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 $\dots 0\to \operatorname{Coker}(C^{-1}\to C^0)\to C^1\to C^2\to \dots$ 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 $\varinjlim R_i$ also satisfies my condition if $R_i$ are of global dimension at most $n$ (where $n$ is fixed) and the transition homomorphisms are flat; is this correct?

  • 1
    $\begingroup$ Don't you only need bounded projective resolutions of finitely presented modules, rather than all finitely generated modules? $\endgroup$ Apr 4, 2019 at 11:14
  • 1
    $\begingroup$ Yes, it suffices to have bounded projective resolutions of finitely projective modules. and I do not know whether these conditions are equivalent. $\endgroup$ Apr 4, 2019 at 16:32
  • $\begingroup$ Maybe almost perfect (sometimes called pseudo-coherent in the literature) instead of perfect is appropriate. On Lurie's Higher Algebra section 7.2.4, especially Proposition, he proved that an $\mathbb E_1$-ring $R$ is left coherent if and only if the canonical $t$-structure on the $\infty$-category of left $R$-modules restricts to a $t$-structure on the $\infty$-category of almost perfect left $R$-modules. $\endgroup$
    – user20948
    Jul 13, 2019 at 13:09
  • $\begingroup$ About the global dimension: Proposition in Higher Algebra claims that an almost perfect left module over a connective $\mathbb E_1$-ring is perfect if and only if it has finite $\operatorname{Tor}$-amplitude. $\endgroup$
    – user20948
    Jul 13, 2019 at 13:25


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy