ON THE HOMOTOPY COFIBRE SPECTRUM OF K(R) → G(R)

GRIGORY GARKUSHA

Abstract. The natural map of K-spectra K(R) → G(R) fits into a homotopy cofibre sequence

K(R) → G(R) → K(wA)

where wA is the Waldhausen category of bounded chain complexes over the category of finitely presented modules, with weak equivalences being stable quasi-isomorphisms.

Let P(R) denote the category of finitely generated projective modules over an associative ring with identity R and let mod R be the category of finitely presented R-modules. In this note we discuss the relationship between the Ktheories K(R) = K(P(R)) and G(R) = K(mod R). More precisely, if we consider P(R) as an exact category with the family of split short exact sequences and mod R as an exact category with the family of short exact sequences, one would like to know the homotopy cofibre of the natural map of spectra K(R) → G(R) induced by the exact inclusion of P(R) into mod R. We will show that it is homotopy equivalent to the Waldhausen K-theory for the category of bounded complexes over mod R with suitable weak equivalences. This identification is, in fact, a consequence of Waldhausen’s fibration theorem.

I should like to thank Paul Balmer for pointing out a mistake in the previous version of the paper.

1. Preliminaries

We assume certain degree of familiarity with the basic facts about exact categories and Waldhausen categories, and their K-theory. We refer the reader for details to [3, 6]. If A is a Waldhausen category, by co(A) and w(A) we denote the corresponding classes of cofibrations and weak equivalences of A. If the choice of w(A) is particular important, we write the category as wA.

Definition. (1) We say that a Waldhausen category A is saturated if the following condition holds: whenever f and g are composable maps and fg is a weak equivalence, f is a weak equivalence if and only if g is.

(2) Suppose that f : E → E′ is a map between cofibration sequences A ↣ B ↠ C and A′ ↣ B′ ↠ C′. We say that A satisfies the extension axiom if whenever the source map and the quotient map of f (A → A′ and C → C′) are weak equivalences, so is the total map of f (B → B′).

(3) A Waldhausen category A has a cylinder functor if there is a functor

T : Ar A ⟶ A

where Ar A is the category of arrows of A, together with three natural transformations p, j_{1}, j_{2}, satisfying the conditions below.

Thus to each morphism f : A → B of A, the functor T assigns an object T f of A. To each commutative diagram in A

\begin{matrix}A & ⟶^{f} & B \\ ↓ & & ↓ \\ A′ & ⟶^{f′} & B′ \end{matrix}

T assigns functorially a morphism T_{a,b} : T f → T f′.

The natural transformations are the maps j_{1} : A → T f, j_{2} : B → T f, and p : T f → B such that pj_{1} = f : A → B, pj_{2} = 1 : B → B, and such that the diagram

\begin{matrix}A ∪ B & ⟶^{j_{1}∪j_{2}} & T f & ⟶^{p} & B \\ ^{a∪b}↓ & & ↓^{T_{a,b}} & ↓^{b} \\ A′ ∪ B′ & ⟶^{j_{1}∪j_{2}} & T f′ & ⟶^{p′} & B′ \end{matrix}  (1.1)

is commutative. We also require the following conditions to hold.

(i) If a and b are in co(A), then not only is T_{a,b} in co(A), but also the map T f ∪_{A∪B} A′ ∪ B′ → T f′ induced by the left square of (1.1) is in co(A).

(ii) If a and b are in w(A), then T_{a,b} is in w(A).

(iii) T (0 → A) = A, with p and j_{2} the identity map.

Below we need also the cylinder axiom:

(iv) For all f, the map p : T f → B is in w(A).

Let A be a Waldhausen category. Suppose A has two classes of weak equivalences v(A) and w(A), and v(A) ⊂ w(A). Assume that w(A) satisfies the saturation and extension axioms, and has a cylinder functor T which satisfies the cylinder axiom.

