The vanishing of non-connective K-theory in negative degrees

Question

In the works of Cisinski, Tabuada, and Schlichting certain non-connective K-theory groups for a differential graded category $C$ are defined. As far as I understand, $K_i(C)$ is not necessarily zero for $i<0$. Yet are there any sufficient conditions on $C$ that ensure that this condition is fulfilled? Are $K_i(C)$ just the 'standard' $K$-groups of a smooth variety $X$ if $C$ is an enhancement of the (derived) category of perfect complexes of sheaves over $X$?

I would be deeply grateful for any explanations and/or precise references here.

Answer 1

This abstract non-connective $K$-theory, when restricted to schemes, is known from a long time: this is the Bass $K$-theory functor $K^B$ considered by Thomason and Trobaugh (the article of Thomason and Trobaugh is a classic on the subject which must be read anyway). The comparison of the abstract construction of non-connective $K$-theory with the Thomason-Trobaugh construction follows straight away from Theorem 5 (Section 8) and Theorem 8 (Section 12) in Schlichting's paper

M. Schlichting, Negative K -theory of derived categories, Math. Z. 253 (2006), 97–134.

The only general way to see negative K-groups vanishing is Theorem 7 of loc. cit: for any noetherian abelian category $A$, $K_i(D^b(A))=0$ for $i<0$ (where $D^b(A)$ stands for a dg version of the bounded derived category of $A$). In particular, if $X$ is a noetherian regular scheme, then the equivalence $Perf(X)\simeq D^b(Coh(X))$ implies that $K_i(X)=K_i(Perf(X))\simeq K_i(D^b(Coh(X)))=0$ for $i<0$.

Weibel's conjecture predicts that, for any noetherian scheme of Krull dimension $\leq d$, $K_{-i}(X)=0$ for $i>d$. This conjecture is proved in characteristic zero in this paper:

G. Cortiñas, C. Häsemeyer, M. Schlichting and C. A. Weibel, Cyclic homology, cdh-cohomology and negative K-theory, Ann. of Math. 167 (2008), 549 - 573.

Assuming a rather strong version of resolution of singularities, the conjecture has been proved in positive characteristic as well in this paper:

T. Geisser, L. Hesselholt, On the vanishing of negative K-groups, Math. Ann. 348 (2010), 707-736.

Finally, we conjecture that $K_i(A)=0$ for $i<0$ for any saturated dg algebra $A$ (possibly such that $A^n=0$ for $n<0$). Such a vanishing would imply the existence of a weight structure à la Bondarko on (an adequate version of) Kontsevich's triangulated category of "non-commutative motives". Evidence for this is given by the fact that this is known if $A$ is Morita equivalent to $Perf(X)$ (for a smooth and projective variety $X$), as well as by the degeneration of the non-commutative version of the Hodge-to-de Rham spectral sequence, see

D. Kaledin, Non-commutative Hodge-to-de Rham degeneration via the method of Deligne-Illusie, Pure Appl. Math. Q. 4 (2008), 785–875.