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Grothendieck groups for categories of complexes

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Abstract

The new intersection theorem states that, over a Noetherian local ring R , for any non-exact complex concentrated in degrees n ,…,0 in the category P(length) of bounded complexes of finitely generated projective modules with finite-length homology, we must have n ≥ d = dim R . One of the results in this paper is that the Grothendieck group of P(length) in fact is generated by complexes concentrated in the minimal number of degrees: if P d (length) denotes the full subcategory of P(length) consisting of complexes concentrated in degrees d ,…0, the inclusion P d (length) → P(length) induces an isomorphism of Grothendieck groups. When R is Cohen–Macaulay, the Grothendieck groups of P d (length) and P(length) are naturally isomorphic to the Grothendieck group of the category M(length) of finitely generated modules of finite length and finite projective dimension. This and a family of similar results are established in this paper.
GROTHENDIECK GROUPS FOR CATEGORIES OF COMPLEXES
HANS-BJØRN FOXBY AND ESBEN BISTRUP HALVORSEN
Abstract. The new intersection theorem states that, over a Noetherian local
ring R, for any non-exact complex concentrated in degrees n, . . . , 0 in the cat-
egory P(length) of bounded complexes of finitely generated projective modules
with finite length homology, we must have nd= dim R.
One of the results in this paper is that the Grothendieck group of P(length)
in fact is generated by complexes concentrated in the minimal number of de-
grees: if Pd(length) denotes the full subcategory of P(length) consisting of com-
plexes concentrated in degrees d, . . . , 0, the inclusion Pd(length) P(length)
induces an isomorphism of Grothendieck groups. When Ris Cohen–Macaulay,
the Grothendieck groups of Pd(length) and P(length) are naturally isomor-
phic to the Grothendieck group of the category M(length) of finitely generated
modules of finite length and finite projective dimension. This and a family of
similar results are established in this paper.
1. Introduction
In this paper, we will prove the existence of isomorphisms between Grothendieck
groups of various related categories of complexes. The paper presents a family of
results that can all be formulated in a similar way. This introduction discusses only
one of the results (as did the abstract); the remaining results can be obtained by
replacing the property of “having finite length” with other properties of modules—
see the next section for further details.
Let Rbe a commutative, Noetherian, local ring of dimension d. Let P(length) de-
note the category of bounded complexes of finitely generated pro jective R-modules
and finite length homology, and let Pd(length) denote the full subcategory of com-
plexes concentrated in degrees d, . . . , 0. We shall denote the Grothendieck groups of
these two categories by K0P(length) and K0Pd(length), respectively. The inclusion
of categories Pd(length) P(length) naturally induces a homomorphism
Id:K0Pd(length) K0P(length),
given by Id([X]) = [X] for a complex XPd(length); here, the two [X]’s are
different, since one is an element of K0Pd(length) and the other is an element of
K0P(length). One of the results of this paper (Corollary 6) is that the above is
an isomorphism. This is particularly interesting when comparing with the new
intersection theorem (cf. [6, Theorem 13.4.1]), which states that, if a complex in
P(length) is non-exact and concentrated in degrees n, . . . , 0, then nd. Thus,
the Grothendieck group K0P(length) is generated by complexes with the minimal
possible amplitude.
Next let M(length) denote the category of R-modules of finite length and fi-
nite projective dimension. We denote the Grothendieck group of M(length) by
K0M(length). Any module in M(length) has a pro jective resolution in P(length),
and there is a natural homomorphism
R:K0M(length) K0P(length),
2000 Mathematics Subject Classification. Primary 13D15, 19A99; Secondary 13D25.
Key words and phrases. Grothendieck group, algebraic K-theory, bounded complexes of finitely
generated projective modules.
1
2 HANS-BJØRN FOXBY AND ESBEN BISTRUP HALVORSEN
given by R([M]) = [X] for a module MM(length) with projective resolution
XP(length).
Now, suppose further that Ris Cohen-Macaulay. The acyclicity lemma by Pesk-
ine and Szpiro (see [6, Theorem 4.3.2]) implies that the complexes in Pd(length) are
acyclic: that is, they are pro jective resolutions of their zeroth homology module.
Taking the homology of a complex induces a natural homomorphism
Hd:K0Pd(length) K0M(length),
given by Hd([X]) = [H0(X)] for a complex XPd(length).
The three homomorphisms that we have introduced so far fit together in a com-
mutative diagram:
K0Pe(length) Id//
Hd''
K0P(length)
K0M(length)
R
88
p
p
p
p
p
p
p
p
p
p
Here, Hdis dotted to emphasize the fact that it required an extra assumption to be
defined. The fact that Idis an isomorphism yields that so are Rand Hd, whenever
defined (Corollary 11).
When replacing the property of “having finite length” with other module prop-
erties, the same picture will emerge. The next section presents all the results of this
paper in a general way—including the results mentioned in this introduction.
Historical note: This paper builds on the first author’s incomplete preprint [2]
whose results have been generalized and completely proven by the second author.
The paper will become part of the second author’s Ph.D. thesis. The results are
generalizations of a result by Roberts and Srinivas [7, Proposition 2]
2. Grothendieck groups for categories of complexes
Notation. Throughout this paper, Rdenotes a non-trivial, unitary, commutative
ring. All modules are, unless otherwise stated, assumed to be R-modules, and all
complexes are, unless otherwise stated, assumed to be complexes of R-modules.
Modules are considered to be complexes concentrated in degree zero.
Let dbe a non-negative integer and let S= (S1,...,Sd) be a family of multi-
plicative systems of R. A module Mis said to be Si-torsion if S1
iM= 0, and M
is said to be S-torsion if it is Si-torsion for i= 1,...,d. The grade of Mis the
number
gradeRM= inf{nN0|Extn
R(M, R)6= 0}.
If M= 0, we set gradeRM=. When Ris Noetherian and Mis non-trivial
and finitely generated, gradeRMis the maximal length of a regular sequence in
AnnRM.Mis said to be d-perfect if M= 0 or gradeRM=d= pdRM.
We shall use the following abbreviations for properties of modules.
S-tor: being S-torsion;
length: having finite length;
gr d: having grade larger than or equal to d; and
d-perf: being d-perfect.
Let ebe a non-negative integer, and let the symbol # denote any of the module
properties above. We define the following categories.
GROTHENDIECK GROUPS FOR CATEGORIES OF COMPLEXES 3
M: the category of finitely generated modules of finite projective dimension;
P: the category of bounded complexes of finitely generated projective mod-
ules;
Pe: the full subcategory of Pconsisting of complexes concentrated in degrees
e, . . . , 0;
M(#): the full subcategory of Mconsisting of modules satisfying #;
P(#): the full subcategory of Pconsisting of complexes whose homology mod-
ules satisfy #;
Pe(#): the intersection of Peand P(#).
So, for example, Pe(S-tor) denotes the category of complexes concentrated in
degrees e, . . . , 0 with finitely generated projective modules and S-torsion homology
modules. We will allow the symbol # to be “empty” so that M(#), P(#) and Pe(#)
also can denote M,Pand Pe, respectively. Similarly, we shall occasionally write
P(#), where the symbol either denotes a non-negative integer eor is “empty”,
in which case we are back with the category P(#).
The isomorphism classes of any of the categories M(#) and P(#) form a set.
We shall occasionally abuse notation and use M(#) and P(#) to denote the sets
of isomorphism classes of the corresponding categories.
Definition 1. The Grothendieck group of a category M(#) is the Abelian group
K0M(#) presented by generators [M], one for each isomorphism class in M(#), and
relations
[M] = [L] + [N] whenever 0 LMN0
is a short exact sequence in M(#).
The Grothendieck group of a category P(#) is the Abelian group K0P(#)
presented by generators [X], one for each isomorphism class in P(#), and relations
[X] = 0 whenever Xis exact,
and
[Y] = [X] + [Z] whenever 0 XYZ0
is a short exact sequence in P(#).
