K0-THEORY OF n-POTENTS IN RINGS AND
ALGEBRAS
EFTON PARK AND JODY TROUT
Abstract. Let n ≥ 2 be an integer. An n-potent is an element
e of a ring R such that en = e. In this paper, we study n-potents
in matrices over R and use them to construct an abelian group
(R). If A is a complex algebra, there is a group isomorphism
(A) ∼=
K0(A)
for all n ≥ 2. However, for algebras over
cyclotomic fields, this is not true, in general. We consider Kn
as a covariant functor, and show that it is also functorial for a
generalization of homomorphism called an n-homomorphism.
1. Introduction
For more than forty years, K-theory has been an essential tool in
studying rings and algebras [1, 7]. Given a ring R, a simple functorial
object associated to R is the abelian group K0(R). There are multi-
ple ways of defining K0(R), but the most useful characterization when
working with operator algebras is to define K0(R) in terms of idempo-
tents (or projections, if an involution is present) in matrix algebras over
R; i.e., elements e in Mk(R) for some k with the feature that e
2 = e
(p = p∗ = p2 in the involutive case). In this paper, we define, for each
natural number n ≥ 2, a group which we denote Kn0 (R). This group
is constructed from matrices e over R with the property that en = e;
we call such matrices n-potents. We define Kn0 (R) for all rings, unital
or not, and show that Kn0 determines a covariant functor from rings to
abelian groups.
Let Q(n− 1) be the cyclotomic field obtained from the rationals by
adjoining the (n− 1)-th roots of unity. We show that Kn0 is half-exact
on the subcategory of Q(n − 1)-algebras, and given any such algebra
A, we show that Kn0 (A) is isomorphic to a direct sum of n − 1 copies
of K0(A). Since a C-algebra A is a Q(n − 1)-algebra for all n, what-
ever invariants are contained in Kn0 (A) are already contained in K0(A).
However, K
0 for p 6= n may generate new groups for cyclotomic alge-
bras, e.g., K40(Q(4))
∼= Z⊕2Z (Theorem 3.15) which is not isomorphic
2010 Mathematics Subject Classification: 18F30, 19A99, 19K99.
http://arxiv.org/abs/0704.0775v2
2 EFTON PARK AND JODY TROUT
to K40 (Q(3))
∼= Z3. Thus, K40 distinguishes between the fields Q(3) and
Q(4), but idempotent, and also tripotent (n = 3), K-theory does not.
The paper is organized as follows. In Section 2, we define various
notions of equivalence on the set of n-potents, and explore the rela-
tionships between these equivalence relations. Most of our results in
this section mirror analogous facts about idempotents, but in many
cases the proofs differ or are more delicate for n-potents. In Section
3, we define n-potent K-theory and study its properties and compute
some examples. Finally, in Section 4, we consider n-homomorphisms
on rings and algebras [2, 3, 4], and show that n-potent K-theory is
functorial for such maps; this is a phenomenon that does not appear
in ordinary idempotent K-theory.
The authors thank Dana Williams and Tom Shemanske for their
helpful comments and suggestions.
Note: Unless stated otherwise, all rings and algebras have a unit;
i.e., a multiplicative identity, and all ring and algebra homomorphisms
are unital.
2. Equivalence of n-potents
Fix a natural number n ≥ 2. In this section, we develop the ba-
sic theory of n-potents, including various equivalence relations among
them. We begin by looking at n-potents over general rings, but even-
tually we will specialize to get a well-behaved theory.
Definition 2.1. Let R be a ring. An element e in R is called an
n-potent if en = e. For n = 2, 3, 4, we use the terms idempotent,
tripotent, and quadripotent, respectively. The set of all n-potents in
R is denoted Pn(R).
We begin with a very simple but useful fact about n-potents:
Lemma 2.2. Suppose e is an n-potent. Then en−1 is an idempotent.
Proof. (en−1)2 = en−1en−1 = enen−2 = een−2 = en−1. �
Definition 2.3. Let e and f be n-potents in a ring R. We say that
e and f are algebraically equivalent and write e ∼a f if there exist
elements a and b in R such that e = ab and f = ba. We say that e and
f are similar and write e ∼s f if there exists an invertible element z
in R with the property that f = zez−1.
Lemma 2.4. Suppose that e and f are algebraically equivalent n-
potents in a ring R. Then the elements a and b described in Definition
K0-THEORY WITH n-POTENTS 3
2.3 can be chosen so that
a = en−1a = afn−1 = en−1afn−1
b = fn−1b = ben−1 = fn−1ben−1.
Proof. Choose elements ã and b̃ in R so that ãb̃ = e and b̃ã = f . Set
a = en−1ãfn−1 and b = fn−1b̃en−1. Using Lemma 2.2, we have
ab = (en−1ãfn−1)(fn−1b̃en−1) = en−1ãfn−1b̃en−1
= en−1(ãb̃)nen−1 = en−1enen−1 = (en−1)2en = en−1e = en = e.
Similarly, ba = f . The two strings of equalities in the statement of the
lemma then follow easily. �
Proposition 2.5. The relations ∼a and ∼s are equivalence relations
on Pn(R).
Proof. The only nonobvious point to establish is that ∼a is transitive.
Let e, f , and g be elements of Pn(R), and suppose that e ∼a f ∼a g.
Choose elements a, b, c and d in R so that e = ab, f = ba = cd, and
g = dc, and set s = afn−2c and t = db. Then
st = afn−2cdb = afn−1b = a(ba)n−1b = (ab)n = en = e
ts = dbafn−2c = dfn−1c = d(cd)n−1c = (dc)n = gn = g.
Proposition 2.6. If e and f are similar n-potents in a ring R, then
they are algebraically equivalent.
Proof. Choose an invertible element z in R such that f = zez−1, and set
a = ez−1 and b = zen−1. Then ab = en = e and ba = zenz−1 = f . �
As is the case with idempotents, algebraic equivalence does not imply
similarity in general. However, we do have the following result, just as
for idempotents:
Proposition 2.7. Suppose that e and f are algebraically equivalent
n-potents in a ring R. Then
in the ring M2(R) of 2× 2 matrices over R.
4 EFTON PARK AND JODY TROUT
Proof. Choose elements a and b in R so that e = ab and f = ba; without
loss of generality, we assume that a and b satisfy the conclusions of
Lemma 2.4. Define
1− fn−1 b
afn−2 1− en−1
1− en−1 en−1
en−1 1− en−1
Straightforward computation yields that both u2 and v2 equal the iden-
tity matrix in M2(R), and thus each is its own inverse. Set z = uv.
Then we compute that
z−1 =
1− fn−1 b
afn−2 1− en−1
1− fn−1 b
afn−2 1− en−1
beafn−2 0
since beafn−2 = b(ab)a(ba)n−2 = (ba)n = fn = f. �
Definition 2.8. We say n-potents e and f in a ring R are orthogonal
if ef = fe = 0, in which case we write e ⊥ f .
The next result follows immediately by mathematical induction.
Proposition 2.9. Let e and f be orthogonal n-potents in a ring R.
Then (e+ f)k = ek + fk. In particular, e+ f is an n-potent.
Proposition 2.10. For i = 1, 2, let ei and fi be algebraically equivalent
n-potents in a ring R. Suppose that e1 and f1 are orthogonal to e2 and
f2, respectively. Then e1 + e2 and f1 + f2 are algebraically equivalent.
Proof. For i = 1, 2, choose ai and bi so that ei = aibi, fi = biai, and so
that ai and bi satisfy the conclusion of Lemma 2.4. Then
a1b2 = a1f
2 b2 = 0.
Similarly, b2a1, a2b1, and b1a2 are also zero. Thus
(a1 + a2)(b1 + b2) = a1b1 + a2b2 = e1 + e2
(b1 + b2)(a1 + a2) = b1a1 + b2a2 = f1 + f2,
whence e1 + e2 is algebraically equivalent to f1 + f2. �
Proposition 2.11. Let e and f be n-potents in a ring R.
