汇报标题 (Title)：Hartshorne's question on cofiniteness of complexes（Hartshorne关于复形余有限性的有关问题）
汇报人 (Speaker)：杨晓燕 教授（浙江科技大学）
汇报功夫 (Time)：2025年6月10日（周二）14:30-15:30
汇报地址 (Place)：校本部 GJ303
约请人(Inviter)：毛雪峰
主办部门：理学院数学系
汇报提要：设d是一个正整数，I是互换诺特环R的梦想。我们回覆了Hartshorne在[Invent. Math. 9 (1970) 145-164]中提出的关于复形余有限性的问题，在dim R=d或dim R/I=d-1或ara(I)=d-1的情况下，证明若d<=2，则复形X\in D_(R)是I-cofinite当且仅当每个同调模H_i(X)是I-有限的 ；若R是一正则部门环，I是perfect的并且d<=2，则复形X\in D(R)是I-cofinite 当且仅当每个同调模H_i(X)是I-有限的 ；若d>=3,则X\in D_(R)是I-cofinite并且对肆意j<=d-2,i\in Z, j<=d-2，Ext_R^i(R/I,H_i(X))是有限天生确当且仅当每个H_i(X)是I-cofinite的。这项钻研是和沈静雯合作实现的。
Abstract：Let d be a positive integer and I an ideal of a commutative noetherian ring R . We answer Hartshorne's question on cofiniteness of complexes posed in [Invent. Math. 9 (1970) 145-164] in the cases dim R=d or dim R/I=d-1 or ara(I)=d-1 show that if d<=2，then a complex X\in D_(R)is I-cofinite if and only if each homology module H_i(X) is I-cofinite; if R is regular local, I is perfect and d<=2 then X\in D(R) is I-cofinite if and only if every H_i(X) is I-cofinite; if d>=3 then X\in D(R) is -cofinite and Ext_R^i(R/I,H_i(X)) is finitely generated for all j<=d-2 and i\in Z if and only if every H_i(X) is I-cofinite. This is joint work with Jingwen Shen.
