Shortening binary complexes and commutativity of K-theory with infinite products
By Daniel Kasprowski and Christoph Winges
Abstract
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.
1. 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 Reference Bar16Reference BGT13.
One of the more elusive properties of algebraic $K$-theory is its compatibility with infinite products. This question was studied by Carlsson Reference Car95 in connection to work of Carlsson–Pedersen on the split injectivity of the $K$-theoretic assembly map Reference CP95, and permeates the literature adapting their “descent” argument to prove more general cases of the $K$-theoretic Novikov conjecture Reference BR07Reference RTY14Reference 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 Reference 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 Reference Nen98 Nenashev gave a different presentation $K^N_1(\mathcal{N})$ of $K_1(\mathcal{N})$ whose generators are binary acyclic complexes of length two. Regarding a binary acyclic complex of length two as a class in $K_0(\Omega \mathcal{N})$ defines a natural homomorphism
Unpublished work of Grayson shows that $\Phi$ is a surjection, cf. Reference 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 Building on Grayson’s unpublished argument (see Remark 5.5), we improve this to a bijectivity statement.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 $\Phi$ is an isomorphism.
We use this to show the following theorem.
Theorem 1.2.
For every family $\{\mathcal{N}_i\}_{i\in I}$ of exact categories, the natural map
is a $\pi _*$-isomorphism. Here $\mathbb{K}^{-\infty }$ denotes non-connective algebraic $K$-theory.
Since Grayson’s results in Reference Gra12 rely only on the fundamental properties of $K$-theory, our proof is not only elementary, but also exhibits Theorem 1.2 as a consequence of the universal property of algebraic $K$-theory.
As a corollary of Theorem 1.1, we obtain the following theorem.
Theorem 1.3.
The canonical map $K_0(\Omega ^n_{[0,2]}\mathcal{N})\to K_0(\Omega ^n\mathcal{N})$ is a surjection.
We also obtain the following theorem.
Theorem 1.4.
For every $n\in \mathbb{N}$ the canonical map $K_0(\Omega _{[0,3]}^n\mathcal{N}) \to K_0(\Omega ^n\mathcal{N}) \cong K_n(\mathcal{N})$ admits a natural section.
The existence of a section to $K_0(\Omega _{[0,4]}^n\mathcal{N})$ is a direct consequence of Theorem 1.1, but our proof actually implies this stronger statement.
Since the proof of Theorem 1.1 is technical, we will begin by showing versions of Theorems 1.3 and 1.4 in Section 3. Here the proofs are considerably easier and they suffice to deduce Theorem 1.2. In Section 4, we use the right inverse to show Theorem 1.2. Finally, we give the proof of Theorem 1.1 in Section 5.
The results were adapted to the setting of stable $\infty$-categories in Reference KW19.
2. Binary complexes
In this section, we give a quick review of Grayson’s description of the higher algebraic $K$-groupsReference Gra12. In the following $\mathcal{N}$ will always denote an exact category. Chain complexes in $\mathcal{N}$ will always be assumed to be bounded. Denote by $C\mathcal{N}$ the category of (bounded) chain complexes in $\mathcal{N}$.
Definition 2.1.
A chain complex $(P_*,d)$ in $\mathcal{N}$ is called acyclic if each differential admits a factorization into an admissible epimorphism followed by an admissible monomorphism
such that $J_n \rightarrowtail P_n \twoheadrightarrow J_{n-1}$ is a short exact sequence.
Denote by $C^q\mathcal{N}\subseteq C\mathcal{N}$ the full subcategory of acyclic chain complexes.
Definition 2.2.
A binary acyclic complex$(P_*, d, d')$ is a graded object $P_*$ over $\mathcal{N}$ together with two degree $-1$ maps $d, d' \colon P_* \to P_*$ such that both $(P_*,d)$ and $(P_*,d')$ are acyclic chain complexes. The differentials $d$ and $d'$ are called the top and bottom differential.
A morphism of binary acyclic complexes is a degree $0$ map of underlying graded objects which is a chain map with respect to both differentials. The resulting category of binary acyclic complexes is denoted by $B^q\mathcal{N}$. There is a natural exact functor $\Delta \colon C^q(\mathcal{N}) \to B^q(\mathcal{N})$ which duplicates the differential of a given acyclic chain complex.
