Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Maximal Cohen-Macaulay approximations and Serre's condition   This paper studies the relationship between Serre's condition $(\R_n)$ and
Auslander--Buchweitz's maximal Cohen--Macaulay approximations. It is proved
that a Gorenstein local ring satisfies $(\R_n)$ if and only if every maximal
Cohen--Macaulay module is a direct summand of a maximal Cohen--Macaulay
approximation of a (Cohen--Macaulay) module of codimension $n+1$.

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The categories of this paper are: cs
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The categories of this paper are: funct-an
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The categories of this paper are: astro-ph
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The categories of this paper are: hep-ex
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The categories of this paper are: bayes-an
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The categories of this paper are: supr-con
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The categories of this paper are: dg-ga
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The categories of this paper are: alg-geom
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The categories of this paper are: physics
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The categories of this paper are: cmp-lg
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The categories of this paper are: adap-org
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The categories of this paper are: hep-lat
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The categories of this paper are: cond-mat
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-ph
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The categories of this paper are: patt-sol