Nontrivial vector bundles with trivial Chern classes
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866918379397316608 |
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| author | Mandal, Satya |
| author_facet | Mandal, Satya |
| contents | Let ${\mathbb F}_0$ be an algebraically closed field, with $char({\mathbb F}_0)=0$. In this article, for prime numbers $p\geq 2$, we construct smooth affine algebras $B$ over ${\mathbb F}_0$, with $\dim B=p+2$. Further, we construct projective $B$-modules $Q$ with $rank(Q)=p$, such that $x=[Q] -[B^p]\neq 0$ in $K_0(B)$ and the total Chern class $C(Q)=1+\sum_{i=1}^{p}C^k(Q) =1$ is trivial. We use the splitting theorem in \cite{ABH} that for projective $B$-modules $P$ with $rank(P)=r=\dim B-1$, vanishing $C^r(P)=0 \Longrightarrow P\cong Q\oplus B$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_01761 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Nontrivial vector bundles with trivial Chern classes Mandal, Satya K-Theory and Homology Commutative Algebra Algebraic Geometry Let ${\mathbb F}_0$ be an algebraically closed field, with $char({\mathbb F}_0)=0$. In this article, for prime numbers $p\geq 2$, we construct smooth affine algebras $B$ over ${\mathbb F}_0$, with $\dim B=p+2$. Further, we construct projective $B$-modules $Q$ with $rank(Q)=p$, such that $x=[Q] -[B^p]\neq 0$ in $K_0(B)$ and the total Chern class $C(Q)=1+\sum_{i=1}^{p}C^k(Q) =1$ is trivial. We use the splitting theorem in \cite{ABH} that for projective $B$-modules $P$ with $rank(P)=r=\dim B-1$, vanishing $C^r(P)=0 \Longrightarrow P\cong Q\oplus B$. |
| title | Nontrivial vector bundles with trivial Chern classes |
| topic | K-Theory and Homology Commutative Algebra Algebraic Geometry |
| url | https://arxiv.org/abs/2601.01761 |