[proofplan] We apply Quillen's localization theorem to the Serre subcategory of finitely generated $S$-torsion $A$-modules inside the category of finitely generated $A$-modules. This gives a long exact sequence in $G$-theory whose quotient term is the category of finitely generated $S^{-1}A$-modules. Since $A$ is regular Noetherian, and localization preserves this regular Noetherian property, the resolution theorem identifies the relevant $G$-groups with the corresponding projective $K$-groups. Taking the segment in degrees $1$ and $0$ gives the displayed exact sequence. [/proofplan] [step:Identify the torsion subcategory and the localized quotient] Let $\operatorname{mod}_{\mathrm{fg}}(A)$ denote the abelian category of finitely generated $A$-modules. Define \begin{align*} \mathcal T_S(A) := \{M \in \operatorname{mod}_{\mathrm{fg}}(A) : \text{for every } m \in M \text{ there exists } s \in S \text{ such that } sm=0_M\}. \end{align*} This is a Serre subcategory of $\operatorname{mod}_{\mathrm{fg}}(A)$: submodules, quotients, and extensions of finitely generated $S$-torsion modules are again finitely generated and $S$-torsion. Let \begin{align*} L:\operatorname{mod}_{\mathrm{fg}}(A) \to \operatorname{mod}_{\mathrm{fg}}(S^{-1}A) \end{align*} be the exact functor given on objects by $L(M)=S^{-1}M$ and on morphisms by localization. The kernel of $L$ is precisely $\mathcal T_S(A)$, because $S^{-1}M=0$ holds exactly when every element of the finitely generated module $M$ is killed by some element of $S$. The Gabriel-Quillen localization theorem for finitely generated modules over a Noetherian ring applies to this exact localization functor: its kernel is the Serre subcategory $\mathcal T_S(A)$, every finitely generated $S^{-1}A$-module is isomorphic to $S^{-1}M$ for some $M \in \operatorname{mod}_{\mathrm{fg}}(A)$, and morphisms in the target are obtained from localized morphisms after clearing denominators. Hence it identifies the quotient exact category \begin{align*} \operatorname{mod}_{\mathrm{fg}}(A)/\mathcal T_S(A) \end{align*} with $\operatorname{mod}_{\mathrm{fg}}(S^{-1}A)$. [guided] The first task is to put the algebraic localization $A \to S^{-1}A$ into the language of exact categories. We write $\operatorname{mod}_{\mathrm{fg}}(A)$ for the abelian category whose objects are finitely generated $A$-modules and whose morphisms are $A$-linear maps. Inside it, define $\mathcal T_S(A)$ to be the full subcategory consisting of those finitely generated $A$-modules $M$ such that every element is killed by some element of $S$: \begin{align*} \mathcal T_S(A) := \{M \in \operatorname{mod}_{\mathrm{fg}}(A) : \text{for every } m \in M \text{ there exists } s \in S \text{ such that } sm=0_M\}. \end{align*} We need this subcategory to be Serre in order to apply the localization theorem. If $N \subset M$ and $M$ is $S$-torsion, then every $n \in N$ is also an element of $M$, so some $s \in S$ satisfies $sn=0_N$. Hence $N$ is $S$-torsion. If $M$ is $S$-torsion and $Q$ is a quotient of $M$, let $\pi:M\to Q$ denote the quotient map. Then an element $q \in Q$ has the form $q=\pi(m)$ for some $m \in M$, and if $sm=0_M$, then $sq=\pi(sm)=0_Q$. Hence $Q$ is $S$-torsion. Finally, if \begin{align*} 0 \to M' \to M \to M'' \to 0 \end{align*} is exact with $M'$ and $M''$ $S$-torsion, take $m \in M$. Its image in $M''$ is killed by some $s_1 \in S$, so $s_1m$ lies in $M'$. Since $M'$ is $S$-torsion, some $s_2 \in S$ kills $s_1m$. Because $S$ is multiplicative, $s_2s_1 \in S$, and \begin{align*} (s_2s_1)m=0_M. \end{align*} Thus $M$ is $S$-torsion. This proves that $\mathcal T_S(A)$ is a Serre subcategory. Now define the localization functor \begin{align*} L:\operatorname{mod}_{\mathrm{fg}}(A) \to \operatorname{mod}_{\mathrm{fg}}(S^{-1}A) \end{align*} by