On the characterizations of cofinite complexes over affine curves and hypersurfaces

Let A be a homomorphic image of a Gorenstein ring of finite Krull dimension, J an ideal of A of dimension one, and N
- a bounded-below complex of A-modules. Suppose that A is complete with respect to a J-adic topology. In this paper, we prove that N
- is a J-cofinite complex if and only if Hi(N
- ) is a J-cofinite module for all i. The same result is also proved for principal ideals J. Consequently, for the fourth question given by R. Hartshorne (cf. [6, Fourth Question, p. 149]), we obtain an answer over the ring, on affine curves and hypersurfaces.