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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{Lambda-ring} \begin{quote}% There is also a notion of [[special lambda-ring]]. But in most cases by `'$\lambda$-ring'` is meant `'special $\lambda$-ring''. \end{quote} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra} [[!include higher algebra - contents]] \hypertarget{rings}{}\section*{{$\Lambda$-rings}}\label{rings} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{motivation_from_representation_theory}{Motivation from representation theory}\dotfill \pageref*{motivation_from_representation_theory} \linebreak \noindent\hyperlink{in_terms_of_universal_algebra}{In terms of universal algebra}\dotfill \pageref*{in_terms_of_universal_algebra} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{the_orthodox_definition}{The ``orthodox'' definition}\dotfill \pageref*{the_orthodox_definition} \linebreak \noindent\hyperlink{HeterodoxDefinition}{The ``heterodox'' definition}\dotfill \pageref*{HeterodoxDefinition} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{FreeAndCofreeLambdaRings}{Free and co-free $\Lambda$-rings -- Symmetric function and Witt vectors}\dotfill \pageref*{FreeAndCofreeLambdaRings} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} A \emph{$\lambda$-ring} is a [[commutative ring]] which is in addition equipped with operations that behave as the operations of forming [[exterior powers]] (of [[vector spaces]]/[[representations]]) in a [[representation ring]]. The name derives from the common symbol $\Lambda^n$ for the $n$th [[exterior power]]. Hence $\lambda$-rings are one incarnation of the [[representation theory]] of the [[symmetric groups]]. Equivalently, it turns out (\hyperlink{Wilkerson82}{Wilkerson 82}) that a $\lambda$-ring is a commutative ring equipped with an [[endomorphism]] that lifts the [[Frobenius endomorphism]] after reduction mod $p$ at each [[prime number]] $p$. As such, $\lambda$-rings appear in [[Borger's absolute geometry]] and are related, in some way, to [[power operations]] (see there for more) in [[stable homotopy theory]]. \hypertarget{motivation_from_representation_theory}{}\subsubsection*{{Motivation from representation theory}}\label{motivation_from_representation_theory} Typically one can form [[direct sums]] of [[representations]] of some [[algebra|algebraic]] [[structure]]. The [[decategorification]] to [[isomorphism classes]] of such [[representations]] then inherits the structure of a [[commutative monoid]]. But nobody likes commutative monoids: we all have an urge to subtract. So, we throw in formal negatives and get an [[abelian group]] --- the \emph{[[Grothendieck group]]}. In many situations, we can also take [[tensor product|tensor products]] of representations. Then the Grothendieck group becomes something better than an abelian group. It becomes a [[ring]]: the [[representation ring]]. Moreover, in many situations we can also take [[exterior power|exterior]] and [[symmetric power|symmetric]] powers of representations; indeed, we can often apply any [[Young diagram]] to a representation and get a new representation. Then the [[representation ring]] becomes something better than a ring: it becomes a \emph{$\lambda$-ring}. More generally, the [[Grothendieck group]] of a [[monoidal category|monoidal]] [[abelian category]] is always a ring, called a [[Grothendieck ring]]. If we start with a [[braided monoidal category|braided monoidal]] abelian category, this ring is commutative. But if we start with a [[symmetric monoidal category|symmetric monoidal]] abelian category, we get a $\lambda$-ring. So, $\lambda$-rings are all about getting the most for your money when you [[decategorify|decategorify]] a [[symmetric monoidal category|symmetric monoidal]] [[abelian category]] --- for example the category of [[representations]] of a [[group]], or the category of [[vector bundle|vector bundles]] on a [[topological space]]. Unsurprisingly, the [[Grothendieck group]] of the free symmetric monoidal abelian