»ã±¨±êÌâ (Title)£ºHartshorne's question on cofiniteness of complexes£¨Hartshorne¹ØÓÚ¸´ÐÎÓàÓÐÏÞÐÔµÄÓйØÎÊÌ⣩

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»ã±¨ÌáÒª£ºÉèdÊÇÒ»¸öÕýÕûÊý£¬IÊÇ»¥»»ÅµÌØ»·RµÄÃÎÏë¡£ÎÒÃǻظ²ÁËHartshorneÔÚ[Invent. Math. 9 (1970) 145-164]ÖÐÌá³öµÄ¹ØÓÚ¸´ÐÎÓàÓÐÏÞÐÔµÄÎÊÌ⣬ÔÚdim R=d»òdim R/I=d-1»òara(I)=d-1µÄÇé¿öÏ£¬Ö¤Ã÷Èôd<=2£¬Ôò¸´ÐÎX\in D_(R)ÊÇI-cofiniteµ±ÇÒ½öµ±Ã¿¸öͬµ÷Ä£H_i(X)ÊÇI-ÓÐÏ޵ģ»ÈôRÊÇÒ»ÕýÔò²¿ÃÅ»·£¬IÊÇperfectµÄ²¢ÇÒd<=2£¬Ôò¸´ÐÎX\in D(R)ÊÇI-cofinite µ±ÇÒ½öµ±Ã¿¸öͬµ÷Ä£H_i(X)ÊÇI-ÓÐÏ޵ģ»Èôd>=3,ÔòX\in D_(R)ÊÇI-cofinite²¢ÇÒ¶ÔËÁÒâj<=d-2,i\in Z, j<=d-2£¬Ext_R^i(R/I,H_i(X))ÊÇÓÐÏÞÌìÉúÈ·µ±ÇÒ½öµ±Ã¿¸öH_i(X)ÊÇI-cofiniteµÄ¡£ÕâÏî×êÑÐÊǺÍÉò¾²ö©ºÏ×÷ʵÏֵġ£

Abstract£ºLet 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.