So, for example, K0Pe(S-tor) denotes the Grothendieck group of the category
Pe(S-tor), whereas the usual zeroth algebraic K-group of Ris the group K0(R) =
K0P0: that is, the Grothendieck group of the category of P0.
In the following three propositions, we list some useful properties of Grothendieck
groups, which will be used throughout this paper. The properties can easily be
verified and are stated without proof; for more details, the reader is referred to
Halvorsen [4, page 8-10].
Proposition 2. Any element in K0M(#) can be written in the form [M][M]for
modules M, M M(#), and any element in K0P(#) can be written in the form
[X][X]for complexes X, XP(#).
If Xis a complex, it can be shifted ndegrees to the left, thereby yielding the
complex ΣnXwith modules (ΣnX)=Xnand differentials ΣnX
= (1)nX
n.
In the case that n= 1, the operator Σ1() is simply denoted by Σ().
Proposition 3. Suppose that Xis a complex in P(#) such that ΣnXis in P(#).
Then nX] = (1)n[X]in K0P(#).
Proposition 4. Suppose that φ:XYis a quasi-isomorphism in P(#) such
that ΣXis in P(#). Then [X] = [Y]in K0P(#).
Note that, since quasi-isomorphisms become identities in the Grothendieck group,
we might as well have modelled the Grothendieck groups on derived categories rather
than usual categories.
4 HANS-BJØRN FOXBY AND ESBEN BISTRUP HALVORSEN
Since Pe(#) is a subcategory of P(#), the inclusion of categories induces a natural
group homomorphism
Ie:K0Pe(#) K0P(#)
given by Ie([X]) = [X]. Note that the two [X]’s here are different: one is an element
of K0Pe(#), whereas the other is an element of K0P(#). Note also that the fact
that Ieis induced by an inclusion of the underlying categories does not mean that
Ieis injective—it only ensures that Ieis well defined.
When Ris Noetherian, a module in M(#) always has a projective resolution in
P(#). It follows from Proposition 4 that different projective resolutions of the same
module always represent the same element in the Grothendieck group K0P(#).
Thus, we can, to each module MM(#) with projective resolution XP(#),
associate the element [X] in K0P(#). Since the modules in a short exact sequences
have projective resolutions that fit together in a short exact sequence, this associa-
tion induces a group homomorphism
R:K0M(#) K0P(#)
given by R([M]) = [X] where XP(#) is a pro jective resolution of MM(#).
As we shall see in the next section, certain additional assumptions on the ring
together with a sufficiently small choice of ecan force the homology of complexes
in Pe(#) to be concentrated in degree zero and hence be modules in M(#). Thus,
in this case, we can, to every complex XPe(#), associate the element [H(X)]
in K0M(#), where H denotes the homology functor. Since this association clearly
preserves the relations in K0Pe(#), it induces a group homomorphism
He:K0Pe(#) K0M(#)
given by He([X]) = [H(X)].
The homomorphisms Ie,Heand Rare connected in a commutative diagram as
shown below.
K0Pe(#) Ie//
He$$
K0P(#)
K0M(#)
R
;;
v
v
v
v
v
v
v
v
v
Heis here dotted to underline the fact that it required an extra assumption to be
defined. The homomorphism Ralways requires Rto be Noetherian in order to be
defined.
Let x= (x1, . . . , xd) denote a regular sequence, and let S(x) denote the family
(S(x1),...,S(xd)) of multiplicative systems S(xi) = {xn
i|nN0}. Further, let
Tdenote a (single) multiplicative system such that TZd R=. In the next
section we shall prove that the homomorphisms Heand Rare defined under the
assumptions on eand Rdescribed in the table below.
#eassumption on R
S(x)-tor dNoetherian, local
T-tor 1 Noetherian, local
0 Noetherian
length dim RNoetherian, local, Cohen–Macaulay
gr d d Noetherian, local
d-perf dNoetherian, local
In this paper we will show that Ie,Heand Rin all but the last of the above
situations are isomorphisms and that, in the last situation, Ieand Rare monomor-
phism and Heis an isomorphism. These results will be derived as corollaries to the
theorem below, which shall henceforth be referred to as the “Main Theorem”. As
GROTHENDIECK GROUPS FOR CATEGORIES OF COMPLEXES 5
the proof of the Main Theorem will show, the Grothendieck group K0Pe(#), where
# is any of the properties in the table above, is, in fact, isomorphic to K0P(#)
whenever eis larger than or equal to the corresponding number in the table and
trivial otherwise.
Main Theorem. Suppose that dis a non-negative integer and that S= (S1,...,Sd)
is a d-tuple of multiplicative systems of R. Then the homomorphism
Id:K0Pd(S-tor) K0P(S-tor)
given by Id([X]) = [X]is an isomorphism.
Note that, in the setting of the Main Theorem, there are no additional require-
ments on R, and the homomorphisms Hdand Ris not necessarily defined. However,
when Hdand Rare defined, we can immediately infer that Hdis injective and that
Ris surjective, and as it is not hard to see that Hdis surjective, it follows that all
three homomorphisms are isomorphisms.
The Main Theorem says that any element of K0P(S-tor) can be represented by
a linear combination of complexes concentrated in degrees d, . . . , 0. As we shall see,
the inverse map I1
d:K0P(S-tor) K0Pd(S-tor) is basically constructed from a
procedure describing how to “make complexes smaller”. When Hdis defined, the
complexes become so small that they are forced to be resolutions of modules with
projective dimension at most d.
When Ris Noetherian and local, d= 1 and the multiplicative set Tcontains no
zero-divisors, H1:K0P1(T-tor) K0M(T-tor) is, as we shall see, defined and all of
I1,H1and Rare isomorphisms. So in this case, the elements of K0M(T-tor) can be
represented by elements in the form [Rn/AR], where Ais an injective n×n-matrix.
Using the localization sequence
K1(R)K1(T1R)K0M(T-tor) K0(R)K0(T1R)
of algebraic K-groups, it is not hard to see that [Rn/ARn] = [R/(det A)R] in
K0M(T-tor). Thus, K0P1(T-tor) (and hence K0P(T-tor)) is in fact generated by
Koszul complexes. This property was fundamental in Foxby’s proof in [3] of Serre’s
intersection conjectures in the case where one module has dimension 1.
The rather tedious proof of the Main Theorem is postponed until Section 4. For
now, we will assume that it has been established and use it to derive all the other
results.
3. Isomorphisms between Grothendieck groups
Definition 5. If xis an element of R,S(x) denotes the multiplicative system
{xn|nN0}, and if x= (x1,...,xd) is a d-tuple of elements from R,S(x) denotes
the d-tuple (S(x1), . . . , S(xd)) of multiplicative systems.
We begin our collection of corollaries to the Main Theorem with the result dis-
cussed in the abstract and the introduction.
Corollary 6. If Ris Noetherian and local with dim R=d, then the group homo-
morphism Id:K0Pd(length) K0P(length) given by Id([X]) = [X]is an isomor-
phism.
Proof. Let x= (x1, . . . , xd) be a system of parameters, and notice that a finitely
generated module has finite length if and only if it is S(x)-torsion. Consequently,
K0P(length) = K0P(S(x)-tor) and K0Pd(length) = K0Pd(S(x)-tor), and the result
follows from the Main Theorem
Lemma 7. Suppose that Ris Noetherian and let x= (x1,...,xd)be a regular se-
quence of length d > 0. Then any complex Xin Pd(S(x)-tor) satisfies the condition
6 HANS-BJØRN FOXBY AND ESBEN BISTRUP HALVORSEN
that its homology complex H(X)is concentrated in degree 0: that is, H(X)is a
module in M(S(x)-tor).
Proof. Let Xbe a non-exact complex in Pd(S(x)-tor) and let tdenote the largest
integer such that Ht(X)6= 0; this exists since H(X)6= 0 and Xis bounded. We
already know that t0, so let us assume that t > 0 and try to reach a contradiction.
