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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 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 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 X∈Pd(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 n≥d. 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 M∈M(length) with projective resolution

X∈P(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 X∈Pd(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 S−1

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{n∈N0|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 →L→M→N→0

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 →X→Y→Z→0

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, X′∈P⋆(#).

If Xis a complex, it can be shifted ndegrees to the left, thereby yielding the

complex ΣnXwith modules (ΣnX)ℓ=Xℓ−nand diﬀerentials ∂ΣnX

ℓ= (−1)n∂X

ℓ−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 [ΣnX] = (−1)n[X]in K0P⋆(#).

Proposition 4. Suppose that φ:X→Yis 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 M∈M(#) with projective resolution X∈P(#),

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 X∈P(#) is a pro jective resolution of M∈M(#).

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 X∈Pe(#), 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|n∈N0}. Further, let

Tdenote a (single) multiplicative system such that T∩Zd 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 I−1

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(T−1R)→K0M(T-tor) →K0(R)→K0(T−1R)

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|n∈N0}, 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 t≥0, 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,...,Nd∈Nsuch that xN1

1,...,xNd

d∈AnnRHt(X)⊆p. Consequently,

(x1/1,...,xd/1) is an Rp-sequence in pp, so depth Rp≥d≥1.

Now, the projective resolution

0→(Xd)p→ · · · → (Xt+1)p→(im ∂X

t+1)p→0

of (im ∂X

t+1)pas an Rp-module shows that pdRp(im ∂X

t+1)p≤d−(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 Rp−pdRp(im ∂X

t+1)p≥t+ 1 ≥2.

Since (ker ∂X

t)pis a submodule of the non-trivial free Rp-module (Xt)pwhich has

depthRp(Xt)p= depth Rp≥d≥1, we must also have depthRp(ker ∂X

t)p≥1. From

the short exact sequence

0→(im ∂X

t+1)p→(ker ∂X

t)p→(Ht(X))p→0,

it now follows that depthRp(Ht(X))p≥1 (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 p≤d, 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:F→M. Next,

using the fact that Mis S(x)-torsion, choose N1,...,Nd∈Nso that xN1

1,...,xNd

d∈

AnnRM, and let F=F/(xN1

1,...,xNd

d)F. The surjection finduces a surjection

f:F→M. Letting Kdenote the kernel of f, we then have an exact sequence

0→K→F→M→0,

and since pdRF=d < p = pdRM, it follows that pdRK=d−1. 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 K0P→Zis given by taking an element

[X]∈K0Pto the integer Pℓ∈Z(−1)ℓrankRXℓ, whereas the isomorphism K0M→Z

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|n∈N0}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 T∩Zd 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,

x4x′def

⇐⇒ RadR(x1,...,xd)⊇RadR(x′

1,...,x′

d)

for x, x′∈E. (It is of course the equivalence classes of xand x′that belong to E,

but this unimportant technicality will be ignored here.) E= (E, 4) is a directed

set, for if xand x′are 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, x′4x′′.

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}∃Nν∈N0:xNν

ν∈AnnRM

⇐⇒ (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, x′∈Ewith 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], x∈E, 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,x′now

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 x∈E, 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 M∈M(d-perf)

and τ([X]) = [X] for X∈P(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 R′are 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 Y∈P(S-tor),

we choose n∈Zso that the shifted complex ΣnYis in Pe(S-tor) for some e > d.

To this complex we associate an element we(ΣnY)∈K0Pe−1(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 we(ΣnY), which, as we

shall see, is represented by the diﬀerence of two complexes of amplitude (at most)

e−1. Repeating this process a ﬁnite number of times, we end up with an element

wd+1 ···we(ΣnY) 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

ℓ)ℓ∈Zof homomorphisms αν

ℓ:Xℓ→Xℓ+1 is an S-contraction of Xwith weight

s= (s1,...,sd)∈S1× · · · × Sdif

∂X

ℓ+1αν

ℓ+αν

ℓ−1∂X

ℓ=sνidXℓ

for all ℓ∈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:X→Xfor ν= 1,...,d are null-homotopic.

