Which series are Hilbert series of graded modules over standard multigraded polynomial rings?

Katth?n Lukas; Moyano‐Fernández Julio José; Uliczka Jan

摘要

Abstract > Consider a polynomial ring R with the <mat:math display="inline" altimg="urn:x-wiley:0025584X:media:mana201800436:mana201800436-math-0001" wiley:location="equation/mana201800436-math-0001.png"> <mat:msup> <mat:mi mathvariant="double-struck">Z</mat:mi> <mat:mi>n</mat:mi> </mat:msup> </mat:math> ‐grading where the degree of each variable is a standard basis vector. In other words, R is the homogeneous coordinate ring of a product of n projective spaces. In this setting, we characterize the formal Laurent series which arise as Hilbert series of finitely generated R ‐modules. We also provide necessary conditions for a formal Laurent series to be the Hilbert series of a finitely generated module with a given depth. In the bigraded case (corresponding to the product of two projective spaces), we completely classify the Hilbert series of finitely generated modules of positive?depth. </abstract> </span> <span class="z_kbtn z_kbtnclass hoverxs" style="display: none;">展开▼</span> </div> <div class="translation abstracttxt"> <span class="zhankaihshouqi fivelineshidden" id="abstract"> <span>机译：</span><Abstract Type =“Main”XML：Lang =“en”> <标题类型=“main”>抽象</ title> > 考虑多项式环 R </ I> 与之 <mat：数学显示=“内联”Altimg =“URN：X-Wiley：0025584X：MANA201800436：MANA201800436-MANA201800436-MATH-0001”WIley：Location =“等式/ MANA201800436-MATH-0001.PNG”> <mat：msup> <mat：mi mathvariant =“双击”> z </ mat：mi> <mat：mi> n </ mat：mi> </ mat：msup> </ mat：math> - 在每个变量的程度是标准基础向量的地方。换句话说， R </ I> 是产品的均匀坐标环 n </ i> 投影空间。在这个设置中，我们表征了正式的Laurent系列，它是主要产生的Hilbert系列 R </ I> - 模块。我们还为正式的Laurent系列提供了必要的条件，以具有给定深度的有限生成模块的Hilbert系列。在Bigraded案例（对应于两个投射空间的产品）中，我们完全将希尔伯特系列的有限产生的正产生模块进行了分类的积极呢？深度。 </ p> </摘要> </span> <span class="z_kbtn z_kbtnclass hoverxs" style="display: none;">展开▼</span> </div> </div> <div class="record"> <h2 class="all_title" id="enpatent33" >著录项</h2> <ul> <li> <span class="lefttit">来源</span> <div style="width: 86%;vertical-align: text-top;display: inline-block;"> <a href='/journal-foreign-24615/'>《Mathematische Nachrichten》</a> <b style="margin: 0 2px;">|</b><span>2020年第1期</span><b style="margin: 0 2px;">|</b><span>共18页</span> </div> </li> <li> <div class="author"> <span class="lefttit">作者</span> <p id="fAuthorthree" class="threelineshidden zhankaihshouqi"> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Katth?n Lukas&option=202" target="_blank" rel="nofollow">Katth?n Lukas;</a> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Moyano‐Fernández Julio José&option=202" target="_blank" rel="nofollow">Moyano‐Fernández Julio José;</a> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Uliczka Jan&option=202" target="_blank" rel="nofollow">Uliczka Jan;</a> </p> <span class="z_kbtnclass z_kbtnclassall hoverxs" id="zkzz" style="display: none;">展开▼</span> </div> </li> <li> <div style="display: flex;"> <span class="lefttit">作者单位</span> <div style="position: relative;margin-left: 3px;max-width: 639px;"> <div class="threelineshidden zhankaihshouqi" id="fOrgthree"> <p>Goethe‐Universit?t FrankfurtFB Informatik und Mathematik60054 Frankfurt am Main Germany;</p> <p>Universitat Jaume I Campus de Riu SecDepartamento de Matemáticas &</p> <p>Institut Universitari de Matemàtiques i Aplicacions de Castelló12071 Castellón de la Plana Spain;</p> <p>Universit?t OsnabrückFB Mathematik/Informatik49069 Osnabrück Germany;</p> </div> <span class="z_kbtnclass z_kbtnclassall hoverxs" id="zhdw" style="display: none;">展开▼</span> </div> </div> </li> <li > <span class="lefttit">收录信息</span> <span style="width: 86%;vertical-align: text-top;display: inline-block;"></span> </li> <li> <span class="lefttit">原文格式</span> <span>PDF</span> </li> <li> <span class="lefttit">正文语种</span> <span>eng</span> </li> <li> <span class="lefttit">中图分类</span> <span><a href="https://www.zhangqiaokeyan.com/clc/156.html" title="数学">数学;</a></span> </li> <li class="antistop"> <span class="lefttit">关键词</span> <p style="width: 86%;vertical-align: text-top;"> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Hilbert polynomial&option=203" rel="nofollow">Hilbert polynomial;</a> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Hilbert series&option=203" rel="nofollow">Hilbert series;</a> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=multigrading&option=203" rel="nofollow">multigrading;</a> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=polynomial ring&option=203" rel="nofollow">polynomial ring;</a> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Primary: 13C05&option=203" rel="nofollow">Primary: 13C05;</a> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Secondary: 05E40&option=203" rel="nofollow">Secondary: 05E40;</a> <a style="color: #3E7FEB;" href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=13A02&option=203" rel="nofollow">13A02;</a> </p> <div class="translation"> 机译：希尔伯特多项式;希尔伯特系列;多项式;多项式环;初级：13C05;二次：05E40;13A02; </div> </li> </ul> </div> </div> <div class="literature cardcommon"> <div class="similarity "> <h3 class="all_title" id="enpatent66">相似文献</h3> <div class="similaritytab clearfix"> <ul> <li class="active" >外文文献</li> <li >中文文献</li> <li >专利</li> </ul> </div> <div class="similarity_details"> <ul > <li> <div> <b>1. </b><a class="enjiyixqcontent" href="/journal-foreign-detail/0704023596149.html">Which series are Hilbert series of graded modules over standard multigraded polynomial rings?</a> <b>[J]</b> . <span> <a href="/search.html?doctypes=4_5_6_1-0_4-0_1_2_3_7_9&sertext=Katth?n Lukas&option=202" target="_blank" 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href="/patent-detail/06130416522726.html">Modular technical system for manufacturing modular, patient data-specific, individualized foot sole-mold, has standard-modules, which are mounted zone-wise successively in series or partially adjacent to modular foot sole-mold</a> <b>[P]</b> . <span> 外国专利：  DE102012017881A1 </span> <span> . 2014-03-13</span> </div> <p class="zwjiyix translation" style="max-width: initial;height: auto;word-break: break-all;white-space: initial;text-overflow: initial;overflow: initial;"> <span>机译：用于制造模块化的，针对患者数据的个性化脚底模制件的模块化技术系统，具有标准模块，这些模块按区域顺序串联或部分邻近模块化脚底模制地安装。 </span> </p> </li> </ul> </div> </div> </div> <div class="theme cardcommon" style="overflow: auto;display:none"> <h3 class="all_title" id="enpatent55">相关主题</h3> <ul id="subject"> </ul> </div> </div> </div> </div> <div class="right rightcon"> <div class="details_img cardcommon clearfix" style="margin-bottom: 10px;display:none;" > </div> </div> </div> <div id="thesis_get_original1" class="downloadBth" 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