Table Of ContentEQUIVALENCE OF λ-STRUCTURES AND ψ-STRUCTURES
ON Q-ALGEBRAS
JOSE MALAGON-LOPEZ
Abstract. We extend to Q-algebras the known result from graded rings
which states the equivalence of λ-structures and ψ-structures.
1. Introduction
This paper deals with additional structures on certain commutative rings with
unit, λ-structures and structures arising from natural operations in the cate-
gory of λ-rings. The notion of λ-ring was introduced by Grothendieck in his
work on the theory of Chern classes [3], and developed later by him and Berth-
elot in [2], and by Atiyah and Tall in [1]. In brief, the notion of λ-structure in
a ring encapsulates the formal properties of exterior powers, giving an axiom-
atization of such structures. An important class of natural operations in the
category of λ-rings are the Adams operations, introduced by Adams, which
can be used to handle the non-additivity of the λ-operations.
Our main result can be stated as follows:
Theorem 1.1. Let R be a commutative ring containing a subring isomorphic
to Q. Then the existence of a special λ-structure on R is equivalent to the
existence of a ψ-structure on R. More precisely, for all n ≥ 1,
nλ = ψ λ −ψ λ +···+(−1)n−2ψ λ +(−1)n−1ψ λ
n 1 n−1 2 n−2 n−1 1 n 0
ψ = ψ λ −ψ λ +···+(−1)nψ λ +(−1)n+1nλ .
n n−1 1 n−2 2 1 n−1 n
Note1.2. SuchresultwaspreviouslyprovenbyWilkerson[4]usingtopological
tools.
Key words and phrases. Lambda Rings, Adams Operations.
1
In the first two section we recall the basic notions of λ-rings and Adams oper-
ations. In the last section we give a proof of the equivalence of such structures
for Q-algebras.
Conventions. Throughout the present work a ring will mean a commutative
ring with unit.
Acknowledgments. The present work is partoftheauthor’s thesis atNorth-
eastern University under the guidance of M. Levine. I would like to express
my gratitude to him for his advice and suggestions.
2. λ-Structures
In this section we will see how the λ-operations resemble the elementary sym-
metric functions.
2.1. λ-Rings. Let R be a ring and denote by W (R) := 1+tR[[t]] the multi-
t
plicative Abelian group of formal power series in t with constant term 1. Let
+ denote the addition on W (R).
w t
A ring R is called a λ-ring if there is a group morphism
(1) λ : R −→ W (R)
t t
x (cid:55)−→ 1+xt+ a tn
n
n≥2
(cid:88)
Set λ (x) = λ (x)tn, for any x ∈ R. The set of maps {λ : R → R}
t n≥0 n n n≥0
is called a λ-structure on R.
(cid:80)
Remark 2.1. λ is a monomorphism, having as left inverse the morphism
t
W (R) → R given by a tn (cid:55)−→ a .
t n≥0 n 1
(cid:80)
Aringmorphismf : R → S betweentwoλ-ringsisaλ-morphism,ormorphism
of λ-rings, if for all n ≥ 0, f ◦ λ = λ ◦ f, i.e., if the following diagram
n n
commutes
f
R (cid:47)(cid:47) S
λt λt
(cid:15)(cid:15) (cid:15)(cid:15)
W (R) (cid:47)(cid:47) W (S)
t t
Wt(f)
2
where W (f) 1+ a tn = 1 + f(a )tn. The category of λ-rings
t n≥1 n n≥1 n
will be denoted by λ-Ring.
(cid:0) (cid:80) (cid:1) (cid:80)
2.1.1. Examples.
(1) The map λ : Z → W (Z), n (cid:55)→ (1+t)n, defines a structure of λ-ring
t t
on Z, called the canonical structure. The λ-operations are given by
λ (n) = n . This power series also induces a structure of λ-ring on Q
k k
and R.
(cid:0) (cid:1)
(2) LetX beascheme. LetK0(X)denotetheGrothendieckgroupoflocally
free O -sheaves. Then K0(X) is a λ-ring by setting λ (E) = [∧n(E)],
X n
for all E ∈ K0(X). If X = Spec(k), with k a field, the λ-structure on
K0(X) is precisely the canonical λ-structure on Z.
