Article scheme conventions

PDF p. 1: Introduction: vector bundles of rank equal to dimension; PDF p. 3: Notation and conventions

Fix a base field $k$, assumed algebraically closed unless a statement specifies otherwise: every nonconstant polynomial in $k[t]$ has a root in $k$. A $k$-scheme is a scheme equipped with a morphism to $\mathrm{Spec}k$; a morphism of $k$-schemes commutes with these structure morphisms.

Write ${\mathrm{S}\mathrm{c}\mathrm{h}}_k$ for the category whose objects are quasi-compact, quasi-separated $k$-schemes and whose morphisms are $k$-morphisms. Quasi-compact means that every open cover has a finite subcover; quasi-separated means that the intersection of any two affine open subschemes is quasi-compact. The opposite category ${\mathrm{S}\mathrm{c}\mathrm{h}}_k^{\mathrm{o}\mathrm{p}}$ has the same objects and reverses all arrows. Thus a functor with domain ${\mathrm{S}\mathrm{c}\mathrm{h}}_k^{\mathrm{o}\mathrm{p}}$ is contravariant on ${\mathrm{S}\mathrm{c}\mathrm{h}}_k$: a $k$-morphism $X\to Y$ induces a map from the value at $Y$ to the value at $X$.

In the article, an affine $k$-scheme $\mathrm{Spec}A$ is assumed to be of finite type over $k$, meaning that its coordinate $k$-algebra $A$ is finitely generated. The dimension of a scheme is the supremum of the lengths of chains of strict inclusions of irreducible closed subsets. A vector bundle is a finite locally free ${\mathcal{O}}_X$-module. Its rank is the well-defined locally constant function $x\mapsto {\dim }_{\kappa (x)}(E{\otimes }_{{\mathcal{O}}_X}\kappa (x))$; it may vary between connected components. Having rank $r$ means this function is constantly $r$, equivalently that the bundle is locally isomorphic everywhere to ${\mathcal{O}}_X^r$. For a finite-dimensional scheme, critical rank means rank equal to the dimension of the scheme.

The definitions of a scheme over a base and of its morphisms are those of Schemes and morphisms over a base. The two compactness conditions are those of Quasi-compact and quasi-separated schemes; they select objects of the category and do not require schemes to be separated. Identity maps and composites of $k$-morphisms again commute with the structure maps, so these objects and morphisms form a category. Reversing identities and composition defines its opposite, and the stated direction of a contravariant functor is the direction of its action on the reversed arrow. Finite generation specifies the article’s convention for affine schemes. The well-definedness and local constancy of rank are supplied by Locally free sheaves of finite rank. The constant-rank and dimension convention is a comparison of two integers and makes no splitting or classification assertion.