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.](https://link.springer.com/book/10.1007/978-1-4757-1858-4). 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](http://www.numdam.org/item/BSMF_1958__86__97_0/) is for Grothendieck-Riemann-Roch.



Original reference for Adams Riemann-Roch theorem
This seems both interesting and hard to pinpoint. Köck ([1991](//zbmath.org/?q=an:0731.19005); [1998](//zbmath.org/?q=an:0951.14029), 4.6b) credits Fulton-Lang ([1985](//ams.org/mathscinet-getitem?mr=88h:14011)) and Soulé ([1985](//ams.org/mathscinet-getitem?mr=87b:18013), *cf.* Thm 7), but your suggestion that the theorem must have been “known” earlier is also well-supported. For one thing, Rössler ([1999](//zbmath.org/?q=an:0961.14006), [§1](http://people.maths.ox.ac.uk/rossler/mypage/publi.html)) finds it in Manin ([1969](http://mi.mathnet.ru/eng/umn/v24/i5/p3), Thm 16.6). For another, Dyer ([1962](//zbmath.org/?q=an:0171.43901)) (cited by Eckmann at ICM ([1963](//zbmath.org/?q=an:0129.38802)), Adams ([1965](//ams.org/mathscinet-getitem?mr=33:6626), (iii) p. 152), Fuchs ([1973](http://mi.mathnet.ru/eng/inta/v8/p71), pp. 349–350), and reprinted in Adams ([1972](//zbmath.org/?q=an:0234.55002))) starts:



> 
> This lecture is principally an exposition of a folk theorem of
>  a Riemann-Roch type for general cohomology theories known to
>  Adams, Atiyah, Hirzebruch... .
> 
> 
> 


While these authors don’t spell out how this “folk” theorem includes yours, Panin ([2004](//zbmath.org/?q=an:1066.14010), [§0.1](https://faculty.math.illinois.edu/K-theory/0552/); [2004](//zbmath.org/?q=an:1077.14501), [p. 823](//www.mfo.de/occasion/0412/www_view)) says at least that his Theorem 2.5.3 “inspired by a Riemann-Roch theorem from [[Dy](//zbmath.org/?q=an:0171.43901)]”, does. See also Smirnov ([2006](//ams.org/mathscinet-getitem?mr=2008h:14009), 2.5.3).


So my impression is that specialists understood the Adams Riemann-Roch theorem as an instance of the more general **Dyer**-Riemann-Roch theorem, long before any of them bothered to name it.


