheme X of finite type over a field k has a finite surjective morphism to affine space An over k, where n is the dimension of X. Likewise, every projective scheme X over a field has a finite surjective morphism to projective space Pn, where n is the dimension of X.

See also

Notes

  1. Hartshorne (1977), section II.3.
  2. Stacks Project, Tag 01WG.
  3. Stacks Project, Tag 01WG.
  4. Stacks Project, Tag 01WG.
  5. Grothendieck, EGA IV, Part 4, Corollaire 18.12.4.
  6. Grothendieck, EGA IV, Part 3, Théorème 8.11.1.
  7. Stacks Project, Tag 01WG.

References

External links

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