K0-THEORY WITH n-POTENTS 5
(b) If e ⊥ f then
e+ f 0
Proof. Define
and b =
0 fn−1
en−1 0
in M2(R). Then
0 fn−1
en−1 0
0 fn−1
en−1 0
which establishes the first part of (a); to obtain the second part, simply
take f to be zero.
To prove (b), first observe that if e ⊥ f , then e + f is an n-potent
by Proposition 2.9. Define
and b =
en−1 fn−1
en−1 fn−1
en efn−1
fen−1 fn
en−1 fn−1
en + fn 0
e + f 0
whence the result follows. �
Later in this paper we will restrict our attention to n-potent K-
theory of cyclotomic algebras:
Definition 2.12. For each integer n ≥ 2, the cyclotomic field Q(n−1)
is the field obtained by adjoining the (n − 1)st primitive root of unity
ζn−1 = e
2πi/(n−1) to the field Q of rational numbers. A cyclotomic
algebra is a Q(n− 1)-algebra for some n ≥ 2.
Observe that Q(n− 1) ⊂ C, and therefore every C-algebra is canon-
ically a Q(n− 1)-algebra for all n.
Definition 2.13. Let F be a field and let A be an F-algebra with unit.
An n-partition of unity is an ordered n-tuple (e0, e1, . . . , en−1) of idem-
potents in A such that
(1) e0 + e1 + · · ·+ en−1 = 1;
6 EFTON PARK AND JODY TROUT
(2) e0, e1, . . . , en−1 are pairwise orthogonal; i.e., ejek = δjkek for all
0 ≤ j, k ≤ n− 1.
Note that e0 = 1 − (e1 + · · · + en−1) is completely determined by
e1, e2, . . . , en−1 and is thus redundant in the notation for an n-partition
of unity.
Cyclotomic algebras admit a distinguished n-partition of unity. Set
ω0 = 0 and let ωk = e
2πi(k−1)/(n−1) for 1 ≤ k ≤ n − 1. Note that
ω1, . . . , ωn−1 are the (n−1)st roots of unity, and Ωn = {ω0, ω1, . . . , ωn−1}
is the set of roots of the polynomial equation xn − x = 0.
Theorem 2.14. Let A be a Q(n − 1)-algebra with unit, and suppose
e is an n-potent in A. Then there exists a unique n-partition of unity
(e0, e1, . . . , en−1) in A such that
ωkek.
Proof. Let p0, p1, . . . , pn−1 ∈ Q(n− 1)[x] be the Lagrange polynomials
pk(x) =
j 6=k(x− ωj)
j 6=k(ωk − ωj)
In particular, p0(x) = 1 − xn−1. Each polynomial pk has degree n − 1
and satisfies pk(ωk) = 1 and pk(ωj) = 0 for all j 6= k. We claim that
for all numbers α ∈ Q(n− 1) ⊆ C,
pk(α) = p0(α) + · · ·+ pn−1(α) = 1
and that
(2) α =
ωkpk(α).
Indeed, these identities follow from the fact that these polynomial equa-
tions have degree n− 1 but are satisfied by the n distinct points in Ωn.
Now, given any ωni = ωi in Ωn it follows that pk(ωi)
2 = pk(ωi).
Hence, for any n-potent e ∈ A, if we define ek = pk(e), then each ek is
an idempotent in A, and Equation (1) implies that
pk(e) = 1.
These idempotents are pairwise orthogonal, because
ejek = pj(e)pk(e) = 0
K0-THEORY WITH n-POTENTS 7
for j 6= k. Finally,
ωkpk(e) =
by Equation (2). �
3. K0-theory with n-potents
We can now proceed to construct our n-potent K-theory groups.
Definition 3.1. Let R be a ring. For all k ≥ 1, let Pnk (R) denote the
set of n-potents in Mk(R), and let ik denote the inclusion
ik(a) =
ofMk(R) intoMk+1(R), as well as its restriction as a map from Pnk (R)
to Pnk+1(R). Define M∞(R) and Pn∞(R) to be the (algebraic) direct
limits
M∞(R) =
Mk(R), Pn∞(R) =
Pnk (R) = Pn(M∞(R)).
We define a binary operation ⊕ on Pn∞(R) as follows: let e and f be
elements of Pn∞(R), choose the smallest natural numbers k and ℓ such
that e ∈Mk(R) and f ∈Ml(R), and set
e⊕ f = diag(e, f) =
∈ Pnk+l(R) ⊂ Pn∞(R).
Definition 3.2. Let R be a ring, and define an equivalence relation ∼
on Pn∞(R) as follows: take e and f in Pn∞(R), and choose a natural
number k sufficiently large that e and f are elements of Pnk (R). Then
e ∼ f if e ∼a f in Mk(R). We let Vn(R) denote the set of equivalence
classes of ∼.
Note that if e = ab and f = ba in Mk(R), then
and therefore the equivalence relation described in Definition 3.2 is
well-defined.
8 EFTON PARK AND JODY TROUT
Note that for any n-potent e, f in M∞(R), we get
Thus, the binary operation ⊕ induces a binary operation + on Vn∞(R)
as follows: take e and f in Pn∞(R), and define
[e] + [f ] = [e⊕ f ] =
This operation is well-defined and commutative by Propositions 2.9
and 2.11.
The next proposition is straightforward and left to the reader.
Proposition 3.3. For every ring R and natural number n ≥ 2, Vn(R)
is an abelian monoid under the addition defined above, and whose iden-
tity element is the class of the zero n-potent. If α : R −→ S is a unital
ring homomorphism, then the induced map Vn(α) : Vn(R) −→ Vn(S)
given by
Vn(α)([(aij)]) = [(α(aij))]
is a well-defined homomorphism of abelian semigroups. The correspon-
dences R 7→ Vn(R) and α 7→ Vn(α) induce a covariant functor from the
category of rings and ring homomorphisms to the category of abelian
monoids and monoid homomorphisms.
Definition 3.4. Let R be a ring and let n ≥ 2 be a natural number.
We define Kn0 (R) to be the Grothendieck completion [6] of the abelian
monoid Vn(R). Given an n-potent e in Pn∞(R), we denote its class in
Kn0 (R) by [e].
In light of Propositions 2.6 and 2.7, we could have alternatively used
similarity to define Vn(R), and hence Kn0 (R).
Proposition 3.5. The assignments R 7→ Kn0 (R) determines a covari-
ant functor from the category of rings and ring homomorphisms to the
category of abelian groups and group homomorphisms.
Proof. Proposition 3.3 states that V is a covariant functor from the
category of rings to the category of abelian monoids, and Grothendieck
completion determines a covariant functor from the category of abelian
monoids to the category of abelian groups; we get the desired result by
composing these two functors. �
The following result shows that for (unital) algebras over a field of
characteristic 6= 2, the tripotent K-theory functor K30 offers us no new
invariants over ordinary idempotent K-theory. However, we will see
later (Theorem 3.15) that the situation is subtly different for K40 .
K0-THEORY WITH n-POTENTS 9
Theorem 3.6. Let F be a field with characteristic 6= 2. If A is a unital
algebra over F then there is a natural isomorphism
K30 (A)
K0(A)
of abelian groups.
Proof. If e = e3 ∈M∞(A) is a tripotent, then one can easily check that
(e2 + e) and e2 =
(e2 − e)
are (unique) idempotents in M∞(A) such that e = e1 − e2. It follows
that we have a natural bijection of abelain monoids
V3(A) → V2(A)⊕ V2(A)
[e] 7→ [e1]⊕ [e2]
with inverse map [e1]⊕[e2] 7→ [e1⊕−e2]. Since these maps are additive,
the result easily follows. �
While Kn0 (R) is well-defined for any ring R, to obtain a well-behaved
theory where the usual exact sequences exist, we must restrict our
attention to a smaller class of rings. The problem is that unlike the
situation for idempotents, it is not generally true that if e is an n-
potent, then so is 1− e. However, given an n-potent in an algebra over
the cyclotomic field Q(n− 1), there is an adequate substitute:
Definition 3.7. Let e be an n-potent in a Q(n − 1)-algebra A, and
write
as in the conclusion of Theorem 2.14. We define an n-potent
ω1(1− e1), ω2(1− e2), . . . , ωn−1(1− en−1)
∈Mn−1(A)
and call e⊥ the complementary n-potent of e.
