On the Grothendieck ring and the relation of its group of units with the Picard group
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866918450074484736 |
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| author | Tarizadeh, Abolfazl |
| author_facet | Tarizadeh, Abolfazl |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2210_02951 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On the Grothendieck ring and the relation of its group of units with the Picard group Tarizadeh, Abolfazl Commutative Algebra Algebraic Geometry K-Theory and Homology 13B02, 13C10, 11R29, 14C22, 13D15, 16E20, 19A49, 14C35 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. |
| title | On the Grothendieck ring and the relation of its group of units with the Picard group |
| topic | Commutative Algebra Algebraic Geometry K-Theory and Homology 13B02, 13C10, 11R29, 14C22, 13D15, 16E20, 19A49, 14C35 |
| url | https://arxiv.org/abs/2210.02951 |