Proposition 15. Each complex X∈P(S-tor) has an S-contraction.

Proof. For each νthe S−1

νR-complex S−1

νXis exact, bounded and consists of

ﬁnitely generated projective S−1

νR-modules, so the identity morphism idS−1

νXon

S−1

νXis null-homotopic (see, for example, [5, Theorem IV.4.1]). Thus, we can ﬁnd

S−1

νR-homomorphisms bν

ℓ:S−1

νXℓ→S−1

νXℓ+1 such that

∂S−1

νX

ℓ+1 bν

ℓ+bν

ℓ−1∂S−1

νX

ℓ= idS−1

νXℓ

for all ℓ∈Z. Writing each bν

ℓin the form βν

ℓ/tνfor an R-homomorphism βν

ℓ:Xℓ→

Xℓ+1 and some common denominator tν∈Sν, we now have in S−1

νXℓthat, for any

x∈Xℓ,

(∂X

ℓ+1βν

ℓ+βν

ℓ−1∂X

ℓ)(x)/tν=x/1.

Consequently, we can ﬁnd uν,x ∈Sνdepending on xso that in Xℓ,

uν,x(∂X

ℓ+1βν

ℓ+βν

ℓ−1∂X

ℓ)(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 uν∈Sν, independent of xand of

ℓ, such that uν(∂X

ℓ+1βν

ℓ+βν

ℓ−1∂X

ℓ)(x) = uνtνxfor all ℓ∈Zand all x∈Xℓ. Setting

αν

ℓ=uνβν

ℓand sν=uνtν, we see that α= (α1,...,αd), where αν= (αν

ℓ)ℓ∈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:X→Ybe a morphism of complexes. Then αand βare

said to be compatible with φif they have the same weight and φℓ+1αν

ℓ=βν

ℓφℓfor

all ℓ∈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 e−1. Suppose further that e

Xe= 0 and that a morphism φ:X→e

Xexists

such that φℓ= idXℓfor ℓ= 0,...,e−2and such that φe−1is 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

e−2=φe−1αν

e−2and eαν

ℓ=αν

ℓfor ℓ= 0,...,e−3.

0//Xe

0

∂X

e//Xe−1

φe−1

∂X

e−1//

αν

e−1

ooXe−2

idXe−2

∂X

e−2//

αν

e−2

oo··· //

αν

e−3

ooX1

idX1

∂X

1//

ooX0

idX0

αX

0

oo//0

0//00//e

Xe−1

0

oo

∂f

X

e−1//Xe−2

∂X

e−2//

φe−1αν

e−2

oo··· //

αν

e−3

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 tα of Xwith weight

st = (s1t1,...,sdtd) by setting tα = (t1α1,...,tdαd) where tναν= (tναν

ℓ)ℓ∈Z. We

can also shift α n degrees to the left for some n∈Zto form an S-contraction

Σnαof ΣnXwith weight sby setting Σnα= (Σnα1,...,Σnαd), where Σnαν=

((Σnαν)ℓ)ℓ∈Z= ((−1)nαν

ℓ−n)ℓ∈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:X→Yis the complex C(φ) deﬁned

by C(φ)ℓ=Yℓ⊕Xℓ−1= (Y⊕ΣX)ℓand

∂C(φ)

ℓ=

∂Y

ℓφℓ−1

0−∂X

ℓ−1

:

Yℓ

⊕

Xℓ−1

→

Yℓ−1

⊕

Xℓ−2

for all ℓ∈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

0→Y→C(φ)→ΣX→0.

Theorem 18. Let φ:X→Ybe 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 ℓ∈Zthe homomorphism

(β∗α)ν

ℓ=

sνβν

ℓβν

ℓφℓαν

ℓ−1

0−tναν

ℓ−1

:C(φ)ℓ=

Yℓ

⊕

Xℓ−1

→

Yℓ+1

⊕

Xℓ

=C(φ)ℓ+1.

