Canonical mapping

If T is a finite locally free OslpkOs module (k < n) then Tv is a finite locally free Os/pkOs-module too. The purpose of this section is to define a canonical map... 

Remark 6.4. In particular this proposition implies that the canonical map M G) Ms Gyy is an isomorphism always (use Remark 6.2). In other words, the sheaves Ms(G) for G ObC(n)s0 are reflexive Os-modules. 

If the residue field k of R is perfect then there is a unique section of the canonical map Wn+i(k) — R compatible with F on R and Frobenius on the Witt vectors. 

Theorem. Let G be an algebraic affine group scheme. Then 7c0(Jc[G]) represents an etale group n0 G, and all maps from G to etale groups factor through the canonical map G - jr0 G. The kernel G° of this map is a connected closed normal subgroup represented by the factor ofk[G] on which s is nonzero. The construction of ic0G and G° commutes with base extension. 

Actually, all the groups used in the following are Abelian. Thus for each subgroup // of G, a factor group G/H is defined. The function that maps each element of G to its class in G/H is called the canonical mapping. 

In particular, if Xg = Xf, we have canonical maps Rf - Rg and Rg -> Rf which are inverse to each other, so we can identify Rf and Rg. Therefore we really can associate a ring Rf to each open set Xf. Furthermore, whenever Xk C Xg C Xf, we get a commutative diagram of canonical maps ... 

This is independent of the choice of dt because changing dt multiplies the triple by a scalar. This procedure works in any genus and defines the so-called canonical map... 

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