Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: An inequality involving tight closure and parameter ideals   We establish an inequality involving colengths of the tight closure of ideals
of systems of parameters in local rings with some mild conditions. As an
application, we prove and refine a result by Goto and Nakamura, conjectured by
Watanabe and Yoshida, which states that the Hilbert-Samuel multiplicity of a
parameter ideal is greater than or equal to the colength of the tight closure
of the ideal.

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The categories of this paper are: supr-con
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The categories of this paper are: atom-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: hep-ph
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The categories of this paper are: hep-th
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The categories of this paper are: nucl-th
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The categories of this paper are: cmp-lg
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The categories of this paper are: math-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: nlin
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The categories of this paper are: ao-sci
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The categories of this paper are: solv-int
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The categories of this paper are: plasm-ph
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The categories of this paper are: q-bio
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The categories of this paper are: gr-qc
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The categories of this paper are: eess