Observe that if n = 2, this definition agrees with the usual one for
idempotents; i.e., e⊥ = 1− e. Note also that e⊕ e⊥ ∼s ω, where
ω = diag(ω11A, . . . , ωn1A) ∈Mn−1(Q(n− 1)) ⊆Mn−1(A).
Proposition 3.8 (Standard Picture of Kn0 (A)). Let n ≥ 2 be a natural
number and let A be a Q(n−1)-algebra. Then every element of Kn0 (A)
can be written in the form [e]−[ω], where e in an n-potent inMk(A) for
some natural number k and ω is a diagonal n-potent in Mk(Q(n− 1)).
10 EFTON PARK AND JODY TROUT
Proof. Start with an element [ẽ]− [f̃ ] in Kn0 (A), and take f̃⊥ to be the
complementary n-potent of f as defined in Definition 3.7. Then
[ẽ]− [f̃ ] =
[ẽ] + [f̃⊥]
[f̃ ] + [f̃⊥]
The n-potents f̃ and f̃⊥ are orthogonal, and therefore
[f̃ ] + [f̃⊥] = [f̃ + f̃⊥] = [ω],
where ω has the desired form. Finally we take e to be ẽ⊕ f̃⊥, and by
enlarging the matrix ω, we obtain the desired result. �
Proposition 3.9. Let n ≥ 2 and let A be a Q(n− 1)-algebra. Suppose
e and f are n-potents in M∞(A). Then [e] = [f ] in K
0 (A) if and only
if e⊕ ω is similar to f ⊕ ω for some n-potent ω in M∞(Q(n− 1)).
Proof. The “only if” direction is obvious. To show the inference in the
opposite direction, suppose that [e] = [f ] in Kn0 (A). By the definition
of the Grothendieck completion, e ⊕ ẽ is similar to f ⊕ ẽ for some n-
potent ẽ in M∞(A). Then e ⊕ ẽ ⊕ ẽ⊥ is similar to f ⊕ ẽ ⊕ ẽ⊥. But if
we write ẽ =
k=1 ωkẽk as in Theorem 2.14, then Proposition 2.11(b)
implies that
ẽ ∼s diag
ω1ẽ1, ω2ẽ2, . . . , ωn−1ẽn−1
Therefore ẽ ⊕ ẽ⊥ is similar to an n-potent in M∞(Q(n − 1)), and the
proposition follows. �
We next turn our attention to n-potent K-theory for nonunital alge-
bras. Given a nonunital Q(n − 1)-algebra A, we define its unitization
A+ as the unital Q(n−1)-algebra A+ = {(a, λ) : a ∈ A, λ ∈ Q(n−1)},
where addition and scalar multiplication are defined componentwise,
and multiplication is given by (a, λ)(b, τ) = (ab+ aτ + bλ, λτ).
Definition 3.10. Let A be a nonunital Q(n−1)-algebra, and let A+ be
its unitization. Let π : A+ −→ Q(n− 1) be the algebra homomorphism
π(a, λ) = λ. Then we define Kn0 (A) = ker π∗.
It is easy to see that π∗ is surjective, so by definition of K
0 (A) we
have a short exact sequence
0 // Kn0 (A)
// Kn0 (A
// Kn0 (Q(n− 1)) //
with splitting induced by the map ψ : Q(n − 1) −→ A+ defined by
ψ(λ) = (0, λ). In addition, it is easy to check that if A already has a
unit and we form A+, then ker π∗ is naturally isomorphic to our original
definition of Kn0 (A).
K0-THEORY WITH n-POTENTS 11
Proposition 3.11. Let A be a nonunital Q(n−1)-algebra. Then every
element of Kn0 (A) can be written in the form [e]− [s(e)], where e is an
n-potents in Mk(A
+) for some integer k ≥ 1, and s = ψ ◦π : A+ → A+
is the scalar mapping [6, Sect. 4.2.1].
Proof. Follows directly from Proposition 3.8 and Definition 3.10. �
Proposition 3.12 (Half-exactness). Every short exact sequence
0 // I
// A/I // 0
of Q(n− 1)-algebras, with A unital, induces an exact sequence
Kn0 (I)
// Kn0 (A)
// Kn0 (A/I)
of abelian n-potent K-theory groups.
Proof. Since q ◦ i = 0, we have by functoriality that q∗ ◦ i∗ = 0 and
so the image of Kn0 (I) under i∗ in K
0 (A) is contained in the kernel of
q∗. To show the reverse inclusion, suppose we have [e]− [λ] in Kn0 (A)
such that q∗
[e]− [λ]
= 0. Then [q(e)] = [q(λ)] = [λ] in Kn0 (A/I). By
Proposition 3.9, there exists an n-potent τ in M∞(Q(n− 1)) so that
q(e)⊕ τ ∼s λ⊕ τ.
Choose N sufficiently large so that we may view e, λ, and τ as N by
N matrices, and choose z in GL2N (A/I) so that
q(e)⊕ τ
z−1 = λ⊕ τ.
By Proposition 3.4.2 and Corollary 3.4.4 in [1], we can lift diag(z, z−1)
to an element u in GL4N(A). Set f = u(e⊕ τ)u−1. Then
q(f) = diag(z, z−1)(q(e)⊕ τ)diag(z−1, z) = λ⊕ τ,
and thus f and λ⊕ τ are in M4N (I+). Therefore
[e]− [λ] = [e⊕ τ ]− [λ⊕ τ ] = i∗([f ]− [λ⊕ τ ])
is in the image of Kn0 (I) under i∗ as desired. �
Note that our proof of Proposition 3.12 relies critically on Proposi-
tion 3.9, which in turn is proved using the standard picture of Kn0 (A).
We do not have a standard picture for Kk0 (A) when k 6= n, and it
seems likely to the authors that Kk0 is, in fact, not half-exact in this
case. However, we do not have a counterexample where half-exactness
fails to hold.
While it is not at all obvious from its definition, Kn0 (A) can be iden-
tified with a more familiar object.
12 EFTON PARK AND JODY TROUT
Theorem 3.13. Let n ≥ 2 be a natural number and let A be a not nec-
essarily unital Q(n− 1)-algebra. Then there is a natural isomorphism
Kn0 (A)
K0(A)
of abelian groups.
Proof. First consider the case where A is unital. We define a homo-
morphism ψ̃ : Vn(A) −→
V0(A)
in the following way: for each
n-potent e =
ωkek in M∞(A), set
ψ̃[e] =
[e1], [e2], . . . , [en−1]
It is easy to check that ψ̃ is additive and well-defined. Next, define a
homomorphism φ̃ :
)n−1 −→ Vn(A) by the formula
[f1], [f2], . . . , [fn−1]
ω1diag(f1, 0, 0, . . . , 0) + ω2diag(0, f2, 0, . . . , 0) + · · ·
+ ωn−1diag(0, 0, . . . , 0, fn−1)
Note that
[f1], [f2], . . . , [fn−1]
ω1diag(f1, 0, . . . , 0) + · · ·+ ωn−1diag(0, 0, . . . , fn−1)
[diag(f1, 0, . . . , 0)], [diag(0, f2, . . . , 0)] . . . [diag(0, 0, . . . , fn−1)]
[f1], [f2], . . . , [fn−1]
φ̃ψ̃[e] = φ
[e1], [e2], . . . , [en−1]
ω1diag(e1, 0, . . . , 0) + · · ·+ ωn−1diag(0, 0, . . . , en−1)
diag(ω1e1, ω2e2, . . . , ωn−1en−1)
= [e],
where the last equality is a consequence of Proposition 2.11(b). The
universal mapping property of the Grothendieck completion implies
that ψ̃ extends uniquely to an abelian group isomorphism
ψ : Kn0 (A) −→
K0(A)
and thus the theorem is true for unital Q(n− 1)-algebras.
