Dévissage
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== Gruson and Raynaud's relative dévissages == |
== Gruson and Raynaud's relative dévissages == |
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Suppose that {{nowrap|f : ''X'' → ''S''}} is a finitely presented morphism of affine schemes, ''s'' is a point of ''S'', and ''M'' is a finite type ''O''''X''-module. If ''n'' is a natural number, then Gruson and Raynaud define an '''''S''-dévissage in dimension ''n''''' to consist of: |
Suppose that {{nowrap|f : ''X'' → ''S''}} is a finitely presented morphism of affine schemes, ''s'' is a point of ''S'', and ''M'' is a finite type ''O''''X''-module. If ''n'' is a [[natural number]], then Gruson and Raynaud define an '''''S''-dévissage in dimension ''n''''' to consist of: |
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# A closed finitely presented subscheme ''X''′ of ''X'' containing the closed subscheme defined by the annihilator of ''M'' and such that the dimension of {{nowrap|''X''′ ∩ f−1(''s'')}} is less than or equal to ''n''. |
# A closed finitely presented subscheme ''X''′ of ''X'' containing the closed subscheme defined by the annihilator of ''M'' and such that the dimension of {{nowrap|''X''′ ∩ f−1(''s'')}} is less than or equal to ''n''. |
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# A scheme ''T'' and a factorization {{nowrap|''X''′ → ''T'' → ''S''}} of the restriction of ''f'' to ''X''′ such that {{nowrap|''X''′ → ''T''}} is a finite morphism and {{nowrap|''T'' → ''S''}} is a smooth affine morphism with geometrically integral fibers of dimension ''n''. Denote the generic point of {{nowrap|''T'' ×''S'' ''k''(''s'')}} by τ and the pushforward of ''M'' to ''T'' by ''N''. |
# A scheme ''T'' and a factorization {{nowrap|''X''′ → ''T'' → ''S''}} of the restriction of ''f'' to ''X''′ such that {{nowrap|''X''′ → ''T''}} is a [[finite morphism]] and {{nowrap|''T'' → ''S''}} is a smooth affine morphism with geometrically integral fibers of dimension ''n''. Denote the generic point of {{nowrap|''T'' ×''S'' ''k''(''s'')}} by τ and the pushforward of ''M'' to ''T'' by ''N''. |
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# A free finite type ''O''''T''-module ''L'' and a homomorphism {{nowrap|α : ''L'' → ''N''}} such that {{nowrap|α ⊗ ''k''(τ)}} is bijective. |
# A free finite type ''O''''T''-module ''L'' and a homomorphism {{nowrap|α : ''L'' → ''N''}} such that {{nowrap|α ⊗ ''k''(τ)}} is bijective. |
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If ''n''1, ''n''2, ..., ''n''''r'' is a strictly decreasing sequence of natural numbers, then an ''S''-dévissage in dimensions ''n''1, ''n''2, ..., ''n''''r'' is defined recursively as: |
If ''n''1, ''n''2, ..., ''n''''r'' is a strictly decreasing sequence of natural numbers, then an ''S''-dévissage in dimensions ''n''1, ''n''2, ..., ''n''''r'' is defined recursively as: |
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# An ''S''-dévissage in dimension ''n''1. Denote the cokernel of α by ''P''1. |
# An ''S''-dévissage in dimension ''n''1. Denote the [[cokernel]] of α by ''P''1. |
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# An ''S''-dévissage in dimensions ''n''2, ..., ''n''''r'' of ''P''1. |
# An ''S''-dévissage in dimensions ''n''2, ..., ''n''''r'' of ''P''1. |
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The dévissage is said to lie between dimensions ''n''1 and ''n''''r''. ''r'' is called the '''length''' of the dévissage. The last step of the recursion consists of a dévissage in dimension ''n''''r'' which includes a morphism {{nowrap|α''r'' : ''L''''r'' → ''N''''r''}}. Denote the cokernel of this morphism by ''P''''r''. The dévissage is called '''total''' if ''P''''r'' is zero.{{Harvnb|Gruson|Raynaud|1971|pp=7–8}} |
The dévissage is said to lie between dimensions ''n''1 and ''n''''r''. ''r'' is called the '''length''' of the dévissage. The last step of the recursion consists of a dévissage in dimension ''n''''r'' which includes a morphism {{nowrap|α''r'' : ''L''''r'' → ''N''''r''}}. Denote the cokernel of this morphism by ''P''''r''. The dévissage is called '''total''' if ''P''''r'' is zero.{{Harvnb|Gruson|Raynaud|1971|pp=7–8}} |
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