2.2. Special λ-Ring. So far we know how the λ-operations should interact
with addition. Now we will describe the interaction with product and the
λ-operations itself.
2.2.1. λ-Ring Structure on W (R). The group W (R) has a structure of λ-
t t
ring: let ξ ,...,ξ , ζ ,...,ζ be indeterminate. Let σ and ς be the n-th
1 r 1 s n n
elementary symmetric functions on the ξ ’s and ζ ’s respectively. Consider the
i i
polynomials P and P determined on the symmetric function σ ’s and ς ’s
n n,m i i
by the identities
1+ P (σ ,...,σ ;ς ,...,ς )tn := (1+ξ ζ t)
n 1 n 1 n i j
n≥1 i,j
(cid:88) (cid:89)
1+ P (σ ,...,σ ) tn := (1+ξ ···ξ t).
n,m 1 nm i1 im
(cid:88)n≥1 1≤i1<(cid:89)···<im≤r
Let a = 1+ a tn and b = 1+ b tn be any two elements in W (R).
n≥1 n n≥1 n t
Then we have a product ∗ given by
(cid:80) w (cid:80)
a∗ b := 1+ P (a ,...,a ;b ,...,b )tn
w n 1 n 1 n
n≥1
(cid:88)
and λ-operations defined as
Λ (a) := 1+ P (a ,...,a )tn.
m n,m 1 nm
n≥1
(cid:88)
3
2.2.2. Special λ-Ring. A λ-ring R is called special if λ is a λ-morphism.
t
A morphism of special λ-rings, or special λ-morphism is just a λ-morphism.
WewilldenotebySpλ-Ringthecategoryofspecialλ-ringswithλ-morphisms.
Remark 2.2. (1) A necessary condition on R to be a special λ-rings is that
it has to be of characteristic zero. This follows from the chain of iden-
tities
n
λ (n) = λ (1)+ ···+ λ (1) = (λ (1))n = (1+t)n = tn
t t w w t t
i=1
(cid:88)
and the fact that n tn (cid:54)= 0 in R[t], for any ring R.
i=1
(2) If R is a special λ-ring, then R contains a λ-subring isomorphic to Z.
(cid:80)
To see that, it suffices to show that the λ-subring generated by 1 is
not finite. Assume that 1 has order m, for some m ∈ N. Since R is a
special λ-ring, λ (0) = 1, but
t
1 = λ (0) = λ (m·1) = (1+t)m (cid:54)= 1
t t
Theorem 2.3 (Grothendieck). Let R be a ring, then W (R) is a special λ-ring.
t
Proof. For a proof, see [1, Theorem 1.4]. (cid:3)
2.3. Some Functorial Properties of Special λ-Ring. In this section we
willseesomeofthefunctorialpropertiesofW ,aswellassomeofthestructures
t
that can be carried on from the category of rings. The proof of the following
lemma is clear.
Lemma 2.4. Let f : R → S be a ring morphism. Then the induced morphism
W (f) : W (R) → W (S), given by 1+ a tn (cid:55)→ 1 + f(a )tn, is a
t t t n≥1 n n≥1 n
λ-morphism.
(cid:80) (cid:80)
Adirectcomputationshows thatthefunctorW : Ring → Spλ−Ring, given
t
by R (cid:55)→ 1+tR[[t]] preserves monomorphisms and epimorphisms.
It is also well known [1] that the category Spλ−Ring has finite products.
4
3. ψ-Structures
In this section we will see that the Adams operations resemble the symmet-
ric power functions. The section is (integrally) taken from basic theory of
symmetric functions.
3.1. ψ-Rings. Given a ring R, let A (R) denote the set of elements of the
t
form a tn, with a ∈ R for all n ≥ 1. It has a ring structure where
n≥1 n n
addition and product are defined term-wise.