K0-THEORY WITH n-POTENTS 13
Now suppose that A does not have a unit. Then we have the following
commutative diagram with exact rows:
0 // Kn0 (A)
// Kn0 (A
+) //
Kn0 (Q(n− 1)) //
0 // K0(A)
n−1 // K0(A
+)n−1 // K0(Q(n− 1))n−1 // 0
An easy diagram chase shows that there is a unique group iso-
morphism from Kn0 (A) to
K0(A)
that makes the diagram com-
mute. �
Since a complex algebra is a Q(n− 1)-algebra for all values of n, we
have the following immediate corollary.
Corollary 3.14. If A is a C-algebra, there are natural isomorphisms
Kn0 (A)
K0(A)
of abelian groups for all natural numbers n ≥ 2.
We now arrive at the result that suggests why we should consider all
Kn0 -functors for algebras over a cyclotomic field.
Theorem 3.15. Let Q(4) = Q[i] be the 4th cyclotomic field. Then we
have the following isomorphisms of abelian groups:
K20(Q(4))
∼= Z,
K30(Q(4))
∼= Z2,
K40(Q(4))
∼= Z⊕ 2Z,
K50(Q(4))
∼= Z4.
Thus, K40 (Q(4)) 6∼= Z3 ∼= K40 (Q(3)).
Proof. Since Q(4) is a field [7], we have K20 (Q(4)) = K0(Q(4))
∼= Z.
The field Q(4) has characteristic 0 6= 2, so Theorem 3.6 implies that
K30(Q(4))
K0(Q(4)
)2 ∼= Z2. Theorem 3.13 implies that we have an
isomorphism K50(Q(4))
K0(Q(4)
)4 ∼= Z4.
However, the spectrum of 4-potents is contained in
0, 1,−1
which is not contained inQ(4) since the two primitive 3rd roots of unity
ω = ζ3 = −12 +
i and ω̄ = ζ̄3 = −12 −
i are not in Q(4) = Q[i].
Given any 4-potent e ∈Mn(Q(4)) ⊂Mn(C) we can uniquely write
e = e1 + ωe2 + ω̄e3,
14 EFTON PARK AND JODY TROUT
where e1, e2, e3 are orthogonal idempotents in Mn(C) that sum to an
idempotent e1+e2+e3 = e
3 inMn(Q(4)) by Lemma 2.2. We thus have
e2 = e1 + ω̄e2 + ωe3
e3 = e1 + e2 + e3
because ω2 = ω̄, ω̄2 = ω, and ω3 = ω̄3 = 1. Since ω + ω̄ = −1, this
implies that the first idempotent
e1 = (e + e
2 + e3)/3 ∈Mn(Q(4))
and the sum of the last two idempotents
e2 + e3 = e
3 − e1 ∈Mn(Q(4))
are both inMn(Q(4)). Using a simple trace argument and the fact that
ω, ω̄ 6∈ Q(4), we conclude that
rank(e2) = trace(e2) = trace(e3) = rank(e3),
and so rank(e2 + e3) = trace(e2 + e3) = 2trace(e2) is even. We then
have a well-defined map
V4(Q(4)) → V2(Q(4))⊕ 2V2(Q(4)) ∼= N⊕ 2N
[e] 7→ [e1]⊕ [e2 + e3] ∼= trace(e1)⊕ 2 trace(e2);
this is because the classes of e1 and e1 + e2 are preserved by (stable)
similarity, and the K0-class of an idempotent in a matrix ring over a
number field (or a PID) is the rank (= trace). It is easy to check that
this map is injective (using e1 ⊥ e2 + e3 in Mn(Q(4))) and additive.
The only question is surjectivity. It suffices to show that there is a
4-potent e over Q(4) whose stable similarity class is mapped to the
generator 1⊕ 2 of N⊕ 2N. Consider the block diagonal matrix
1 0 0
0 0 i
0 i −1
 ∈M3(Q(4)),
which is easily checked to be quadripotent. The lower right quadripo-
tent 2× 2 invertible block has the desired eigenvalues ω and ω̄, and so
does not diagonalize over Q(4). The result now follows easily. �
4. n-Homomorphisms and Kn0 Functorality
We know from Proposition 3.5 that Kn0 is a covariant functor from
the category of (unital) rings and ring homomorphisms to the category
of abelian groups and group homomorphisms. However, Kn0 is actually
functorial for a more general class of ring mappings.
K0-THEORY WITH n-POTENTS 15
Definition 4.1. Let R and S be rings. An additive map (not neces-
sarily unital) φ : R −→ S is called an n-homomorphism if
φ(a1a2 · · · an) = φ(a1)φ(a2) · · ·φ(an)
for all a1, a2, . . . , an in R.
Obviously every (ring) homomorphism is an n-homomorphism, but
the converse is false in general. For example, an AEn-ring is a ring
R such that every additive map φ : R → R is an n-homomorphism.
Feigelstock [2, 3] classified all unital AEn-rings. The algebraic version
of n-homomorphism was introduced for complex algebras in [4] and has
been carefully studied in the case of C∗-algebras in [5].
Proposition 4.2. Let φ : R → S be an n-homomorphism between
unital rings. Then φ induces a group homomorphism
φ∗ : K
0 (R) −→ Kn0 (S).
Furthermore, the assignment R 7→ Kn0 (R) is a covariant functor from
the category of unital rings and n-homomorphisms to the category of
abelian groups and ordinary group homomorphisms.
Proof. For each natural number k, we extend φ to a map from Mk(R)
to Mk(S) by applying φ to each matrix entry; it is easy to check this
also gives us an n-homomorphism. Moreover, φ is compatible with
stabilization of matrices; the only nonobvious point to check is that φ
respects algebraic equivalence.
Let e and f be algebraically equivalent n-potents in Mk(R) for some
k, and choose a and b in Mk(R) so that e = ab and f = ba. Define
elements a′ = φ(ea)φ(f)n−2 and b′ = φ(b) in Mk(S). We compute:
a′b′ = φ(ea)φ(f)n−2φ(b) = φ((ea)fn−2b) =
φ(ea(ba)n−2b) = φ(e(ab)n−1) = φ(en) = φ(e).
A similar argument shows that b′a′ = φ(f). Therefore φ determines
a monoid homomorphism from Vn(R) to Vn(S), and hence a group
homomorphism φ∗ : K
0 (R) −→ Kn0 (S). We leave it to the reader to
make the straightforward computations to show that we have a covari-
ant functor. �
Note that while we have an isomorphism Kn0 (A)
K0(A)
Q(n− 1)-algebras, it is not at all clear from the right hand side of this
isomorphism that Kn0 (A) is functorial for n-homomorphisms.
16 EFTON PARK AND JODY TROUT
References
[1] B. Blackadar,K-theory for Operator Algebras, 2nd ed., MSRI Publication Series
5, Springer-Verlag, New York, 1998.
[2] S. Feigelstock, Rings whose additive endomorphisms are N -multiplicative, Bull.
Austral. Math. Soc. 39 (1989), no. 1, 11–14.
[3] S. Feigelstock, Rings whose additive endomorphisms are n-multiplicative. II,
Period. Math. Hungar. 25 (1992), no. 1, 21–26.
[4] M. Hejazian, M. Mirzavaziri, M.S. Moslehian, n-homomorphisms, Bull. Iranian
Math. Soc. 31 (2005), no. 1, 13-23.
[5] E. Park and J. Trout, On the Nonexistence of Nontrivial Involutive n-
homomorphisms of C∗-algebras, Trans. Amer. Math. Soc. 361 (2009), no. 4,
1949–1961
[6] M. Rordam, F. Larsen, N. Laustsen, An Introduction to K-theory for C∗-
algebras, London Mathematical Society Student Texts, vol. 49. Cambridge Uni-
versity Press, Cambridge, 2000.
[7] J. Rosenberg, Algebraic K-theory and Its Applications, Graduate Texts in Math-
ematics, vol. 147, Springer-Verlag, New York, 1994.
