{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T04:28:22Z","timestamp":1750220902979,"version":"3.41.0"},"reference-count":8,"publisher":"Association for Computing Machinery (ACM)","issue":"4","license":[{"start":{"date-parts":[[2019,5,30]],"date-time":"2019-05-30T00:00:00Z","timestamp":1559174400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.acm.org\/publications\/policies\/copyright_policy#Background"}],"content-domain":{"domain":["dl.acm.org"],"crossmark-restriction":true},"short-container-title":["ACM Commun. Comput. Algebra"],"published-print":{"date-parts":[[2019,5,30]]},"abstract":"\n Let\n R<\/jats:italic>\n :=\n K<\/jats:italic>\n [\n x<\/jats:italic>\n 1<\/jats:sub>\n ,...,\n \n x\n n<\/jats:sub>\n <\/jats:italic>\n ] be a polynomial ring over an infinite field\n K<\/jats:italic>\n , and let\n I<\/jats:italic>\n \u2282\n R<\/jats:italic>\n be a homogeneous ideal with respect to a weight vector \u03c9 = (\u03c9\n 1<\/jats:sub>\n , ..., \u03c9\n \n n<\/jats:italic>\n <\/jats:sub>\n ) \u2208 (Z\n +<\/jats:sup>\n )\n \n n<\/jats:italic>\n <\/jats:sup>\n such that dim(\n R\/I<\/jats:italic>\n ) =\n d.<\/jats:italic>\n We consider the minimal graded free resolution of\n R\/I<\/jats:italic>\n as\n A<\/jats:italic>\n -module, that we call the Noether resolution of\n R\/I<\/jats:italic>\n , whenever\n A<\/jats:italic>\n :=\n K<\/jats:italic>\n [\n x<\/jats:italic>\n \n n-d<\/jats:italic>\n +1\n <\/jats:sub>\n ,...,\n \n x\n n<\/jats:sub>\n <\/jats:italic>\n ] is a Noether normalization of\n R\/I.<\/jats:italic>\n When\n d<\/jats:italic>\n = 2 and\n I<\/jats:italic>\n is saturated, we give an algorithm for obtaining this resolution that involves the computation of a minimal Gr\u00f6bner basis of\n I<\/jats:italic>\n with respect to the weighted degree reverse lexicographic order. In the particular case when\n R\/I<\/jats:italic>\n is a 2-dimensional semigroup ring, we also describe the multigraded version of this resolution in terms of the underlying semigroup. Whenever we have the Noether resolution of\n R\/I<\/jats:italic>\n or its multigraded version, we obtain formulas for the corresponding Hilbert series of\n R\/I<\/jats:italic>\n , and when\n I<\/jats:italic>\n is homogeneous, we obtain a formula for the Castelnuovo-Mumford regularity of\n R\/I.<\/jats:italic>\n <\/jats:p>\n As an application of the results for 2-dimensional semigroup rings, we provide a new upper bound for the Castelnuovo-Mumford regularity of the coordinate ring of a projective monomial curve. Finally, we describe the multigraded Noether resolution of either the coordinate ring of a projective monomial curve [EQUATION] associated to an arithmetic sequence or of any of its canonical projections [EQUATION].<\/jats:p>","DOI":"10.1145\/3338637.3338638","type":"journal-article","created":{"date-parts":[[2019,5,31]],"date-time":"2019-05-31T12:37:11Z","timestamp":1559306231000},"page":"114-116","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["Noether resolutions in dimension 2"],"prefix":"10.1145","volume":"52","author":[{"given":"I.","family":"Bermejo","sequence":"first","affiliation":[]},{"given":"E.","family":"Garc\u00eda-Llorente","sequence":"additional","affiliation":[]},{"given":"I.","family":"Garc\u00eda-Marco","sequence":"additional","affiliation":[]},{"given":"M.","family":"Morales","sequence":"additional","affiliation":[]}],"member":"320","published-online":{"date-parts":[[2019,5,30]]},"reference":[{"key":"e_1_2_1_1_1","volume-title":"A criterion for detecting m-regularity. Invent. Math. 87<\/b>","author":"Bayer D.","year":"1987","unstructured":"D. Bayer , M. Stillman , A criterion for detecting m-regularity. Invent. Math. 87<\/b> ( 1987 ), no. 1, 1--11 D. Bayer, M. Stillman, A criterion for detecting m-regularity. Invent. Math. 87<\/b> (1987), no. 1, 1--11"},{"key":"e_1_2_1_2_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jsc.2016.11.001"},{"key":"e_1_2_1_3_1","doi-asserted-by":"publisher","DOI":"10.1016\/j.jalgebra.2017.03.026"},{"key":"e_1_2_1_4_1","volume-title":"Effective methods in algebraic geometry (Bath","author":"Bermejo I.","year":"2000","unstructured":"I. Bermejo , Ph. Gimenez , Computing the Castelnuovo-Mumford regularity of some subschemes of Pn<\/sup>K<\/sub> using quotients of monomial ideals , Effective methods in algebraic geometry (Bath , 2000 ). J. Pure Appl. Algebra bf 164 (2001), no. 1-2, 23--33. I. Bermejo, Ph. Gimenez, Computing the Castelnuovo-Mumford regularity of some subschemes of Pn<\/sup>K<\/sub> using quotients of monomial ideals, Effective methods in algebraic geometry (Bath, 2000). J. Pure Appl. Algebra bf 164 (2001), no. 1-2, 23--33."},{"key":"e_1_2_1_5_1","first-page":"2","article-title":"J","volume":"303","author":"Bermejo I.","year":"2006","unstructured":"I. Bermejo , Ph. Gimenez , Saturation and Castelnuovo-Mumford regularity. J . Algebra 303 ( 2006 ), no. 2 , 592--617. I. Bermejo, Ph. Gimenez, Saturation and Castelnuovo-Mumford regularity. J. Algebra 303 (2006), no. 2, 592--617.","journal-title":"Algebra"},{"key":"e_1_2_1_6_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01398398"},{"key":"e_1_2_1_7_1","doi-asserted-by":"publisher","DOI":"10.1007\/BF01445273"},{"key":"e_1_2_1_8_1","volume-title":"On the Hilbert function of certain rings of monomial curves. J. Pure Appl. Algebra 101<\/b>","author":"Molinelli S.","year":"1995","unstructured":"S. Molinelli , G. Tamone , On the Hilbert function of certain rings of monomial curves. J. Pure Appl. Algebra 101<\/b> ( 1995 ), no. 2, 191--206 S. Molinelli, G. Tamone, On the Hilbert function of certain rings of monomial curves. J. Pure Appl. Algebra 101<\/b> (1995), no. 2, 191--206"}],"container-title":["ACM Communications in Computer Algebra"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3338637.3338638","content-type":"unspecified","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/dl.acm.org\/doi\/pdf\/10.1145\/3338637.3338638","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,17]],"date-time":"2025-06-17T23:44:47Z","timestamp":1750203887000},"score":1,"resource":{"primary":{"URL":"https:\/\/dl.acm.org\/doi\/10.1145\/3338637.3338638"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,5,30]]},"references-count":8,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2019,5,30]]}},"alternative-id":["10.1145\/3338637.3338638"],"URL":"https:\/\/doi.org\/10.1145\/3338637.3338638","relation":{},"ISSN":["1932-2240"],"issn-type":[{"type":"print","value":"1932-2240"}],"subject":[],"published":{"date-parts":[[2019,5,30]]},"assertion":[{"value":"2019-05-30","order":2,"name":"published","label":"Published","group":{"name":"publication_history","label":"Publication History"}}]}}