Let A^{w} be the full subcategory of A whose objects are the objects A of A such that 0 → A is in w(A). Then A^{w} becomes a Waldhausen category with co(A^{w}) = co(A) ∩ A^{w} and v(A^{w}) = v(A) ∩ A^{w}. If T is a cylinder functor both for v(A) and for w(A), T induces a cylinder functor on A^{w}.

1.1. Theorem (Waldhausen’s fibration theorem). With the notation and hypotheses above, suppose also that A has a cylinder functor T which is a cylinder functor both for v(A) and for w(A). Then the exact inclusion functors vA^{w} → vA, vA → wA, induce a homotopy fibre sequence of spectra

K(vA^{w}) ⟶ K(vA) ⟶ K(wA).

Proof. [6, 1.6.4]. □

We also use the theorem due to Waldhausen and Gillet. It allows to convert the K-theory of an exact category to the K-theory of chain complexes. We use it only for exact categories with split idempotents. We refer the reader to [5] for details.

Fix an exact category E with split idempotents. Let C^{b}(E) be the category of bounded chain complexes of objects of E and let K^{b}(E) denote the homotopy category of E. Let A(E) be the full subcategory of K^{b}(E) consisting of acyclic complexes. A complex

X^{n} → X^{n+1} → X^{n+2}

is called acyclic if each map X^{n} → X^{n+1} decomposes in E as X^{n} ↠^{e_{n}} D^{n} ↣^{m_{n}} X^{n+1} where e_{n} is an admissible epimorphism and m_{n} is an admissible monomorphism; furthermore, D^{n} ↣^{m_{n}} X^{n+1} ↠^{e_{n+1}} D^{n+1} must be an exact sequence.

Neeman [1] has shown that A(E) is a thick subcategory of K^{b}(E). The derived category D^{b}(E) for E is, by definition, the quotient of K^{b}(E) by the thick subcategory A(E). A morphism of complexes from C^{b}(E) is called a quasi-isomorphism provided that its image in D^{b}(E) is an isomorphism.

Remark. There is a fully faithful embedding of E into an abelian category A such that E ⊂ A is closed under extensions and the exact functor E → A reflects exact sequences (see, e.g., [5]). Then a complex in C^{b}(E) is acyclic if and only if so is its image in C^{b}(A) [1]. In particular, a morphism of C^{b}(E) is a quasiisomorphism if and only if so is its image in C^{b}(A).

Let cofibrations co(C^{b}(E)) for C^{b}(E) be the degree-wise admissible monomorphisms and weak equivalences v(C^{b}(E)) be quasi-isomorphisms. Then C^{b}(E) is a Waldhausen category. Moreover, C^{b}(E) has a cylinder functor T: for a map of chain complexes f : A → B we put

T f_{n} = A_{n} ⊕ A_{n−1} ⊕ B_{n}.  (1.2)

Then the functor T satisfies the cylinder axiom.

1.2. Theorem (Waldhausen-Gillet). Let E be an exact category with split idempotents. Then the exact functor E → C^{b}(E) sending E of E to the complex which is E in degree zero and zero in other degrees induces a homotopy equivalence of K-theory spectra

K(E) ≈ K(C^{b}(E)).

Proof. [5, 1.11.7]. □

2. Perfect complexes and stable quasi-isomorphisms

Throughout this section we denote the category of bounded chain complexes C^{b}(mod R) over mod R by A. The family of weak equivalences consisting of quasi-isomorphisms we denote by v(A).

Definitions. (1) An object of A is called perfect if it is quasi-isomorphic to a complex with components in P(R). The full subcategory of perfect complexes we denote by perf R.

(2) A complex C from the unbounded derived category D(Mod R) of right Rmodules Mod R is said to be compact if the functor Hom(C, −) commutes with arbitrary set-indexed coproducts. Denote the full subcategory of D(Mod R) of compact objects by comp R.

Let K^{b}(P(R)) be the homotopy category of bounded complexes of P(R). Then the canonical functor

K^{b}(P(R)) ⟶ comp R

is an equivalence of triangulated categories (see [4, section 6]). Since a complex of A is acyclic if and only if it is acyclic in the category of unbounded chain complexes C(Mod R) of Mod R, one has the following relations:

comp R ⊂ D^{b}(mod R) ⊂ D(Mod R).

