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Demazure, lectures on p-divisible groups, I.1, k-functors

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Definition

(k-ring, k-functor,affine k-scheme)

For a ring kk the category of kk-rings, denoted by M k:=k/RingM_k:=k/Ring is defined to be the category of commutative associative kk-algebras with unit which are rings. This is equivalently the category of pairs (R,f:kR)(R,f:k\to R) where RR is a Ring and ff is a morphism of kk-algebras.

The category of kk-functors, denoted by coPsh(M k)co Psh (M_k), is defined to be the category of covariant functors M kSetM_k\to Set.

The forgetful functor O k:RRO_k:R\to R sending a kk-ring to its underlying set is called affine line.

For the full and faithful contravariant functor

Sp k:{M k coPsh(M k) A M k(A,)Sp_k:\begin{cases} M_k&\to& co Psh(M_k) \\ A&\to& M_k(A,-) \end{cases}

Sp kASp_k A (and every isomorphic functor) is called an affine kk-scheme. (In most modern texts one uses the notation SpecSpec” instead of SpSp”.) Sp kSp_k restricts to an equivalence between the categories of kk-rings and the category AffSch kAff Sch_k of affine kk-schemes. We think of this category as of M k opM_k^{op}. The functor Sp kSp_k commutes with limits and skalar extension (see below). Consequently AffSch kAff Sch_k is closed under limits and base change.

The affine line O k=M k(k[t],)O_k=M_k(k[t],-) is an affine kk-scheme.

A function on a kk-scheme XX is defined to be an object fO(X):=coPsh(M k)(X,O k)f\in O(X):=co Psh (M_k)(X,O_k). O(X)O(X) is a kk-ring by component-wise addition and -multiplication.

Proposition

There is an adjoint equivalence

(SpO):Sch affORing k(Sp\dashv O):Sch_{aff}\stackrel{O}{\to}Ring_k

of the categories of affine k-schemes and kk-rings.

Remark

The category of kk-functors has limits.

The terminal object is e:R{}e:R\mapsto\{\varnothing\}. Products and pullbacks are computed component-wise.

Remark

For ϕ:kk \phi:k\to k^\prime the ”base change” functor () kk :coPsh(M k)coPsh(M k )(-)\otimes_k k^\prime:co Psh(M_k)\to co Psh(M_{k^\prime}) induced by ()ϕ:M kM k (-)\circ \phi:M_k\to M_{k^\prime} given by postcompositions with ϕ\phi is called skalar extension.

Revised on June 1, 2012 16:09:47 by Stephan Alexander Spahn (178.195.221.252)