nLab relative affine n-space


In algebraic geometry, for an algebraic scheme SS and positive integer nn, the relative affine nn-space A S n\mathbf{A}^n_S (or: affine nn-space over SS) is the generalization of an affine nn-space A k n\mathbf{A}^n_k (considered as an algebraic kk-scheme) from a ground ring kk to the category of relative schemes over the base scheme SS.

There are several equivalent definitions.

Definition using affine covers

In the affine case, if S=Spec(B)S = Spec(B) this is simply the spectrum A S n=Spec(B[x 1,,x n])\mathbf{A}^n_S = Spec(B[x_1,\ldots,x_n]), and the morphism to the base scheme SS is induced by the ring inclusion BB[x 1,,x n]B\hookrightarrow B[x_1,\ldots,x_n].

For general scheme, one takes an open cover of SS by affine schemes, and notices that the inclusions of double intersections into single intersection induce morphisms of ringed spaces from the double intersection to the single intersection, satisfying the cocycle condition allowing for gluing of such relative affine nn-spaces over affines. Finally one checks that this definition does not depend on cover.

Definition via fiber product

A S n=A Z n× SpecZS \mathbf{A}^n_S = \mathbf{A}^n_{\mathbf{Z}}\times_{Spec{Z}} S

Definition via relative spectrum

There is a general notion of a relative spectrum (sometimes called global spectrum) of a sheaf of 𝒪 T\mathcal{O}_T-algebras over a scheme TT. The affine nn-space over a scheme SS may be viewed as the nn-dimensional vector bundle over SS, but not viewed as a locally free sheaf but as a total space. Fiberwise, to make a scheme from a vector space, one needs to take the spectrum of a symmetric algebra.

Affine nn-space over SS is simply the spectrum of the symmetric algebra of the direct sum of nn-copies of the structure sheaf of SS:

A S n=Spec(Sym(𝒪 S n)) \mathbf{A}^n_S = Spec(Sym(\mathcal{O}^n_S))

Created on July 31, 2023 at 12:08:27. See the history of this page for a list of all contributions to it.