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For a -functor and a set of functions on , Definition in k-ring?, we define
(The ”unit ideal of ” is just itself.) For a transformation of -functors and a subfunctor we define
A subfunctor is called open subfunctor resp. closed subfunctor if for every transformation we have is of the form resp. .
A -functor is called a -scheme if the following two conditions hold:
( is a sheaf for the Zarisky Grothendieck topology on ) For all -rings and all families such that we have: if for all such that the images of and coincide in there is a unique mapping to the .
( has a cover of Zarisky open immersions of affine -schemes) There exists a family of open affine subfunctors of such that for all fields we have that .
The category of -schemes is closed under limits, forming open- and closed subfunctors and skalar extension.