Let pbe an associated prime of Ht(X). Since H(X) is S(x)-torsion, we can
find N1,...,NdNsuch that xN1
1,...,xNd
dAnnRHt(X)p. Consequently,
(x1/1,...,xd/1) is an Rp-sequence in pp, so depth Rpd1.
Now, the projective resolution
0(Xd)p · · · (Xt+1)p(im X
t+1)p0
of (im X
t+1)pas an Rp-module shows that pdRp(im X
t+1)pd(t+ 1). From
the Auslander–Buchsbaum formula (see, for example, [1, Theorem 1.3.3]), it now
follows that
depthRp(im X
t+1)p= depth RppdRp(im X
t+1)pt+ 1 2.
Since (ker X
t)pis a submodule of the non-trivial free Rp-module (Xt)pwhich has
depthRp(Xt)p= depth Rpd1, we must also have depthRp(ker X
t)p1. From
the short exact sequence
0(im X
t+1)p(ker X
t)p(Ht(X))p0,
it now follows that depthRp(Ht(X))p1 (see, for example, [1, Proposition 1.2.9]).
This is a contradiction, however, because depthRp(Ht(X))p= 0, since pis associated
to Ht(X). Thus, t= 0 as desired.
Corollary 8. If Ris Noetherian and local, and x= (x1,...,xd)is a regular se-
quence of length d > 0, then there is a commutative diagram
K0Pd(S(x)-tor) Id//
Hd((
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
QK0P(S(x)-tor)
K0M(S(x)-tor)
R
66
m
m
m
m
m
m
m
m
m
m
m
m
in which Id,Hdand Rare isomorphisms.
Proof. Lemma 7 shows that Id,Hdand Rare well-defined homomorphisms, and
the Main Theorem states that Idis an isomorphism.Thus, we already know that
Hdis injective and Ris surjective.
Now, let Mbe a module in M(S(x)-tor), and let us show by induction on p=
pdRMthat [M]im Hd. If pd, it is clear that [M]im H, since Min this case
has a projective resolution in Pd(S(x)-tor). So assume that p > d, and choose a
finitely generated free module Fand a surjective homomorphism f:FM. Next,
using the fact that Mis S(x)-torsion, choose N1,...,NdNso that xN1
1,...,xNd
d
AnnRM, and let F=F/(xN1
1,...,xNd
d)F. The surjection finduces a surjection
f:FM. Letting Kdenote the kernel of f, we then have an exact sequence
0KFM0,
and since pdRF=d < p = pdRM, it follows that pdRK=d1. By construction,
Fand Kare S(x)-torsion, so Fand Kare modules in M(S(x)-tor), and the induc-
tion hypothesis yields [M] = [F][K]im Hd. Consequently Hdis surjective, and
it follows that Hdas well as Rare isomorphisms.
Corollary 8 also holds in the case d= 0, where the requirement of being S(x)-
torsion drops out, even without the assumption that Ris local. We state this as a
separate corollary and leave the straightforward proof to the reader.
GROTHENDIECK GROUPS FOR CATEGORIES OF COMPLEXES 7
Corollary 9. If Ris Noetherian, then there is a commutative diagram
K0(R) = K0P0
I0//
H0""
F
F
F
F
F
F
F
FK0P
K0M
R
==
z
z
z
z
z
z
z
z
in which I0,H0and Rare isomorphisms.
When Rin addition is local, the Grothendieck groups in Corollary 9 are all
isomorphic to Zthrough the rank on K0P0. As the proof of the Main Theorem
(Theorem 42) will show, the isomorphism K0PZis given by taking an element
[X]K0Pto the integer PZ(1)rankRX, whereas the isomorphism K0MZ
is given by taking and element [M]K0Mto the Euler characteristic χR(M),
defined as the alternating sum of the ranks in a finite free resolution of M.
The proofs of Lemma 7 and Corollary 8 in the case d= 1 clearly show that the
multiplicative system S(x) = S(x1) = {xn
1|nN0}can be replaced by any mul-
tiplicative system Scontaining only non-zerodivisors. This is because any element
of such a multiplicative system in itself constitutes a regular sequence of length 1.
We state this as a separate corollary.
Corollary 10. If Ris Noetherian and Tis a multiplicative system with TZd R=
, then there is a commutative diagram
K0P1(T-tor) I1//
H1''
O
O
O
O
O
O
O
O
O
O
OK0P(T-tor)
K0M(T-tor)
R
77
p
p
p
p
p
p
p
p
p
p
p
in which I1,H1and Rare isomorphisms.
Another special case of Corollary 8 that we would like to point out is the case
d= dim R, which is only possible when Ris Cohen–Macaulay. In this case, the
property of being S-torsion is identical to the property of having finite length. The
result in this case was discussed in the abstract and the introduction, and we also
state it as a separate corollary.
Corollary 11. If Ris a Noetherian, local Cohen-Macaulay ring of dimension d,
then there is a commutative diagram
K0Pd(length) Id//
Hd''
P
P
P
P
P
P
P
P
P
P
PK0P(length)
K0M(length)
R
77
o
o
o
o
o
o
o
o
o
o
o
in which Id,Hdand Rare isomorphisms.
As we shall see in Corollary 12 below, Corollary 8 can also be used to derive
results concerning the property of having grader larger than or equal to d.
Corollary 12. If Ris Noetherian and local, and dis a positive integer, then there
is a commutative diagram
K0Pd(gr d)Id//
Hd''
N
N
N
N
N
N
N
N
N
N
NK0P(gr d)
K0M(gr d)
R
88
p
p
p
p
p
p
p
p
p
p
p
in which Id,Hdand Rare isomorphisms.
8 HANS-BJØRN FOXBY AND ESBEN BISTRUP HALVORSEN
Proof. If dis so large that there are no regular sequences in Rof length d, then
the involved Grothendieck groups are all trivial and the theorem holds. We can
therefore assume that regular sequences of length ddo exist.
If Xis a complex in Pd(gr d), we can find a regular sequence x= (x1,...,xd)
of length dcontained in the annihilator of all the homology modules of X. Then X
will be homologically S(x)-torsion, and it follows from Lemma 7 that the homology
of Xis concentrated in degree 0. Consequently, Id,Hdand Rare well-defined
homomorphisms.
We define an equivalence relation on the set of regular sequences, letting a regular
sequence x= (x1,...,xd) be equivalent to a regular sequence x= (x
1,...,x
d)
whenever
RadR(x1,...,xd) = RadR(x
1,...,x
d),
where the radical RadRIof an ideal Iis the intersection of all prime ideals con-
taining I. It is clear that this, indeed, is an equivalence relation. Denote the set
of equivalence classes by E, and partially order Eby reversed inclusion of radical
ideals: that is,
x4xdef
RadR(x1,...,xd)RadR(x
1,...,x
d)
for x, xE. (It is of course the equivalence classes of xand xthat belong to E,
but this unimportant technicality will be ignored here.) E= (E, 4) is a directed
set, for if xand xare regular sequences of length d, then we can find a regular
sequence x′′ of length dcontained in (x)(x) and hence satisfying the condition
that x, x4x′′.
Now, the category M(S(x)-tor) is uniquely determined by the equivalence class
of xin E, since, for any finitely generated module M,
Mis S(x)-torsion ⇒ ∀ν∈ {1,...,d}∃NνN0:xNν
νAnnRM
(x1,...,xd)RadR(AnnRM)
RadR(x1,...,xd)RadR(AnnRM).
Thus, we can consider the family of Grothendieck groups K0M(S(x)-tor) indexed by
the equivalence classes in E. Given x, xEwith x4x, there is a homomorphism
Ix,x:K0M(S(x)-tor) K0M(S(x)-tor)
given by Ix,x([M]) = [M]; this is well defined, since it is induced by an inclusion of
categories as seen from the bi-implications above. Consequently, (K0M(S(x)-tor),Ix,x)x4x
is a direct system, and it is straightforward to see that the Grothendieck group
K0M(gr d) together with the natural homomorphisms τx:K0M(S(x)-tor)
K0M(gr d) induced by the inclusion of the underlying categories and given by
τx([M]) = [M], xE, satisfy the universal property required by a direct limit of
this system.