Fix $n > 0$. Since both $C^q\mathcal{N}$ and $B^q\mathcal{N}$ are exact categories, these constructions can be iterated. For any finite sequence $\mathbb{W}= (W_1,\dots ,W_n)$ in $\{B,C\}$, denote by $\mathbb{W}^q\mathcal{N}$ the category $W_1^q\dots W_n^q\mathcal{N}$. If $\mathbb{W}$ is the constant sequence on the letter $B$, we also write $(B^q)^n\mathcal{N}$. Letting $\mathbb{W}$ vary over all possible choices defines a commutative $n$-cube of exact categories which induces a commutative $n$-cube of spectra upon taking algebraic $K$-theory. The spectrum $\mathbb{K}(\Omega ^n\mathcal{N})$ is defined to be the total homotopy cofiber of this cube.
We rely on the following result about $\mathbb{K}(\Omega ^n\mathcal{N})$.
The abelian groups $K_n(\mathcal{N})$ and $K_0(\Omega ^n\mathcal{N})$ are naturally isomorphic.
This theorem facilitates a completely algebraic description of higher $K$-theoryReference Gra12, Corollary 7.4. For example, it implies that $K_1(\mathcal{N})$ can be described as the Grothendieck group of the category of binary acyclic complexes $B^q\mathcal{N}$ with the additional relation that a binary acyclic complex represents the trivial class if its top and bottom differential coincide. We use this description of $K_1(\mathcal{N})$ extensively in Section 3.
Throughout this article, we employ the following variations of this construction: Let $J \subseteq \mathbb{Z}$ be an interval, i.e., $J = \{ z \in \mathbb{Z}\mid a \leq z \leq b \}$ for some $a,b \in \mathbb{Z}\cup \{\pm \infty \}$. Then we denote by $B^q_J\mathcal{N}$ and $C^q_J\mathcal{N}$ the categories of (binary) acyclic complexes supported on $J$. Thus, any sequence of intervals $\mathbb{J}= (J_1,\dots ,J_n)$ in $\mathbb{Z}$ gives rise to an abelian group $K_0(\Omega _\mathbb{J}\mathcal{N}) \coloneq K_0(\Omega _{J_1}\dots \Omega _{J_n}\mathcal{N})$. If $\mathbb{J}' = (J_1',\dots ,J_n')$ is another such sequence satisfying $J_k \subseteq J_k'$ for all $k$, we have a natural homomorphism
Note that $\Delta \colon C^q_J\mathcal{N}\to B^q_J\mathcal{N}$ admits two natural splits $\top$ and $\bot$ which forget the bottom, respectively, top, differential of a binary acyclic complex. Using one of these, we see that $i_{\mathbb{J},\mathbb{J}'}$ is naturally a retract of the homomorphism
It is notationally convenient to work with $\mathbb{N}$-graded bounded chain complexes instead of $\mathbb{Z}$-graded chain complexes. The following lemma justifies this convention.
Lemma 2.4.
The natural map $K_0(\Omega _{[0,\infty )}^n\mathcal{N}) \to K_0(\Omega ^n\mathcal{N})$ is an isomorphism for all $n \geq 1$.
Proof.
We begin with the case $n=1$. The map $K_0(\Omega _{[0,\infty )}\mathcal{N}) \to K_0(\Omega \mathcal{N})$ 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.
We will now prove the lemma by induction. Assume that it holds for $n-1$. The map $K_0(\Omega ^n_{[0,\infty )}\mathcal{N})\to K_0(\Omega \Omega ^{n-1}_{[0,\infty )}\mathcal{N})$ is a retract of $K_0(\Omega _{[0,\infty )}(B^q_{[0,\infty )})^{n-1}\mathcal{N})\to K_0(\Omega (B^q_{[0,\infty )})^{n-1}\mathcal{N})$, which is an isomorphism by the induction beginning. Hence, $K_0(\Omega ^n_{[0,\infty )}\mathcal{N})\to K_0(\Omega \Omega ^{n-1}_{[0,\infty )}\mathcal{N})$ is an isomorphism as well. Using that $\Omega$ and $\Omega _{[0,\infty )}$ commute, it suffices to show that $K_0(\Omega ^{n-1}_{[0,\infty )}\Omega \mathcal{N})\to K_0(\Omega ^n\mathcal{N})$ is an isomorphism. This map is a retract of $K_0(\Omega ^{n-1}_{[0,\infty )}B^q\mathcal{N})\to K_0(\Omega ^{n-1}B^q\mathcal{N})$, which is an isomorphism by assumption.
