FRINK Linked Data Fragments server

Matches in SemOpenAlex for { <https://semopenalex.org/work/W2963484736> ?p ?o ?g. }

• W2963484736 endingPage "47" @default.
• W2963484736 startingPage "35" @default.
• W2963484736 abstract "Given an essentially finite type morphism of schemes <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f colon upper X right-arrow upper Y> <mml:semantics> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:<!-- : --></mml:mo> <mml:mi>X</mml:mi> <mml:mo stretchy=false>→<!-- → --></mml:mo> <mml:mi>Y</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>fcolon Xto Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and a positive integer <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding=application/x-tex>d</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, let <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f Superscript StartSet d EndSet Baseline colon upper X Superscript StartSet d EndSet Baseline right-arrow upper Y> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>f</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:mi>d</mml:mi> <mml:mo fence=false stretchy=false>}</mml:mo> </mml:mrow> </mml:msup> <mml:mo>:<!-- : --></mml:mo> <mml:msup> <mml:mi>X</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:mi>d</mml:mi> <mml:mo fence=false stretchy=false>}</mml:mo> </mml:mrow> </mml:msup> <mml:mo stretchy=false>→<!-- → --></mml:mo> <mml:mi>Y</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>f^{{d}}colon X^{{d}}to Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> denote the natural map from the <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d> <mml:semantics> <mml:mi>d</mml:mi> <mml:annotation encoding=application/x-tex>d</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-fold fiber product <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X Superscript StartSet d EndSet Baseline equals upper X times Subscript upper Y Baseline midline-horizontal-ellipsis times Subscript upper Y Baseline upper X> <mml:semantics> <mml:mrow> <mml:msup> <mml:mi>X</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:mi>d</mml:mi> <mml:mo fence=false stretchy=false>}</mml:mo> </mml:mrow> </mml:msup> <mml:mo>=</mml:mo> <mml:mi>X</mml:mi> <mml:msub> <mml:mo>×<!-- × --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>Y</mml:mi> </mml:mrow> </mml:msub> <mml:mo>⋯<!-- ⋯ --></mml:mo> <mml:msub> <mml:mo>×<!-- × --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>Y</mml:mi> </mml:mrow> </mml:msub> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>X^{{d}}= Xtimes _{Y}cdots times _{Y}X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=pi Subscript i Baseline colon upper X Superscript StartSet d EndSet Baseline right-arrow upper X> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>π<!-- π --></mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>:<!-- : --></mml:mo> <mml:msup> <mml:mi>X</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:mi>d</mml:mi> <mml:mo fence=false stretchy=false>}</mml:mo> </mml:mrow> </mml:msup> <mml:mo stretchy=false>→<!-- → --></mml:mo> <mml:mi>X</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>pi _icolon X^{{d}}to X</mml:annotation> </mml:semantics> </mml:math> </inline-formula> the <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=i> <mml:semantics> <mml:mi>i</mml:mi> <mml:annotation encoding=application/x-tex>i</mml:annotation> </mml:semantics> </mml:math> </inline-formula>th canonical projection. When <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper Y> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding=application/x-tex>Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is smooth over a field and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper F> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>F</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a coherent sheaf on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper X> <mml:semantics> <mml:mi>X</mml:mi> <mml:annotation encoding=application/x-tex>X</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, it is proved that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper F> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>F</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal F</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is flat over <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper Y> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding=application/x-tex>Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula> if (and only if) <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=f Superscript StartSet d EndSet> <mml:semantics> <mml:msup> <mml:mi>f</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mo fence=false stretchy=false>{</mml:mo> <mml:mi>d</mml:mi> <mml:mo fence=false stretchy=false>}</mml:mo> </mml:mrow> </mml:msup> <mml:annotation encoding=application/x-tex>f^{{d}}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> maps the associated points of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=circled-times Underscript i equals 1 Overscript d Endscripts pi Subscript i Superscript asterisk Baseline script upper F> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:munderover> <mml:mo>⨂<!-- ⨂ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mi>d</mml:mi> </mml:munderover> </mml:mrow> <mml:msubsup> <mml:mi>π<!-- π --></mml:mi> <mml:mi>i</mml:mi> <mml:mo>∗<!-- ∗ --></mml:mo> </mml:msubsup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>F</mml:mi> </mml:mrow> </mml:mrow> </mml:mrow> <mml:annotation encoding=application/x-tex>{bigotimes _{i=1}^d}pi _i^*{mathcal F}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> to generic points of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper Y> <mml:semantics> <mml:mi>Y</mml:mi> <mml:annotation encoding=application/x-tex>Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, for some <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=d greater-than-or-equal-to dimension upper Y> <mml:semantics> <mml:mrow> <mml:mi>d</mml:mi> <mml:mo>≥<!-- ≥ --></mml:mo> <mml:mi>dim</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mi>Y</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>dge dim Y</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The equivalent statement in commutative algebra is an analog—but not a consequence—of a classical criterion of Auslander and Lichtenbaum for the freeness of finitely generated modules over regular local rings." @default.