Then (β∗α) = ((β∗α)1,...,(β∗α)d), where (β∗α)ν= ((β∗α)ν

ℓ)ℓ∈Z, is an S-

contraction of the mapping cone C(φ)of φwith weight st = (s1t1, . . . , sdtd), and

the S-contractions sβ,(β∗α)and Σtα are compatible with the morphisms in the

canonical exact sequence

0→Y→C(φ)→ΣX→0.

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 Σe−dK(s, Xe): that is, the Koszul com-

plex of the sequence s= (s1,...,sd) with coeﬃcients in Xeand shifted e−ddegrees

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 ℓ∈Z, let Υ(ℓ) denote the set of ℓ-element subsets of {1,...,d}: that

is, the set of subsets in the form i={i1,...,iℓ}where 1 ≤i1<··· < iℓ≤d. In

particular, Υ(0) = {∅}, Υ(d) = {{1,...,d}} and Υ(ℓ) = ∅for all ℓ /∈ {0,...,d}.

Thus, in any case, Υ(ℓ) contains d

ℓelements. An object i∈Υ(ℓ) is called a multi-

index and its elements are always denoted by i1,...,iℓin increasing order, so that

i={i1,...,iℓ}, where 1 ≤i1<··· < iℓ≤d.

Deﬁnition 19. ∆e(X, s) denotes the complex whose ℓ’th module is given by

∆e(X, s)ℓ=a

i∈Υ(e−ℓ)

∆e(X, s)i

ℓ,where ∆e(X, s)i

ℓ=Xe,

and whose ℓ’th diﬀerential ∂∆e(X,s)

ℓ: ∆e(X, s)ℓ→∆e(X, s)ℓ−1is given by the fact

that its (j, i)-entry (∂∆e(X,s)

ℓ)j,i : ∆e(X, s)i

ℓ→∆e(X, s)j

ℓ−1for i∈Υ(e−ℓ) and

j∈Υ(e−ℓ+ 1) is

(∂∆e(X,s)

ℓ)j,i =(−1)u+1sjuidXe,if j\i={ju}

0,if j+i

So ∆e(X, s) is a complex whose ℓ’th module ∆e(X, s)ℓconsists of d

e−ℓcopies

of Xeand whose ℓ’th diﬀerential as a map from the i’th copy of Xein ∆e(X, s)ℓ

to the j’th copy of Xein ∆e(X, s)ℓ+1 is non-zero only when i⊆j, 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 ℓ∈Zand each ν= 1,...,d, let the homomorphism

δe(X, s)ν

ℓ: ∆e(X, s)ℓ→∆e(X, s)ℓ+1 be given by the fact that its (j, i)-entry for

i∈Υ(e−ℓ)and j∈Υ(e−ℓ−1) is

(δe(X, s)ν

ℓ)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

ℓ)ℓ∈Z, is an

S-contraction of ∆e(X, s)with weight s

Proof. This is a matter of veriﬁcation. For each ν∈ {1,...,d},ℓ∈Zand i, i′∈

Υ(d−ℓ), the (i′, i)-entry of ∂∆e(X,s)

ℓ+1 δe(X, s)ν

ℓis

sνidXe,if i=i′and ν∈i,

(−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

ℓ−1∂∆e(X,s)

ℓis

sνidXe,if i=i′and 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)

ℓ+1 δe(X, s)ν

ℓ+δe(X, s)ν

ℓ−1∂∆e(X,s)

ℓis

sνidXeif i=i′and 0 otherwise. This is what we wanted to show.

Deﬁnition 22. Let φe(X, α) denote the family (φe(X, α)ℓ)ℓ∈Zof homomorphisms

φe(X, α)ℓ:Xℓ→∆e(X, s)ℓ=`i∈Υ(e−ℓ)Xegiven by the fact that their i’th entries

for i∈Υ(e−ℓ) are

φe(X, α)i

ℓ=αie−ℓ

e−1αie−ℓ−1

e−2···αi1

ℓ.

