We prove the classical Riemann–Roch theorems for the Adams operations ${\psi }^j$ on $K$-theory: a statement with coefficients on $\mathbb{Z}\left[j^{-1}\right]$, that holds for arbitrary projective morphisms, as well as another statement with integral coefficients, that is valid for closed immersions. In presence of rational coefficients, we also analyze the relation between the corresponding Riemann–Roch formula for one Adams operation and the analogous formula for the Chern character. To do so, we complete the elementary exposition of the work of Panin–Smirnov that was initiated by the first author in a previous paper. Their notion of oriented cohomology theory on algebraic varieties allows to use classical arguments to prove general and neat statements, which imply all the aforementioned results as particular cases.

我们证明亚当斯运算的经典黎曼-罗赫定理${\psi }^j$在$K$-理论：系数的陈述$\mathbb{Z}\left[j^{-1}\right]$，这适用于任意射影态射，以及另一个具有积分系数 的陈述，这对闭浸没有效。在有理系数存在的情况下，我们还分析了一个 Adams 运算的相应黎曼-罗赫公式与陈省性的类似公式之间的关系。为此，我们完成了对帕宁-斯米尔诺夫工作的基本阐述，该阐述是由第一作者在上一篇论文中发起的。他们关于代数簇的定向上同调理论的概念允许使用经典论证来证明一般和简洁的陈述，这意味着所有上述结果都是特殊情况。