(cid:80)
We say that R is a ψ-ring if there is a ring morphism ψ : R → A (R).
t t
Set ψ (a) = ψ (a)tn, for any a ∈ R. The collection of endomorphisms
t n≥1 n
{ψ : R → R} is called a ψ-structure on R. For any n ≥ 1, ψ is called
n (cid:80)n≥1 n
n-th Adams operation. A ψ-morphism is a ring morphism that commutes with
the Adams operations. We will denote by ψ-Ring the category of ψ-rings.
Example 3.1. Given a scheme X, the n-th Adams operations on K0(X) are
characterized by ψ ([L]) = [L⊗n], with L → X a line bundle.
n
The proof of the following lemma is clear.
Lemma 3.2. Given any ring R, the set of ring morphisms
Ψ : A (R) → A (R) a tn (cid:55)→ a tn; m ∈ Z, m ≥ 1
m t t n nm
(cid:40) (cid:12) (cid:41)
(cid:12) (cid:88)n≥1 (cid:88)n≥1
(cid:12)
provides a ψ-structure for A ((cid:12)R).
t(cid:12)
Remark 3.3. (1) Let R be a ψ-ring. ψ is a monomorphism and a ψ-
t
morphism.
(2) In fact, any ring monomorphism R → A (R) induces a ψ-structure on
t
R.
We can think of R (cid:55)→ A (R) as a functor Ring → ψ-Ring, by sending a
t
ring morphism f : R → S to the ψ-morphism A (f) : A (R) → A (S), which
t t t
is given by a tn (cid:55)→ f(a )tn. It is clear from definition that this
n≥1 n n≥1 n
functor preserves monomorphisms and epimorphisms. From now on, A (R)
(cid:80) (cid:80) t
will denote the ψ-ring whose underlying ring is Aℵ.
5
3.2. Case of Special λ-Ring. Let R be a special λ-ring. For all integer
n ≥ 1, define ψ : R → R by ψ (x) = ψ (x)tn, for any x ∈ R, where
n t n≥1 n
(2) ψ (x) := −tD log((cid:80)λ (x)) ∈ R[[t]]
t t −t
We say that ψ is the n-th Adams operation on R.
n
Lets see that the Adams operations ψ are polynomials on the λ-operations,
n
so they are characterized by their action on the linear elements. First notice
that equation (2) is equivalent to
ψ (x) = −tD log(λ (x))
−t t t
Thus, we have that as maps (keeping track of the degree)
−t nλ tn−1
(−1)nψ tn = −tD log λ tn = n≥0 n
n t n
λ tn
n≥1 (cid:32)n≥0 (cid:33) (cid:0)(cid:80)n≥0 n (cid:1)
(cid:88) (cid:88)
Hence (cid:80)
λ tn (−1)n−1ψ tn−1 = nλ tn−1
n n n
(cid:32) (cid:33)(cid:32) (cid:33)
n≥0 n≥1 n≥0
(cid:88) (cid:88) (cid:88)
Since this is an equality of power series, we have for all n ≥ 1
(3) ψ λ −ψ λ +···+(−1)n−2ψ λ +(−1)n−1ψ λ = nλ
1 n−1 2 n−2 n−1 1 n 0 n
3.2.1. Regarding (3) as an equation with values on the polynomial ring of λ-
operations with integral coefficients, Z[λ ] , we get for any n ≥ 1 the system
n n≥0
of equations
λ 0 ··· 0 ψ λ
0 1 1
λ −λ ··· 0 ψ 2λ
1 0 2 2
... ... ... ... ... = ...
λ −λ ··· (−1)n−1λ ψ nλ
n−1 n−2 0 n n
This implies that the Adams operations are polynomials in the λ-operations.
It can be given a different characterization of the Adams operations in term
of the Newton Polynomials. Let N (σ ,...,σ ) := ξn +···+ξn be the n-th
n 1 r 1 r
Newton polynomial, where σ denotes the i-th elementary symmetric function
i
in the {ξ }. Then
j
ψ (x) = N (λ (x),...,λ (x))
n n 1 r
6
The following result is a consequence of such characterization and summarizes
the basic properties of the Adams operations.
Proposition 3.4. (1) If x ∈ R is a linear element, then ψ (x) = xn, for
n
all n ≥ 1.
