SHORTENING BINARY COMPLEXES AND COMMUTATIVITY OF K-THEORY WITH INFINITE PRODUCTS

. We show that in Grayson’s model of higher algebraic K-theory using binary acyclic complexes, the complexes of length two suﬃce to generate the whole group. Moreover, we prove that the comparison map from Nenashev’s model for K 1 to Grayson’s model for K 1 is an isomorphism. It follows that algebraic K -theory of exact categories commutes with inﬁnite products.


Introduction
On a conceptual level, the algebraic K-theory functor is by now well understood in terms of a universal property, which encapsulates the known fundamental properties of Quillen's or Waldhausen's construction [Bar16,BGT13].
One of the more elusive properties of algebraic K-theory is its compatibility with infinite products. This question was studied by Carlsson [Car95] in connection to work of Carlsson-Pedersen on the split injectivity of the K-theoretic assembly map [CP95], and permeates the literature adapting their "descent" argument to prove more general cases of the K-theoretic Novikov conjecture [BR07,RTY14,Kas15]. Carlsson's proof, while relying on the Additivity theorem, is for the most part concerned with simplicial techniques involving what he calls quasi-Kan complexes.
The present article aims to provide a different perspective on the question. In [Gra12], Grayson showed that the higher algebraic K-theory of an exact category can be expressed in terms of binary acyclic complexes. See Section 2 for a quick review.
In [Nen98] Nenashev gave a different presentation K N 1 (N ) of K 1 (N ) whose generators are binary acyclic complexes of length two. Regarding a binary acyclic complex of length two as a class in K 0 (ΩN ) defines a natural homomorphism Φ: K N 1 (N ) → K 0 (ΩN ); see [Gra12,Remark 8.1] and the beginning of Section 5.
Unpublished work of Grayson shows that Φ is a surjection, cf. [Gra12, Remark 8.1]. Building on Grayson's unpublished argument (see Remark 5.5), we improve this to a bijectivity statement.
Theorem 1.1. The map Φ is an isomorphism.
Proof. We begin with the case n = 1. The map K 0 (Ω [0,∞) N ) → K 0 (ΩN ) is an isomorphism since the class group of a filtered union is isomorphic to the colimit of the class groups and shifting induces an isomorphism in K-theory.
From now on, we write K 0 (ΩN ) for K 0 (Ω [0,∞) N ). All chain complexes considered in what follows will be assumed to be positive.
In the remainder of this section, we record some important properties of K 0 (ΩN ).
Definition 2.5. Let P = (P * , d, d ) be a binary acyclic complex and let i ∈ N.
(1) The ith shift P[i] is defined to be the binary acyclic complex with underlying graded object P [i] * = P * −i and differentials (2) The ith suspension Σ i P is defined to be the binary acyclic complex with underlying graded object Σ i P * = P * −i and differentials Σ i P n = P n−i P n−i−1 = Σ i P n−1 .
Remark 2.6. Our terminology is in disagreement with [Gra12], where the suspension is called a shift.
As for ordinary chain complexes, we have the following lemma.
Proof. The first equality holds since P[1] ∼ = ΣP. The second equality holds since P and ΣP fit into a short exact sequence with the cone of P.
Definition 2.8. A binary double complex is a bounded bigraded object (P k,l ) k,l∈N in N together with morphisms d h k,l : P k,l → P k−1,l , d v k,l : P k,l → P k,l−1 , and d ,h k,l : P k,l → P k−1,l , d ,v k,l : P k,l → P k,l−1 , such that (P * , * , d h , d v ) and (P * , * , d ,h , d ,v ) are double complexes in the sense that (P * ,l , d h ) and (P * ,l , d ,h ) are chain complexes for all l, (P k, * , d v ) and (P k, * , d ,v ) are chain complexes for all k, and  Let J be an object in N and denote by In particular, the element has order at most two.
