A remark on the Chern class of a tensor product (Q1911201)

Let \(X\) be any algebraic scheme over a field, and let \(\alpha\in K^0(X)\). Then \(\alpha\) has a well-defined rank \(\text{rk} \alpha\), and Chern classes \(c_k(\alpha)\). We consider \(\alpha\otimes [{\mathcal L}]\), where \({\mathcal L}\) is an arbitrary line bundle on \(X\). Theorem. \(c_{\text{rk} \alpha+1} (\alpha)= c_{\text{rk} \alpha+1} (\alpha\otimes [{\mathcal L}])\).