Box 298900, Texas Christian University, Fort Worth, TX 76129
E-mail address : e.park@tcu.edu
6188 Kemeny Hall, Dartmouth College, Hanover, NH 03755
E-mail address : jody.trout@dartmouth.edu
	1. Introduction
	2. Equivalence of n-potents
	3. K0-theory with n-potents
	4. n-Homomorphisms and K0n Functorality
	References

  Let $n \geq 2$ be an integer. An \emph{$n$-potent} is an element $e$ of a
ring $R$ such that $e^n = e$. In this paper, we study $n$-potents in matrices
over $R$ and use them to construct an abelian group $K_0^n(R)$. If $A$ is a
complex algebra, there is a group isomorphism $K_0^n(A) \cong
\bigl(K_0(A)\bigr)^{n-1}$ for all $n \geq 2$. However, for algebras over
cyclotomic fields, this is not true in general. We consider $K_0^n$ as a
covariant functor, and show that it is also functorial for a generalization of
homomorphism called an \emph{$n$-homomorphism}.

Introduction
For more than forty years, K-theory has been an essential tool in
studying rings and algebras [1, 7]. Given a ring R, a simple functorial
object associated to R is the abelian group K0(R). There are multi-
ple ways of defining K0(R), but the most useful characterization when
working with operator algebras is to define K0(R) in terms of idempo-
tents (or projections, if an involution is present) in matrix algebras over
R; i.e., elements e in Mk(R) for some k with the feature that e
2 = e
(p = p∗ = p2 in the involutive case). In this paper, we define, for each
natural number n ≥ 2, a group which we denote Kn0 (R). This group
is constructed from matrices e over R with the property that en = e;
we call such matrices n-potents. We define Kn0 (R) for all rings, unital
or not, and show that Kn0 determines a covariant functor from rings to
abelian groups.
Let Q(n− 1) be the cyclotomic field obtained from the rationals by
adjoining the (n− 1)-th roots of unity. We show that Kn0 is half-exact
on the subcategory of Q(n − 1)-algebras, and given any such algebra
A, we show that Kn0 (A) is isomorphic to a direct sum of n − 1 copies
of K0(A). Since a C-algebra A is a Q(n − 1)-algebra for all n, what-
ever invariants are contained in Kn0 (A) are already contained in K0(A).
However, K
0 for p 6= n may generate new groups for cyclotomic alge-
bras, e.g., K40(Q(4))
∼= Z⊕2Z (Theorem 3.15) which is not isomorphic
2010 Mathematics Subject Classification: 18F30, 19A99, 19K99.
http://arxiv.org/abs/0704.0775v2
2 EFTON PARK AND JODY TROUT
to K40 (Q(3))
∼= Z3. Thus, K40 distinguishes between the fields Q(3) and
Q(4), but idempotent, and also tripotent (n = 3), K-theory does not.
The paper is organized as follows. In Section 2, we define various
notions of equivalence on the set of n-potents, and explore the rela-
tionships between these equivalence relations. Most of our results in
this section mirror analogous facts about idempotents, but in many
cases the proofs differ or are more delicate for n-potents. In Section
3, we define n-potent K-theory and study its properties and compute
some examples. Finally, in Section 4, we consider n-homomorphisms
on rings and algebras [2, 3, 4], and show that n-potent K-theory is
functorial for such maps; this is a phenomenon that does not appear
in ordinary idempotent K-theory.
The authors thank Dana Williams and Tom Shemanske for their
helpful comments and suggestions.
Note: Unless stated otherwise, all rings and algebras have a unit;
i.e., a multiplicative identity, and all ring and algebra homomorphisms
are unital.
2. Equivalence of n-potents
Fix a natural number n ≥ 2. In this section, we develop the ba-
sic theory of n-potents, including various equivalence relations among
them. We begin by looking at n-potents over general rings, but even-
tually we will specialize to get a well-behaved theory.
Definition 2.1. Let R be a ring. An element e in R is called an
n-potent if en = e. For n = 2, 3, 4, we use the terms idempotent,
tripotent, and quadripotent, respectively. The set of all n-potents in
R is denoted Pn(R).
We begin with a very simple but useful fact about n-potents:
Lemma 2.2. Suppose e is an n-potent. Then en−1 is an idempotent.
Proof. (en−1)2 = en−1en−1 = enen−2 = een−2 = en−1. �
Definition 2.3. Let e and f be n-potents in a ring R. We say that
e and f are algebraically equivalent and write e ∼a f if there exist
elements a and b in R such that e = ab and f = ba. We say that e and
f are similar and write e ∼s f if there exists an invertible element z
in R with the property that f = zez−1.
Lemma 2.4. Suppose that e and f are algebraically equivalent n-
potents in a ring R. Then the elements a and b described in Definition
K0-THEORY WITH n-POTENTS 3
2.3 can be chosen so that
a = en−1a = afn−1 = en−1afn−1
b = fn−1b = ben−1 = fn−1ben−1.
Proof. Choose elements ã and b̃ in R so that ãb̃ = e and b̃ã = f . Set
a = en−1ãfn−1 and b = fn−1b̃en−1. Using Lemma 2.2, we have
ab = (en−1ãfn−1)(fn−1b̃en−1) = en−1ãfn−1b̃en−1
= en−1(ãb̃)nen−1 = en−1enen−1 = (en−1)2en = en−1e = en = e.
Similarly, ba = f . The two strings of equalities in the statement of the
lemma then follow easily. �
Proposition 2.5. The relations ∼a and ∼s are equivalence relations
on Pn(R).
Proof. The only nonobvious point to establish is that ∼a is transitive.
Let e, f , and g be elements of Pn(R), and suppose that e ∼a f ∼a g.
Choose elements a, b, c and d in R so that e = ab, f = ba = cd, and
g = dc, and set s = afn−2c and t = db. Then
st = afn−2cdb = afn−1b = a(ba)n−1b = (ab)n = en = e
ts = dbafn−2c = dfn−1c = d(cd)n−1c = (dc)n = gn = g.
Proposition 2.6. If e and f are similar n-potents in a ring R, then
they are algebraically equivalent.
Proof. Choose an invertible element z in R such that f = zez−1, and set
a = ez−1 and b = zen−1. Then ab = en = e and ba = zenz−1 = f . �
As is the case with idempotents, algebraic equivalence does not imply
similarity in general. However, we do have the following result, just as
for idempotents:
Proposition 2.7. Suppose that e and f are algebraically equivalent
n-potents in a ring R. Then
in the ring M2(R) of 2× 2 matrices over R.
4 EFTON PARK AND JODY TROUT
Proof. Choose elements a and b in R so that e = ab and f = ba; without
loss of generality, we assume that a and b satisfy the conclusions of
Lemma 2.4. Define
1− fn−1 b
afn−2 1− en−1
1− en−1 en−1
en−1 1− en−1
Straightforward computation yields that both u2 and v2 equal the iden-
tity matrix in M2(R), and thus each is its own inverse. Set z = uv.
Then we compute that
z−1 =
1− fn−1 b
afn−2 1− en−1
1− fn−1 b
afn−2 1− en−1
beafn−2 0
since beafn−2 = b(ab)a(ba)n−2 = (ba)n = fn = f. �
Definition 2.8. We say n-potents e and f in a ring R are orthogonal
if ef = fe = 0, in which case we write e ⊥ f .
The next result follows immediately by mathematical induction.
Proposition 2.9. Let e and f be orthogonal n-potents in a ring R.
Then (e+ f)k = ek + fk. In particular, e+ f is an n-potent.
Proposition 2.10. For i = 1, 2, let ei and fi be algebraically equivalent
n-potents in a ring R. Suppose that e1 and f1 are orthogonal to e2 and
f2, respectively. Then e1 + e2 and f1 + f2 are algebraically equivalent.
Proof. For i = 1, 2, choose ai and bi so that ei = aibi, fi = biai, and so
that ai and bi satisfy the conclusion of Lemma 2.4. Then
a1b2 = a1f
2 b2 = 0.
Similarly, b2a1, a2b1, and b1a2 are also zero. Thus
(a1 + a2)(b1 + b2) = a1b1 + a2b2 = e1 + e2
(b1 + b2)(a1 + a2) = b1a1 + b2a2 = f1 + f2,
whence e1 + e2 is algebraically equivalent to f1 + f2. �
Proposition 2.11. Let e and f be n-potents in a ring R.
