When does the canonical tt-structure restrict to perfect complexes?

I am interested in non-Noetherian(!) rings such that the canonical t-structure on D(R) (the derived category of left R-modules) restricts to perfect complexes i.e. to the subcategory of complexes of finitely generated projective R-modules; i.e., for a complex C of this sort the complex 0Coker(C1C0)C1C2 should be quasi-isomorphic to a perfect complex.

This is equivalent to the existence of perfect resolutions for cohomology of perfect complexes. It also appears to be equivalent to the (left) coherence of R (cf. https://mathoverflow.net/a/325943/2191) along with the existence of bounded projective resolutions of all finitely presented R-modules. However, I do not know whether the latter assumption is equivalent to the existence of a uniform bound on the length of projective resolutions for arbitrary R-modules, i.e., to the finiteness of the left global dimension of R. Also, how one can obtain (non-Noetherian) examples? I can certainly take a semi-hereditary R, but this is only the “finitely generated” dimension 1 case.

P.S. I also suspect the the ring limRi also satisfies my condition if Ri are of global dimension at most n (where n is fixed) and the transition homomorphisms are flat; is this correct?


Source : Link , Question Author : Mikhail Bondarko , Answer Author : Community

Leave a Comment