(2) ψ is a λ-morphism, for all n ≥ 1.
n
(3) ψ ◦ψ = ψ = ψ ◦ψ , for all n,m.
n m mn m n
(4) ψ (x) = x, for all x ∈ R.
1
(5) ψpr(x) ≡ xpr mod p, for all x ∈ R, and any prime number p
4. λ-structures and ψ-structures
Consider the maps
D ln : W (R) −→ A (R) eR dt : A (R) −→ W (R)
t t t t t
a tn (cid:55)−→ −tD log a (−t)n a tn (cid:55)−→ b tn
n t n n n
(cid:32) (cid:33)
n≥0 n≥0 n≥1 n≥0
(cid:88) (cid:88) (cid:88) (cid:88)
where
t −1
(−1)nb tn = exp a tndt
n n
t
n≥0 (cid:32)(cid:90)0 n≥1 (cid:33)
(cid:88) (cid:88)
The goal of this section is to give a proof of the following theorem.
Theorem 4.1. Let R be a commutative ring with unit such that it contains a
subring isomorphic to Q. Then the existence of a special λ-structure on R is
equivalent to the existence of a ψ-structure on R. Such equivalence is given by
ψ := D lnλ λ := eR dtψ
t t −t −t t
Corollary 4.2. Under the assumptions of the theorem,
nλ = ψ λ −ψ λ +···+(−1)n−2ψ λ +(−1)n−1ψ λ
n 1 n−1 2 n−2 n−1 1 n 0
ψ = ψ λ −ψ λ +···+(−1)nψ λ +(−1)n+1nλ
n n−1 1 n−2 2 1 n−1 n
for all integer n ≥ 1.
7
4.1. Before giving our proof, we list a few facts taking R as in the theorem.
The first lemma follows directly from the definitions.
Lemma 4.3. The maps D ln and eR dt are natural transformations.
t
Lemma 4.4. D ln : W (R) → A (R) is a ring morphism.
t t t
Proof. Recall that the additive structure on W (R) is given by the usual prod-
t
uct of formal power series, so the additive identity is given by 1+t. Its image
under D ln is
t
t
D ln(1+t) = −tD ln(1−t) = = tn
t t
1−t
n≥1
(cid:88)
which is the additive identity on A (R).
t
Let a = a tn and b = b tn be any two elements in W (R). Then
n≥0 n n≥0 n t
(cid:80) (cid:80)
(4) D ln(a+ b) = D ln(ab) = D (ln(a)+ln(b)) = D ln(a)+D ln(b)
t Wt t t t t
where (cid:48)(cid:48)+(cid:48)(cid:48) stands for the usual sum of formal power series. Since the ring
structure on A (R) is the induced by R coordinate wise, equation 4 says that
t
D ln is compatible with the group structure.
t
Finally, since the multiplicative structure on W (R) is given by the universal
t
polynomials P , to show that is compatible with D ln it is sufficient to
m,n t
consider the case
r s
a = (1+x t) b = (1+y t)
i j
i=1 j=1
(cid:89) (cid:89)
First notice that
r
D ln (1+x t) = D ln(1+x t)+ ···+ D ln(1+x t)
t i t 1 At At t r
(cid:32) (cid:33)
i=1
(cid:89)
r
= xntn + ···+ xntn = xn tn
1 At At r i
(cid:32) (cid:33) (cid:32) (cid:33) (cid:32) (cid:33)
n≥1 n≥1 n≥1 i=1
(cid:88) (cid:88) (cid:88) (cid:88)
8
Thus
r s
D ln (1+x t)∗ (1+y t) = D ln (1+x y t)
t i Wt j t i j
(cid:32) (cid:33) (cid:32) (cid:33)
i=1 j=1 i,j
(cid:89) (cid:89) (cid:89)
r s
= xnyn tn = xn yn tn
i j i j
(cid:32) (cid:33) (cid:32) (cid:33)(cid:32) (cid:33)
n≥1 i,j n≥1 i=1 j=1
(cid:88) (cid:88) (cid:88) (cid:88) (cid:88)
r s
= xn tn ∗ yn tn
i At j
(cid:32) (cid:32) (cid:33) (cid:33) (cid:32) (cid:32) (cid:33) (cid:33)
n≥1 i=1 n≥1 j=1
(cid:88) (cid:88) (cid:88) (cid:88)
r s
= D ln (1+x t) ∗ (1+y t)
t i At j
(cid:32) (cid:33) (cid:32) (cid:33)
i=1 j=1
(cid:89) (cid:89)
(cid:3)
which finish the proof of the lemma.
