From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/5379 Path: news.gmane.org!not-for-mail From: =?iso-8859-1?Q?Joyal=2C_Andr=E9?= Newsgroups: gmane.science.mathematics.categories Subject: a conjecture Date: Thu, 17 Dec 2009 13:58:12 -0500 Message-ID: References: Reply-To: =?iso-8859-1?Q?Joyal=2C_Andr=E9?= NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1261101766 31700 80.91.229.12 (18 Dec 2009 02:02:46 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 18 Dec 2009 02:02:46 +0000 (UTC) To: "Andrew Stacey" , Original-X-From: categories@mta.ca Fri Dec 18 03:02:38 2009 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from mailserv.mta.ca ([138.73.1.1]) by lo.gmane.org with esmtp (Exim 4.50) id 1NLSAr-0007x5-Qj for gsmc-categories@m.gmane.org; Fri, 18 Dec 2009 03:02:34 +0100 Original-Received: from Majordom by mailserv.mta.ca with local (Exim 4.61) (envelope-from ) id 1NLRpi-0000Zu-K9 for categories-list@mta.ca; Thu, 17 Dec 2009 21:40:42 -0400 Original-Sender: categories@mta.ca Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:5379 Archived-At: 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.=20 >Let Z[]:CMon ---> CRing be the functor which associates to a = commutative monoid M the=20 >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=20 >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]]=20 >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=3DZ[-]=20 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.=20 The natural transformation exibits Z as the colimit of the functor Z[-], = since we have Z[1]=3DZ 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=3D{n.1}. The decomposition Z[-]=3Ddisjoint union_n Z[-]_n=20 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=3DZ[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[-])=3DEnd(Z\times Z[-]_0)=3DZ^Z\times End(Z[-]_0)^Z Hence the set End(Z[-]) is uncountable, since the set Z^Z is = uncountable.=20 It seems that the basic idea can be saved by making a slight modification to the functor V. =20 Let me say that an element z in a monoid M is a ZERO ELEMENT=20 if we have zx=3Dxz=3Dz 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'].=20 The additive group of Z[M'] is the free abelian group on = M'=3DM\setminus{0}.=20 If we compose the functot Z['] with the forgetful functor U:CRing = --->Set we obtain a functor V':CMonz --->Set. I conjecture that=20 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]]=20 is the group homomorphism Z[M']--->1+tZ[M'][[t]]=20 which takes an element x\in M to the power series 1+tx (notices that = 1+t0=3D1). Best,=20 Andr=E9 -------- Message d'origine-------- De: Joyal, Andr=E9 Date: mer. 16/12/2009 08:08 =C0: Andrew Stacey; categories@mta.ca Objet : RE=A0: categories: Re: A well kept secret =20 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, Andr=E9 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]