We have now shown that K0M(gr d) is the direct limit of the direct system
(K0M(S(x)-tor),Ix,x)x4x. By the same methods one can show that K0Pd(gr d)
and K0P(gr d) are the direct limits of the direct systems (K0Pd(S(x)-tor),Ix,x)x4x
and (K0P(S(x)-tor),Ix,x)x4x, respectively, where the homomorphisms Ix,xnow
are given by Ix,x([X]) = [X] for complexes Xin Pd(S(x)-tor) and P(S(x)-tor),
respectively. Now, we already know from Corollary 8 that there is a commutative
diagram of isomorphisms
K0Pd(S(x)-tor) Id//
Hd''
P
P
P
P
P
P
P
P
P
P
P
PK0P(S(x)-tor)
K0M(S(x)-tor)
R
77
o
o
o
o
o
o
o
o
o
o
o
o
GROTHENDIECK GROUPS FOR CATEGORIES OF COMPLEXES 9
for all xE, and hence there must also be a commutative diagram of isomorphisms
K0Pd(gr d)Id//
Hd''
N
N
N
N
N
N
N
N
N
N
NK0P(gr d)
K0M(gr d)
R
88
p
p
p
p
p
p
p
p
p
p
p
involving the direct limits.
Because of Lemma 7, the homology of any complex in Pd(gr d) must be a
d-perfect module. Thus, Pd(gr d) = Pd(d-perf), and hence K0Pd(gr d) =
K0Pd(d-perf). It follows that the isomorphisms Hd:K0Pd(gr d)K0M(gr d)
and Id:K0Pd(gr d)K0P(gr d) from Corollary 12 must factor through
K0M(d-perf) and K0P(d-perf), respectively. This is discussed in Corollary 13 below,
which extends Corollary 12, and where we let τ:K0M(d-perf) K0M(gr d) and
τ:K0P(d-perf) K0P(gr d) denote the natural homomorphisms induced by the
inclusion of the underlying categories and given by τ([M]) = [M] for MM(d-perf)
and τ([X]) = [X] for XP(d-perf).
Corollary 13. If Ris Noetherian and local and dis a positive integer, then there
is a commutative diagram
K0M(d-perf)
R
&&
N
N
N
N
N
N
N
N
N
N
N
τ
K0Pd(d-perf)
H
d
77
p
p
p
p
p
p
p
p
p
p
pI
d//
K0P(d-perf)
τ
K0Pd(gr d)Id//
Hd''
N
N
N
N
N
N
N
N
N
N
NK0P(gr d)
K0M(gr d)
R
88
p
p
p
p
p
p
p
p
p
p
p
in which Id,Hd,R,H
dand τare isomorphisms, I
dand Rare monomorphisms
and τis an epimorphism.
Proof. Commutativity of the diagram is clear, and we have already seen in Corol-
lary 12 that Id,Hdand Rare isomorphisms. From this it follows that I
dand H
d
are injective, and that τand τare surjective. However, H
dis clearly also surjec-
tive, since any finitely generated d-perfect module has a resolution in Pd(d-perf),
and hence H
dand τare isomorphisms.
Note that Corollary 13 (and hence Corollary 12) actually holds when d= 0, but
that including this case is unnecessary, as it is already stated in Corollary 9.
4. Proving the Main Theorem
Establishing the Main Theorem is a cumbersome task. We will construct an in-
verse to Id:K0Pd(S-tor) K0P(S-tor) as follows. Given a complex YP(S-tor),
we choose nZso that the shifted complex ΣnYis in Pe(S-tor) for some e > d.
To this complex we associate an element wenY)K0Pe1(S-tor); this is the cru-
cial step, in which we “make a complex smaller”, starting with the complex ΣnY
of amplitude (at most) eand ending up with the element wenY), which, as we
shall see, is represented by the difference of two complexes of amplitude (at most)
e1. Repeating this process a finite number of times, we end up with an element
wd+1 ···wenY) in K0Pd(S-tor). This is the image of [Y] under the inverse of Id.
10 HANS-BJØRN FOXBY AND ESBEN BISTRUP HALVORSEN
4.1. Contractions.
Notation. Throughout Section 4.1, ddenotes a non-negative integer and S=
(S1,...,Sd) denotes a d-tuple of multiplicative systems of R.
Definition 14. Let Xbe a complex. A d-tuple α= (α1,...,αd) of families αν=
(αν
)Zof homomorphisms αν
:XX+1 is an S-contraction of Xwith weight
s= (s1,...,sd)S1× · · · × Sdif
X
+1αν
+αν
1X
=sνidX
for all Zand ν= 1,...,d.
In the case that d= 0, the concept of S-contractions is meaningless, and the
property of having an S-contraction is trivially satisfied. In any case, the existence
of an S-contraction of Xwith weight s= (s1,...,sd) is equivalent to the condition
that the morphisms sνidX:XXfor ν= 1,...,d are null-homotopic.
Proposition 15. Each complex XP(S-tor) has an S-contraction.
Proof. For each νthe S1
νR-complex S1
νXis exact, bounded and consists of
finitely generated projective S1
νR-modules, so the identity morphism idS1
νXon
S1
νXis null-homotopic (see, for example, [5, Theorem IV.4.1]). Thus, we can find
S1
νR-homomorphisms bν
:S1
νXS1
νX+1 such that
S1
νX
+1 bν
+bν
1S1
νX
= idS1
νX
for all Z. Writing each bν
in the form βν
/tνfor an R-homomorphism βν
:X
X+1 and some common denominator tνSν, we now have in S1
νXthat, for any
xX,
(X
+1βν
+βν
1X
)(x)/tν=x/1.
Consequently, we can find uν,x Sνdepending on xso that in X,
uν,x(X
+1βν
+βν
1X
)(x) = uν,xtνx.
Since Xis bounded and consists of finitely generated modules, by multiplying a
finite number of uν,x’s, we can obtain an element uνSν, independent of xand of
, such that uν(X
+1βν
+βν
1X
)(x) = uνtνxfor all Zand all xX. Setting
αν
=uνβν
and sν=uνtν, we see that α= (α1,...,αd), where αν= (αν
)Z, is an
S-contraction of Xwith weight s= (s1,...,sd).
Definition 16. Let Xand Ybe complexes in Pwith S-contractions αand β,
respectively, and let φ:XYbe a morphism of complexes. Then αand βare
said to be compatible with φif they have the same weight and φ+1αν
=βν
φfor
all Zand ν= 1,...,d.
Theorem 17 below provides an example of a situation where an S-contraction of
a complex induces an S-contraction of another complex. Although the hypotheses
of the theorem are very specific, the theorem turns out to be applicable in several
situations.
Theorem 17. Let Xbe a complex in Pe, where e > 1, and suppose that αis an
S-contraction of Xwith weight s. Let e
Xbe another complex in Pe, and suppose that
the complex e
Xis identical to Xexcept for the modules and differentials in degrees
eand e1. Suppose further that e
Xe= 0 and that a morphism φ:Xe
Xexists
such that φ= idXfor = 0,...,e2and such that φe1is surjective. Then the
S-contraction αon Xinduces an S-contraction eαon e
Xwith weight ssuch that α
GROTHENDIECK GROUPS FOR CATEGORIES OF COMPLEXES 11
and eαare compatible with the morphism φ; for ν= 1,...,d,eανis defined by setting
eαν
e2=φe1αν
e2and eαν
=αν
for = 0,...,e3.
0//Xe
0
X
e//Xe1
φe1
X
e1//
αν
e1
ooXe2
idXe2
X
e2//
αν
e2
oo··· //
αν
e3
ooX1
idX1
X
1//
ooX0
idX0
αX
0
oo//0
0//00//e
Xe1
0
oo
f
X
e1//Xe2
X
e2//
φe1αν
e2
oo··· //
αν
e3
ooX1
X
1//
ooX0//
αν
0
oo0
Proof. By inspection.