2.1. Lemma. The subcategory of compact objects is thick in D^{b}(mod R).

Proof. We must show that any direct summand of a compact object is a compact object [2]. Let C ∈ comp R and C = A ⊕ B in D^{b}(mod R). Then C = A ⊕ B in D(Mod R). Since the representable functor Hom(C, −) commutes with direct sums, so do the functors Hom(A, −) and Hom(B, −). Thus both A and B are compact, and hence comp R is a thick subcategory of D^{b}(mod R). □

Definitions. (1) The stable derived category of bounded complexes \underline{D}^{b}(mod R) is the quotient category of D^{b}(mod R) with respect to the thick subcategory of compact objects comp R.

(2) A morphism of chain complexes in A is called a stable quasi-isomorphism provided that its image in \underline{D}^{b}(mod R) is an isomorphism. The family of stable quasi-isomorphisms of A is denoted by w(A).

2.2. Proposition. w(A) forms a set of weak equivalences and satisfies the saturation and extension axioms.

Proof. Clearly, each isomorphism is a morphism of w(A). To show that w(A) is a set of weak equivalences we must check the “gluing axiom”. Given a commutative diagram in A

B ⟵^{i} A ⟶^{j} C
b↓  ↓^{a}  ↓^{c}
B′ ⟵_{i′} A′ ⟶_{j′} C′

with i, i′ being cofibrations and with a, b, c being in w(A), we must show the induced map B ∐_{A} C → B′ ∐_{A′} C′ to be in w(A). We have the following commutative diagram of chain complexes with exact rows:

0 ⟶ A ⟶ B ∐ C ⟶ B ∐_{A} C ⟶ 0
↓  ↓  ↓
0 ⟶ A′ ⟶ B′ ∐ C′ ⟶ B′ ∐_{A′} C′ ⟶ 0

Since the functor A → \underline{D}^{b}(mod R) takes finite direct coproducts to coproducts and short exact sequences to triangles, we obtain the following commutative diadram in \underline{D}^{b}(mod R):

A ⟶ B ∐ C ⟶ B ∐_{A} C ⟶ A[1]
↓  ↓  ↓  ↓
A′ ⟶ B′ ∐ C′ ⟶ B′ ∐_{A′} C′ ⟶ A′[1].

Here the morphisms A → A′, B ∐ C → B′ ∐ C′, and A[1] → A′[1] are isomorphisms. Hence B ∐_{A} C → B′ ∐_{A′} C′ is also an isomorphism. This follows from elementary properties of triangles (see, e.g., [2]). Hence A together with the family of stable quasi-isomorphisms is a Waldhausen category.

The saturation axiom is trivial for w(A). It remains to check the extension axiom. Consider a commutative diagram whose rows are cofibration sequences

0 ⟶ A ⟶ B ⟶ C ⟶ 0
a↓  ↓^{b}  ↓^{c}
0 ⟶ A′ ⟶ B′ ⟶ C′ ⟶ 0

and both a and c are in w(A). As above, we get a morphism of triangles in \underline{D}^{b}(mod R)

C[−1] ⟶ A ⟶ B ⟶ C
c[−1]↓  ^{a}↓  ^{b}↓  ↓^{c}
C′[−1] ⟶ A′ ⟶ B′ ⟶ A′

with c[−1], a, c being isomorphisms. Then b is also an isomorphism, and so b is in w(A). □

Next, let T be the cylinder functor (1.2). Then T satisfies the cylinder axiom both for v(A) and for w(A). Let us verify that if a and b are in w(A), then also T_{a,b} is in w(A).

