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

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For a kk-functor XcoPsh(M k)X\in coPsh(M_k) and EO(X)=M k(X,O k)E\subseteq O(X)=M_k(X,O_k) a set of functions on XX, Definition in k-ring?, we define

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


D(E)(R):={xX(R)|fE,thef(x)generate the unit ideal ofR}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 RR’‘ is just RR itself.) For a transformation u:YXu:Y\to X of kk-functors and ZXZ\subseteq X a subfunctor we define

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

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


A kk-functor XX is called a kk-scheme if the following two conditions hold:

  1. (XX is a sheaf for the Zarisky Grothendieck topology on M k opM_k^{op}) For all kk-rings and all families (f i) i(f_i)_i such that R= iRf iR=\coprod_i R f_i we have: if for all x iR[f i 1]x_i\in R[f_i^{-1}] such that the images of x ix_i and x jx_j coincide in X(R[f i 1f j 1])X(R[f_i^{-1} f_j^{-1}]) there is a unique xX(R)x\in X(R) mapping to the x ix_i.

  2. (XX has a cover of Zarisky open immersions of affine kk-schemes) There exists a family (U i) i(U_i)_i of open affine subfunctors of XX such that for all fields KM kK\in M_k we have that X(K)= iU i(K)X(K)=\coprod_i U_i(K).


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

Last revised on September 25, 2012 at 16:19:35. See the history of this page for a list of all contributions to it.