Given an S-contraction αof Xwith weight s= (s1,...,sd) and a d-tuple t=
(t1,...,td)S1× · · · × Sd, we can construct an S-contraction of Xwith weight
st = (s1t1,...,sdtd) by setting = (t1α1,...,tdαd) where tναν= (tναν
)Z. We
can also shift α n degrees to the left for some nZto form an S-contraction
Σnαof ΣnXwith weight sby setting Σnα= (Σnα1,...,Σnαd), where Σnαν=
((Σnαν))Z= ((1)nαν
n)Z.
The following theorem shows how to construct a natural S-contraction of the
mapping cone of a morphism between two complexes that both have S-contractions.
Recall that the mapping cone of a morphism φ:XYis the complex C(φ) defined
by C(φ)=YX1= (YΣX)and
C(φ)
=
Y
φ1
0X
1
:
Y
X1
Y1
X2
for all Z. The (degreewise) inclusion Y ֒Cφ and the (degreewise) projection
C(φ)։ΣXare both morphisms of complexes, and together they form the canonical
short exact sequence
0YC(φ)ΣX0.
Theorem 18. Let φ:XYbe a morphism of complexes and let αand βbe
S-contractions of Xand Y, respectively, with weights sand t, respectively. Define
for ν= 1,...,d and Zthe homomorphism
(βα)ν
=
sνβν
βν
φαν
1
0tναν
1
:C(φ)=
Y
X1
Y+1
X
=C(φ)+1.
Then (βα) = ((βα)1,...,(βα)d), where (βα)ν= ((βα)ν
)Z, is an S-
contraction of the mapping cone C(φ)of φwith weight st = (s1t1, . . . , sdtd), and
the S-contractions ,(βα)and Σare compatible with the morphisms in the
canonical exact sequence
0YC(φ)ΣX0.
Proof. By inspection.
4.2. The idea behind the proof of the Main Theorem.
Notation. Throughout Section 4.2, ddenotes a non-negative integer and S=
(S1,...,Sd) denotes a d-tuple of multiplicative systems of R. Furthermore, X
denotes a fixed complex in Pe(S-tor) for some integer e > d, and αdenotes an
S-contraction of Xwith weight s= (s1,...,sd)S1× · · · × Sd.
Proving the Main Theorem involves the introduction of a complex ∆e(X, s).
More specifically, ∆e(X, s) is the complex ΣedK(s, Xe): that is, the Koszul com-
plex of the sequence s= (s1,...,sd) with coefficients in Xeand shifted eddegrees
to the left. For convenience we will now present an explicit description of ∆e(X, s).
12 HANS-BJØRN FOXBY AND ESBEN BISTRUP HALVORSEN
For any Z, let Υ() denote the set of -element subsets of {1,...,d}: that
is, the set of subsets in the form i={i1,...,i}where 1 i1<··· < id. In
particular, Υ(0) = {∅}, Υ(d) = {{1,...,d}} and Υ() = for all ℓ /∈ {0,...,d}.
Thus, in any case, Υ() contains d
elements. An object iΥ() is called a multi-
index and its elements are always denoted by i1,...,iin increasing order, so that
i={i1,...,i}, where 1 i1<··· < id.
Definition 19. e(X, s) denotes the complex whose ’th module is given by
e(X, s)=a
iΥ(e)
e(X, s)i
,where ∆e(X, s)i
=Xe,
and whose ’th differential e(X,s)
: ∆e(X, s)e(X, s)1is given by the fact
that its (j, i)-entry (e(X,s)
)j,i : ∆e(X, s)i
e(X, s)j
1for iΥ(e) and
jΥ(e+ 1) is
(e(X,s)
)j,i =(1)u+1sjuidXe,if j\i={ju}
0,if j+i
So ∆e(X, s) is a complex whose ’th module ∆e(X, s)consists of d
ecopies
of Xeand whose ’th differential as a map from the i’th copy of Xein ∆e(X, s)
to the j’th copy of Xein ∆e(X, s)+1 is non-zero only when ij, in which case
it is multiplication by (1)u+1sjufor the unique juwhich is in jand not in i. In
particular, if d= 0 the sequence sis empty and ∆e(X, s) is the complex concentrated
in degree ewith ∆e(X, s)e=Xe.
Proposition 20. The complex e(X, s)is in P(S-tor) and is concentrated in de-
grees e, . . . , e d.
Proof. The definition clearly implies that ∆e(X, s) is concentrated in degrees
e, . . . , e dand consists of finitely generated projective modules. Since ∆e(X, s) is
the Koszul complex of the sequence s1,...,sd, the homology modules of ∆e(X, s)
are annihilated by the ideal (s1,...,sd) (see, for example, [1, Proposition 1.6.5]); in
particular, the homology modules must be Sν-torsion for ν= 1, . . . , d.
The complex ∆e(X, s) comes naturally equipped with an S-contraction.
Theorem 21. For each Zand each ν= 1,...,d, let the homomorphism
δe(X, s)ν
: ∆e(X, s)e(X, s)+1 be given by the fact that its (j, i)-entry for
iΥ(e)and jΥ(e1) is
(δe(X, s)ν
)j,i =(1)w+1 idXe,if i\j={iw}={ν},
0,if i+j.
Then δe(X, s) = (δe(X, s)1,...,δe(X, s)d), where δe(X, s)ν= (δe(X, s)ν
)Z, is an
S-contraction of e(X, s)with weight s
Proof. This is a matter of verification. For each ν∈ {1,...,d},Zand i, i
Υ(d), the (i, i)-entry of e(X,s)
+1 δe(X, s)ν
is
sνidXe,if i=iand νi,
(1)u+wsi
uidXe,if i\i={iw}={ν}and i\i={i
u}, and
0,otherwise,
whereas the (i, i)-entry of δe(X, s)ν
1e(X,s)
is
sνidXe,if i=iand ν /i,
(1)u+w+1si
uidXe,if i\i={iw}={ν}and i\i={i
u}, and
0,otherwise.
GROTHENDIECK GROUPS FOR CATEGORIES OF COMPLEXES 13
Overall, we see that the (i, i)-entry of e(X,s)
+1 δe(X, s)ν
+δe(X, s)ν
1e(X,s)
is
sνidXeif i=iand 0 otherwise. This is what we wanted to show.
Definition 22. Let φe(X, α) denote the family (φe(X, α))Zof homomorphisms
φe(X, α):Xe(X, s)=`iΥ(e)Xegiven by the fact that their i’th entries
for iΥ(e) are
φe(X, α)i
=αie
e1αie1
e2···αi1
.
For =e, this means that φe(X, α)e= idXe, and for ℓ /∈ {e, . . . , e d}, it means
that φe(X, α)= 0.
Proposition 23. φe(X, α) : Xe(X, s)is a morphism of complexes.
Proof. Let def
= ∆e(X, s) and φdef
=φe(X, α). To prove that φis a morphism, we
need to show that φ1X
=
φfor all Z: that is, we need to verify that
the j’th entry, αje+1
e1···αj1
1X
, of the left side equals the j’th entry of
φfor
each jΥ(e+ 1). Since the (j, i)-entry of
is (1)u+1sjuidXewhenever iis
a subset of jwith j\i={ju}, that is, whenever i={j1,...,ju1, ju+1,...,je+1 }
for some u∈ {1,...,e+ 1}, we see that the j’th coordinate of
φmust be
e+1
X
u=1
(1)u+1sjuαje+1
e1···αju+1
+u1αju1
+u2···αj1
.
So overall, we need to show that
e+1
X
u=1
(1)u+1sjuαje+1
e1···αju+1
+u1αju1
+u2···αj1
=αje+1
e1···αj1
1X
(1)
for all jΥ(e+ 1). We do this by descending induction on .
