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Demazure, lectures on p-divisible groups, I.6, the four definitions of formal schemes

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Definition and Remark

Let kk be a field. Let Mf kMf_k denote the category of finite dimensional kk-rings.

  1. A kk-scheme is called a kk-formal scheme if it is is equivalent to a codirected colimit of finite (affine) kk-schemes.

  2. A kk-scheme is a kk-formal scheme if it is presented by a profinite kk-ring or -equivalently- by a kk-ring which is the limit of discrete quotients which are finite kk-rings. If AA is such a topological kk-ring Spf k(A)(R)Spf_k(A)(R) denotes the set of continous morphisms from AA to the topological discrete ring RR. We have Spf kSpf_k is a contravariant equivalence between the category of profinite kk-rings and the category fSch kfSch_k of formal kk-schemes.

  3. Instead of defining fSch kfSch_k as the opposite of Mf kMf_k define it covariantly on the category of finite dimensional kk-corings.

  4. A formal kk-scheme is precisely a left exact (commutig with finite limits) functor X:Mf kSetX:Mf_k\to Set.

The inclusion Mf kM kMf_k\hookrightarrow M_k induces a functor

^:Sch kfSch k{}^\hat\; :Sch_k\to fSch_k

called completion functor.

3

k-coalgebra? and k-coring?

A kk-coalgebra is a kk-vector space CC equipped with a kk-linear map Δ:CC kC\Delta:C\to C\otimes_k C.

A kk-coalgebra CC is called a kk-coring if Δ\Delta is

  1. coassociative in that (Δ1 C)Δ=(1 CΔ)Δ(\Delta\otimes 1_C)\circ \Delta=(1_C\otimes \Delta)\circ \Delta

  2. cocommutative in that the image of Δ\Delta consists only of symmetric tensors.

  3. has a counit ϵ\epsilon to Δ\Delta satisfying Δ(1 Cϵ)=Δ(ϵ1 C)=1 C\Delta\circ (1_C\otimes \epsilon)=\Delta\circ (\epsilon\otimes 1_C)=1_C.

spectrum of a k-coring?

Let AA and RR be two finite kk-rings, let A *A^* denote the dual k-coring? of AA.

Linear maps ARA\to R correspond bijectively to elements of the tensor product A *RA^*\otimes R. The kk-linear maps Δ A *\Delta_{A^*} and ϵ A *\epsilon_{A^*} extends to RR-linear maps

A *R(A *R) R(A *R)A^*\otimes R\to (A^*\otimes R)\otimes_R(A^*\otimes R)

and

A *RRA^*\otimes R\to R

denoted by Δ\Delta and ϵ\epsilon.

Lemma and Definition
  1. A kk-linear map ARA\to R associated to uA *Ru\in A^*\otimes R is a ring morphism iff Δu=uu\Delta u= u\otimes u and ϵu=1\epsilon u=1.

  2. There is a functorial isomorphism

Sp(A(R)={uA *R|Δu=uu,ϵu=1}Sp(A(R)=\{u\in A^*\otimes R|\Delta u=u\otimes u, \epsilon u =1\}

and the kk-formal spectrum of the coring CC is defined by

Sp *C(R)={uCR|Δu=uu,ϵu=1}Sp^* C(R)=\{u\in C\otimes R|\Delta u =u\otimes u, \epsilon u =1\}

in particular Sp *:M k opcoPsh(M k)Sp^*: M_k^{op}\to coPsh(M_k) is a covariant functor from the category of kk-corings to the category of kk-formal functors.

References

  • Michel Demazure, lectures on p-divisible groups web
Revised on May 27, 2012 13:09:49 by Stephan Alexander Spahn (79.227.168.80)