Given a commutative ring $A$ with unity, Grothendieck used universal polynomials to define a *special* $\lambda$-ring structure on $\Lambda(A):=1+t\:A[[t]]$. Suppose $A$ is graded, say $A=\bigoplus\_{i=0}^\infty A\_i$. In [**Riemann–Roch Algebra**](https://doi.org/10.1007/978-1-4757-1858-4), p. 11, Fulton and Lang define $\Lambda^{\circ}(A):=\{1+a\_1t+a\_2t^2\dotsb\mid a\_i\in A\_i\}$. Then on page 15 they state that since the product and $\lambda$ operations of $\Lambda(A)$ take $\Lambda^\circ(A)$ to itself, $\Lambda^\circ(A)$ becomes a $\lambda$-ring (without unit). They use this $\lambda$-ring structure of $\Lambda^\circ(A)$ in the proof of Theorem 3.1 on p. 16.


However, a straightforward computation shows that the product in $\Lambda(A)$ does *not* take $\Lambda^\circ(A)$ to itself. For example, if $1+a\_1t+a\_2t^2\dotsb$ and $1+b\_1t+b\_2t^2\dotsb$ are elements in $\Lambda^\circ(A)$, then their product using the product of $\Lambda(A)$ is given by $1+P\_1(a\_1;b\_1)t+P\_2(a\_1,a\_2;b\_1,b\_2)t^2+\dotsb$, where $P\_1,P\_2,\dotsc$ are certain universal polynomials. But $P\_1(a\_1;b\_1)$ turns out to be $a\_1b\_1$ ([see here](https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsdGRvbWFpbnxkYXJpamdyaW5iZXJnfGd4OjEwNzljZThlNDcwMGE5YmU), p. 22) and $a\_1b\_1$ is not in $A\_1$, which shows the product is not in $\Lambda^\circ(A)$.


**Question.** Is there an error in the book? If yes, can it be fixed?


**Edit.** If you know other errors in this book that one should be aware of, please share them here.



$\lambda$-ring structure defined for a graded ring in Fulton–Lang's book
As others have said, the definition of the Chern ring there is wrong. But if memory serves, the only mistake is that they forgot to introduce the right multiplication law on the sets of power series they consider. The usual one in the theory is given by the universal formulas for exterior powers of tensor products $\Lambda^n(E\otimes F)$, but the one they want is for Chern classes $c\_n(E\otimes F)$. When $n=1$ and $E$ and $F$ are line bundles, the first is multiplication and the second is addition. So it's obviously just an oversight, but one that can be confusing if you're seeing these things for the first time. (In the copy at U Chicago, someone mercifully added a warning note in the margin. There are a few obvious suspects.)


If you want a reference where the details are correct, I'd recommend SGA6. Grothendieck's introduction in expose 0 is very clear. Page 28 is where the discussion of the Chern ring starts. If I remember, Berthelot's expose goes into more depth, but I found Grothendieck's easier to read. Berthelot gets to the Chern ring on page 344. Atiyah-Tall is also generally a good reference, but I think they don't cover the Chern ring (although they do introduce the gamma-filtration).


