{"title":"论完全相交的局部除数类群","authors":"Daniel Windisch","doi":"10.1016/j.jpaa.2024.107804","DOIUrl":null,"url":null,"abstract":"<div><p>Samuel conjectured in 1961 that a (Noetherian) local complete intersection ring that is a UFD in codimension at most three is itself a UFD. It is said that Grothendieck invented local cohomology to prove this fact. Following the philosophy that a UFD is nothing else than a Krull domain (that is, a normal domain, in the Noetherian case) with trivial divisor class group, we take a closer look at the Samuel–Grothendieck Theorem and prove the following generalization: Let <em>A</em> be a local Cohen–Macaulay ring.</p><ul><li><span>(1)</span><span><p><em>A</em> is a normal domain if and only if <em>A</em> is a normal domain in codimension at most 1.</p></span></li><li><span>(2)</span><span><p>Suppose that <em>A</em> is a normal domain and a complete intersection. Then the divisor class group of <em>A</em> is a subgroup of the projective limit of the divisor class groups of the localizations <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>, where <em>p</em> runs through all prime ideals of height at most 3 in <em>A</em>.</p></span></li></ul> We use this fact to describe for an integral Noetherian locally complete intersection scheme <em>X</em> the gap between the groups of Weil and Cartier divisors, generalizing in this case the classical result that these two concepts coincide if <em>X</em> is locally a UFD.</div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0022404924002019/pdfft?md5=1e4c0d69cfcf0e52b73f98c66deb7d97&pid=1-s2.0-S0022404924002019-main.pdf","citationCount":"0","resultStr":"{\"title\":\"On local divisor class groups of complete intersections\",\"authors\":\"Daniel Windisch\",\"doi\":\"10.1016/j.jpaa.2024.107804\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>Samuel conjectured in 1961 that a (Noetherian) local complete intersection ring that is a UFD in codimension at most three is itself a UFD. It is said that Grothendieck invented local cohomology to prove this fact. Following the philosophy that a UFD is nothing else than a Krull domain (that is, a normal domain, in the Noetherian case) with trivial divisor class group, we take a closer look at the Samuel–Grothendieck Theorem and prove the following generalization: Let <em>A</em> be a local Cohen–Macaulay ring.</p><ul><li><span>(1)</span><span><p><em>A</em> is a normal domain if and only if <em>A</em> is a normal domain in codimension at most 1.</p></span></li><li><span>(2)</span><span><p>Suppose that <em>A</em> is a normal domain and a complete intersection. Then the divisor class group of <em>A</em> is a subgroup of the projective limit of the divisor class groups of the localizations <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>, where <em>p</em> runs through all prime ideals of height at most 3 in <em>A</em>.</p></span></li></ul> We use this fact to describe for an integral Noetherian locally complete intersection scheme <em>X</em> the gap between the groups of Weil and Cartier divisors, generalizing in this case the classical result that these two concepts coincide if <em>X</em> is locally a UFD.</div>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-09-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"https://www.sciencedirect.com/science/article/pii/S0022404924002019/pdfft?md5=1e4c0d69cfcf0e52b73f98c66deb7d97&pid=1-s2.0-S0022404924002019-main.pdf\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0022404924002019\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022404924002019","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
萨缪尔在 1961 年猜想,一个(诺特)局部完全交环在至多三维上是一个 UFD,它本身也是一个 UFD。据说格罗thendieck 发明了局部同调学来证明这一事实。根据 UFD 无非是一个具有微分子类群的 Krull 域(在诺特情况下是一个正域)这一理念,我们仔细研究了塞缪尔-格罗thendieck 定理,并证明了以下概括:(1)当且仅当 A 是标度至多为 1 的正域时,A 是正域。(2)假设 A 是正域和完全交集。我们利用这一事实来描述一个积分诺特局部完全交方案 X 的魏尔组和卡蒂埃分维组之间的差距,并在这种情况下推广经典结果,即如果 X 是局部 UFD,这两个概念是重合的。
On local divisor class groups of complete intersections
Samuel conjectured in 1961 that a (Noetherian) local complete intersection ring that is a UFD in codimension at most three is itself a UFD. It is said that Grothendieck invented local cohomology to prove this fact. Following the philosophy that a UFD is nothing else than a Krull domain (that is, a normal domain, in the Noetherian case) with trivial divisor class group, we take a closer look at the Samuel–Grothendieck Theorem and prove the following generalization: Let A be a local Cohen–Macaulay ring.
(1)
A is a normal domain if and only if A is a normal domain in codimension at most 1.
(2)
Suppose that A is a normal domain and a complete intersection. Then the divisor class group of A is a subgroup of the projective limit of the divisor class groups of the localizations , where p runs through all prime ideals of height at most 3 in A.
We use this fact to describe for an integral Noetherian locally complete intersection scheme X the gap between the groups of Weil and Cartier divisors, generalizing in this case the classical result that these two concepts coincide if X is locally a UFD.