• W2963484736 created "2019-07-30" @default.
• W2963484736 creator A5041807524 @default.
• W2963484736 creator A5056968077 @default.
• W2963484736 date "2012-05-21" @default.
• W2963484736 modified "2023-09-23" @default.
• W2963484736 title "Detecting flatness over smooth bases" @default.
• W2963484736 cites W1508234517 @default.
• W2963484736 cites W1979671175 @default.
• W2963484736 cites W2081448924 @default.
• W2963484736 cites W2319559022 @default.
• W2963484736 cites W2530969432 @default.
• W2963484736 cites W4246932132 @default.
• W2963484736 cites W4248269519 @default.
• W2963484736 cites W4255719948 @default.
• W2963484736 doi "https://doi.org/10.1090/s1056-3911-2012-00579-7" @default.
• W2963484736 hasPublicationYear "2012" @default.
• W2963484736 type Work @default.
• W2963484736 sameAs 2963484736 @default.
• W2963484736 citedByCount "5" @default.
• W2963484736 countsByYear W29634847362012 @default.
• W2963484736 countsByYear W29634847362013 @default.
• W2963484736 countsByYear W29634847362015 @default.
• W2963484736 crossrefType "journal-article" @default.
• W2963484736 hasAuthorship W2963484736A5041807524 @default.
• W2963484736 hasAuthorship W2963484736A5056968077 @default.
• W2963484736 hasBestOaLocation W29634847361 @default.
• W2963484736 hasConcept C11413529 @default.
• W2963484736 hasConcept C154945302 @default.
• W2963484736 hasConcept C41008148 @default.
• W2963484736 hasConceptScore W2963484736C11413529 @default.
• W2963484736 hasConceptScore W2963484736C154945302 @default.
• W2963484736 hasConceptScore W2963484736C41008148 @default.
• W2963484736 hasIssue "1" @default.
• W2963484736 hasLocation W29634847361 @default.
• W2963484736 hasLocation W29634847362 @default.
• W2963484736 hasLocation W29634847363 @default.
• W2963484736 hasOpenAccess W2963484736 @default.
• W2963484736 hasPrimaryLocation W29634847361 @default.
• W2963484736 hasRelatedWork W2003465964 @default.
• W2963484736 hasRelatedWork W2051487156 @default.
• W2963484736 hasRelatedWork W2333698505 @default.
• W2963484736 hasRelatedWork W2351491280 @default.
• W2963484736 hasRelatedWork W2371447506 @default.
• W2963484736 hasRelatedWork W2386767533 @default.
• W2963484736 hasRelatedWork W2748952813 @default.
• W2963484736 hasRelatedWork W2899084033 @default.
• W2963484736 hasRelatedWork W303980170 @default.
• W2963484736 hasRelatedWork W3107474891 @default.
• W2963484736 hasVolume "22" @default.
• W2963484736 isParatext "false" @default.
• W2963484736 isRetracted "false" @default.
• W2963484736 magId "2963484736" @default.
• W2963484736 workType "article" @default.