For ℓ=e, this means that φe(X, α)e= idXe, and for ℓ /∈ {e, . . . , e −d}, it means

that φe(X, α)ℓ= 0.

Proposition 23. φe(X, α) : X→∆e(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 φℓ−1∂X

ℓ=∂∆

ℓφℓfor all ℓ∈Z: that is, we need to verify that

the j’th entry, αje−ℓ+1

e−1···αj1

ℓ−1∂X

ℓ, of the left side equals the j’th entry of ∂∆

ℓφℓfor

each j∈Υ(e−ℓ+ 1). Since the (j, i)-entry of ∂∆

ℓis (−1)u+1sjuidXewhenever iis

a subset of jwith j\i={ju}, that is, whenever i={j1,...,ju−1, ju+1,...,je−ℓ+1 }

for some u∈ {1,...,e−ℓ+ 1}, we see that the j’th coordinate of ∂∆

ℓφℓmust be

e−ℓ+1

X

u=1

(−1)u+1sjuαje−ℓ+1

e−1···αju+1

ℓ+u−1αju−1

ℓ+u−2···αj1

ℓ.

So overall, we need to show that

e−ℓ+1

X

u=1

(−1)u+1sjuαje−ℓ+1

e−1···αju+1

ℓ+u−1αju−1

ℓ+u−2···αj1

ℓ=αje−ℓ+1

e−1···αj1

ℓ−1∂X

ℓ(1)

for all j∈Υ(e−ℓ+ 1). We do this by descending induction on ℓ.

When ℓ > e, the equation clearly holds since both sides are trivial, and in the

case that ℓ=e, (1) states that sj1idXe=αj1

e−1∂X

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 ℓ. We then have

αje−ℓ+1

e−1···αj1

ℓ−1∂X

ℓ=αje−ℓ+1

e−1···αj2

ℓ(sj1idXℓ−∂X

ℓ+1αj1

ℓ)

=sj1αje−ℓ+1

e−1···αj2

ℓ

−(

e−ℓ+1

X

u=2

(−1)usjuαje−ℓ+1

e−1···αju+1

ℓ+u−1αju−1

ℓ+u−2···αj2

ℓ+1)αj1

ℓ

=

e−ℓ+1

X

u=1

(−1)u+1sjuαje−ℓ+1

e−1···αju+1

ℓ+u−1αju−1

ℓ+u−2···αj1

ℓ.

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//Xe„φe

−∂X

e«//

∆e

⊕

Xe−1

∂∆

eφe−1

0−∂X

e−1!//

∆e−1

⊕

Xe−2

//···

··· //

∆e−d

⊕

Xe−d−1

(0−∂X

e−d−1)

//Xe−d−2

−∂X

e−d−2//··· //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

ℓ=

sνδν

ℓδν

ℓφℓαν

ℓ−1

0−sναν

ℓ−1

:

∆ℓ

⊕

Xℓ−1

→

∆ℓ+1

⊕

Xℓ

for each ℓ∈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

0→∆e(X, s)→Ce(X, α)→ΣX→0 (2)

induces the long exact sequence

· · · → Hℓ(∆e(X, s)) →Hℓ(Ce(X, α)) →Hℓ(ΣX)→ · · ·

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

e−1denote the homomorphism

∂D

e−1=

−φe(X, α)e−1

∂X

e−1

:Xe−1−→

∆e(X, s)e−1

⊕

Xe−2

=Ce(X, α)e−1,

and let De(X, α) denote the complex

0//Xe−1

∂D

e−1//Ce(X, α)e−1

−∂Ce(X,α)

e−1//Ce(X, α)e−2

−∂Ce(X,α)

e−2//··· //Ce(X, α)1//0

concentrated in degrees e−1,...,0.

(One veriﬁes easily that De(X, α) indeed is a complex. It is identical to the

shifted mapping cone Σ−1Ce(X, α) except in degrees e+ 1 and e.)

Proposition 28. De(X, α)is an object of Pe−1(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 Σ−1Ce(X, α).