K0-THEORY WITH n-POTENTS 5
(b) If e ⊥ f then
e+ f 0
Proof. Define
and b =
0 fn−1
en−1 0
in M2(R). Then
0 fn−1
en−1 0
0 fn−1
en−1 0
which establishes the first part of (a); to obtain the second part, simply
take f to be zero.
To prove (b), first observe that if e ⊥ f , then e + f is an n-potent
by Proposition 2.9. Define
and b =
en−1 fn−1
en−1 fn−1
en efn−1
fen−1 fn
en−1 fn−1
en + fn 0
e + f 0
whence the result follows. �
Later in this paper we will restrict our attention to n-potent K-
theory of cyclotomic algebras:
Definition 2.12. For each integer n ≥ 2, the cyclotomic field Q(n−1)
is the field obtained by adjoining the (n − 1)st primitive root of unity
ζn−1 = e
2πi/(n−1) to the field Q of rational numbers. A cyclotomic
algebra is a Q(n− 1)-algebra for some n ≥ 2.
Observe that Q(n− 1) ⊂ C, and therefore every C-algebra is canon-
ically a Q(n− 1)-algebra for all n.
Definition 2.13. Let F be a field and let A be an F-algebra with unit.
An n-partition of unity is an ordered n-tuple (e0, e1, . . . , en−1) of idem-
potents in A such that
(1) e0 + e1 + · · ·+ en−1 = 1;
6 EFTON PARK AND JODY TROUT
(2) e0, e1, . . . , en−1 are pairwise orthogonal; i.e., ejek = δjkek for all
0 ≤ j, k ≤ n− 1.
Note that e0 = 1 − (e1 + · · · + en−1) is completely determined by
e1, e2, . . . , en−1 and is thus redundant in the notation for an n-partition
of unity.
Cyclotomic algebras admit a distinguished n-partition of unity. Set
ω0 = 0 and let ωk = e
2πi(k−1)/(n−1) for 1 ≤ k ≤ n − 1. Note that
ω1, . . . , ωn−1 are the (n−1)st roots of unity, and Ωn = {ω0, ω1, . . . , ωn−1}
is the set of roots of the polynomial equation xn − x = 0.
Theorem 2.14. Let A be a Q(n − 1)-algebra with unit, and suppose
e is an n-potent in A. Then there exists a unique n-partition of unity
(e0, e1, . . . , en−1) in A such that
ωkek.
Proof. Let p0, p1, . . . , pn−1 ∈ Q(n− 1)[x] be the Lagrange polynomials
pk(x) =
j 6=k(x− ωj)
j 6=k(ωk − ωj)
In particular, p0(x) = 1 − xn−1. Each polynomial pk has degree n − 1
and satisfies pk(ωk) = 1 and pk(ωj) = 0 for all j 6= k. We claim that
for all numbers α ∈ Q(n− 1) ⊆ C,
pk(α) = p0(α) + · · ·+ pn−1(α) = 1
and that
(2) α =
ωkpk(α).
Indeed, these identities follow from the fact that these polynomial equa-
tions have degree n− 1 but are satisfied by the n distinct points in Ωn.
Now, given any ωni = ωi in Ωn it follows that pk(ωi)
2 = pk(ωi).
Hence, for any n-potent e ∈ A, if we define ek = pk(e), then each ek is
an idempotent in A, and Equation (1) implies that
pk(e) = 1.
These idempotents are pairwise orthogonal, because
ejek = pj(e)pk(e) = 0
K0-THEORY WITH n-POTENTS 7
for j 6= k. Finally,
ωkpk(e) =
by Equation (2). �
3. K0-theory with n-potents
We can now proceed to construct our n-potent K-theory groups.
Definition 3.1. Let R be a ring. For all k ≥ 1, let Pnk (R) denote the
set of n-potents in Mk(R), and let ik denote the inclusion
ik(a) =
ofMk(R) intoMk+1(R), as well as its restriction as a map from Pnk (R)
to Pnk+1(R). Define M∞(R) and Pn∞(R) to be the (algebraic) direct
limits
M∞(R) =
Mk(R), Pn∞(R) =
Pnk (R) = Pn(M∞(R)).
We define a binary operation ⊕ on Pn∞(R) as follows: let e and f be
elements of Pn∞(R), choose the smallest natural numbers k and ℓ such
that e ∈Mk(R) and f ∈Ml(R), and set
e⊕ f = diag(e, f) =
∈ Pnk+l(R) ⊂ Pn∞(R).
Definition 3.2. Let R be a ring, and define an equivalence relation ∼
on Pn∞(R) as follows: take e and f in Pn∞(R), and choose a natural
number k sufficiently large that e and f are elements of Pnk (R). Then
e ∼ f if e ∼a f in Mk(R). We let Vn(R) denote the set of equivalence
classes of ∼.
Note that if e = ab and f = ba in Mk(R), then
and therefore the equivalence relation described in Definition 3.2 is
well-defined.
8 EFTON PARK AND JODY TROUT
Note that for any n-potent e, f in M∞(R), we get
Thus, the binary operation ⊕ induces a binary operation + on Vn∞(R)
as follows: take e and f in Pn∞(R), and define
[e] + [f ] = [e⊕ f ] =
This operation is well-defined and commutative by Propositions 2.9
and 2.11.
The next proposition is straightforward and left to the reader.
Proposition 3.3. For every ring R and natural number n ≥ 2, Vn(R)
is an abelian monoid under the addition defined above, and whose iden-
tity element is the class of the zero n-potent. If α : R −→ S is a unital
ring homomorphism, then the induced map Vn(α) : Vn(R) −→ Vn(S)
given by
Vn(α)([(aij)]) = [(α(aij))]
is a well-defined homomorphism of abelian semigroups. The correspon-
dences R 7→ Vn(R) and α 7→ Vn(α) induce a covariant functor from the
category of rings and ring homomorphisms to the category of abelian
monoids and monoid homomorphisms.
Definition 3.4. Let R be a ring and let n ≥ 2 be a natural number.
We define Kn0 (R) to be the Grothendieck completion [6] of the abelian
monoid Vn(R). Given an n-potent e in Pn∞(R), we denote its class in
Kn0 (R) by [e].
In light of Propositions 2.6 and 2.7, we could have alternatively used
similarity to define Vn(R), and hence Kn0 (R).
Proposition 3.5. The assignments R 7→ Kn0 (R) determines a covari-
ant functor from the category of rings and ring homomorphisms to the
category of abelian groups and group homomorphisms.
Proof. Proposition 3.3 states that V is a covariant functor from the
category of rings to the category of abelian monoids, and Grothendieck
completion determines a covariant functor from the category of abelian
monoids to the category of abelian groups; we get the desired result by
composing these two functors. �
The following result shows that for (unital) algebras over a field of
characteristic 6= 2, the tripotent K-theory functor K30 offers us no new
invariants over ordinary idempotent K-theory. However, we will see
later (Theorem 3.15) that the situation is subtly different for K40 .
K0-THEORY WITH n-POTENTS 9
Theorem 3.6. Let F be a field with characteristic 6= 2. If A is a unital
algebra over F then there is a natural isomorphism
K30 (A)
K0(A)
of abelian groups.
Proof. If e = e3 ∈M∞(A) is a tripotent, then one can easily check that
(e2 + e) and e2 =
(e2 − e)
are (unique) idempotents in M∞(A) such that e = e1 − e2. It follows
that we have a natural bijection of abelain monoids
V3(A) → V2(A)⊕ V2(A)
[e] 7→ [e1]⊕ [e2]
with inverse map [e1]⊕[e2] 7→ [e1⊕−e2]. Since these maps are additive,
the result easily follows. �
While Kn0 (R) is well-defined for any ring R, to obtain a well-behaved
theory where the usual exact sequences exist, we must restrict our
attention to a smaller class of rings. The problem is that unlike the
situation for idempotents, it is not generally true that if e is an n-
potent, then so is 1− e. However, given an n-potent in an algebra over
the cyclotomic field Q(n− 1), there is an adequate substitute:
Definition 3.7. Let e be an n-potent in a Q(n − 1)-algebra A, and
write
as in the conclusion of Theorem 2.14. We define an n-potent
ω1(1− e1), ω2(1− e2), . . . , ωn−1(1− en−1)
∈Mn−1(A)
and call e⊥ the complementary n-potent of e.
