DRESS, ANDREAS W.M.

SIEBENEICHER, CHRISTIAN

Kyoto University Research Information Repository

TitleWitt Vectors Associated with Arbitrary Pro-finiteGroups.(Representation Theory of Finite Groups and FiniteDimensional Algebras)Author(s)DRESS, ANDREAS W.M.; SIEBENEICHER, CHRISTIANCitation数理解析研究所講究録 (1992), 799: 21-26Issue Date1992-08URL http://hdl.handle.net/2433/82827RightType Departmental Bulletin PaperTextversionpublisherKyoto University21Witt Vectors Associated with Arbitrary Pro-finite Groups.ANDREAS W.M. DRESS and CHRISTIAN SIEBENEICHER*Consider, for example, the following covariant functors defined on the category rings ofcommutative rings with a unit element’ and with values in rings :$A\mapsto F(A):=A[X]$$A\}arrow$ $F(A):=A[X]/(X^{2})$$A\mapsto$ $F(A):=A\cross A$$A\mapsto$ $F(A):=A\otimes_{Z}A$These functors share the following property:If $p$ is a prime number and if $p\cdot A=0$ , then $p\cdot F(A)=0$ ,that is, char $A=p\Rightarrow charF(A)=p$ .Question: Do all functors from rings to rings share this property?Answer: No.The simplest counterexample known to us is based on the well known fact that every primenumber $p$ divides the binomial coefficient $(\begin{array}{l}pi\end{array})$ for all $i\in$ $\{1, )p-1\}$ .Indeed, consider for an arbitrary ring $A$ the subset$A_{p}^{(2)}$ $:=\{r_{p}(a, b) :=(a, a^{p}+p\cdot b)|a, b\in A\}\subset A\cross A$ .of the cartesian product $A\cross A$ and observe that with$(\begin{array}{l}pi\end{array})$ $;= \frac{1}{p}\cdot(\begin{array}{l}pi\end{array})$ $(i\in\{1, \ldots,p-1\})$one has$r_{p}(0,0)$ $=$ $(0,0)\in A_{p}^{(2)}$ ,$r_{p}(1,0)$ $=$ $(1,1)\in A_{p}^{(2)}$ ,’Supported by a generous grant of the University of Hokkaido, SapporoiIn the following all rings will be assumed to be commutative and to have a unit element, denoted by 1.799 1992 21-2622as well as$r_{p}(a_{1}, b_{1})\pm r_{p}(a_{2}, b_{2})$ $=$ $(a_{1} \pm a_{2}, (a_{1}\pm a_{2})^{p}+p(b_{1}\pm b_{2}-\sum_{*=1}^{p-1}(\pm 1)^{i}(\begin{array}{l}pi\end{array})\cdot a_{1}^{p-}\cdot a_{2}^{i}))$$=r_{p}(a_{1} \pm a_{2}, b_{1}\pm b_{2}-\sum_{=:1}^{p-1}(\pm 1)^{i}(\begin{array}{l}pi\end{array})\cdot a_{1}^{p-i}\cdot a_{2}^{i})$and$r_{p}(a_{1}, b_{1})\cdot r_{p}(a_{2}, b_{2})$ $=$ $(a_{1}\cdot a_{2}, (a_{1}\cdot a_{2})^{p}+p\cdot(a_{1}^{p}\cdot b_{2}+b_{1}\cdot a_{2}^{p}+p\cdot b_{1}\cdot b_{2}))$$=r_{p}(a_{1}\cdot a_{2}, a_{1}^{p}\cdot b_{2}+b_{1}\cdot a_{2}^{p}+p\cdot b_{1}\cdot b_{2})$for all $a_{1},$ $b_{1},$ $a_{2},$ $b_{2}\in A$ . So the subset $A_{p}^{(2)}$ is a sub-ring of the product ring $A\cross A$ and theabove formulae suggest to define quite formally a new addition and multiplication, $say+p$and $p\circ$ on the set $A\cross A$ by$(a_{1}, b_{1})+p(a_{2}, b_{2})$ $:=(a_{1}+a_{2}, b_{1}+b_{2}- \sum_{i=1}^{p-1}(\begin{array}{l}pi\end{array})\cdot a_{1}^{p-:}\cdot a_{2}^{i})$and$(a_{1}, b_{1})p\circ(a_{2}, b_{2})$ $:=(a_{1}\cdot a_{2}, a_{1}^{p}\cdot b_{2}+b_{1}\cdot a_{2}^{p}+p\cdot b_{1} . b_{2})$ ,so that the map$r_{p}:A\cross Aarrow A\cross A$ $(a, b)\mapsto r_{p}(a, b)$becomes a homomorphism from $(A\cross A, +,\circ)Pp$ int$0$ the product-ring $A\cross A$ .Obviously, if $A$ has no $p-$-torsion, the homomorphism $r_{p}$ maps $(A\cross A, +,\circ)pp$ isomorpicallyonto $A_{p}^{(2)}$ , which establishes in particular that $(A\cross A, +,\circ)pp$ is indeed a ring for such $A$ . Buteven if $A$ has p-torsion, in which case the map $r_{p}$ is no more injective, $(A\cross A, +,\circ)pp$ is a ring.This can be verified either by direct computation or by using a surjective homomorphismfrom some appropriate r–torsion free ring, e.g. some polynomial ring over $Z$ , onto the ring$A$ .In other words, the above construction defines a functor$W_{C_{p}}$ : rings $arrow rings$$A\mapsto W_{C_{p}}(A)$$:=(A\cross A, +,\circ)pp$$(h:Aarrow A’)\mapsto(W_{C_{p}}(h):A\cross Aarrow A’\cross A’ (a, b)->(h(a), h(b)))$for which there exists a canonical natural transformation$\Phi$ : $W_{C_{p}}$ $arrow$ id $\cross id$$\Phi(A):W_{C_{p}}(A)arrow A\cross A$ : $(a, b)\mapsto r_{p}(a, b)$ .$Thisfunctorprovidesacounter-examplefortheassumptionmadeabove,$ $i.e$ . $ifAisaring$for which $p\cdot A=0$ , then $p\cdot W_{C_{p}}(A)\neq 0$ :23Indeed the calculation$r_{p}(p\circ(a, b))=p\cdot r_{p}(a, b)$$=$ $(pa,pa^{p}+p^{2}b)$$=r_{p}(pa,(1-p^{p-1})a^{p}+pb)$shows that$p\circ(1,0)=(p, 1-p^{p-1})$holds at least if $A$ has no -torsion, and therefore, as above, this identity must hold for allrings $A$ .Hence if charA $=p,$ $thenfortheunitelement(1,0)ofW_{C_{p}}(A)onehas$$po(1,0)=(0,1)\neq(0,0)$ .More generally, E. WITT observed that for every ring $A$ the subset$\{(a_{1},a_{1}^{2}+2a_{2}, \ldots, \sum_{d|n}d\cdot a_{d}^{n/d}, \ldots)|a_{1},a_{2}, \ldots\in A\}$of the infinite product ring $A^{N},$ $N=\{1,2,3, \ldots\}$ constitutes a sub-ring of $A^{N}$ and that,as above, this allows to construct a functor$W$ : rings $arrow rings$which is uniquely determined by the following properties:$\bullet W(A)=A^{N}$$\bullet$ $W(h : Aarrow A’)=h^{N}$ : $(a_{1}, a_{2}, \ldots)\mapsto(h(a_{1}), h(a_{2}),$ $\ldots$ )$\bullet$ for every $n\in N$ one has a natural transformation$\Phi_{n}$ : $W$ $arrow$ id$\Phi_{n}(A)$ : $W(A)arrow A$ : $(a_{1}, a_{2}, \ldots)\mapsto\sum_{d|n}d\cdot a_{d}^{n/d}$To understand these constructions from a structural rather than a purely computationalpoint of view, consider even more generally a pro-finite group $G$ and let $\mathcal{O}(G)$ denote theset of open subgroups of $G$ . For every ring $A$ , one considers the ring of functions$A^{O(G)/\sim}:=$ { $f$ : $\mathcal{O}(G)arrow A|f(U)=f(V)$ if $U\sim GV$, i.e. if $U$ is $G$-conjugate to $V$}.Then the subset of all those maps $g:\mathcal{O}(G)arrow A$ for which there exists some $f\in A^{O(G)/\sim}$such that$g(U)= \sum/\#Fix_{U}(G/W)\cdot f(W)^{(W:U)}W\in \mathcal{O}(G)$24(where the symbol $\sum’$ is meant to indicate that for each conjugacy class of open subgroups$W$ of $G$ exactly one summand has to be taken and with $(W : U)$ $:=(G : U)/(G : W)$which is an integer whenever $Fix_{U}(G/W)$ is non empty) can be shown to be a sub-ringof $A^{O(G)/\sim}$ . As above, this allows to construct an associated functor $W_{G}$ from rings torings described inTheorem 1:Let $G$ be a pro-finite group and let $\mathcal{O}(G)$ denote the set of open $sub-gro$ups of $G$ . Thenthere exists