Theorem 29. Let Bdenote the exact complex 0→Xe

id

→Xe→0concentrated in

degrees eand e−1. There is then an exact sequence

0→B→Σ−1Ce(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//Σ−1Ce(X, α)//De(X, α)//0

k k k

degree 0

0

0

e0//Xe

−idXe//

idXe

Xe

„−φe

∂X

e«

//0//

0

e−10//Xe„idXe

−∂X

e«//

∆e

⊕

Xe−1

−∂∆

e−φe−1

0∂X

e−1!

(∂X

eidXe−1)

//Xe−1//

„−φe−1

∂X

e−1«

0

e−20//0

//

∆e−1

⊕

Xe−2

id //

∆e−1

⊕

Xe−2

//0

.

.

..

.

..

.

..

.

.

It is straightforward to verify that the diagram commutes and that all the rows are

exact.

The morphism Σ−1Ce(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 Σ−1µe(X, α) on Σ−1Ce(X, α).

Deﬁnition 30. The S-contraction of De(X , α) induced in the sense of Theorem 17

from Σ−1µe(X, α) through the morphism Σ−1Ce(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

ℓ=

−sνδν

ℓ+1 −δℓ+1φℓ+1 αν

ℓ

0sναν

ℓ

:

∆ℓ+1

⊕

Xℓ

→

∆ℓ+2

⊕

Xℓ+1

whenever ℓ=e−3, . . . , 0, and, as veriﬁed by a small calculation, by

ηe(X, α)ν

e−2=−sν∂X

eδν

e−1αν

e−2∂X

e−1αν

e−2:

∆e−1

⊕

Xe−2

→Xe−1

whenever ℓ=e−2.

From Theorem 29 and (2) in Proposition 26 it follows that

[X] = [Σ−1Ce(X, α)] −[Σ−1∆e(X, s)] = [De(X, α)] −[Σ−1∆e(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 e−1,...,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)] −[Σ−1∆e(X, s)]

in K0Pe−1(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) →

K0Pe−1(S-tor) induces a homomorphism We:K0Pe(S-tor) →K0Pe−1(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

0−→ Yψ

−→ Yψ

−→ e

Y−→ 0

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

0→∆e(Y , t)→∆e(Y , t)→∆e(e

Y , t)→0,(3)

0→Ce(Y , β )→Ce(Y, β)→Ce(e

Y , e

β)→0and (4)

0→De(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 K0Pe−1(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

0−→ Ye

ψe

−→ Ye

ψe

−→ e

Ye−→ 0,

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

ℓ=φe(Y, β )i

ℓψℓand ψeφe(Y, β )i

ℓ=φe(e

Y , e

β)i

ℓψℓfor

each ℓ∈Zand i∈Υ(e−ℓ), 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 →Ye→Ye→0,

0→Ye→Ye→0 and 0 →e

Ye→e

Ye→0 from Theorem 29, concentrated in degrees

eand e−1. These three complexes come together in a short exact sequence 0 →

B→B→e

B→0, induced by the short exact sequence 0 →Ye→Ye→e

Ye→0.

We claim that there is a commutative diagram

0

0

0

0//B

//B

//e

B

//0

0//Σ−1Ce(Y , β )

//Σ−1Ce(Y, β )

//Σ−1Ce(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)] −[Σ−1∆e(Y , t)]

= [De(Y , β )] + [De(e

Y , e

β)] −[Σ−1∆e(Y , t)] −[Σ−1∆e(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 K0Pe−1(S-tor).

Proof. Let e

∂e−1denote the inclusion map im ∂X

e−1֒→Xe−2, and let e

Xdenote the

complex

0−→ im ∂X

e−1e

∂e−1

−→ Xe−2

∂X

e−2

−→ Xe−3−→ · · · −→ X1

∂X

1

−→ X0−→ 0

concentrated in degrees e−1,...0. Since Xis exact, e

Xis exact, and it follows that

im ∂X

e−1is projective, and hence that e

Xis a complex in Pe−1(S-tor).