sending an object $M$ to $S^{-1}M$ and sending an $A$-linear morphism $f:M\to N$ to the induced $S^{-1}A$-linear morphism $S^{-1}f:S^{-1}M\to S^{-1}N$. [Localization is exact](/theorems/2845), so $L$ is an exact functor. Its kernel is exactly $\mathcal T_S(A)$: an $A$-module $M$ localizes to zero precisely when each element $m \in M$ becomes zero in $S^{-1}M$, equivalently when some $s \in S$ satisfies $sm=0_M$. Since $A$ is Noetherian, finitely generated modules localize to finitely generated $S^{-1}A$-modules, and every finitely generated $S^{-1}A$-module is obtained from a finitely generated $A$-module after localization. The same localization statement also controls morphisms: an $S^{-1}A$-[linear map](/page/Linear%20Map) between localized finitely generated modules is represented, after clearing denominators, by the localization of an $A$-linear map. Therefore the Gabriel-Quillen localization theorem for finitely generated modules over a Noetherian ring identifies the quotient exact category \begin{align*} \operatorname{mod}_{\mathrm{fg}}(A)/\mathcal T_S(A) \end{align*} with $\operatorname{mod}_{\mathrm{fg}}(S^{-1}A)$. [/guided] [/step] [step:Apply Quillen localization in $G$-theory] By Quillen's localization theorem for exact categories applied to the Serre inclusion \begin{align*} \mathcal T_S(A) \hookrightarrow \operatorname{mod}_{\mathrm{fg}}(A), \end{align*} and to the quotient identified with $\operatorname{mod}_{\mathrm{fg}}(S^{-1}A)$, there is a long exact sequence \begin{align*} \cdots \to K_1(\mathcal T_S(A)) \to G_1(A) \to G_1(S^{-1}A) \xrightarrow{\partial} K_0(\mathcal T_S(A)) \to G_0(A) \to G_0(S^{-1}A) \to \cdots . \end{align*} Here $G_i(A)$ denotes the $K_i$-group of the exact category $\operatorname{mod}_{\mathrm{fg}}(A)$, and similarly $G_i(S^{-1}A)$ denotes the $K_i$-group of $\operatorname{mod}_{\mathrm{fg}}(S^{-1}A)$. The theorem invoked in this step is Quillen localization for exact categories, cited as a result not yet in the wiki: Quillen localization theorem for exact categories. [/step] [step:Replace $G$-theory by projective $K$-theory using regularity] Since $A$ is regular Noetherian, every finitely generated $A$-module has a finite resolution by finitely generated projective $A$-modules. Therefore the resolution theorem for regular Noetherian rings identifies the inclusion of finitely generated projective $A$-modules into finitely generated $A$-modules with a $K$-theory equivalence in the required low degrees. Thus there are natural isomorphisms \begin{align*} K_i(A) \cong G_i(A) \end{align*} for $i=0,1$. The localization $S^{-1}A$ is again a regular Noetherian commutative ring, because localization preserves Noetherianity and regularity. Applying the same resolution theorem to $S^{-1}A$ gives natural isomorphisms \begin{align*} K_i(S^{-1}A) \cong G_i(S^{-1}A) \end{align*} for $i=0,1$. The external results used here are the resolution theorem for regular Noetherian rings and preservation of regular Noetherian rings under localization, both cited as results not yet in the wiki. [/step] [step:Extract the low-degree exact segment] Substituting the natural identifications $G_1(A)\cong K_1(A)$, $G_1(S^{-1}A)\cong K_1(S^{-1}A)$, $G_0(A)\cong K_0(A)$, and $G_0(S^{-1}A)\cong K_0(S^{-1}A)$ into the long exact localization sequence gives an exact sequence \begin{align*}K_1(\mathcal T_S(A)) \to K_1(A) \to K_1(S^{-1}A) \xrightarrow{\partial} K_0(\mathcal T_S(A)) \to K_0(A) \to K_0(S^{-1}A).\end{align*} Removing the term preceding $K_1(A)$ preserves exactness at every displayed term beginning with $K_1(S^{-1}A)$ and records the asserted low-degree segment: \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*} This is exactly the claimed localization exact sequence in low degree. [/step]