Proof. Apply Lemma 2.9 to the binary acyclic double complex

Shortening binary complexes
The goal of this section is to prove the following weaker versions of Theorems 1.3 and 1.4. These suffice to prove Theorem 1.2 without using the more technical proof of Theorem 1.1. It is also possible to only read Lemma 3.4 and skip the rest of this section, continue with Section 5, and use the splitting obtained there for Section 4.  As before, N denotes an exact category. The basic approach is the same as that of Harris [Har15,Section 2.2] in showing that the canonical map from Bass' K 1 to K 0 (ΩN ) is an isomorphism for split-exact categories. Our arguments rely on a description of equality of classes in K 0 of an exact category which is due to Heller [Hel65, Lemma 2.1]. We include a proof following [Tho97, Lemma 2.4] for the reader's convenience. (1) We call J and K extension-equivalent if there are objects A, B ∈ N such that there exist exact sequences (2) We call J and K stably extension-equivalent if there exists an object S ∈ N such that J ⊕ S and K ⊕ S are extension-equivalent.
Despite its name, extension-equivalence need not be an equivalence relation. On the other hand, the following lemma shows that stable extension-equivalence is always an equivalence relation. Then the sequences formed by taking direct sums are exact, too. Rewriting , so ϕ is an epimorphism. Moreover, it is immediate from the definition of ∼ that the kernel of ϕ is trivial. This proves that ϕ is an isomorphism, and the claim of the lemma follows.
We can now prove Theorem 3.1. Let P := (P * , d, d ) be a binary acyclic complex supported on [0, m] for some m ∈ N. Choose factorizations d n : P n J n−1 P n−1 and d n : P n K n−1 P n−1 for all n. Since J n and K n both fit into an exact sequence with P n−1 , . . . , P 0 , they represent the same class in K 0 (N ). Therefore, there exist A n , B n , S n ∈ N and exact sequences For n ≥ 3, let S n denote the binary acyclic complex Note that S n is zero for almost all n. Furthermore, let S 2 denote the binary acyclic complex Lemma 3.5. The equation Proof. Let P denote the binary acyclic complex . The lemma then follows by iterating this procedure. For this notice that the complex S 2 for P is precisely the complex S 3 for P. Consider the following binary acyclic double complex. All differentials written as a single arrow are the identity on the summand appearing in domain and codomain and zero on all other summands. In particular, both differentials agree in this case. The remaining four non-trivial binary acyclic complexes are P, P , S 2 , and a fourth one explained in the diagram Applying Nenashev's relation (Lemma 2.9) and omitting all summands which are obviously zero, we obtain We will show that the first summand is trivial. Assuming this, it follows from Lemma 2.7 that as claimed. In fact, triviality of the binary acyclic complex in K 0 (ΩN ) follows directly from the existence of the following short exact sequence of binary acyclic complexes: Lemma 3.5 immediately implies Theorem 3.1 since the complexes S n are supported on [0, 4] for all n ≥ 2.
Our next goal is to prove Theorem 3.2.
Proof. Note that all J n and K n are unique up to isomorphism. We first show that ∞ n=2 (−1) n [S n ] is independent of the choices of A n , B n , S n and the extensions Fix k > 2, and let A k , B k , and S k be different choices fitting into extensions as A k , B k , and S k . Denote by S k and S k+1 the same binary acyclic complexes as S k and S k+1 , except that the extensions involving A k , B k , and S k are replaced by those involving A k , B k , and S k . Note that S l is independent of the choice of A k , B k , and S k for l = k, k + 1. The binary acyclic complexes S k ⊕ (S k+1 [2]) and S k ⊕ (S k+1 [2]) have isomorphic underlying graded objects. We regard both as binary acyclic complexes Both the pair of chain complexes given by the top differentials and the pair of chain complexes given by the bottom differentials of S k ⊕ (S k+1 [2]) and S k ⊕ (S k+1 [2]) are isomorphic: The isomorphism for the top differentials has to flip the two copies of K k , while the one for the bottom differentials has to flip the two copies of J k .
That is, there is the following binary acyclic double complex whose upper row ). Here all unmarked downward arrows are the identity, and τ K and τ J denote the automorphisms switching the two copies of K k and J k , respectively, Applying Lemma 2.9, the difference between the classes of S k ⊕ (S k+1 [2]) and by Lemma 2.11. In combination with Lemma 2.7, this shows An analogous argument works for k = 2, so the class ∞ n=2 (−1) n [S n ] is independent of the choices we make.
Next, we show that the map is independent of the choice of the representative P of [P]. First note that if both differentials of the double complex P agree, then K n and J n agree and we can choose the same extension for both. In this case, both differentials for all S n agree, so [S n ] = 0 for all n.
It remains to see that for a short exact sequence P P P we also get short exact sequences S n S n S n for all n ≥ 2. For every n, we have short exact sequences J n J n J n and K n K n K n . As above, the K 0 -classes of K n and J n as well as those of K n and J n agree. By the Additivity theorem [Qui73, Theorem 2], we have where EN is the exact category of exact sequences in N . Therefore, we find short exact sequences A n A n A n , B n B n B n , and S n S n S n fitting into short exact sequences of short exact sequences: Note that the middle vertical exact sequences are direct sums of the given sequences. Using these extensions for the definition of S n , S n , and S n , we get the desired short exact sequence S n S n S n .
[0,7] ) n−1 N ) which admits a natural section by the case n = 1. Hence admits a natural section as well. Since Ω [0,7] and Ω commute, it suffices to show that K 0 (Ω n−1 [0,7] ΩN ) → K 0 (Ω n N ) admits a natural section. But this map is a retract of the map K 0 (Ω n−1 −1 B q N ), which admits a natural section by the induction assumption.

Algebraic K-theory of infinite product categories
The results of Section 3 allow us to show that the comparison map K( i∈I N i ) → i∈I K(N i ) of connective K-theory spectra is a π * -isomorphism. Theorem 4.1. For every family {N i } i∈I of exact categories and every n ∈ N the natural map is an isomorphism.
Proof. Note that the natural map K 0 ( i∈I N i ) → i∈I K 0 (N i ) is clearly surjective, and that injectivity is a consequence of Lemma 3.4.
Recall that K n (N ) is naturally isomorphic to K 0 (Ω n N ). Consider the following diagram, where the vertical maps are the sections from Theorem 3.2 followed by the canonical homomorphisms: 7] (N i ) are isomorphisms, the middle horizontal map is an isomorphism. A diagram chase implies that the natural map K n ( i∈I N i ) → i∈I K n (N i ) is an isomorphism.
In the remainder of this section, we extend this statement to non-connective K-theory. Our model for the non-connective algebraic K-theory K −∞ of an exact category is Schlichting's delooping [Sch06,Section 12].
The argument to extend Theorem 4.1 to non-connective algebraic K-theory is based on a localization sequence of Schlichting [Sch04]. To state it, we need to recall the following definition. (1) An admissible epimorphism N A with N ∈ N and A ∈ A is special if there exists an admissible monomorphism B N with B ∈ A such that the composition B → A is an admissible epimorphism.
(2) The inclusion A ⊆ N is called left s-filtering if the following holds: (a) The subcategory A is closed under admissible subobjects and admissible quotients in N . (b) Every admissible epimorphism N A from an object N ∈ N to an object A ∈ A is special.
Finally, recall the countable envelope FN of an idempotent complete exact category N [Sch04, Section 3] (and the references therein). The concrete definition need not concern us here. It suffices to know that FN is an exact category which contains N as a left s-filtering subcategory, and that K −∞ (FN ) is contractible [Sch04, Lemma 3.2]; the latter claim holds because FN admits countable coproducts. Moreover, FN depends functorially on N . Denote by SN the quotient category FN /N . The category SN is called the suspension of N . Write S n N for the n-fold suspension of N . From Theorem 4.3, it follows directly that Ω n K −∞ (S n N ) is naturally equivalent to K −∞ (N ). In particular, we have K −n (N ) ∼ = K 0 (Idem(S n N )) for all n > 0, where Idem(−) denotes the idempotent completion functor.
Proof of Theorem 1.2. Let {N i } i∈I be a family of exact categories.
Since the natural map K −∞ (N ) N )) is an equivalence for every exact category and Idem( i∈I N i ) ∼ = i∈I Idem(N i ), we may assume that N i is idempotent complete for all i ∈ I.
Consider the left s-filtering inclusion i∈I N i ⊆ F( i∈I N i ). The various projection functors i∈I N i → N j induce an exact functor F( i∈I N i ) → i∈I FN i . Moreover, the inclusion i∈I N i ⊂ i∈I FN i is left s-filtering since it is left sfiltering on each factor. Since SN i is obtained from FN i by a calculus of left fractions, we can identify i∈I FN i / i∈I N i ∼ = i∈I SN i (for an explicit description of the morphism sets making this obvious, cf. [GZ67, Chapter I, Section 2.2]). Therefore, we have by Theorem 4.3 a map of homotopy fiber sequences of spectra: Since both F( i∈I N i ) and i∈I FN i admit countable coproducts, the K-theory of both vanishes and the middle vertical arrow is a π * -isomorphism. Hence, the right vertical map is a π * -isomorphism. By induction, it follows that the canonical map is a π * -isomorphism for every family of idempotent complete exact categories. Let n > 0. We have the commutative diagram The map c is an isomorphism as we have just discussed. Since the diagonal map is an isomorphism by Theorem 4.1, the theorem follows.
Remark 4.4. Note that the proof for negative K-groups only used that K 0 commutes with infinite products, which was a direct consequence of Lemma 3.4.

The relation to Nenashev's K 1
The abelian group K 0 (ΩN ) is not the first algebraic description of K 1 of an exact category. Nenashev gave the following description of K 1 (N ).
Definition 5.1. Define K N 1 (N ) as the abelian group generated by binary acyclic complexes of length two P = P 2 P 1 P 0 subject to the following relations: (1) If the top and bottom differential of a binary acyclic complex coincide, that complex represents zero.
(2) For any binary acyclic double complex (see Remark 2.10) The main result of [Nen98] states that K N 1 (N ) is isomorphic to K 1 (N ). By Lemma 2.9, regarding a binary acyclic complex of length two as a class in K 0 (ΩN ) defines a natural homomorphism Φ: K N 1 (N ) → K 0 (ΩN ), as already remarked in the introduction. In this section, we prove Theorem 1.1. Before doing so, we give the following corollary. N ) is a surjection and the homomorphism K 0 (Ω n [0,4] N ) → K 0 (Ω n N ) admits a natural section.
. By Lemma 2.9, Φ factors as . This exhibits K 0 (ΩN ) as a natural retract of K 0 (Ω [0,4] N ). For n > 1, the claim follows as in the proof of Theorem 3.2 by induction.
Hence, Theorem 1.1 also proves that the algebraic K-theory functor commutes with infinite products.
In the remainder of this section, we give a proof of Theorem 1.1. As in the proof of Theorem 3.2, this will be accomplished by producing an explicit formula that expresses the class of an arbitrary binary acyclic complex in terms of binary acyclic complexes of length two.
Before we start shortening binary acyclic complexes, we make a quick observation about K N 1 (N ), which we will need later in the argument. Lemma 5.3. For any binary acyclic complex of length two, we have Proof. This follows directly from applying the defining relations of K N 1 to the binary acyclic double complex Let P := (P * , d, d ) be a binary acyclic complex. In a first step we will not shorten P but produce a complex P representing the same class in K 0 (ΩN ), which we will then be able to shorten.
