Could we extend the exact sequence $K^0(X)\to K_0(X)\to K_0(D_{sg}(X))\to 0$ to the left?

Question

Let $X$ be a variety over a field $k$. We have the bounded derived category of coherent sheaves $D^b_{coh}(X)$ and the derived category of perfect complex $Perf(X)$. It is clear that $Perf(X)$ is a strictly full triangulated subcategory of $D^b_{coh}(X)$. Then following Orlov 2003 we define the triangulated category of singularities of $X$ as the quotient of $D^b_{coh}(X)$ and $Perf(X)$, i.e. $$ D_{sg}(X)=D^b_{coh}(X)/Perf(X). $$

Recall that we call $\mathcal{A}\to \mathcal{B}\to\mathcal{C}$ an exact sequence of triangulated categories if the composition sends $\mathcal{A}$ to zero, $\mathcal{A}\to \mathcal{B}$ is fully faithful and coincides (up to equivalence) with the subcategory of those objects in $\mathcal{B}$ which are zero in $\mathcal{C}$, and the induced functor $\mathcal{B}/\mathcal{A}\to \mathcal{C}$ is an equivalence. It is easy to verify that $$ Perf(X)\to D^b_{coh}(X)\to D_{sg}(X) $$ is an exact sequence of triangulated categories.

On the other hand we have the Grothendieck groups of the above triangulated categories. In more details we define $K^0(X)$ the Grothendieck groups of $Perf(X)$, $K_0(X)$ the Grothendieck groups of $ D^b_{coh}(X)$, and $K_0( D_{sg}(X))$ the Grothendieck groups of $ D_{sg}(X)$. Then from the exact sequence $Perf(X)\to D^b_{coh}(X)\to D_{sg}(X)$ we get an exact sequence of abelian groups $$ K^0(X)\to K_0(X)\to K_0(D_{sg}(X))\to 0. $$ See Schlichting 2008 Exercise 3.1.6.

We would like to extend the above exact sequence to the left, via higher algebraic K-theory. However, there is not higher K-theory on merely triangulated categories. Nevertheless we have $K^i(X)$ and $K_i(X)$ for $i\geq 1$ in the framework of complicial exact categories. $\textbf{My question}$ is: could we define the higher K-theory of $D_{sg}(X)$ and get a long exact sequence $$ \ldots \to K^i(X)\to K_i(X)\to K_i(D_{sg}(X))\to K^{i-1}(X)\to \ldots ? $$

Answer 1

The exact sequence of triangulated categories $$ Perf(X)\to D^b_{coh}(X)\to D_{sg}(X) $$ may be lifted to an exact sequence of stable $\infty$-categories or dg-categories in the sense of BGT: choose an enhancement of $D^b_{coh}(X)$, take the induced enhancement on the subcategory $Perf(X)$, and define $D_{sg}(X)$ to be the cofibre of the inclusion in the $\infty$-category of small stable $\infty$-categories.

Algebraic K-theory can be defined at the level of stable $\infty$-categories, and it is an additive invariant in the sense of BGT (see Prop. 7.10 of loc. cit.); this means that it sends split exact sequences to (co)fibre sequences of spectra. Its nonconnective version $\mathbb{K}$ has the stronger property of being a localizing invariant, which means that it sends arbitrary exact sequences to cofibre sequences of spectra. Hence applying nonconnective K-theory to the exact sequence above, one gets a (co)fibre sequence of nonconnective K-theory spectra $$ \mathbb{K}(Perf(X))\to \mathbb{K}(D^b_{coh}(X))\to \mathbb{K}(D_{sg}(X)). $$ The first term is identified with the Bass-Thomason-Trobaugh K-theory of $X$, and the second is nonconnective G-theory (nonconnective K-theory of the abelian category of coherent sheaves). These agree with their connective versions on nonnegative homotopy groups, so the induced long exact sequence on homotopy groups gives what you are looking for.