Let

\begin{matrix}A & ⟶^{f} & B \\ ^{a}↓ & & ↓^{b} \\ A′ & ⟶^{f′} & B′ \end{matrix}

be a commutative square with a, b in w(A). We have the following commutative square

T f ⟶^{p} B
T_{a,b}↓  ↓^{b}
T f′ ⟶^{p′} B′

(see definition of the cylinder functor). Since p, p′ are in v(A) ⊂ w(A) by the cylinder axiom, the morphism bp = p′T_{a,b} is a stable quasi-isomorphism, and so T_{a,b} is a stable quasi-isomorphism by the saturation axiom. Hence T is a cylinder functor for w(A).

The full subcategory A^{w} of A whose objects are the complexes stable quasiisomorphic to zero is perf R.

We are now in a position to prove the main result of the paper.

2.3. Theorem. The natural map K(R) → G(R) fits into a homotopy cofibre sequence of spectra

K(R) ⟶ G(R) ⟶ K(wA)

where wA is the Waldhausen category of bounded chain complexes over mod R with weak equivalences being stable quasi-isomorphisms. In particular, there is a long exact sequence:

… K_{n+1}(wA) → K_{n}(R) → G_{n}(R) → K_{n}(wA) …,

ending in the obvious surjection G_{0}(R) → K_{0}(wA).

Proof. The Waldhausen category vA^{w} is the category of perfect complexes perf R in A. By the Gillet-Waldhausen theorem we have two homotopy equivalences K(R) ≈ K(perf R) and G(R) ≈ K(vA). It remains to apply the Waldhausen fibration theorem to the exact functors perf R = vA^{w} → vA and vA → wA. This implies the claim. □

Let fg R denote the category of finitely generated right R-modules. Then mod R ⊆ fg R and mod R = fg R if and only if R is right noetherian. The category fg R is exact and small. We define B to be the category of bounded chain complexes over fg R. As above, we can introduce the class w(B) of stable quasi-isomorphisms and then wB becomes a Waldhausen category. Also, all the above arguments we stated for the map K(R) → G(R) remain the same for the relationship between the K-theories K(R) and K(fg R). More precisely, the following theorem is true.

2.4. Theorem. The natural map K(R) → K(fg R) induced by the exact inclusion P(R) → fg R fits into a homotopy cofibre sequence of spectra

K(R) ⟶ K(fg R) ⟶ K(wB)

where wB is the Waldhausen category of bounded chain complexes over fg R with weak equivalences being stable quasi-isomorphisms. In particular, there is a long exact sequence:

… K_{n+1}(wB) → K_{n}(R) → K_{n}(fg R) → K_{n}(wB) …,

ending in the obvious surjection K_{0}(fg R) → K_{0}(wB).

In conclusion, we remark that in some special cases one can obtain a similar relationship between K- and G-theories of a scheme X. For this subject we recommend the reader to consult the work [5].

References

[1] Neeman, A., The derived category of an exact category, J. Algebra 135 (1990), 388-394.

[2] Neeman, A., Triangulated categories, Annals of Mathematics Studies 148, Princeton University Press, Princeton, 2001.

[3] Quillen, D., Higher algebraic K-theory. I, In Algebraic K-theory I, Lecture Notes in Mathematics, No. 341, Springer–Verlag, 1973, pp. 85-147.

[4] Rickard, J., Morita theory for derived categories, J. London Math. Soc. (2) 39 (1989), 436-456.

[5] Thomason, R. W., Trobaugh, T., Higher algebraic K-theory of schemes and of derived categories, The Grothendieck Festschrift III, Collect. Artic. in Honor of the 60th Birthday of A. Grothendieck, Progress in Mathematics 88, Birkhäuser, 1990, pp. 247-435.

[6] Waldhausen, F., Algebraic K-theory of spaces, In Algebraic and geometric topology, Proc. Conf., New Brunswick/USA 1983, Lecture Notes in Mathematics, No. 1126, Springer-Verlag, 1985, pp. 318-419.

St. Petersburg State University, Department of Mathematics and Mechanics, Universitetsky pr. 28, 198504, Stary Peterhof, Russia

E-mail address: ggarkusha@mail.ru