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• W2066757435 abstract "We investigate commutative Noetherian rings of prime characteristic such that the Frobenius functor applied to any injective module is again injective. We characterize the class of one-dimensional local rings with this property and show that it includes all one-dimensional <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper F> <mml:semantics> <mml:mi>F</mml:mi> <mml:annotation encoding=application/x-tex>F</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-pure rings. We also give a characterization of Gorenstein local rings in terms of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper T normal o normal r Subscript i Superscript upper R Baseline left-parenthesis upper R Superscript f Baseline comma upper E right-parenthesis> <mml:semantics> <mml:mrow> <mml:msubsup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>T</mml:mi> <mml:mi mathvariant=normal>o</mml:mi> <mml:mi mathvariant=normal>r</mml:mi> </mml:mrow> <mml:mi>i</mml:mi> <mml:mi>R</mml:mi> </mml:msubsup> <mml:mo stretchy=false>(</mml:mo> <mml:msup> <mml:mi>R</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>f</mml:mi> </mml:mrow> </mml:msup> <mml:mo>,</mml:mo> <mml:mi>E</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>mathrm {Tor}_i^R(R^{f},E)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper E> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding=application/x-tex>E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the injective hull of the residue field and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper R Superscript f> <mml:semantics> <mml:msup> <mml:mi>R</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>f</mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding=application/x-tex>R^{f}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the ring <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper R> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding=application/x-tex>R</mml:annotation> </mml:semantics> </mml:math> </inline-formula> whose right <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper R> <mml:semantics> <mml:mi>R</mml:mi> <mml:annotation encoding=application/x-tex>R</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-module action is given by the Frobenius map." @default.
• W2066757435 created "2016-06-24" @default.
• W2066757435 creator A5019258920 @default.
• W2066757435 date "2014-03-03" @default.
• W2066757435 modified "2023-09-28" @default.
• W2066757435 title "The Frobenius functor and injective modules" @default.
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• W2066757435 doi "https://doi.org/10.1090/s0002-9939-2014-11924-1" @default.
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