Observe that if n = 2, this definition agrees with the usual one for
idempotents; i.e., e⊥ = 1− e. Note also that e⊕ e⊥ ∼s ω, where
ω = diag(ω11A, . . . , ωn1A) ∈Mn−1(Q(n− 1)) ⊆Mn−1(A).
Proposition 3.8 (Standard Picture of Kn0 (A)). Let n ≥ 2 be a natural
number and let A be a Q(n−1)-algebra. Then every element of Kn0 (A)
can be written in the form [e]−[ω], where e in an n-potent inMk(A) for
some natural number k and ω is a diagonal n-potent in Mk(Q(n− 1)).
10 EFTON PARK AND JODY TROUT
Proof. Start with an element [ẽ]− [f̃ ] in Kn0 (A), and take f̃⊥ to be the
complementary n-potent of f as defined in Definition 3.7. Then
[ẽ]− [f̃ ] =
[ẽ] + [f̃⊥]
[f̃ ] + [f̃⊥]
The n-potents f̃ and f̃⊥ are orthogonal, and therefore
[f̃ ] + [f̃⊥] = [f̃ + f̃⊥] = [ω],
where ω has the desired form. Finally we take e to be ẽ⊕ f̃⊥, and by
enlarging the matrix ω, we obtain the desired result. �
Proposition 3.9. Let n ≥ 2 and let A be a Q(n− 1)-algebra. Suppose
e and f are n-potents in M∞(A). Then [e] = [f ] in K
0 (A) if and only
if e⊕ ω is similar to f ⊕ ω for some n-potent ω in M∞(Q(n− 1)).
Proof. The “only if” direction is obvious. To show the inference in the
opposite direction, suppose that [e] = [f ] in Kn0 (A). By the definition
of the Grothendieck completion, e ⊕ ẽ is similar to f ⊕ ẽ for some n-
potent ẽ in M∞(A). Then e ⊕ ẽ ⊕ ẽ⊥ is similar to f ⊕ ẽ ⊕ ẽ⊥. But if
we write ẽ =
k=1 ωkẽk as in Theorem 2.14, then Proposition 2.11(b)
implies that
ẽ ∼s diag
ω1ẽ1, ω2ẽ2, . . . , ωn−1ẽn−1
Therefore ẽ ⊕ ẽ⊥ is similar to an n-potent in M∞(Q(n − 1)), and the
proposition follows. �
We next turn our attention to n-potent K-theory for nonunital alge-
bras. Given a nonunital Q(n − 1)-algebra A, we define its unitization
A+ as the unital Q(n−1)-algebra A+ = {(a, λ) : a ∈ A, λ ∈ Q(n−1)},
where addition and scalar multiplication are defined componentwise,
and multiplication is given by (a, λ)(b, τ) = (ab+ aτ + bλ, λτ).
Definition 3.10. Let A be a nonunital Q(n−1)-algebra, and let A+ be
its unitization. Let π : A+ −→ Q(n− 1) be the algebra homomorphism
π(a, λ) = λ. Then we define Kn0 (A) = ker π∗.
It is easy to see that π∗ is surjective, so by definition of K
0 (A) we
have a short exact sequence
0 // Kn0 (A)
// Kn0 (A
// Kn0 (Q(n− 1)) //
with splitting induced by the map ψ : Q(n − 1) −→ A+ defined by
ψ(λ) = (0, λ). In addition, it is easy to check that if A already has a
unit and we form A+, then ker π∗ is naturally isomorphic to our original
definition of Kn0 (A).
K0-THEORY WITH n-POTENTS 11
Proposition 3.11. Let A be a nonunital Q(n−1)-algebra. Then every
element of Kn0 (A) can be written in the form [e]− [s(e)], where e is an
n-potents in Mk(A
+) for some integer k ≥ 1, and s = ψ ◦π : A+ → A+
is the scalar mapping [6, Sect. 4.2.1].
Proof. Follows directly from Proposition 3.8 and Definition 3.10. �
Proposition 3.12 (Half-exactness). Every short exact sequence
0 // I
// A/I // 0
of Q(n− 1)-algebras, with A unital, induces an exact sequence
Kn0 (I)
// Kn0 (A)
// Kn0 (A/I)
of abelian n-potent K-theory groups.
Proof. Since q ◦ i = 0, we have by functoriality that q∗ ◦ i∗ = 0 and
so the image of Kn0 (I) under i∗ in K
0 (A) is contained in the kernel of
q∗. To show the reverse inclusion, suppose we have [e]− [λ] in Kn0 (A)
such that q∗
[e]− [λ]
= 0. Then [q(e)] = [q(λ)] = [λ] in Kn0 (A/I). By
Proposition 3.9, there exists an n-potent τ in M∞(Q(n− 1)) so that
q(e)⊕ τ ∼s λ⊕ τ.
Choose N sufficiently large so that we may view e, λ, and τ as N by
N matrices, and choose z in GL2N (A/I) so that
q(e)⊕ τ
z−1 = λ⊕ τ.
By Proposition 3.4.2 and Corollary 3.4.4 in [1], we can lift diag(z, z−1)
to an element u in GL4N(A). Set f = u(e⊕ τ)u−1. Then
q(f) = diag(z, z−1)(q(e)⊕ τ)diag(z−1, z) = λ⊕ τ,
and thus f and λ⊕ τ are in M4N (I+). Therefore
[e]− [λ] = [e⊕ τ ]− [λ⊕ τ ] = i∗([f ]− [λ⊕ τ ])
is in the image of Kn0 (I) under i∗ as desired. �
Note that our proof of Proposition 3.12 relies critically on Proposi-
tion 3.9, which in turn is proved using the standard picture of Kn0 (A).
We do not have a standard picture for Kk0 (A) when k 6= n, and it
seems likely to the authors that Kk0 is, in fact, not half-exact in this
case. However, we do not have a counterexample where half-exactness
fails to hold.
While it is not at all obvious from its definition, Kn0 (A) can be iden-
tified with a more familiar object.
12 EFTON PARK AND JODY TROUT
Theorem 3.13. Let n ≥ 2 be a natural number and let A be a not nec-
essarily unital Q(n− 1)-algebra. Then there is a natural isomorphism
Kn0 (A)
K0(A)
of abelian groups.
Proof. First consider the case where A is unital. We define a homo-
morphism ψ̃ : Vn(A) −→
V0(A)
in the following way: for each
n-potent e =
ωkek in M∞(A), set
ψ̃[e] =
[e1], [e2], . . . , [en−1]
It is easy to check that ψ̃ is additive and well-defined. Next, define a
homomorphism φ̃ :
)n−1 −→ Vn(A) by the formula
[f1], [f2], . . . , [fn−1]
ω1diag(f1, 0, 0, . . . , 0) + ω2diag(0, f2, 0, . . . , 0) + · · ·
+ ωn−1diag(0, 0, . . . , 0, fn−1)
Note that
[f1], [f2], . . . , [fn−1]
ω1diag(f1, 0, . . . , 0) + · · ·+ ωn−1diag(0, 0, . . . , fn−1)
[diag(f1, 0, . . . , 0)], [diag(0, f2, . . . , 0)] . . . [diag(0, 0, . . . , fn−1)]
[f1], [f2], . . . , [fn−1]
φ̃ψ̃[e] = φ
[e1], [e2], . . . , [en−1]
ω1diag(e1, 0, . . . , 0) + · · ·+ ωn−1diag(0, 0, . . . , en−1)
diag(ω1e1, ω2e2, . . . , ωn−1en−1)
= [e],
where the last equality is a consequence of Proposition 2.11(b). The
universal mapping property of the Grothendieck completion implies
that ψ̃ extends uniquely to an abelian group isomorphism
ψ : Kn0 (A) −→
K0(A)
and thus the theorem is true for unital Q(n− 1)-algebras.
K0-THEORY WITH n-POTENTS 13
Now suppose that A does not have a unit. Then we have the following
commutative diagram with exact rows:
0 // Kn0 (A)
// Kn0 (A
+) //
Kn0 (Q(n− 1)) //
0 // K0(A)
n−1 // K0(A
+)n−1 // K0(Q(n− 1))n−1 // 0
An easy diagram chase shows that there is a unique group iso-
morphism from Kn0 (A) to
K0(A)
that makes the diagram com-
mute. �
Since a complex algebra is a Q(n− 1)-algebra for all values of n, we
have the following immediate corollary.
Corollary 3.14. If A is a C-algebra, there are natural isomorphisms
Kn0 (A)
K0(A)
of abelian groups for all natural numbers n ≥ 2.
We now arrive at the result that suggests why we should consider all
Kn0 -functors for algebras over a cyclotomic field.
Theorem 3.15. Let Q(4) = Q[i] be the 4th cyclotomic field. Then we
have the following isomorphisms of abelian groups:
K20(Q(4))
∼= Z,
K30(Q(4))
∼= Z2,
K40(Q(4))
∼= Z⊕ 2Z,
K50(Q(4))
∼= Z4.
Thus, K40 (Q(4)) 6∼= Z3 ∼= K40 (Q(3)).
Proof. Since Q(4) is a field [7], we have K20 (Q(4)) = K0(Q(4))
∼= Z.
The field Q(4) has characteristic 0 6= 2, so Theorem 3.6 implies that
K30(Q(4))
K0(Q(4)
)2 ∼= Z2. Theorem 3.13 implies that we have an
isomorphism K50(Q(4))
K0(Q(4)
)4 ∼= Z4.
However, the spectrum of 4-potents is contained in
0, 1,−1
which is not contained inQ(4) since the two primitive 3rd roots of unity
ω = ζ3 = −12 +
i and ω̄ = ζ̄3 = −12 −
i are not in Q(4) = Q[i].
Given any 4-potent e ∈Mn(Q(4)) ⊂Mn(C) we can uniquely write
e = e1 + ωe2 + ω̄e3,
14 EFTON PARK AND JODY TROUT
where e1, e2, e3 are orthogonal idempotents in Mn(C) that sum to an
idempotent e1+e2+e3 = e
3 inMn(Q(4)) by Lemma 2.2. We thus have
e2 = e1 + ω̄e2 + ωe3
e3 = e1 + e2 + e3
because ω2 = ω̄, ω̄2 = ω, and ω3 = ω̄3 = 1. Since ω + ω̄ = −1, this
implies that the first idempotent
e1 = (e + e
2 + e3)/3 ∈Mn(Q(4))
and the sum of the last two idempotents
e2 + e3 = e
3 − e1 ∈Mn(Q(4))
are both inMn(Q(4)). Using a simple trace argument and the fact that
ω, ω̄ 6∈ Q(4), we conclude that
rank(e2) = trace(e2) = trace(e3) = rank(e3),
and so rank(e2 + e3) = trace(e2 + e3) = 2trace(e2) is even. We then
have a well-defined map
V4(Q(4)) → V2(Q(4))⊕ 2V2(Q(4)) ∼= N⊕ 2N
[e] 7→ [e1]⊕ [e2 + e3] ∼= trace(e1)⊕ 2 trace(e2);
this is because the classes of e1 and e1 + e2 are preserved by (stable)
similarity, and the K0-class of an idempotent in a matrix ring over a
number field (or a PID) is the rank (= trace). It is easy to check that
this map is injective (using e1 ⊥ e2 + e3 in Mn(Q(4))) and additive.
The only question is surjectivity. It suffices to show that there is a
4-potent e over Q(4) whose stable similarity class is mapped to the
generator 1⊕ 2 of N⊕ 2N. Consider the block diagonal matrix
1 0 0
0 0 i
0 i −1
 ∈M3(Q(4)),
which is easily checked to be quadripotent. The lower right quadripo-
tent 2× 2 invertible block has the desired eigenvalues ω and ω̄, and so
does not diagonalize over Q(4). The result now follows easily. �
4. n-Homomorphisms and Kn0 Functorality
We know from Proposition 3.5 that Kn0 is a covariant functor from
the category of (unital) rings and ring homomorphisms to the category
of abelian groups and group homomorphisms. However, Kn0 is actually
functorial for a more general class of ring mappings.
K0-THEORY WITH n-POTENTS 15
Definition 4.1. Let R and S be rings. An additive map (not neces-
sarily unital) φ : R −→ S is called an n-homomorphism if
φ(a1a2 · · · an) = φ(a1)φ(a2) · · ·φ(an)
for all a1, a2, . . . , an in R.
Obviously every (ring) homomorphism is an n-homomorphism, but
the converse is false in general. For example, an AEn-ring is a ring
R such that every additive map φ : R → R is an n-homomorphism.
Feigelstock [2, 3] classified all unital AEn-rings. The algebraic version
of n-homomorphism was introduced for complex algebras in [4] and has
been carefully studied in the case of C∗-algebras in [5].
Proposition 4.2. Let φ : R → S be an n-homomorphism between
unital rings. Then φ induces a group homomorphism
φ∗ : K
0 (R) −→ Kn0 (S).
Furthermore, the assignment R 7→ Kn0 (R) is a covariant functor from
the category of unital rings and n-homomorphisms to the category of
abelian groups and ordinary group homomorphisms.
Proof. For each natural number k, we extend φ to a map from Mk(R)
to Mk(S) by applying φ to each matrix entry; it is easy to check this
also gives us an n-homomorphism. Moreover, φ is compatible with
stabilization of matrices; the only nonobvious point to check is that φ
respects algebraic equivalence.
Let e and f be algebraically equivalent n-potents in Mk(R) for some
k, and choose a and b in Mk(R) so that e = ab and f = ba. Define
elements a′ = φ(ea)φ(f)n−2 and b′ = φ(b) in Mk(S). We compute:
a′b′ = φ(ea)φ(f)n−2φ(b) = φ((ea)fn−2b) =
φ(ea(ba)n−2b) = φ(e(ab)n−1) = φ(en) = φ(e).
A similar argument shows that b′a′ = φ(f). Therefore φ determines
a monoid homomorphism from Vn(R) to Vn(S), and hence a group
homomorphism φ∗ : K
0 (R) −→ Kn0 (S). We leave it to the reader to
make the straightforward computations to show that we have a covari-
ant functor. �
Note that while we have an isomorphism Kn0 (A)
K0(A)
Q(n− 1)-algebras, it is not at all clear from the right hand side of this
isomorphism that Kn0 (A) is functorial for n-homomorphisms.
16 EFTON PARK AND JODY TROUT
References
[1] B. Blackadar,K-theory for Operator Algebras, 2nd ed., MSRI Publication Series
5, Springer-Verlag, New York, 1998.
[2] S. Feigelstock, Rings whose additive endomorphisms are N -multiplicative, Bull.
Austral. Math. Soc. 39 (1989), no. 1, 11–14.
[3] S. Feigelstock, Rings whose additive endomorphisms are n-multiplicative. II,
Period. Math. Hungar. 25 (1992), no. 1, 21–26.
[4] M. Hejazian, M. Mirzavaziri, M.S. Moslehian, n-homomorphisms, Bull. Iranian
Math. Soc. 31 (2005), no. 1, 13-23.
[5] E. Park and J. Trout, On the Nonexistence of Nontrivial Involutive n-
homomorphisms of C∗-algebras, Trans. Amer. Math. Soc. 361 (2009), no. 4,
1949–1961
[6] M. Rordam, F. Larsen, N. Laustsen, An Introduction to K-theory for C∗-
algebras, London Mathematical Society Student Texts, vol. 49. Cambridge Uni-
versity Press, Cambridge, 2000.
[7] J. Rosenberg, Algebraic K-theory and Its Applications, Graduate Texts in Math-
ematics, vol. 147, Springer-Verlag, New York, 1994.
Box 298900, Texas Christian University, Fort Worth, TX 76129
E-mail address : e.park@tcu.edu
6188 Kemeny Hall, Dartmouth College, Hanover, NH 03755
E-mail address : jody.trout@dartmouth.edu
	1. Introduction
	2. Equivalence of n-potents
	3. K0-theory with n-potents
	4. n-Homomorphisms and K0n Functorality
	References
