a conjecture

["Joyal, André" <joyal.andre@uqam.ca>]

Dear Andrew,

My statement about the theory of lambda ring was wrong.
I would like to correct it.
Let me first recall the original statement.


>What is a lambda-ring?
>In their book "Riemann-Roch-Algebra" Fulton and Lang define a lambda-ring
>to be a pre-lambda-ring satisfying two complicated identities [(1.4) and (1.5)]
>[Beware that F&L are using an old terminology: they call a lambda-ring a "special lambda-ring"
>and they call a pre-lambda-ring a "lambda-ring"]
>The notion of lambda-ring (ie of "special lambda-ring" in the terminology of F&L)
>can be defined in a natural way if we use category theory. 
>Let Z[]:CMon ---> CRing be the functor which associates to a commutative monoid M the 
>ring Z[M] freely generated by M (it is the left adjoint to the forgetful functor in the
>opposite direction). If we compose the functot Z[] with the forgetful functor U:CRing --->Set
>we obtain a functor V:CMon --->Set. The algebraic theory of lambda-rings 
>can be defined to be the theory of natural operations on the functor V.
>The total lambda operation V(M)--->V(M)[[t]] is the group homomorphism Z[M]--->1+tZ[M][[t]] 
>which takes an element x\in M to the power series 1+tx.

I woke up in the middle of the following night
with the strong impression that it could not work.
Here is the problem: the theory of natural operations on the functor V=Z[-] 
is uncountable, whilst the theory of lambda-rings is countable.
To see this, let analyse the set of unary operations on the functor Z[-].
For any monoid M, the algebra Z[M] is equipped with a natural augmentation e:Z[M]--->Z
from which we obtain a natural transformation e:Z[-]--->Z
from the functor Z[-] to the constant functor Z. 
The natural transformation exibits Z as the colimit of the functor Z[-], 
since we have Z[1]=Z and since the monoid 1 is terminal in
 the category of commutative monoids CMon.
Let me denote by Z[M]_n the fiber of the map e:Z[M]--->Z at n\in Z.
The (set valued) functor Z[-]_n is connected, since Z[1]_n={n.1}.
The decomposition
Z[-]=disjoint union_n Z[-]_n 
coincide with the canonical decomposition of the functor Z[-]
as a disjoint union of connected components.
Notices that the functor Z[-]_n is isomorphic to the functor Z[-]_0
since we have n.1+Z[M]_0=Z[M]_n for any monoid M.
Hence the functor Z[-] is isomorphic to the functor Z\times Z[-]_0.
It then follows from the connectedness of the functor Z[-]_0
that we have a bijection

End(Z[-])=End(Z\times Z[-]_0)=Z^Z\times End(Z[-]_0)^Z

Hence the set End(Z[-]) is uncountable, since the set Z^Z is uncountable. 


It seems that the basic idea can be saved by
making a slight modification to the functor V.  
Let me say that an element z in a monoid M is a ZERO ELEMENT 
if we have zx=xz=z for every x\in M. A zero element is unique
when it exists.  I will denote a zero element by 0.
Let me denote by CMonz the category of commutative
monoids with zero element, where a map M--->N
should preserve the zero elements. Then the obvious
forgetful functor CRing--->CMonz has a left
adjoint which associates to M a commutative ring Z[M']. 
The additive group of Z[M'] is the free abelian group on M'=M\setminus{0}. 
If we compose the functot Z['] with the forgetful functor U:CRing --->Set
we obtain a functor V':CMonz --->Set. I conjecture that 
the theory of natural operations on the functor V' is
the algebraic theory of lambda-rings.
The total lambda operation V'(M)--->V'(M)[[t]] 
is the group homomorphism Z[M']--->1+tZ[M'][[t]] 
which takes an element x\in M to the power series 1+tx (notices that 1+t0=1).


Best, 
André




-------- Message d'origine--------
De: Joyal, André
Date: mer. 16/12/2009 08:08
À: Andrew Stacey; categories@mta.ca
Objet : RE : categories: Re: A well kept secret
 
Dear Andrew,

Please disregard my suggestion about a new definition of lambda ring!
My memory may have failed me!
I am not sure the new definition is right!

Best,
André



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