On the Grothendieck ring and the relation of its group of units with the Picard group

As the first main result of this article, we prove that if $e$ and $e^{\mathrm{\prime }}$ are idempotents of a commutative ring $A$, then there is a canonical isomorphism of $A$-modules: $$Ae\oplus A{e}^{\mathrm{\prime }}\simeq Ae\mathrm{/}Ae(1-e^{\mathrm{\prime }})\oplus A{e}^{\mathrm{\prime }}\mathrm{/}A{e}^{\mathrm{\prime }}(1-e)\oplus A(e+e^{\mathrm{\prime }}-2e{e}^{\mathrm{\prime }})\mathrm{.}$$ This result plays an important role in proving several results on the Grothendieck ring $K_0(A)$. Especially, we first show that for any ring $A$ there is a complex of Abelian groups which is exact at the beginning and end: \xymatrix{0\ar[r]&\Pic(A)\ar[r]&K_{0}(A)^{\ast} \ar[r]&\mathscr{B}(A)\ar[r]&0.} Then we show that the above sequence is split exact for some certain rings $A$ (including Dedekind domains or more generally Noetherian one dimensional rings). The next main result asserts that for any ring $A$ we have the canonical isomorphisms of Abelian groups $\mathcal{B}(A)\simeq \mathcal{B}(K_0(A))\simeq H_0(A{)}^{\ast }$. As an application, we show that a morphism of rings $A\to B$ lifts idempotents if and only if the induced ring map $K_0(A)\to K_0(B)$ lifts idempotents. If moreover, $B$ has finitely many maximal ideals then the map $K_0(A)\to K_0(B)$ is surjective. Finally, we show that the support of a finitely generated projective module is the whole prime spectrum if and only if its trace ideal is the whole unit ideal.