18 HANS-BJØRN FOXBY AND ESBEN BISTRUP HALVORSEN

Letting Bdenote the exact complex 0 →Xe

id

→Xe→0 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

e−10//Xe

∂X

e−1//

Xe−1

∂X

e−1//

∂X

e−1

im ∂X

e−1//

e

∂e−1

0

e−20//0//

Xe−2

idXe−2//

Xe−2//

0

.

.

..

.

..

.

..

.

.

and we claim that there is a commutative diagram

0

0

0

0//0//

B//

B//

0

0//Σ−1∆e(X, s)//

Σ−1Ce(X, α)//

X//

0

0//Σ−1∆e(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

0→Σ−1∆e(X, s)→De(X, α)→e

X→0 (7)

of complexes in Pe−1(S-tor). Since e

Xis exact, it follows that

we(X, α) = [De(X, α)] −[Σ−1∆e(X, s)] = [ e

X] = 0

in K0Pe−1(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 r∈S1, let ∆(r)def

= ∆e(X, (r, s2,...,sd)); hence, in particular,

∆(s1) = ∆e(X, s).

Lemma 35. Suppose r, r′∈S1, and deﬁne homomorphisms

π(r, r′)ℓ: ∆(rr′)ℓ→∆(r)ℓand ξ(r, r′): ∆(r)ℓ→∆(rr′)ℓ

for each ℓ∈Zby the fact that their (i′, i)-entries for i, i′∈Υ(e−ℓ)are

(π(r, r′)ℓ)i′,i =

0,if i6=i′,

idXe,if i=i′and 1 ∈i,

r′idXe,if i=i′and 1 /∈i,

and

(ξ(r, r′)ℓ)i′,i =

0,if i6=i′,

r′idXe,if i=i′and 1 ∈i,

idXe,if i=i′and 1 /∈i.

Then π(r, r′) = (π(r, r′)ℓ)ℓ∈Zis a morphism of complexes ∆(rr′)→∆(r)and

ξ(r, r′) = (ξ(r, r′)ℓ)ℓ∈Zis a morphism of complexes ∆(r)→∆(rr′).

Proof. Assume that i∈Υ(e−ℓ) and j∈Υ(e−ℓ+ 1). A direct calculation then

shows that the (j, i)-entries of ∂∆(r)

ℓπ(r, r′)ℓand π(r, r′)ℓ−1∂∆(rr′)

ℓare both given

by

0,if j+i,

(−1)u+1sjuidXe,if j\i={ju}and 1 ∈i,

(−1)u+1sjur′idXe,if j\i={ju} 6={1}and 1 /∈i, and

rr′idXe,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′)

ℓξ(r, r′)ℓand

ξ(r, r′)ℓ−1∂∆(r)

ℓare both given by

0,if j+i,

(−1)u+1sjur′idXe,if j\i={ju}and 1 ∈i,

(−1)u+1sjuidXe,if j\i={ju} 6={1}and 1 /∈i, and

rr′idXe,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 tα = (t1α1,...,tdαd)of Xwith weight st = (s1t1, . . . , sdtd). Then

we(X, tα) = we(X, α)in K0Pe−1(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

tα = (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 K0Pe−1(S-tor).

[Σ−1∆(s1t1)] = [Σ−1∆(s1)] + [Σ−1∆(t1)].(8)

[De(X, tα)] = [De(X, α)] + [Σ−1∆(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, r′∈S1according 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, r′∈S1. Furthermore, ξ(1, t1)π(1, s1) as well

as π(t1, s1)ξ(s1, t1) are deﬁned in degree ℓby the fact that their (i, i′)-entries for

i, i′∈Υ(e−ℓ) are

0,if i6=i′,

t1idXe,if i=i′and 1 ∈i, and

s1idXe,if i=i′and 1 /∈i.