When ℓ > e, the equation clearly holds since both sides are trivial, and in the
case that =e, (1) states that sj1idXe=αj1
e1X
e, which is satisfied since αis an
S-contraction of Xwith weight s. Suppose now that ℓ < e is arbitrarily chosen and
that (1) holds for larger values of . We then have
αje+1
e1···αj1
1X
=αje+1
e1···αj2
(sj1idXX
+1αj1
)
=sj1αje+1
e1···αj2
(
e+1
X
u=2
(1)usjuαje+1
e1···αju+1
+u1αju1
+u2···αj2
+1)αj1
=
e+1
X
u=1
(1)u+1sjuαje+1
e1···αju+1
+u1αju1
+u2···αj1
.
Here the second equality follows from the induction hypothesis. This proves (1) by
induction, so φis a morphism of complexes.
Definition 24. The mapping cone of φe(X, α) is denoted by Ce(X, α).
Letting ∆ def
= ∆e(X, s) and φdef
=φe(X, α), Ce(X, α) is the complex
0//Xeφe
X
e«//
e
Xe1
eφe1
0X
e1!//
e1
Xe2
//···
··· //
ed
Xed1
(0X
ed1)
//Xed2
X
ed2//··· //X0//0
14 HANS-BJØRN FOXBY AND ESBEN BISTRUP HALVORSEN
concentrated in degrees e+ 1,...,1.
Since Xand ∆e(X, s) are equipped with S-contractions, Theorem 18 provides
an S-contraction of Ce(X, α).
Definition 25. The S-contraction δe(X, s)αof Ce(X, α) is denoted by µe(X, α).
Letting ∆ def
= ∆e(X, s), φdef
=φe(X, α) and δdef
=δe(X, s), µe(X, α) is given by
µe(X, α)ν
=
sνδν
δν
φαν
1
0sναν
1
:
X1
+1
X
for each Zand ν∈ {1,...,d}. The weight of µe(X, α) is s2= (s2
1,...,s2
d).
Proposition 26. Ce(X, α)is an object of Pe+1(S-tor) concentrated in degrees e+
1,...,1.
Proof. Ce(X, α) is clearly concentrated in degrees e+ 1,...,1 and composed of
finitely generated projective modules. To see that Ce(X, α) is homologically S-
torsion, recall that the canonical short exact sequence
0e(X, s)Ce(X, α)ΣX0 (2)
induces the long exact sequence
· · · H(∆e(X, s)) H(Ce(X, α)) HX) · · ·
on homology. By localizing at Sνfor ν= 1,...,d it follows that, since ∆e(X, s) as
well as ΣXare homologically S-torsion, Ce(X, α) must be homologically S-torsion
as well.
Definition 27. Let D
e1denote the homomorphism
D
e1=
φe(X, α)e1
X
e1
:Xe1
e(X, s)e1
Xe2
=Ce(X, α)e1,
and let De(X, α) denote the complex
0//Xe1
D
e1//Ce(X, α)e1
Ce(X,α)
e1//Ce(X, α)e2
Ce(X,α)
e2//··· //Ce(X, α)1//0
concentrated in degrees e1,...,0.
(One verifies easily that De(X, α) indeed is a complex. It is identical to the
shifted mapping cone Σ1Ce(X, α) except in degrees e+ 1 and e.)
Proposition 28. De(X, α)is an object of Pe1(S-tor).
Proof. De(X, α) is clearly composed of finitely generated projective modules. The
fact that De(X, α) is homologically S-torsion is a consequence of Theorem 29 below,
from which it follows that De(X, α) is quasi-isomorphic to Σ1Ce(X, α).
Theorem 29. Let Bdenote the exact complex 0Xe
id
Xe0concentrated in
degrees eand e1. There is then an exact sequence
0BΣ1Ce(X, α)De(X, α)0.
GROTHENDIECK GROUPS FOR CATEGORIES OF COMPLEXES 15
Proof. Let def
= ∆e(X, s) and φdef
=φe(X, α), and recall that e=Xeand φe=
idXe. The situation is as follows.
0//B//Σ1Ce(X, α)//De(X, α)//0
k k k
degree 0
0
0
e0//Xe
idXe//
idXe
Xe
φe
X
e«
//0//
0
e10//XeidXe
X
e«//
e
Xe1
eφe1
0X
e1!
(X
eidXe1)
//Xe1//
φe1
X
e1«
0
e20//0
//
e1
Xe2
id //
e1
Xe2
//0
.
.
..
.
..
.
..
.
.
It is straightforward to verify that the diagram commutes and that all the rows are
exact.
The morphism Σ1Ce(X, α)De(X, α) from Theorem 29 is clearly in the form
described in Theorem 17, so we are able to induce an S-contraction of De(X, α)
with weight s2from the S-contraction Σ1µe(X, α) on Σ1Ce(X, α).
Definition 30. The S-contraction of De(X , α) induced in the sense of Theorem 17
from Σ1µe(X, α) through the morphism Σ1Ce(X, α)De(X, α) from The-
orem 29 is denoted by ηe(X, α).
Letting ∆ def
= ∆e(X, s), φdef
=φe(X, α) and δdef
=δe(X, s), ηe(X, α) from the above
definition is given by
ηe(X, α)ν
=
sνδν
+1 δ+1φ+1 αν
0sναν
:
+1
X
+2
X+1
whenever =e3, . . . , 0, and, as verified by a small calculation, by
ηe(X, α)ν
e2=sνX
eδν
e1αν
e2X
e1αν
e2:
e1
Xe2
Xe1
whenever =e2.
From Theorem 29 and (2) in Proposition 26 it follows that
[X] = [Σ1Ce(X, α)] 1e(X, s)] = [De(X, α)] 1e(X, s)]
16 HANS-BJØRN FOXBY AND ESBEN BISTRUP HALVORSEN
in K0Pe(S-tor). The complexes involved in the right end are both concentrated in
degrees e1,...,0. This gives us the idea of how to construct the inverse of the
homomorphism Idfrom the Main Theorem.
Definition 31. By we(X, α) we denote the element
we(X, α) = [De(X, α)] 1e(X, s)]
in K0Pe1(S-tor).
The next section is devoted to showing that we(X, α) is independent of the
choice of αsuch that we can simply write we(X); that the map we:Pe(S-tor)
K0Pe1(S-tor) induces a homomorphism We:K0Pe(S-tor) K0Pe1(S-tor); and
that the We’s for different e’s can be combined to form an inverse of Id.
4.3. Proving the Main Theorem.
Notation. Throughout Section 4.3, ddenotes a non-negative integer and S=
(S1,...,Sd) denotes a d-tuple of multiplicative systems of R. Furthermore, X
denotes a fixed complex in Pe(S-tor) for some integer e > d, and αdenotes an
S-contraction of Xwith weight s= (s1,...,sd)S1× · · · × Sd.
We begin with a collection of useful lemmas.
Lemma 32. If
0Yψ
Yψ
e
Y0
is an exact sequence in Pe(S-tor), and if β,βand e
βare S-contractions of Y,Y
and e
Y, respectively, compatible with the morphisms in the above exact sequence (and
thereby all having the same weight t), then there are exact sequences
0e(Y , t)e(Y , t)e(e
Y , t)0,(3)
0Ce(Y , β )Ce(Y, β)Ce(e
Y , e
β)0and (4)
0De(Y , β )De(Y, β)De(e
Y , e
β)0,(5)
proving that we(Y, β) = we(Y , β ) + we(e
Y , e
β)in K0Pe1(S-tor). Furthermore, the
S-contractions δe(Y , t),δe(Y, t)and δe(e
Y , t)are compatible with the morphisms in
(3); the S-contractions µe(Y , β),µe(Y , β)and µe(e
Y , e
β)are compatible with the mor-
phisms in (4); and the S-contractions ηe(Y , β ),ηe(Y, β )and ηe(e
Y , e
β)are compatible
with the morphisms in (5).
