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On top local cohomology modules, Matlis duality and tensor products

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M.Y. Sadeghi, M. Eghbali, Kh. Ahmadi-Amoli

Commutative Algebra · Submitted 6 Feb 2013

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim In the case that $c=\dim R$, we first give a bound for depth~$D(H^c_\mathfrak{a}(R))$, where $c>2$ and $(R,\mathfrak{m})$ is complete.

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arXiv ID1302.1274 (version 2)
Pith IDZ4RVIJ4L, Pith's permanent record of this paper and its review
CitationKH. Ahmadi-amoli, M. Eghbali, M.Y. Sadeghi. "On top local cohomology modules, Matlis duality and tensor products." arXiv:1302.1274 (2013). https://pith.science/paper/Z4RVIJ4L
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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,966 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.29 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.63 · compares with prior results 0.06 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

Let $\mathfrak{a}$ be an ideal of a local ring $(R, \mathfrak{m})$ with $c = \mathrm{cd}(\mathfrak{a},R)$ the cohomological dimension of $\mathfrak{a}$ in $R$. In the case that $c=\dim R$, we first give a bound for depth~$D(H^c_\mathfrak{a}(R))$, where $c>2$ and $(R,\mathfrak{m})$ is complete. Later, $H^c_\mathfrak{a}(R) \otimes_R H^c_\mathfrak{a}(R)$, $D(H^c_\mathfrak{a}(R)) \otimes_R D(H^c_\mathfrak{a}(R))$ and $H^c_\mathfrak{a}(R) \otimes_R D(H^c_\mathfrak{a}(R))$ are examined. In the case $c=\dim R-1$, the set Att$_R H^c_\mathfrak{a}(R)$ is considered.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Let $\mathfrak{a}$ be an ideal of a local ring $(R, \mathfrak{m})$ with $c = \mathrm{cd}(\mathfrak{a},R)$ the cohomological dimension of $\mathfrak{a}$ in $R$. background 1.00 0.08 0.02 asserted 0.57
s1 In the case that $c=\dim R$, we first give a bound for depth~$D(H^c_\mathfrak{a}(R))$, where $c>2$ and $(R,\mathfrak{m})$ is complete. contribution 0.61 0.92 0.16 asserted 1.00
s2 Later, $H^c_\mathfrak{a}(R) \otimes_R H^c_\mathfrak{a}(R)$, $D(H^c_\mathfrak{a}(R)) \otimes_R D(H^c_\mathfrak{a}(R))$ and $H^c_\mathfrak{a}(R) \otimes_R D(H^c_\mathfrak{a}(R))$ are examined. contribution 0.99 0.88 0.11 asserted 0.89
s3 In the case $c=\dim R-1$, the set Att$_R H^c_\mathfrak{a}(R)$ is considered. contribution 0.99 0.85 0.07 asserted 0.79

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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sources read
  • arXiv listing abstract v2
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
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  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

Let $\mathfrak{a}$ be an ideal of a local ring $(R, \mathfrak{m})$ with $c = \mathrm{cd}(\mathfrak{a},R)$ the cohomological dimension of $\mathfrak{a}$ in $R$. In the case that $c=\dim R$, we first give a bound for depth~$D(H^c_\mathfrak{a}(R))$, where $c>2$ and $(R,\mathfrak{m})$ is complete. Later, $H^c_\mathfrak{a}(R) \otimes_R H^c_\mathfrak{a}(R)$, $D(H^c_\mathfrak{a}(R)) \otimes_R D(H^c_\mathfrak{a}(R))$ and $H^c_\mathfrak{a}(R) \otimes_R D(H^c_\mathfrak{a}(R))$ are examined. In the case $c=\dim R-1$, the set Att$_R H^c_\mathfrak{a}(R)$ is considered.

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Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

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