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

TL;DR

This paper explores the structure of the Grothendieck ring of a commutative ring, establishing new isomorphisms and exact sequences involving the Picard group, and analyzing how ring morphisms affect idempotents and projective modules.

Contribution

It introduces a canonical isomorphism related to idempotents, constructs an exact sequence involving the Picard group and Grothendieck ring, and characterizes support of projective modules.

Findings

Established a canonical isomorphism for modules from idempotents.

Proved an exact sequence involving Picard group, units of the Grothendieck ring, and a new group.

Showed conditions under which ring morphisms lift idempotents and the surjectivity of induced maps.

Abstract

As the first main result of this article, we prove that if $e$ and $e'$ are idempotents of a commutative ring $A$, then there is a canonical isomorphism of $A$-modules: $$Ae\oplus Ae'\simeq Ae/Ae(1-e')\oplus Ae'/Ae'(1-e)\oplus A(e+e'-2ee').$$ 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 $\mathscr{B}(A)\simeq\mathscr{B}\big(K_{0}(A)\big)\simeq H_{0}(A)^{\ast}$. As an application, we show that a morphism of rings $A\rightarrow B$ lifts idempotents if and only if the induced ring map $K_{0}(A)\rightarrow K_{0}(B)$ lifts idempotents. If moreover, $B$ has finitely many maximal ideals then the map $K_{0}(A)\rightarrow K_{0}(B)$ 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.