, Let S be a Dedekind scheme, with ring of rational functions K. Let A be a K-scheme. A Néron lft-model over S for A is the datum of a smooth separated scheme A ? S and a K-isomorphism ? : A × S K ? A satisfying the following universal property: for any smooth map of schemes
, A Néron lft-model differs from a Néron model in that the former is not required to be quasi-compact
, Let S be a trait and G a smooth separated S-group scheme. The following are equivalent: i) G is a Néron lft-model of its generic fibre
, the map G(S ) ? G(K ) is surjective. K : Spec K ? Pic X K /K . Then the fppf-quotient sheaf N = Pic X /S / cl(e K ) is representable by a smooth separated S-group scheme. Moreover, the quotient morphism
, K ) is a group algebraic space, smooth over S because Pic X /S is; as cl(e K ) is closed in Pic X /S , N is separated over S. In particular, N is a separated group algebraic space locally of finite type over S, so it is a group scheme by, Proof. As cl(e K ) is flat over S, the fppf-quotient of sheaves N = Pic X /S / cl
, 2, ii)) is injective, hence the intersection of cl(e K ) with the identity component Pic 0
, S is trivial and it follows that cl
, Let S sh ? S be a strict henselization of S with respect to some algebraic closure of the residue field, and denote by K sh its fraction field. If (?, l) is not circuit-coprime, the map Pic(X S sh ) ? Pic
, Now, as the special fibre of X S sh /S sh is generically smooth, X S sh ? S sh admits a section
, As the quotient Pic X /S ? N is anétale surjective morphism of S sh -algebraic spaces (lemma 13.5), the map Pic X /S (S sh ) ? N (S sh ) is surjective. We deduce that N (S sh ) ? Pic X K /K (K sh ) is not surjective
, ) is not surjective, hence N is not a Néron model of Pic X K /K . Now assume that (?, l) is circuit coprime. Assume first that S is strictly henselian. By proposition 13.4 it is enough to prove that for all essentially smooth local extensions R ? R of discrete valuation rings
, The two maps p * 1 g, p * 2 g : q * T ? q * N both coincide with q * f when restricted to q * T K . As q * T ? S is flat, q * T K is schematically dense in q * T . Since moreover q * N is separated, we have that p * 1 g = p * 2 g. Hence g descends to a morphism T ? N extending f . Again, the extension is unique because N ? S is separated and T K is schematically dense in T, is surjective. As X ? S admits a section, we may apply lemma 13.2 and just show that Pic(X R ) ? Pic(X K ) is surjective. The map R ? R has ramification index 1, vol.1
, It is enough to check that the labelled graph ( ?, l) of X ? S is circuitcoprime, by the previous Theorem. As labelled graphs are preserved unde? etale extensions of the base trait, we may assume that X ? S has special fibre with split singularities
, Let k be a separable closure of the residue field of S and suppose that the graph of X k is a tree. Then N = Pic X /S / cl(e K ) is a Néron lft-model for Pic X K /K over S. We have shown how to construct Néron lft-models for the group scheme Pic X K /K , without ever imposing bounds on the degree of line bundles; the following lemma allows us to retrieve lft-Néron models for subgroup schemes of Pic X K /K , and applies in particular to subgroup schemes that are open and closed, Corollary 13.8. Let X ? S be a nodal curve over a trait with perfect fraction field K
,
, Let X /S be a nodal curve over a trait, and H ? Pic X K /K a K-smooth closed subgroup scheme of Pic X K /K . Let N ? S be the Néron model of Pic X K /K . Then H admits a Néron lft-model H over S
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