To show the exactness of the sequence above, it therefore only remains to show

that, for each ℓ∈Z, the kernel in degree ℓof the second morphism is contained in

the image in degree ℓof 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

ℓ⊕∆(s1t1)i

ℓof the ℓ’th module of ∆(1) ⊕∆(s1t1). So suppose that such an

element is in the kernel of the map in degree ℓof 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 ℓof 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 ℓof the ﬁrst morphism. In either

case, (x, y) is in the image of the map in degree ℓof 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 ℓ∈Za homomorphism

γℓ−1:Xℓ−1→∆(1)ℓby letting its i’th entry for i∈Υ(e−ℓ) be

γi

ℓ−1=0,if 1 ∈i,

αie−ℓ

e−1···αi1

ℓα1

ℓ−1,if 1 /∈i.

Another way of writing this is

γℓ−1=a

i∈Υ(e−ℓ)

1/∈i

αie−ℓ

e−1···αi1

ℓα1

ℓ−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 ℓby

Φℓ=

π(1, s1)ℓγℓ−1

ξ(s1, t1)ℓ0

0 idXℓ−1

:

∆(s1)ℓ

⊕

Xℓ−1

−→

∆(1)ℓ

⊕

∆(s1t1)ℓ

⊕

Xℓ−1

and

Ψℓ=−ξ(1, t1)ℓπ(t1, s1)ℓξ(1, t1)ℓγℓ−1:

∆(1)ℓ

⊕

∆(s1t1)ℓ

⊕

Xℓ−1

−→ ∆(t1)ℓ.

GROTHENDIECK GROUPS FOR CATEGORIES OF COMPLEXES 21

Proving that Φ and Ψ indeed are morphisms of complexes means proving that

∂∆(1)

ℓ+1 0 0

0∂∆(s1t1)

ℓ+1 φe(X, tα)ℓ

0 0 −∂X

ℓ

Φℓ+1 = Φℓ∂∆(s1)

ℓ+1 φe(X, α)ℓ

0−∂X

ℓ

and

∂∆(t1)

ℓ+1 Ψℓ+1 = Ψℓ

∂∆(1)

ℓ+1 0 0

0∂∆(s1t1)

ℓ+1 φe(X, tα)ℓ

0 0 −∂X

ℓ

for all ℓ∈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 ℓ∈Z:

π(1, s1)ℓφe(X, α)ℓ=∂∆(1)

ℓ+1 γℓ+γℓ−1∂X

ℓ; (10)

φe(X, tα)ℓ=ξ(s1, t1)ℓφe(X, α)ℓ; and (11)

π(t1, s1)ℓφe(X, tα)ℓ=ξ(1, t1)ℓγℓ−1∂X

ℓ+∂∆(t1)

ℓ+1 ξ(1, t1)ℓ+1γℓ.(12)

We verify (10) by brute force, calculating on the right hand side of the equation:

∂∆(1)

ℓ+1 γℓ+γℓ−1∂X

ℓ=∂∆(1)

ℓ+1 a

j∈Υ(e−ℓ−1)

1/∈j

αje−ℓ−1

e−1···αj1

ℓ+1α1

ℓ

+a

i∈Υ(e−ℓ)

1/∈i

αie−ℓ

e−1···αi1

ℓα1

ℓ−1∂X

ℓ

=a

i∈Υ(e−ℓ)

1∈i

αie−ℓ

e−1···αi2

ℓ+1α1

ℓ

+a

i∈Υ(e−ℓ)

1/∈i

(

e−ℓ

X

u=1

(−1)u+1siuαie−ℓ

e−1···αiu+1

ℓ+uαiu−1

ℓ+u−1···αi1

ℓ+1α1

ℓ)

+a

i∈Υ(e−ℓ)

1/∈i

αie−ℓ

e−1···αi1

ℓα1

ℓ−1∂X

ℓ

=a

i∈Υ(e−ℓ)

1∈i

αie−ℓ

e−1···αi1

ℓ

+a

i∈Υ(e−ℓ)

1/∈i

αie−ℓ

e−1···αi1

ℓ(∂X

ℓ+1α1

ℓ+α1

ℓ−1∂X

ℓ)

=a

i∈Υ(e−ℓ)

1∈i

αie−ℓ

e−1···αi1

ℓ+a

i∈Υ(e−ℓ)

1/∈i

s1αie−ℓ

e−1···αi1

ℓ

=π(1, s1)ℓφe(X, α)ℓ.

