The Stacks project

Use e78014e7 for etale loc structure nodal curves

There probably is a direct proof of of the etale local structure of a
family of nodal curves along the following lines, avoiding the use of
Popescu's theorem:

-- reduce to finite type over Z
-- take affines so we have A ---> B flat and some maximal ideal m of B
such that A ---> B has a node at m in the special fibre,
-- after shrinking B we may assume A ---> B is lci (already in SP)
-- write A = P/I as a quotient of a polynomial ring P
-- write B = B' / IB' for some P ---> B' lci (possible by result in SP)

This reduces us to the case where the base ring A is regular. This
remains true if we etale localize

-- using etale local structure of unramified maps reduce to the case
that Sing(B/A) maps isomorphically to a closed subscheme Z of Spec(A)
by etale localization
-- show that either Z = Spec(A) or Z is an effective Cartier divisor
(because true after completions)
-- conclude if Z = Spec(A), so may assume Z is an effective Cartier divisor
-- after localization may assume Z = V(f) for some nonzerodivisor f of A
-- show that, after etale localization, the spectrum of B/fB is the
scheme theoretic union of two closed subschemes Z_1 and Z_2 both
smooth of relative dimension 1 over Z and such that Z_1 \cap Z_2 maps
isomorphically to Z (again use that this is true after completion +
use that you may assume there is a section of Spec(B) --> \Spec(A)
into the smooth locus so you can pick out which component is the one
containing the section; also first reduce to the case where all
geom fibres are connected and have at most two irreducible components)
-- Show that Z_i are effective Cartier divisors in Spec(B) (as this is
true upon completion)
-- after localization Z_i = V(g_i) with g_i in B
-- conclude that g_1 g_2 = (unit in B) f in B because they scheme
theoretically define the same locus (bc true after completion)
-- conclude by multiplying g_2 by inverse of unit

Tags: Added new tags

Add missing reference

Etale local structure at-worst-nodal relative dim 1

Question: Is there a proof of this avoiding the use of
complete local rings and Artin approximation...