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

This entry is about a section of the text

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

Definition and Remark

Let $k$ be a field. Let ${\mathrm{Mf}}_k$ denote the category of finite dimensional $k$-rings.

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

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

3. Instead of defining ${\mathrm{fSch}}_k$ as the opposite of ${\mathrm{Mf}}_k$ define it covariantly on the category of finite dimensional $k$-corings.

4. A formal $k$-scheme is precisely a left exact (commutig with finite limits) functor $X:{\mathrm{Mf}}_k\to \mathrm{Set}$.

The inclusion ${\mathrm{Mf}}_k\hookrightarrow M_k$ induces a functor

$${}^{\hat{}}:{\mathrm{Sch}}_k\to {\mathrm{fSch}}_k$${}^\hat\; :Sch_k\to fSch_k

called completion functor.

3

k-coalgebra? and k-coring?

A $k$-coalgebra is a $k$-vector space $C$ equipped with a $k$-linear map $\Delta :C\to C{\otimes }_{k}C$.

A $k$-coalgebra $C$ is called a $k$-coring if $\Delta$ is

1. coassociative in that $(\Delta \otimes 1_C)\circ \Delta =(1_C\otimes \Delta )\circ \Delta$

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

3. has a counit $ϵ$ to $\Delta$ satisfying $\Delta \circ (1_C\otimes ϵ)=\Delta \circ (ϵ\otimes 1_C)=1_C$.

spectrum of a k-coring?

Let $A$ and $R$ be two finite $k$-rings, let $A^{*}$ denote the dual k-coring? of $A$.

Linear maps $A\to R$ correspond bijectively to elements of the tensor product $A^{*}\otimes R$. The $k$-linear maps ${\Delta }_{A^{*}}$ and ${ϵ}_{A^{*}}$ extends to $R$-linear maps

$$A^{*}\otimes R\to (A^{*}\otimes R){\otimes }_R(A^{*}\otimes R)$$A^*\otimes R\to (A^*\otimes R)\otimes_R(A^*\otimes R)

and

$$A^{*}\otimes R\to R$$A^*\otimes R\to R

denoted by $\Delta$ and $ϵ$.

Lemma and Definition

1. A $k$-linear map $A\to R$ associated to $u\in A^{*}\otimes R$ is a ring morphism iff $\Delta u=u\otimes u$ and $ϵu=1$.

2. There is a functorial isomorphism

$$\mathrm{Sp}(A(R)=\{u\in A^{*}\otimes R\mid \Delta u=u\otimes u,ϵu=1\}$$Sp(A(R)=\{u\in A^*\otimes R|\Delta u=u\otimes u, \epsilon u =1\}

and the $k$-formal spectrum of the coring $C$ is defined by

$${\mathrm{Sp}}^{*}C(R)=\{u\in C\otimes R\mid \Delta u=u\otimes u,ϵu=1\}$$Sp^* C(R)=\{u\in C\otimes R|\Delta u =u\otimes u, \epsilon u =1\}

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

References

• Michel Demazure, lectures on p-divisible groups web