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)ℓφe(X, α)ℓ=a

i∈Υ(e−ℓ)

1∈i

t1φe(X, α)i

ℓ+a

i∈Υ(e−ℓ)

1/∈i

φe(X, α)i

ℓ=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)ℓγℓ−1∂X

ℓ+∂∆(t1)

ℓ+1 ξ(1, t1)ℓ+1γℓ

=ξ(1, t1)ℓ(γℓ−1∂X

ℓ+∂∆(1)

ℓ+1 γℓ)

=ξ(1, t1)ℓπ(1, s1)ℓφe(X, α)ℓ.

In contrast, applying (11) to the left side of (12) yields

π(t1, s1)ℓφe(X, tα)ℓ=π(t1, s1)ℓξ(s1, t1)ℓφe(X, α)ℓ,

so proving equation (12) merely requires showing that

ξ(1, t1)ℓπ(1, s1)ℓ=π(t1, s1)ℓξ(s1, t1)ℓ.(13)

This, however, follows since, for i, i′∈Υ(e−ℓ), both sides of (13) have (i, i′)-entries

given by

0,if i6=i′,

s1idXe,if i=i′and 1 /∈i, and

t1idXe,if i=i′and 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

0−→ Ce(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 ℓ∈Z, the

element (x, y)∈∆(s1)ℓ⊕Xℓ−1=Ce(X, α)ℓmaps to 0 under Φℓ: that is,

0 =

π(1, s1)ℓγℓ−1

ξ(s1, t1)ℓ0

0 idXℓ−1

x

y=

π(1, s1)ℓ(x) + γℓ−1(y)

ξ(s1, t1)ℓ(x)

y

.

It immediately follows that y= 0, and we are left with the equations π(1, s1)ℓ(x) =

ξ(s1, t1)ℓ(x) = 0 which imply that x= 0. Thus, Φℓis injective and (14) is exact at

Ce(X, α).

To see that the sequence is exact at ∆(t1), suppose that x∈∆(t1)i

ℓfor some

ℓ∈Zand i∈Υ(e−ℓ). Then, if 1 ∈i,

−ξ(1, t1)ℓπ(t1, s1)ℓξ(1, t1)ℓγℓ−1

0

x

0

=x,

and if 1 /∈i,

−ξ(1, t1)ℓπ(t1, s1)ℓξ(1, t1)ℓγℓ−1

−x

0

0

=x.

In either case, xis in the image of Ψℓ, and we conclude that Ψℓis 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 Ψℓis contained in the image of Φℓfor all

ℓ∈Z. So suppose that (x, y, z )∈∆(1)ℓ⊕∆(s1t1)ℓ⊕Xℓ−1= (∆(1) ⊕Ce(X, tα))ℓ

maps to 0 under Ψℓ: that is,

−ξ(1, t1)ℓ(x) + π(t1, s1)ℓ(y) + ξ(1, t1)ℓγℓ−1(z) = 0.

GROTHENDIECK GROUPS FOR CATEGORIES OF COMPLEXES 23

Here x= (xi)i∈Υ(e−ℓ)and y= (yi)i∈Υ(e−ℓ)are Υ(e−ℓ)-tuples, so the above equation

states that, for i∈Υ(e−ℓ),

−t1xi+yi= 0,when 1 ∈i, and

−xi+s1yi+γi

ℓ−1(z) = 0,when 1 /∈i.

Now let w= (wi)i∈Υ(e−ℓ)∈∆(s1)ℓbe deﬁned by wi=xiwhenever 1 ∈iand

wi=yiwhenever 1 /∈i. Then

π(1, s1)ℓγℓ−1

ξ(s<