{"title":"根堆栈和周期分解","authors":"A. Bodzenta, W. Donovan","doi":"10.1007/s00229-024-01574-y","DOIUrl":null,"url":null,"abstract":"<p>For an effective Cartier divisor <i>D</i> on a scheme <i>X</i> we may form an <span>\\({n}^{\\text {th}}\\)</span> root stack. Its derived category is known to have a semiorthogonal decomposition with components given by <i>D</i> and <i>X</i>. We show that this decomposition is <span>\\(2n\\)</span>-periodic. For <span>\\(n=2\\)</span> this gives a purely triangulated proof of the existence of a known spherical functor, namely the pushforward along the embedding of <i>D</i>. For <span>\\(n > 2\\)</span> we find a higher spherical functor in the sense of recent work of Dyckerhoff et al. (<i>N</i>-spherical functors and categorification of Euler’s continuants. arXiv:2306.13350, 2023). We use a realization of the root stack construction as a variation of GIT, which may be of independent interest.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Root stacks and periodic decompositions\",\"authors\":\"A. Bodzenta, W. Donovan\",\"doi\":\"10.1007/s00229-024-01574-y\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>For an effective Cartier divisor <i>D</i> on a scheme <i>X</i> we may form an <span>\\\\({n}^{\\\\text {th}}\\\\)</span> root stack. Its derived category is known to have a semiorthogonal decomposition with components given by <i>D</i> and <i>X</i>. We show that this decomposition is <span>\\\\(2n\\\\)</span>-periodic. For <span>\\\\(n=2\\\\)</span> this gives a purely triangulated proof of the existence of a known spherical functor, namely the pushforward along the embedding of <i>D</i>. For <span>\\\\(n > 2\\\\)</span> we find a higher spherical functor in the sense of recent work of Dyckerhoff et al. (<i>N</i>-spherical functors and categorification of Euler’s continuants. arXiv:2306.13350, 2023). We use a realization of the root stack construction as a variation of GIT, which may be of independent interest.</p>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-06-15\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00229-024-01574-y\",\"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://doi.org/10.1007/s00229-024-01574-y","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
对于方案 X 上的有效卡蒂埃除数 D,我们可以形成一个 \({n}^{text {th}}\) 根堆栈。我们证明这个分解是 \(2n\)-periodic 的。对于(n=2),这给出了一个已知球形函子存在的纯三角证明,即沿着 D 的嵌入的推演。对于(n >2),我们找到了 Dyckerhoff 等人最近工作意义上的更高球形函子(N-球形函子和欧拉连续体的分类。arXiv:2306.13350, 2023)。我们将根栈构造的实现作为 GIT 的一种变体,这可能会引起独立的兴趣。
For an effective Cartier divisor D on a scheme X we may form an \({n}^{\text {th}}\) root stack. Its derived category is known to have a semiorthogonal decomposition with components given by D and X. We show that this decomposition is \(2n\)-periodic. For \(n=2\) this gives a purely triangulated proof of the existence of a known spherical functor, namely the pushforward along the embedding of D. For \(n > 2\) we find a higher spherical functor in the sense of recent work of Dyckerhoff et al. (N-spherical functors and categorification of Euler’s continuants. arXiv:2306.13350, 2023). We use a realization of the root stack construction as a variation of GIT, which may be of independent interest.