category on one generator is the free $\lambda$-ring on one generator. This category is very important in representation theory. Objects in this category are called [[Schur functor|Schur functors]], because for obvious reasons they act as functors on \emph{any} symmetric monoidal abelian category. The irreducible objects in this category are called `Young diagrams'. Elements of the free $\lambda$-ring on one generator are called [[symmetric function|symmetric functions]]. \hypertarget{in_terms_of_universal_algebra}{}\subsubsection*{{In terms of universal algebra}}\label{in_terms_of_universal_algebra} A \emph{$\lambda$-ring} $L$ is a [[P-ring]] presented by the [[polynomial ring]] $Symm=\mathbb{Z}[h_1,h_1\dots]$ in countably many indeterminates over the [[integer|integers]] or, equivalently, $Symm$ is the ring of [[symmetric function|symmetric functions]] in countably many [[variables]]. This means that (the underlying set valued functor of) $L$ is a [[copresheaf]] presented by $Symm$ such that \begin{enumerate}% \item $L:\CRing \to CRing$ defines an endofunctor on the category of commutative rings. \item $L$ gives rise to a [[comonad]] on $C Ring$. \end{enumerate} A $\lambda$-ring is hence a commutative ring equipped with a [[co-action]] of this comonad. As always is the case with [[monad|monads]] and comonads this definition can be formulated in terms of an [[adjunction]]. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} \hypertarget{the_orthodox_definition}{}\subsubsection*{{The ``orthodox'' definition}}\label{the_orthodox_definition} \begin{defn} \label{}\hypertarget{}{} A \emph{$\lambda$-structure} on a [[commutative ring|commutative unital ring]] $R$ is defined to be a sequence of maps $\lambda^n$ for $n\ge 0$ satisfying \begin{enumerate}% \item $\lambda^0(r)=1$ for all $r\in R$ \item $\lambda^1=id$ \item $\lambda^n(1)=0$, for $n\gt 1$ \item $\lambda^n(r+s)=\sum_{k=0}^n \lambda^k(r)\lambda^{n-k}(s)$, for all $r,s\in R$ \item $\lambda^n(r s)=P_n(\lambda^1(r),\dots,\lambda^n(r),\lambda^1(s),\dots,\lambda^n(s))$ for all $r,s\in R$ \item $\lambda^m(\lambda^n(r)):=P_{m,n}(\lambda^1(r),\dots,\lambda^{m n}(r))$, for all $r\in R$ \end{enumerate} where $P_n$and $P_{m,n}$ are certain (see the reference for their calculation) [[universal polynomial|universal polynomials]] with integer coefficients. $R$ is in this case called a \emph{$\lambda$-ring}. Note that the $\lambda^n$ are not required to be morphisms of rings. A [[homomorphism]] of $\lambda$-structures is defined to be a [[homomorphism]] of [[rings]] commuting with all $\lambda^n$ maps. \end{defn} \begin{example} \label{}\hypertarget{}{} There exists a $\lambda$-ring structure on the ring $1+ R[ [t] ]$ of [[power series]] with constant term $1$ where a) addition on $1+R[ [t] ]$ is defined to be multiplication of power series b) multiplication is defined by \begin{displaymath} (1+\sum_{n=1}^\infty r_n t^n)(1+\sum_{n=1}^\infty s_n t^n):=1+\sum_{n=1}^\infty P_n(r_1,\dots,r_n,s_1,\dots,s_n)t^n \end{displaymath} c) the $\lambda$-operations are defined by \begin{displaymath} \lambda^n (1+\sum_{m=1}^\infty r_m t^m)=1+\sum_{m=1}^\infty P_{m,n}(r_1,\dots,r_{mn})t^m \end{displaymath} \end{example} (\hyperlink{Hopkinson}{Hopkinson}) \begin{prop} \label{}\hypertarget{}{} Let $\Lambda$ denote the ring of [[symmetric function|symmetric functions]], let $R$ be a $\lambda$-ring. Then for every $x\in R$ there is a unique [[homomorphism]] of $\lambda$-rings \begin{displaymath} \Phi_x:\Lambda\to R \end{displaymath} sending $e_1\mapsto x$, $e_n\mapsto \lambda^n(x)$, $p_n\mapsto\psi^n(x)$ where $e_n$ denotes the $n$-th [[elementary symmetric function]] and $\psi^n$ denotes the $n$-th [[Adams operation]] (explained in the reference). Equivalently this result asserts that $\Lambda$ is the [[free construction|free]] $\lambda$-ring in the single variable $e_1$. \end{prop} This is due to (\hyperlink{Hopkinson}{Hopkinson}) \begin{proof} We define $\Phi_x(e_1)=x$, then the assumption on $\Phi_x$ to be a morphism of $\lambda$-rings