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On Grothendieck groups and rings with exact sequences for the Picard, $K_{0}(R)^{\ast}$, idempotents and ideal class groups

Abolfazl Tarizadeh
Oct 2022
The main goal of this article is to investigate the Grothendieck groups,especially the Grothendieck ring $K_{0}(R)$, the Picard group $\Pic(R)$ and theideal class group $\Cl(R)$ of a given commutative ring $R$. Among the mainresults, we obtain a general theorem which asserts that for any commutativering $R$ we have the exact sequence of groups:$$\xymatrix{0\ar[r]&\Cl(R)\ar[r]&\Pic(R)\ar[r]&\Pic(T(R))}$$ where $T(R)$denotes the total ring of fractions of $R$. As an application of this result,if $T(R)$ has finitely many maximal ideals (e.g. $R$ is reduced with finitelymany minimal primes), then we obtain a canonical isomorphism of groups$\Cl(R)\simeq\Pic(R)$. The latter result generalizes several classical theoremsin the literature. Next we show that for any ring $R$, we have the canonicalisomorphisms of groups:$\mathcal{B}(R)\simeq\mathcal{B}\big(K_{0}(R)\big)\simeq H_{0}(R)^{\ast}$ where$H_{0}(R)$ denotes the ring of all continuous functions$\Spec(R)\rightarrow\mathbb{Z}$, and $\mathcal{B}(R)$ denotes the additivegroup of the Boolean ring of idempotents of $R$. It is proved that if a ring$R$ has the line bundle property (e.g. a Dedekind domain or mor generally aNoetherian one dimensional ring), then we have the split exact sequence ofgroups: $$\xymatrix{0\ar[r]&\Pic(R)\ar[r]&K_{0}(R)^{\ast}\ar[r]&\mathcal{B}(R)\ar[r]&0.}$$ The identification$K_{0}(R)_{\mathrm{red}}\simeq H_{0}(R)$ has several consequences. Especially,we show that a morphism of rings $R\rightarrow R'$ lifts idempotents if andonly if the induced ring map $K_{0}(R)\rightarrow K_{0}(R')$ lifts idempotents.Finally, we prove that the localization of the monoid-ring $R[M]$ with respectto its multiplicative set of the unit vectors (monomials) is canonicallyisomorphic to the group-ring $R[G]$ where $G$ is the Grothendieck group of thecommutative monoid $M$.