2022/10/06 by Abolfazl Tarizadeh, Tarizadeh, Abolfazl
Mathematics · #11R29 #13B02 #13C10 #13D15 #14C22 #14C35 #16E20 #19A49 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #FOS: Mathematics #K-Theory and Homology (math.KT) #Rings, Modules, and Algebras
paper · pdf · doi:10.48550/arxiv.2210.02951
openalex publication_date 2022/10/06 · openalex created_date 2022/10/08 · openalex updated_date 2026/08/01
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⊕ Ae'≃ Ae/Ae(1-e')⊕ Ae'/Ae'(1-e)⊕ A(e+e'-2ee'). This result plays an important role in proving several results on the Grothendieck ring K0(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: \xymatrix0\ar[r]amp;\Pic(A)\ar[r]amp;K0(A)∗ \ar[r]amp;\mathscrB(A)\ar[r]amp;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 \mathscrB(A)≃\mathscrB(K0(A))≃ H0(A)∗. As an application, we show that a morphism of rings A→ B lifts idempotents if and only if the induced ring map K0(A)→ K0(B) lifts idempotents. If moreover, B has finitely many maximal ideals then the map K0(A)→ K0(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.