On the depth and Stanley depth of the integral closure of powers of monomial ideals

• Autores: S. A. Seyed Fakhari
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 70, Fasc. 3, 2019, págs. 447-459
• Idioma: inglés
• DOI: 10.1007/s13348-019-00240-x
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• Resumen

  • Let ? be a field and ?=?[?1,…,??] be the polynomial ring in n variables over ? . For any monomial ideal I, we denote its integral closure by ?⎯⎯⎯ . Assume that G is a graph with edge ideal I(G). We prove that the modules ?/?(?)?⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ and ?(?)?⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯/?(?)?+1⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ satisfy Stanley’s inequality for every integer ?≫0 . If G is a non-bipartite graph, we show that the ideals ?(?)?⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯ satisfy Stanley’s inequality for all ?≫0 . For every connected bipartite graph G (with at least one edge), we prove that sdepth(?(?)?)≥2 , for any positive integer ?≤girth(?)/2+1 . This result partially answers a question asked in Seyed Fakhari (J Algebra 489:463–474, 2017). For any proper monomial ideal I of S, it is shown that the sequence {depth(??⎯⎯⎯⎯⎯⎯/??+1⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯)}∞?=0 is convergent and lim?→∞depth(??⎯⎯⎯⎯⎯⎯/??+1⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯⎯)=?−ℓ(?) , where ℓ(?) denotes the analytic spread of I. Furthermore, it is proved that for any monomial ideal I, there exists an integer s such that depth(?/???)≤depth(?/?⎯⎯⎯), for every integer ?≥1 . We also determine a value s for which the above inequality holds. If I is an integrally closed ideal, we show that depth(?/??)≤depth(?/?) , for every integer ?≥1 . As a consequence, we obtain that for any integrally closed monomial ideal I and any integer ?≥1 , we have Ass(?/?)⊆Ass(?/??) .