{"title":"相干正则环上复合物的同调维数","authors":"James Gillespie, Alina Iacob","doi":"arxiv-2409.08393","DOIUrl":null,"url":null,"abstract":"We show that Iacob-Iyengar's answer to a question of Avromov-Foxby extends\nfrom Noetherian to coherent rings. In particular, a coherent ring R is regular\nif and only if the injective (resp. projective) dimension of each complex X of\nR-modules agrees with its graded-injective (resp. graded-projective) dimension.\nThe same is shown for the analogous dimensions based on FP-injective R-modules,\nand on flat R-modules.","PeriodicalId":501475,"journal":{"name":"arXiv - MATH - Commutative Algebra","volume":"15 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Homological dimensions of complexes over coherent regular rings\",\"authors\":\"James Gillespie, Alina Iacob\",\"doi\":\"arxiv-2409.08393\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We show that Iacob-Iyengar's answer to a question of Avromov-Foxby extends\\nfrom Noetherian to coherent rings. In particular, a coherent ring R is regular\\nif and only if the injective (resp. projective) dimension of each complex X of\\nR-modules agrees with its graded-injective (resp. graded-projective) dimension.\\nThe same is shown for the analogous dimensions based on FP-injective R-modules,\\nand on flat R-modules.\",\"PeriodicalId\":501475,\"journal\":{\"name\":\"arXiv - MATH - Commutative Algebra\",\"volume\":\"15 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-12\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Commutative Algebra\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.08393\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Commutative Algebra","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.08393","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
我们证明，伊阿科布-伊延格尔对阿夫罗莫夫-福克斯比问题的回答从诺特环扩展到了相干环。特别是，如果且只有当 R 模块的每个复数 X 的注入（或投影）维度与其分级注入（或分级投影）维度一致时，相干环 R 才是正则的。
Homological dimensions of complexes over coherent regular rings
We show that Iacob-Iyengar's answer to a question of Avromov-Foxby extends
from Noetherian to coherent rings. In particular, a coherent ring R is regular
if and only if the injective (resp. projective) dimension of each complex X of
R-modules agrees with its graded-injective (resp. graded-projective) dimension.
The same is shown for the analogous dimensions based on FP-injective R-modules,
and on flat R-modules.
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