Corollary 4.5. The maps eR dt : A (R) → W (R) and D ln : W (R) → A (R)
t t t t t
are ring isomorphisms.
Corollary 4.6. The following diagram is commutative.
eRdt Dsln
A A (R) (cid:47)(cid:47) W A (R) (cid:47)(cid:47) A A (R)
s t s t s t
As(eRdt) Ws(eRdt) As(eRdt)
(cid:15)(cid:15) eRdt (cid:15)(cid:15) Dsln (cid:15)(cid:15)
A W (R) (cid:47)(cid:47) W W (R) (cid:47)(cid:47) A W (R)
s t s t s t
As(Dtln) Ws(Dtln) As(Dtln)
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
A A (R) (cid:47)(cid:47) W A (R) (cid:47)(cid:47) A A (R)
s t s t s t
eRds Dsln
Proof. Given eR dt and D ln as ring morphisms, the corollary follows from the
t
naturality of D ln and eR ds. (cid:3)
s
Lemma 4.7. Let R be a ring. Consider A (R) with the λ-structure induced
t
by
eR ds ◦Ψ : A −→ A A (R) −→ W A (R)
s t s t s t
9
Then D ln : W (R) → A (R) is a λ-morphism, i.e., the following diagram
t t t
commutes:
Λ−s
W (R) (cid:47)(cid:47) W W (R)
t s t
Dtln Ws(Dtln)
(cid:15)(cid:15) (cid:15)(cid:15)
A (R) (cid:47)(cid:47) W A (R)
t s t
eRds◦Ψs
Proof. Since the λ-structure on W (R) is given by the universal polynomials
t
P , it is sufficient to consider only the elements of the form r (1−a t).
m,n i=1 i
(Wecanassumethatweare“lifting”theuniversalpolynomialstotheuniversal
(cid:81)
special λ-ring Z[λ ] ). Since Λ : W (R) → W W (R) is a morphism of
n {n≥0} s t s t
special λ-rings
r
Λ (1−a t) = Λ (1−a t)+ ···+ (1−a t)
−s i −s 1 Wt Wt r
(cid:32) (cid:33)
(cid:89)i=1 (cid:16) (cid:17)
= Λ (1−a t)+ ···+ Λ (1−a t)
−s 1 Ws Ws −s r
= (1−t)−(1−a t)s + ···+ (1−t)−(1−a t)s
1 Ws Ws r
(cid:16) (cid:17) (cid:16) (cid:17)
From the other side
r
Ψ ◦D ln (1−a t) = Ψ (D ln(1−a t)+ ···+ D ln(1−a t))
s t i s t 1 At At t r
(cid:32) (cid:33)
i=1
(cid:89)
= Ψ antn + ···+ antn
s 1 At At r
(cid:32)(cid:32) (cid:33) (cid:32) (cid:33)(cid:33)
n≥1 n≥1
(cid:88) (cid:88)
= anmtn sm + ···+ anmtn sm
1 As As r
(cid:32) (cid:32) (cid:33) (cid:33) (cid:32) (cid:32) (cid:33) (cid:33)
m≥1 n≥1 m≥1 n≥1
(cid:88) (cid:88) (cid:88) (cid:88)
The ring structure on A (R) is given coordinate wise, so for all 1 ≤ i ≤ r
s
∗Asm
anmtn sm = antn sm
i i
(cid:32) (cid:32) (cid:33) (cid:33) (cid:32) (cid:32) (cid:33) (cid:33)
m≥1 n≥1 m≥1 n≥1
(cid:88) (cid:88) (cid:88) (cid:88)
where ∗ denotes the product in A .
As s
10
