8
$\begingroup$

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.

$\endgroup$
7
$\begingroup$

This seems both interesting and hard to pinpoint. Köck (1991; 1998, 4.6b) credits Fulton-Lang (1985) and Soulé (1985, cf. Thm 7), but your suggestion that the theorem must have been “known” earlier is also well-supported. For one thing, Rössler (1999, §1) finds it in Manin (1969, Thm 16.6). For another, Dyer (1962) (cited by Eckmann at ICM (1963), Adams (1965, (iii) p. 152), Fuchs (1973, pp. 349–350), and reprinted in Adams (1972)) 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, §0.1; 2004, p. 823) says at least that his Theorem 2.5.3 “inspired by a Riemann-Roch theorem from [Dy]”, does. See also Smirnov (2006, 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.

$\endgroup$
  • $\begingroup$ 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! $\endgroup$ – Tintin Jun 7 at 7:35
  • $\begingroup$ @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 $\mathcal T_\tau$.”􏰎 $\endgroup$ – Francois Ziegler Jun 7 at 16:09

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.