Choose factorizations d 2 : P 2 J P 1 and d 2 : P 2 K P 1 . Since J and K both are the kernel of an admissible epimorphism P 1 P 0 , they represent the same class in K 0 (N ). Therefore, there exist by Lemma 3.4 A, B, S ∈ N , and exact sequences

A J⊕ S B and A K⊕ S B.
Let S denote the binary acyclic complex Let P denote the binary acyclic complex For an object M ∈ N we denote by Δ M the binary acyclic complex Consider the following binary acyclic double complex. All differentials written as a single arrow are the identity on the summand appearing in domain and codomain and zero on all other summands. In particular, both differentials agree in this case. The remaining four non-trivial binary acyclic complexes are P ⊕ Δ B , P , and S A A Let P denote the binary acyclic complex We can build the following binary acyclic double complex involving , and a third non-trivial complex.
The non-trivial vertical complex differs from Δ J [1] ⊕ S ⊕ Δ K [1] by identifying J ⊕ P 1 ⊕ K ⊕ B ∼ = J ⊕ B ⊕ P 1 ⊕ K and using τ K and τ J in the bottom differential. Since K and J represent the same class in K 0 , we have and thus the vertical complex represents the same class as S by Lemma 2.11. Applying Nenashev's relation (Lemma 2.9) and omitting all summands which are obviously zero, we obtain Let Q denote the binary acyclic complex Let us fix the following notation: If M is an object containing N as a direct summand, denote by e N the obvious idempotent M → M whose image is N . Consider the following double complex involving P and Q[1]. Note that only the rows are acyclic. This suffices to see that the total complex shifted down by one represents the same class as Let T denote the total complex shifted down by one. Then T is Assume that P was supported on [0, m], then T admits a projection onto Δ P m [m−1]. The kernel of this projection admits a projection to Δ P m−1 [m−2] and so on until we take the kernel of the projection to Δ P 2 [1]. The remaining acyclic binary complex T is Since T is supported on [0, 2] and Q has length one shorter than P, iterating this argument already shows that K 0 (Ω [0,2] N ) → K 0 (ΩN ) is surjective.
Remark 5.5. The idea to use the complexes P, T, and T is from the aforementioned, unpublished result of Grayson. He uses a different argument to show that [ P]−[P] is contained in the image of Φ. Grayson's argument avoids the use of Heller's lemma, but the computation of the appearing correction terms is more complicated.
We now want to simplify T . Let T triv denote the binary acyclic complex whose underlying graded object is that of T , but with both differentials equal to the top differential of T . Then the following diagram, where the upper row is T and the second row is T triv , commutes: Both differentials of T triv agree and thus it represents the trivial class. Since τ K and τ J are of order two, we conclude from Lemma 2.9 that ].
Since J P 1 P 0 is exact, this is the same as ]. This shows that (5.6) We are now going to iterate this argument. Choose factorizations d n : P n J n−1 P n−1 and d n : P n K n−1 P n−1 for all n ≥ 2 such that J n P n J n−1 and K n P n K n−1 are exact for all n. Set J 0 := P 0 and K 0 := P 0 . For any natural number k, fix the following auxiliary notation: First of all, we define for every natural number k a binary acyclic complex P k of the form For even natural numbers k, we equip P k with the top differential Note that P 0 is precisely the complex P. For odd natural numbers k, we equip P k with the top differential Note that P 1 is precisely the complex Q appearing in (5.6). Moreover, if k is sufficiently large so that P n ∼ = 0 for all n > k, then P k+1 is obtained from P k by interchanging the top and bottom differential. For every k let Q k denote the complex obtained from P k by the same procedure as Q is obtained from P.
