Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$(R,{\mathfrak {m}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>R</mi> <mo>,</mo> <mi mathvariant="fraktur">m</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be a Noetherian local ring and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${\mathfrak {a}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">a</mi> </math></EquationSource> </InlineEquation> be an ideal of R. Suppose that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\operatorname {height}\,({\mathfrak {a}}\hat{R}+\mathfrak P)/{\mathfrak {P}}\le 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>height</mo> <mspace width="0.166667em" /> <mo stretchy="false">(</mo> <mi mathvariant="fraktur">a</mi> <mover accent="true"> <mi>R</mi> <mo stretchy="false">^</mo> </mover> <mo>+</mo> <mi mathvariant="fraktur">P</mi> <mo stretchy="false">)</mo> <mo stretchy="false">/</mo> <mi mathvariant="fraktur">P</mi> <mo>≤</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, for every <InlineEquation ID="IEq4"> <EquationSource Format="TEX">${\mathfrak {P}}\in {{\,\textrm{Spec}\,}}\hat{R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="fraktur">P</mi> <mo>∈</mo> <mrow> <mspace width="0.166667em" /> <mtext>Spec</mtext> <mspace width="0.166667em" /> </mrow> <mover accent="true"> <mi>R</mi> <mo stretchy="false">^</mo> </mover> </mrow> </math></EquationSource> </InlineEquation>. In this paper, it is shown that the category of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">${\mathfrak {a}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="fraktur">a</mi> </math></EquationSource> </InlineEquation>-cofinite modules forms an Abelian subcategory of the category of all R-modules. This assertion provides a partially affirmative answer to a question raised by R. Hartshorne in [Affine duality and cofiniteness, Invent. Math. 9 (1970), 145-164].

An Abelian category of cofinite modules

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2025年10月19日

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Moharram Aghapournahr,
MirYousef Sadeghi

摘要

Let $(R,{\mathfrak {m}})$ $(R, m)$ be a Noetherian local ring and ${\mathfrak {a}}$ $a$ be an ideal of R. Suppose that $\operatorname {height}\,({\mathfrak {a}}\hat{R}+\mathfrak P)/{\mathfrak {P}}\le 1$ $height (a \hat{R} + P) / P \leq 1$ , for every ${\mathfrak {P}}\in {{\,\textrm{Spec}\,}}\hat{R}$ $P \in Spec \hat{R}$ . In this paper, it is shown that the category of ${\mathfrak {a}}$ $a$ -cofinite modules forms an Abelian subcategory of the category of all R-modules. This assertion provides a partially affirmative answer to a question raised by R. Hartshorne in [Affine duality and cofiniteness, Invent. Math. 9 (1970), 145-164].