# Original reference for Adams Riemann-Roch theorem

Let $$f\colon Y\to X$$ be a proper morphism between smooth quasiprojective $$k$$-algebraic varieties. Denote by $$\psi^j$$ the $$j$$-th Adams operation on the Grothendieck group of vector bundles and $$\theta^j(T_f)$$ the $$j$$-th cannibalistic class of the relative tangent bundle $$T_f$$. The Adams Riemann-Roch theorem states that for any $$j$$ the diagram $$\require{AMScd}$$ $$\begin{CD} K_0(Y)@>f_*>>K_0(X)\\ @V\theta^j(T_f)\cdot \psi^jVV@VV\psi^jV\\ K_0(Y)\otimes\mathbb{Z}[\tfrac{1}{j}]@>f_*>>K_0(X) \otimes \mathbb{Z}[\tfrac{1}{j}] \end{CD}$$ commutes.

The oldest reference I know is Theorem 7.6 of Chapter V in W. Fulton; S. Lang: Riemann-Roch algebra. Grundlehren der Mathematischen Wissenschaften , 277. Springer-Verlag, New York, 1985. x+203 pp.. However, that reference is already very general (it does not even require schemes to be over a field) and 1985 is very "late" for such a Riemann-Roch type statement. Therefore my question:

$$\phantom{aaaaaaaa}$$What is the original reference for the Adams Riemann-Roch theorem?

I am looking for something as Borel-Serre's paper is for Grothendieck-Riemann-Roch.

• Thanks for the great answer, @Francois, it has helped me a lot. Definitely, the Adams R-R theorem was known before Fulton's 85 and you have given two partial references: Manin's for closed immersions, and Dyer's 62 in Differential Geometry context (Dyer's proof of R-R type theorems could not be directly transferred to algebraic geometry at that time). It now seems to me that the theorem was not named explicitly and completely written down until it later gained relevance in higher $K$-theory context. In any, case: Thanks a lot again! – Tintin Jun 7 at 7:35
• @Tintin Thanks and you’re welcome. One might add the discussion in Adams’ survey (1971, p. 14) and in Fomenko & Fuchs’ nice new book (2016), both framing R-R as a statement about oriented bundles (Dyer not credited): “It is this obvious statement that is called the general Riemann-Roch theorem. It stops being trivial as soon as we consider specific examples, since then it includes an ... explicit computation of $\mathcal T_\tau$.”􏰎 – Francois Ziegler Jun 7 at 16:09