Suppose now that k is odd. Substituting appropriately in (5.6), we obtain the equation where X denotes the kernel of the first top differential of P k . By the definition of the binary acyclic complex P k , we may choose

SHORTENING AND K-THEORY OF INFINITE PRODUCTS 21
As in the proof of Proposition 3.6, since J n and K n represent the same class in K 0 for all n, we have by Lemma 2.11 ]. Similarly, is the kernel of the first bottom differential of P k . Note that the complement of J k+1 in X and the complement of K k+1 in Y are the same; let Z denote that complement.
Unwinding the definition of Q k , we see that, up to automorphisms flipping the two copies of Z in the three lowest degrees of Q k , Q k coincides with the sum of P k+1 with some complexes in the image of the diagonal functor Δ. Since, by (5.7), ]. Hence, The argument for k even is completely analogous. Therefore, we have for every k ≥ 0 the equation Proof of Theorem 1.1. Define a map Ψ : K 0 (ΩN ) → K N 1 (N ) by the rule [P] → x(P, k(P)), where k(P) is defined to be k(P) := min{n ∈ N | P n ∼ = 0 for all n > n}.
We have to show that this is a well-defined homomorphism. By our definition of k(P), the complex P k(P) has length two.
Let P P P be an exact sequence of binary acyclic complexes. Evidently, x(P , k(P)) + x(P , k(P)) = x(P, k(P)). Note that k(P ), k(P ) ≤ k(P) and at least one of k(P ) and k(P ) equals k(P). If k(P ) = k(P) = k(P ), we already have Ψ([P ]) + Ψ([P ]) = Ψ([P]). Suppose k(P ) < k(P). Then P k(P) arises from P k(P ) by interchanging the role of top and bottom differential k(P) − k(P ) times. Since interchanging the top and bottom differential results only in a change of sign (Lemma 5.3) and J n ∼ = 0 for n > k(P ), we have x(P , k(P )) = x(P , k(P)). The case k(P ) < k(P) is analogous. Suppose now that P lies in the image of the diagonal functor Δ : C q N → B q N . Then we may choose J n = K n for all n. In this case, the top and bottom differential 22 D. KASPROWSKI AND C. WINGES of P k(P) are isomorphic. However, the two differentials do not agree on the nose but only after flipping all appearing K n = J n . Since each one of these appears three times in P k(P) , applying the Nenashev relation we see that Consequently, Ψ([P]) = 0 by Lemma 2.11. This shows that Ψ is a well-defined homomorphism K 0 (ΩN ) → K N 1 (N ). Our previous discussion implies that Φ • Ψ = id K 0 (ΩN ) . What is left to do is to show that Ψ • Φ = id K N 1 (N ) . To do so, it suffices to establish equation 5.6 in K N 1 (N ) for all binary acyclic complexes of length two. Let be a binary acyclic complex of length two. Then P 1 is the binary acyclic complex P 2 ⊕ P 2 P 2 ⊕ P 2 ⊕ P 2 ⊕ P 1 P 2 ⊕ P 2 ⊕ P 0 with the following top and bottom differentials: Consider the following binary acyclic double complex where the upper row is P 1 , the lower row is P with switched differentials plus Δ P 2 ⊕ Δ P 2 ⊕ Δ P 2 [1], and the middle vertical map τ P 2 denotes the flip of the second and third summand: P 2 ⊕ P 2 P 2 ⊕ P 2 ⊕ P 2 ⊕ P 1 P 2 ⊕ P 2 ⊕ P 0 P 2 ⊕ P 2 P 2 ⊕ P 2 ⊕ P 2 ⊕ P 1 P 2 ⊕ P 2 ⊕ P 0 τ P 2 id τ P 2 id τ P 2 id Therefore, we obtain using Lemmas 2.11 and 5.3 This finishes the proof.
Proof of Theorem 1.4. The statement is obtained by considering the first part of the proof of Theorem 1.1 and observing that Lemmas 5.3 and 2.11 and the Nenashev relation used there only require total complexes of length at most three.