Proof. According to the assumption, there is an exact sequence of modules
0Ye
ψe
Ye
ψe
e
Ye0,
which immediately induces the exact sequence in (3), because ψeand ψeclearly
commute with each entry of the differentials in ∆e(Y , t), ∆e(Y , t) and ∆e(e
Y , t).
Since ψeand ψealso commute with each entry of the the S-contractions δe(Y , t),
δe(Y, t) and δe(e
Y , t), these must be compatible with the morphisms in the sequence.
In addition, the compatibility of the S-contractions β,βand e
βwith the morphisms
ψand ψmeans that ψeφe(Y , β )i
=φe(Y, β )i
ψand ψeφe(Y, β )i
=φe(e
Y , e
β)i
ψfor
each Zand iΥ(e), and hence that there is a commutative diagram with
exact rows:
0//Y//
φe(Y ,β)
Y//
φe(Y,β )
e
Y//
φe(e
Y ,e
β)
0
0//e(Y , s)//e(Y , s)//e(e
Y , s)//0
(6)
GROTHENDIECK GROUPS FOR CATEGORIES OF COMPLEXES 17
From this we induce the exact sequence of the mapping cones in (4). Straightforward
calculation easily verifies that the compatibility of the S-contractions β,βand
e
βwith the morphisms ψand ψ, the compatibility of the S-contractions δe(Y , t),
δe(Y, t) and δe(e
Y , t) with the morphisms in (3) and the commutativity of diagram (6)
imply that the S-contractions µe(Y , β ), µe(Y, β) and µe(e
Y , e
β) are compatible with
the morphisms in (4).
We now claim that the exact sequence in (4) induces the exact sequence in (5).
To see this, let B,Band e
Bdenote the exact complexes 0 YeYe0,
0YeYe0 and 0 e
Yee
Ye0 from Theorem 29, concentrated in degrees
eand e1. These three complexes come together in a short exact sequence 0
BBe
B0, induced by the short exact sequence 0 YeYee
Ye0.
We claim that there is a commutative diagram
0
0
0
0//B
//B
//e
B
//0
0//Σ1Ce(Y , β )
//Σ1Ce(Y, β )
//Σ1Ce(e
Y , e
β)
//0
0//De(Y , β )
//De(Y, β )
//De(e
Y , e
β)
//0
0 0 0
The columns are exact according to Theorem 29 and the top rectangles are readily
verified to be commutative. A little diagram chase now shows that we can use the
morphisms in the middle row to induce the morphisms in the bottom row, making
the entire diagram commutative by construction. As we have seen, the two top rows
are exact, so the exactness of the bottom row follows from the 9-lemma applied in
each degree. This establishes the exact sequence in (5). Once again, straightforward
calculation demonstrates that the S-contractions ηe(Y , β ), ηe(Y, β ) and ηe(e
Y , e
β) are
compatible with the morphisms in (5).
From (3) and (5), we now obtain that
we(Y, β ) = [De(Y, β)] 1e(Y , t)]
= [De(Y , β )] + [De(e
Y , e
β)] 1e(Y , t)] 1e(e
Y , t)]
=we(Y , β ) + we(e
Y , e
β),
and the proof is complete.
Lemma 33. If Xis exact, then we(X, α) = 0 in K0Pe1(S-tor).
Proof. Let e
e1denote the inclusion map im X
e1֒Xe2, and let e
Xdenote the
complex
0im X
e1e
e1
Xe2
X
e2
Xe3 · · · X1
X
1
X00
concentrated in degrees e1,...0. Since Xis exact, e
Xis exact, and it follows that
im X
e1is projective, and hence that e
Xis a complex in Pe1(S-tor).
18 HANS-BJØRN FOXBY AND ESBEN BISTRUP HALVORSEN
Letting Bdenote the exact complex 0 Xe
id
Xe0 from Theorem 29, there
is an exact sequence
0//B//X//e
X//0
k k k
degree 0
0
0
e0//Xe
idXe//
idXe
Xe//
X
e
0//
0
e10//Xe
X
e1//
Xe1
X
e1//
X
e1
im X
e1//
e
e1
0
e20//0//
Xe2
idXe2//
Xe2//
0
.
.
..
.
..
.
..
.
.
and we claim that there is a commutative diagram
0
0
0
0//0//
B//
B//
0
0//Σ1e(X, s)//
Σ1Ce(X, α)//
X//
0
0//Σ1e(X, s)//
De(X, α)//
e
X//
0
0 0 0
The columns are exact (the middle one according to Theorem 29), and the top
rectangles are readily verified to be commutative. A little diagram chase shows
that we can use the morphisms in the middle row to induce the morphisms in the
bottom row, so that the entire diagram is commutative by construction. Now, the
two top rows are exact, so the exactness of the bottom row follows from the 9-lemma
applied in each degree. Thus, we have constructed an exact sequence
0Σ1e(X, s)De(X, α)e
X0 (7)
of complexes in Pe1(S-tor). Since e
Xis exact, it follows that
we(X, α) = [De(X, α)] 1e(X, s)] = [ e
X] = 0
in K0Pe1(S-tor) as desired.
In the next lemma and the theorem that follows, we shall work with a number
of similar Koszul complexes. We therefore introduce some convenient notation.
GROTHENDIECK GROUPS FOR CATEGORIES OF COMPLEXES 19
Definition 34. For rS1, let ∆(r)def
= ∆e(X, (r, s2,...,sd)); hence, in particular,
∆(s1) = ∆e(X, s).
Lemma 35. Suppose r, rS1, and define homomorphisms
π(r, r): ∆(rr)∆(r)and ξ(r, r): ∆(r)∆(rr)
for each Zby the fact that their (i, i)-entries for i, iΥ(e)are
(π(r, r))i,i =
0,if i6=i,
idXe,if i=iand 1 i,
ridXe,if i=iand 1 /i,
and
(ξ(r, r))i,i =
0,if i6=i,
ridXe,if i=iand 1 i,
idXe,if i=iand 1 /i.
Then π(r, r) = (π(r, r))Zis a morphism of complexes ∆(rr)∆(r)and
ξ(r, r) = (ξ(r, r))Zis a morphism of complexes ∆(r)∆(rr).
Proof. Assume that iΥ(e) and jΥ(e+ 1). A direct calculation then
shows that the (j, i)-entries of ∆(r)
π(r, r)and π(r, r)1∆(rr)
are both given
by
0,if j+i,
(1)u+1sjuidXe,if j\i={ju}and 1 i,
(1)u+1sjuridXe,if j\i={ju} 6={1}and 1 /i, and
rridXe,if j\i={ju}={1}and 1 /i.
This proves that π(r, r) is a morphism of complexes.
Similarly, a direct calculation shows that the (j, i)-entries of ∆(rr)
ξ(r, r)and
ξ(r, r)1∆(r)
are both given by
0,if j+i,
(1)u+1sjuridXe,if j\i={ju}and 1 i,
(1)u+1sjuidXe,if j\i={ju} 6={1}and 1 /i, and
rridXe,if j\i={ju}={1}and 1 /i.
This proves that ξ(r, r) is a morphism of complexes.
We are now ready to take the first step in proving that we(X, α) is independent
of the S-contraction α.
Theorem 36. Suppose that t= (t1,...,td)S1× · · · × Sdand consider the S-
contraction = (t1α1,...,tdαd)of Xwith weight st = (s1t1, . . . , sdtd). Then
we(X, tα) = we(X, α)in K0Pe1(S-tor).
Proof. If only we can show the equation in the case where tν= 1 for all but one of
the ν’s, then the equation follows since
= (t1,...,td)α= (t1,1. . . , 1) ···(1,...,1, td)α.
We will therefore assume that t= (t1,1,...,1); the other cases follow similarly
(since we can permute the Sν’s).
To show the desired equation, it suffices to prove that the following equations
hold in K0Pe1(S-tor).
