Let $A$ be a regular Noetherian commutative unital ring, in the sense that every finitely generated $A$-module has a finite resolution by finitely generated projective $A$-modules, and let $S \subset A$ be a multiplicative subset with $1_A \in S$. Let $S^{-1}A$ be the localization of $A$ at $S$, and let $\mathcal T_S(A)$ denote the exact category of finitely generated $A$-modules $M$ such that for every $m \in M$ there exists $s \in S$ with $sm=0_M$, equipped with the exact structure inherited from finitely generated $A$-modules. Let $K_i(A)$ and $K_i(S^{-1}A)$ denote Quillen $K$-groups of the exact categories of finitely generated projective modules over the indicated rings, and let $K_i(\mathcal T_S(A))$ denote Quillen $K$-groups of the exact category $\mathcal T_S(A)$ for $i=0,1$. Then the localization homomorphism $A \to S^{-1}A$ and Quillen's boundary map give an exact sequence of abelian groups

\begin{align*} K_1(A) \longrightarrow K_1(S^{-1}A) \xrightarrow{\partial} K_0(\mathcal T_S(A)) \longrightarrow K_0(A) \longrightarrow K_0(S^{-1}A). \end{align*}