yields $\Phi_x(e_n)=\Phi_x(\lambda^n(e_1))=\lambda^n(x)$. \end{proof} \begin{theorem} \label{}\hypertarget{}{} (\hyperlink{Hazewinkel}{Hazewinkel} 1.11, 16.1) a) The [[endofunctor]] of the [[category]] of commutative rings \begin{displaymath} \Lambda:\begin{cases}C Ring \to C Ring\\A\mapsto 1 + A [ [t] ]\end{cases} \end{displaymath} sending a commutative ring to the set of power series with constant term $1$ is representable by the polynomial ring $Symm \coloneqq \mathbb{Z}[h_1, h_2,\dots]$ in an infinity of indeterminates over the integers. b) There is an [[adjunction]] $(forget\, \lambda\dashv \Lambda)$ where $forget\,\lambda: \lambda Ring\to CRing$ is the forgetful functor assigning to a $\lambda$-ring its underlying commutative ring. The left inverse $\g_{S,A}$ of the natural isomorphism $q_{S,A}:hom(forget\,\lambda,A)\to hom(S,\Lambda(A))$ is given by the ghost component $s_1$. (see also (\hyperlink{Borger08}{Borger 08, section 1.8})) \end{theorem} An instructive introduction to the ``orthodox''- and preparation for the ``heterodox'' view (described below) on $\lambda$-rings is Hazewinkel's survey article on [[Witt vectors]], (\hyperlink{Hazewinkel}{Hazewinkel}). There is also a [[Hazewinkel, Witt vectors|reading guide]] to that article. \hypertarget{HeterodoxDefinition}{}\subsubsection*{{The ``heterodox'' definition}}\label{HeterodoxDefinition} There is a second, ``heterodox'' way to approach $\lambda$-rings with a strong connection to [[arithmetic]] discussed in detail in (\hyperlink{Borger08}{Borger 08, section 1}). An survey is in (\hyperlink{Borger09}{Borger 09}) where it says in the abstract: \begin{quote}% The theory of $\Lambda$-rings, in the sense of Grothendieck's Riemann--Roch theory, is an enrichment of the theory of commutative rings. In the same way, we can enrich usual [[algebraic geometry]] over the ring $\mathbf{Z}$ of integers to produce $\Lambda$-algebraic geometry. We show that $\Lambda$-algebraic geometry is in a precise sense an algebraic geometry over a deeper base than $\mathbf{Z}$ and that it has many properties predicted for algebraic geometry over the mythical [[field with one element]]. Moreover, it does this in a way that is both formally robust and closely related to active areas in arithmetic algebraic geometry. \end{quote} First some standard notation: \begin{defn} \label{FrobeniusMorphism}\hypertarget{FrobeniusMorphism}{} For $p$ a [[prime number]] write $\mathbb{F}_p$ for the [[finite field]] whose underlying [[abelian group]] is the [[cyclic group]] $\mathbb{Z}/p\mathbb{Z}$. For $A$ an $\mathbb{F}_p$-[[associative algebra|algebra]], then the [[Frobenius endomorphism]] \begin{displaymath} F_p \colon A \longrightarrow A \end{displaymath} is that given by taking each element to its $p$th power \begin{displaymath} F_p \colon x \mapsto x^p \,. \end{displaymath} \end{defn} \begin{defn} \label{LambdaRingByFrobeniusLifts}\hypertarget{LambdaRingByFrobeniusLifts}{} For $p$ a [[prime number]], then a \emph{$p$-typical $\Lambda$-ring} is \begin{itemize}% \item a [[commutative ring]] $R$ \item equipped with an [[endomorphism]] $F_A \colon A \to A$ \end{itemize} such that under [[tensor product]] with $\mathbb{F}_p$ it becomes the [[Frobenius morphism]], def.\ref{FrobeniusMorphism}: \begin{displaymath} \mathbb{F}_p \otimes_{\mathbb{Z}} F_A = F_p \; \colon \; \mathbb{F}_p \otimes_{\mathbb{Z}} A \longrightarrow \mathbb{F}_p \otimes_{\mathbb{Z}} A \,. \end{displaymath} A \emph{big $\Lambda$-ring} is a commutative ring equipped with commuting endomorphisms, one for each prime number $p$, such that each of them makes the ring $p$-typical, respectively, as above. \end{defn} This is def. 1.7 in (\hyperlink{Borger08}{Borger 08}), formulated for the special case of example 1.15 there (which is stated in terms of Witt vectors) and translated to $\Lambda$-rings in view of prop. 1.10 c) (see \hyperlink{AdjointTriple}{the adjunction}) there. \begin{quote}% the following originates from revision 19 but needs attention \end{quote} \begin{example} \label{}\hypertarget{}{} The $p$-th [[Adams operation]] $\psi_p$ is a Frobenius lift. Moreover given any two prime numbers then their [[Adams operations]] commute with each other. \end{example} The following two theorems are crucial for the ``heterodox'' point of view. We will see later that in fact we do not need the torsion-freeness assumption. \begin{theorem} \label{}\hypertarget{}{} (\hyperlink{Wilkerson82}{Wilkerson 82}) Let $A$ be an additively torsion-free commutative ring. Let $\{\psi_p\}$ be a commuting family of Frobenius lifts. Then there is a unique $\lambda$-ring structure on $A$ whose [[Adams operations]] are the given Frobenius lifts $\{\psi_p\}$. \end{theorem} \begin{theorem} \label{}\hypertarget{}{} A ring morphism $f$ between two $\lambda$-rings is a morphism of $\lambda$-rings (i.e. commuting with the $\lambda$-operations) iff $f$ commutes with the [[Adams operations]]. \end{theorem} \begin{cor} \label{}\hypertarget{}{} There is an equivalence between the category of torsion-free $\lambda$-rings and the category of torsion-free commutative rings equipped with commuting Frobenius lifts.. \end{cor} Now we will argue that these statements hold for arbitrary commutative rings. \begin{remark} \label{}\hypertarget{}{} a) The category $\lambda Ring$ of $\lambda$-rings is [[monadic]] and [[comonadic]] over the category of $C Ring$ of commutative rings. b) The category $\lambda Ring_{\neg tor}$ of $\lambda$-rings is [[monadic]] and [[comonadic]] over the category of $C Ring_{\neg tor}$ of commutative rings. \end{remark} \begin{prop} \label{}\hypertarget{}{} Let $\i:C Ring_{\neg tor}\hookrightarrow C Ring$ be the inclusion. Let $W^\prime$ denote this comonad on $C Ring_{\neg tor}$. Then a) $W \coloneqq Lan_i i\circ W^\prime \colon C Ring\to CRing$ is a comonad. b) The category of coalgebras of $W$ is equivalent to the category of $\lambda$-rings. c) $W$ is the [[big-Witt-vectors functor]]. \end{prop} \begin{remark} \label{}\hypertarget{}{} The ``heterodox'' generalizes to arbitrary [[Dedekind domain|Dedekind domains]] with finite residue field. For instance over $F_p[x]$ (instead of $\mathbb{Z}$), we would look at families of $\psi$-operators indexed by the irreducible monic polynomials $f(x)$, and each $\psi_{f(x)}$ would have to be congruent to the $q$-th power map modulo $f(x)$, where $q$ is the size of $F_p[x]/(f(x))$. \end{remark} \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \begin{example} \label{}\hypertarget{}{} The [[topological K-theory]] ring $K(X)$ of any [[topological space]] carries the structure of a $\lambda$-ring with operations induced from (skew-)symmetrized [[tensor products]] of [[vector bundles]]. \end{example} This is originally due to [[Alexander Grothendieck]]. See for instance \href{Introduction+to+topological+K-Theory#Wirthmuller12}{Wirthmuller 12, section 11} and see at \emph{[[Adams operations]]}. Generalizing to [[equivariant K-theory]], the [[representation ring]] of a [[group]] inherits the structure of a Lambda-ring, see \href{representation+ring#LambdaRingStructure}{there}. \begin{example} \label{}\hypertarget{}{} The [[equivariant elliptic cohomology]] at the [[Tate curve]] $Ell_{Tate}(X//G)$ is a $\lambda$-ring (and even an ``elliptic $\lambda$-ring''). \end{example} (\hyperlink{Ganter07}{Ganter 07}, \hyperlink{Ganter13}{Ganter 13}) \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{FreeAndCofreeLambdaRings}{}\subsubsection*{{Free and co-free $\Lambda$-rings -- Symmetric function and Witt vectors}}\label{FreeAndCofreeLambdaRings} \begin{prop} \label{AdjointTriple}\hypertarget{AdjointTriple}{} The [[forgetful functor]] $U \;\colon\; \Lambda Ring \longrightarrow CRing$ from $\Lambda$-rings to [[commutative rings]] has \begin{itemize}% \item a [[left adjoint]], given by forming the ring $Symm$ of [[symmetric functions]]; \item a [[right adjoint]] given by forming the [[ring of Witt vectors]] $W$. \end{itemize} \begin{displaymath} (Symm \dashv U \dashv W) \;\colon\; \Lambda Ring \stackrel{\overset{Symm}{\leftarrow}}{\stackrel{\overset{U}{\longrightarrow}}{\underset{W}{\leftarrow}}} CRing \,. \end{displaymath} Hence \begin{itemize}% \item [[rings of Witt vectors]] are the \emph{[[co-free functors|co-free]] Lambda-rings;} \item rings of [[symmetric functions]] are the [[free construction|free]] Lambda-rings. \end{itemize} \end{prop} This statement appears in (\hyperlink{Hazewinkel08}{Hazewinkel 08, p. 87, p. 97, 98}). The right adjoint in a more general context is in (\hyperlink{Borger08}{Borger 08, prop. 1.10 (c)}). \begin{remark} \label{}\hypertarget{}{} On the level of [[toposes]] ([[etale toposes]]) over these [[sites]] of rings, this statement reappears as an [[essential geometric morphism]] from the [[etale topos]] of [[Spec(Z)]] to that over ``[[F1]]'' in [[Borger's absolute geometry]] (\hyperlink{Borger08}{Borger 08}, exposition in \hyperlink{Borger09}{Borger 09}). \end{remark} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[Grothendieck group]] \item [[p-divisible group|p-divisible groups]]. \item [[field with one element]] \item [[blueprint|blueprints]], [[blueprint|blue scheme]] \item [[Witt vector]] \item [[Borger's absolute geometry]] \item [[Generalized Lambda-structure]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} The $\lambda$-ring structure on [[topological K-theory]] goes back to [[Alexander Grothendieck]] in the 1960s. The relation to lifts of [[Frobenius homomorphisms]] is due to \begin{itemize}% \item Wilkerson 1982 \end{itemize} Modern accounts include \begin{itemize}% \item [[Michiel Hazewinkel]], \emph{Formal groups and applications} \item [[Michiel Hazewinkel]], \emph{[[Hazewinkel, Witt vectors|Witt vectors]]}, (\href{http://arxiv.org/abs/0804.3888}{arXiv}) \end{itemize} See also \begin{itemize}% \item John Baez, \href{http://golem.ph.utexas.edu/category/2007/12/this_weeks_finds_in_mathematic_19.html#c013821}{comment}. \item John R. Hopkinson, \emph{Universal polynomials in lambda-rings and the K-theory of the infinite loop space $tmf$}, thesis, \href{http://dspace.mit.edu/bitstream/handle/1721.1/34544/71011847.pdf?sequence=1}{pdf} \item Donald Knutson, $\lambda$-Rings and the Representation Theory of the Symmetric Group, Lecture Notes in Mathematics, Vol. 308, Springer, Berlin, 1973. \item \href{http://concretenonsense.wordpress.com/2009/07/23/lambda-rings}{concretenonsense blog} \item Donald Yau, \href{http://www.worldscibooks.com/mathematics/7664.html}{LAMBDA-RINGS}, World Scientific, 2010. \end{itemize} school/conference in Leiden: Frobenius lifts and lambda rings 5-10. October 2009 featuring \begin{itemize}% \item [[Pierre Cartier]]: Lambda-rings and Witt vectors \item [[Lars Hesselholt]]: The de Rham-Witt complex \item Alexandru Buium: Arithmetic differential equations \item [[James Borger]]: Lambda-algebraic geometry \end{itemize} \href{http://www.lorentzcenter.nl/lc/web/2009/342/info.php3?wsid=342}{conference site} \href{http://www.lorentzcenter.nl/lc/web/2009/342/participants.php3?wsid=342}{participants} The $\lambda$-ring structure on [[equivariant elliptic cohomology]] is due to \begin{itemize}% \item [[Nora Ganter]], \emph{Stringy power operations in Tate K-theory}, Homology, Homotopy, Appl., 2013; arXiv:math/0701565 \item [[Nora Ganter]], Power operations in orbifold Tate K-theory``; arXiv:1301.2754 \end{itemize} Discussion in the context of [[Borger's absolute geometry]] is in \begin{itemize}% \item [[James Borger]], section 1 of \emph{The basic geometry of Witt vectors, I: The affine case} (\href{http://arxiv.org/abs/0801.1691}{arXiv:0801.1691}) \item [[James Borger]], \emph{Lambda-rings and the field with one element} (\href{http://arxiv.org/abs/0906.3146}{arXiv/0906.3146}) \end{itemize} [[!redirects Lambda-ring]] [[!redirects Lambda-rings]] [[!redirects Lambda ring]] [[!redirects Lambda rings]] [[!redirects lambda-ring]] [[!redirects lambda-rings]] [[!redirects lambda ring]] [[!redirects lambda rings]] [[!redirects λ-ring]] [[!redirects λ-rings]] [[!redirects Λ-ring]] [[!redirects Λ-rings]] \end{document}