nLab
Demazure, lectures on p-divisible groups, I.3, open- and closed subfunctors; schemes

This entry is about a section of the text

• Michel Demazure, lectures on p-divisible groups, web

For a $k$-functor $X\in \mathrm{coPsh}(M_k)$ and $E\subseteq O(X)=M_k(X,O_k)$ a set of functions on $X$, Definition in k-ring?, we define

$$V(E)(R):=\{x\in X(R)\mid f\in E,f(x)=0\}$$V(E)(R):=\{x\in X(R) | f\in E, f(x)=0\}

and

$$D(E)(R):=\{x\in X(R)\mid f\in E,\text{the}f(x)\text{generate the unit ideal of}R\}$$D(E)(R):=\{x\in X(R)|f\in E, \text{the} f(x) \text{generate the unit ideal of} R\}

(The ”unit ideal of $R$” is just $R$ itself.) For a transformation $u:Y\to X$ of $k$-functors and $Z\subseteq X$ a subfunctor we define

$$u^{-1}(Z)(R):=\{y\in Y(R)\mid u(y)\in Z(R)\}$$u^{-1}(Z)(R):=\{y\in Y(R)|u(y)\in Z(R)\}

A subfunctor $Y\subseteq X$ is called open subfunctor resp. closed subfunctor if for every transformation $u:T\to X$ we have $u^{-1}(Y)$ is of the form $V(E)$ resp. $D(E)$.

Definition

A $k$-functor $X$ is called a $k$-scheme if the following two conditions hold:

1. ($X$ is a sheaf for the Zarisky Grothendieck topology on $M_k^{\mathrm{op}}$) For all $k$-rings and all families $(f_i{)}_i$ such that $R={\coprod }_{i}R{f}_i$ we have: if for all $x_i\in R[f_i^{-1}]$ such that the images of $x_i$ and $x_j$ coincide in $X(R[f_i^{-1}{f}_j^{-1}])$ there is a unique $x\in X(R)$ mapping to the $x_i$.

2. ($X$ has a cover of Zarisky open immersions of affine $k$-schemes) There exists a family $(U_i{)}_i$ of open affine subfunctors of $X$ such that for all fields $K\in M_k$ we have that $X(K)={\coprod }_i{U}_i(K)$.

Remark

The category of $k$-schemes is closed under limits, forming open- and closed subfunctors and skalar extension.