1∆(s1t1)] = [Σ1∆(s1)] + [Σ1∆(t1)].(8)
[De(X, tα)] = [De(X, α)] + [Σ1∆(t1)].(9)
20 HANS-BJØRN FOXBY AND ESBEN BISTRUP HALVORSEN
Since ∆(1) is exact (being the Koszul complex of a sequence involving a unit), the
first equation follows if we can show that there is an exact sequence
0//∆(s1)π(1,s1)
ξ(s1,t1)«//
∆(1)
∆(s1t1)
(ξ(1,t1)π(t1,s1) )
//∆(t1)//0.
The two matrices clearly define morphisms of complexes, since π(r, r) and ξ(r, r)
are morphisms of complexes for r, rS1according to Lemma 35. Exactness
at ∆(s1) and ∆(t1) is clear since there is always one identity map involved in
either of π(r, r) and ξ(r, r) for r, rS1. Furthermore, ξ(1, t1)π(1, s1) as well
as π(t1, s1)ξ(s1, t1) are defined in degree by the fact that their (i, i)-entries for
i, iΥ(e) are
0,if i6=i,
t1idXe,if i=iand 1 i, and
s1idXe,if i=iand 1 /i.
To show the exactness of the sequence above, it therefore only remains to show
that, for each Z, the kernel in degree of the second morphism is contained in
the image in degree of the first. Since all (i, i)-entries of the maps involved are
trivial except when i=i, it suffices to consider an element (x, y) in the i-entry
∆(1)i
∆(s1t1)i
of the ’th module of ∆(1) ∆(s1t1). So suppose that such an
element is in the kernel of the map in degree of the second morphism. If 1 i,
this means that t1x=y, and in this case (x, y ) is the image of xunder the map
in degree of the first morphism. If 1 /i, it means that x=s1y, and in this case
(x, y) is the image of yunder the map in degree of the first morphism. In either
case, (x, y) is in the image of the map in degree of the first morphism, and hence
the sequence is exact and equation (8) has been proven.
Moving on to equation (9), we first define for each Za homomorphism
γ1:X1∆(1)by letting its i’th entry for iΥ(e) be
γi
1=0,if 1 i,
αie
e1···αi1
α1
1,if 1 /i.
Another way of writing this is
γ1=a
iΥ(e)
1/i
αie
e1···αi1
α1
1.
We now claim that there are morphisms
Φ: Ce(X, α)
∆(1)
Ce(X, tα)
and Ψ:
∆(1)
Ce(X, tα)
∆(t1)
given in degree by
Φ=
π(1, s1)γ1
ξ(s1, t1)0
0 idX1
:
∆(s1)
X1
∆(1)
∆(s1t1)
X1
and
Ψ=ξ(1, t1)π(t1, s1)ξ(1, t1)γ1:
∆(1)
∆(s1t1)
X1
∆(t1).
GROTHENDIECK GROUPS FOR CATEGORIES OF COMPLEXES 21
Proving that Φ and Ψ indeed are morphisms of complexes means proving that
∆(1)
+1 0 0
0∆(s1t1)
+1 φe(X, tα)
0 0 X
Φ+1 = Φ∆(s1)
+1 φe(X, α)
0X
and
∆(t1)
+1 Ψ+1 = Ψ
∆(1)
+1 0 0
0∆(s1t1)
+1 φe(X, tα)
0 0 X
for all Z. Since we already know from Lemma 35 that π(r, u) and ξ(r, u) are
morphisms for r, u S1, proving the above equations comes down to showing that
the following hold for all Z:
π(1, s1)φe(X, α)=∆(1)
+1 γ+γ1X
; (10)
φe(X, tα)=ξ(s1, t1)φe(X, α); and (11)
π(t1, s1)φe(X, tα)=ξ(1, t1)γ1X
+∆(t1)
+1 ξ(1, t1)+1γ.(12)
We verify (10) by brute force, calculating on the right hand side of the equation:
∆(1)
+1 γ+γ1X
=∆(1)
+1 a
jΥ(e1)
1/j
αje1
e1···αj1
+1α1
+a
iΥ(e)
1/i
αie
e1···αi1
α1
1X
=a
iΥ(e)
1i
αie
e1···αi2
+1α1
+a
iΥ(e)
1/i
(
e
X
u=1
(1)u+1siuαie
e1···αiu+1
+uαiu1
+u1···αi1
+1α1
)
+a
iΥ(e)
1/i
αie
e1···αi1
α1
1X
=a
iΥ(e)
1i
αie
e1···αi1
+a
iΥ(e)
1/i
αie
e1···αi1
(X
+1α1
+α1
1X
)
=a
iΥ(e)
1i
αie
e1···αi1
+a
iΥ(e)
1/i
s1αie
e1···αi1
=π(1, s1)φe(X, α).
Here, the third equality follows from (1) in Proposition 23. This proves the equation
in (10). The equation in (11) is clear, since
ξ(s1, t1)φe(X, α)=a
iΥ(e)
1i
t1φe(X, α)i
+a
iΥ(e)
1/i
φe(X, α)i
=φe(X, tα).
22 HANS-BJØRN FOXBY AND ESBEN BISTRUP HALVORSEN
To prove that the equation in (12) holds, we apply (10) to the right side of (12):
ξ(1, t1)γ1X
+∆(t1)
+1 ξ(1, t1)+1γ
=ξ(1, t1)(γ1X
+∆(1)
+1 γ)
=ξ(1, t1)π(1, s1)φe(X, α).
In contrast, applying (11) to the left side of (12) yields
π(t1, s1)φe(X, tα)=π(t1, s1)ξ(s1, t1)φe(X, α),
so proving equation (12) merely requires showing that
ξ(1, t1)π(1, s1)=π(t1, s1)ξ(s1, t1).(13)
This, however, follows since, for i, iΥ(e), both sides of (13) have (i, i)-entries
given by
0,if i6=i,
s1idXe,if i=iand 1 /i, and
t1idXe,if i=iand 1 i.
Thus, we have verified equation (12), and we conclude that Φ and Ψ are morphisms
of complexes.
We now claim that there is a short exact sequence
0Ce(X, α)Φ
∆(1)
Ce(X, tα)
Ψ
∆(t1)0.(14)
To see that the sequence is exact at Ce(X, α), suppose that, for some Z, the
element (x, y)∆(s1)X1=Ce(X, α)maps to 0 under Φ: that is,
0 =
π(1, s1)γ1
ξ(s1, t1)0
0 idX1
x
y=
π(1, s1)(x) + γ1(y)
ξ(s1, t1)(x)
y
.
It immediately follows that y= 0, and we are left with the equations π(1, s1)(x) =
ξ(s1, t1)(x) = 0 which imply that x= 0. Thus, Φis injective and (14) is exact at
Ce(X, α).
To see that the sequence is exact at ∆(t1), suppose that x∆(t1)i
for some
Zand iΥ(e). Then, if 1 i,
ξ(1, t1)π(t1, s1)ξ(1, t1)γ1
0
x
0
=x,
and if 1 /i,
ξ(1, t1)π(t1, s1)ξ(1, t1)γ1
x
0
0
=x.
In either case, xis in the image of Ψ, and we conclude that Ψis surjective and
that (14) is exact at ∆(t1).
Equation (13) clearly shows that ΨΦ = 0, so to show the exactness of (14), it
only remains verify that the kernel of Ψis contained in the image of Φfor all
Z. So suppose that (x, y, z )∆(1)∆(s1t1)X1= (∆(1) Ce(X, tα))
maps to 0 under Ψ: that is,
ξ(1, t1)(x) + π(t1, s1)(y) + ξ(1, t1)γ1(z) = 0.
GROTHENDIECK GROUPS FOR CATEGORIES OF COMPLEXES 23
Here x= (xi)iΥ(e)and y= (yi)iΥ(e)are Υ(e)-tuples, so the above equation
states that, for iΥ(e),
t1xi+yi= 0,when 1 i, and
xi+s1yi+γi
1(z) = 0,when 1 /i.
Now let w= (wi)iΥ(e)∆(s1)be defined by wi=xiwhenever 1 iand
wi=yiwhenever 1 /i. Then
π(1, s1)γ1
ξ(s<