a unique functor $W_{G}$ : rings $arrow rings$ with the following properties:$\bullet W_{G}(A)$ $:=A^{\mathcal{O}(G)/\sim}$ ,$\bullet$ for every ring homomorphism $h:Aarrow A’$ one has$W_{G}(h)$ : $W_{G}(A)arrow W_{G}(A’)$ : $f\mapsto h\circ f$ ,$\bullet$ for every open subgroup $U\in \mathcal{O}(G)$ one has a natural transformation$\Phi_{U}$ : $W_{G}arrow id$ ,defined by$\Phi_{U}(A):W_{G}(A)arrow A$ : $f \mapsto V\in \mathcal{O}(G)\sum/\neq Fix_{U}(G/V)\cdot f(V)^{(V;U)}$ .Remarks:(1) Witt’s theorem presents the special case where $G$ is the pro-finite completion $\hat{C}$ of theinfinite cyclic group C.(2) The functor $W_{C_{p}}$ considered in our first example is precisely the functor $W_{C_{p}}$ for $G$the cyclic group $C_{p}$ with $p$ elements.Further results concerning this construction are:Theorem 2:With $F_{p}$ the finite field with $p$ elements, one has $p^{n}\cdot W_{G}(F_{p})=0$ if and only if $p\cdot\neq G_{p}$divides $p^{n}$ , where $G_{p}$ denotes a -Sylow subgroup of $G$ . In particular, if $G_{p}$ is infinite, onehas $p^{n}\cdot W_{G}(F_{p})\neq 0$ for all $n\in N$ .Theorem 3:There exists a canonical isomorphism from $W_{G}(Z)$ onto the (completed) Burnside ring2$\hat{\Omega}.(G)$ . It has the following property: If for $e$very positive integer $q\in N$ and for every$U\in \mathcal{O}(G)$ one denotes by $C_{0}(U, \{1, \ldots, q\}))$ the $U$-set of all continuous maps from $U$ intothe discrete set $\{1, \ldots, q\}$ 3 and if $ind_{U}^{G}(C_{0}(U, \{1, \ldots, q\}))$ denotes the almost finite $G$-set2that is the Grothendieck ring of those discrete $G$-spaces–called almost finite $G-sets$–where for everyopen subgroup $U\in O(G)$ there are only finitely many points which are invariant under $U$ .$3C_{0}(U, \{1, \ldots, q\}))$ is easily seen to be an almost finite $U$-set.25induced from it,4 then the canonical isomorphism maps every $f\in W_{G}(Z)$ with $f(U)\geq 0$for all $U\in \mathcal{O}(G)$ onto the disjoint union$[f]$$:=\cup\prime ind_{U}^{G}(C_{0}(U, \{1, \ldots, f(U)\}))U\in \mathcal{O}(G)$taken over all conjugacy classes in $\mathcal{O}(G)$ .Remark:Using this isomorphism the above formula in Theorem 1 for the natural transformation$\Phi_{U}(A)$ has a rather natural interpretation:for any $f\in W_{G}(Z)$ as in Theorem 3 the number of $U$-invariant elements in the almostfinite $G$-set $[f]$ is precisely $\Sigma_{V\in \mathcal{O}(G)}’\#Fix_{U}(G/V)\cdot f(V)^{(V:U)}$ .Theorem 4:1. For every open subgroup $U\in \mathcal{O}(G)$ there are natural transformations$\bullet F_{U}$ : $W_{G}arrow W_{U}$$\bullet V_{U}$ ; $W_{U}arrow W_{G}$where for every ring $A$$\bullet$ the map $F_{U}(A):W_{G}(A)arrow W_{U}(A)$ is a ring homomorphism,$\bullet$ the map $V_{U}(A):W_{U}(A)arrow W_{G}(A)$ is an additive homomorphism.2. Using the identification from Theorem 3 $F_{U}(Z)$ : $W_{G}(Z)arrow W_{U}(Z)$ coincides withthe restriction map $res_{U}^{G}$ : $\hat{\Omega}(GJarrow\hat{\Omega}(U)$ and $V_{U}(Z)$ : $W_{U}(Z)arrow W_{G}(Z)$ coincideswith the induction map $ind_{U}^{G}$ : $\Omega(U)arrow\hat{\Omega}(G)$ .3. The standard identities relating restriction and induction hold more generally for $F$and $V$ , e.g. for any ring $A$ and any $x\in W_{G}(A)$ and $y\in W_{U}(A)$ one has $x\cdot V_{U}(A)(y)=$$V_{U}(A)(F_{U}(A)(x)\cdot y)$ (Frobenius reciprocity) and for $U_{1},$ $U_{2}\in \mathcal{O}(G)$ and $x\in W_{U_{1}}(A)$one can compute $F_{U_{2}}(A)(V_{U_{1}}(A)(x))\in W_{U_{2}}(A)$ according to an appropriate variantof the Mackey sub-group formula.Remark:In case $G=\hat{C}$ , the natural transformations $F$ and $V$ specialize to the well known Frobe-nius and Verschiebung maps defined for universal Witt vectors. Moreover, the well knownidentities relating the Frobenius and Verschiebung maps follow from the third assertion ofTheorem 4 in this particular case.To prove Witt’s theorem as well as Theorems 1 to 4 one needs to show that certain rationalnumbers–like e.g. $\frac{1}{p}(\begin{array}{l}pi\end{array})$–are indeed integers. In the case $\frac{1}{p}(\begin{array}{l}pi\end{array})$ this, of course, canbe shown by direct computation, but it can also be shown without any computation by4For an almost finite $G$-set $X$ we denote by $ind_{U}^{G}(X)$ the almost finite $G$-set of $U$-orbits $\overline{(g,x)}$ in thecartesian product $G\cross X$ relative to the (free) $U$-action $U\cross(G\cross X)arrow G\cross X$ defined by $(u, (g, x))\mapsto$$(gu^{-1}, ux)$ where of course $g_{1}\cdot\overline{(g_{2},x)}:=(g_{1}g_{2}, x)$ .26realizing that $\frac{1}{p}(\begin{array}{l}pi\end{array})$ is the $nu$mber of orbits of the action of the cyclic group $C_{p}$ of order$p$ on its subsets of cardinality $.It is this way of using group actions to prove integrality results of this type which isfundamental for the proof of our theorems and which–first of all–suggested that a rathergeneral variant of Witt’s construction should exist, based on the equivariant combinatoricsof arbitrary rather than of cyclic pro-finite groups, only.ReferencesCARTIER, $P$ : Groupes formels associ\’ees aux anneaux de Witt g\’en\’emlis\’ees,C.R.Acad.Sc.Paris, vol. 265 (1967), 49-52DRESS, A.W.M., SIEBENEICHER, Ch: The Burnside Ring ofprofinite Groups and the Witt VectorConstruction.Advances in Mathematics, vol. 70 (1988), 87-132.DRESS A.W.M. AND SIEBENEICHER, Ch: The Burnside Ring of the Infinite Cyclic Group andits Relations to the Necklace Algebra, $\lambda$ -Rings and the Universal Ring of Witt Vectors (1987),Advances in Mathematics, vol. 78 (1989), 1-41.METROPOLIS N. AND ROTA G.-C.: Witt Vectors and the Algebra of Necklaces,Advances in Mathematics, vol. 50 (1983), 95-125.WITT E.: Zyklische Korper und Algebren der Charakteristik $p$ vom Gmde $p^{n}$ , J. Reine Angew.Math. (Crelle), vol. 176 (1937), 126-140.Universitat Bielefeld, Fakultat fur MathematikPostfach 8640$D$ 4800 Bielefeld 1, FRGe-mail: dress@mathl.mathematik.uni-bielefeld.dee-mail: sieben@math10.mathematik.uni-bielefeld.de

Witt Vectors Associated with Arbitrary Pro-finite Groups.(Representation Theory of Finite Groups and Finite Dimensional Algebras)

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Witt Vectors Associated with Arbitrary Pro-finite Groups.(Representation Theory of Finite Groups and Finite Dimensional Algebras)

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