■
From now on, we write $K_0(\Omega \mathcal{N})$ for $K_0(\Omega _{[0,\infty )}\mathcal{N})$. 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 $K_0(\Omega \mathcal{N})$.
Definition 2.5.
Let $\mathbb{P}= (P_*, d, d')$ be a binary acyclic complex and let $i \in \mathbb{N}$.
(1)
The $i$th shift$\mathbb{P}[i]$ is defined to be the binary acyclic complex with underlying graded object $P[i]_* = P_{*-i}$ and differentials$$\begin{equation*} \vcenter{\img[][166pt][27pt][{\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \begin{tikzcd} P[i]_n = P_{n-i} \ar[r, shift right, "d_{n-i}'"']\ar[r, shift left, "d_{n-i}"] &[+10pt] P_{n-i-1} = P[i]_{n-1}.\end{tikzcd}}]{Images/img3d8ecc32e71a26cf555c8d94672f525f.svg}} \end{equation*}$$
(2)
The $i$th suspension$\Sigma ^i\mathbb{P}$ is defined to be the binary acyclic complex with underlying graded object $\Sigma ^iP_* = P_{*-i}$ and differentials$$\begin{equation*} \vcenter{\img[][169pt][30pt][{\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \begin{tikzcd} \Sigma^iP_n = P_{n-i} \ar[r, shift right, "(-1)^i d_{n-i}'"']\ar[r, shift left, "(-1)^i d_{n-i}"] &[+20pt] P_{n-i-1} = \Sigma^iP_{n-1}.\end{tikzcd}}]{Images/imgfea85365c0df541ee74ba9da09759f11.svg}} \end{equation*}$$
Remark 2.6.
Our terminology is in disagreement with Reference Gra12, where the suspension is called a shift.
As for ordinary chain complexes, we have the following lemma.
The first equality holds since $\mathbb{P}[1] \cong \Sigma \mathbb{P}$. The second equality holds since $\mathbb{P}$ and $\Sigma \mathbb{P}$ fit into a short exact sequence with the cone of $\mathbb{P}$.
■Definition 2.8.
A binary double complex is a bounded bigraded object $(P_{k,l})_{k,l \in \mathbb{N}}$ in $\mathcal{N}$ together with morphisms
such that $(P_{*,*}, d^h, d^v)$ and $(P_{*,*}, d^{\prime ,h}, d^{\prime ,v})$ are double complexes in the sense that $(P_{*,l}, d^h)$ and $(P_{*,l},d^{\prime ,h})$ are chain complexes for all $l$,$(P_{k,*},d^v)$ and $(P_{k,*},d^{\prime ,v})$ are chain complexes for all $k$, and $d^hd^v =d^vd^h$, respectively, $d^{\prime ,h}d^{\prime ,v} = d^{\prime ,v}d^{\prime ,h}$.
We call $(P_{*,*}, d^h, d^v, d^{\prime ,h}, d^{\prime ,v})$ a binary acyclic double complex if $(P_{*,l}, d^h, d^{\prime ,h})$ is a binary acyclic complex for all $l$ and $(P_{k,*}, d^v, d^{\prime ,v})$ is a binary acyclic complex for all $k$.
Let $(P_{*,*}, d^h, d^v, d^{\prime ,h}, d^{\prime ,v})$ be a binary acyclic double complex. Forming the total complex of $(P_{*,*}, d^h, d^v)$ and $(P_{*,*}, d^{\prime ,h}, d^{\prime ,v})$, using the usual sign trick, produces a binary acyclic complex $\mathbb{T}$. Filtering $\mathbb{T}$ according to the horizontal (respectively, vertical) filtration of the double complexes and applying Lemma 2.7 immediately gives the following lemma.
in $K_0(\Omega _{\operatorname {supp}\mathbb{T}}\mathcal{N})$.
This relation is analogous to the relation used by Nenashev Reference Nen98 to define $K^N_1(\mathcal{N})$, hence its name.
Remark 2.10.
Specifying a binary double complex involves a sizeable amount of data. In order to write down such complexes without occupying too much space, we will follow Nenashev’s convention and depict binary double complexes by diagrams of the form
where it is understood that the left vertical morphisms commute with the top horizontal morphisms (corresponding to $d^h$ and $d^v$), and that the right vertical morphisms commute with the bottom horizontal morphisms (corresponding to $d^{\prime ,h}$ and $d^{\prime ,v}$).
Let $J$ be an object in $\mathcal{N}$ and denote by
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.
Theorem 3.1.
The canonical map $K_0(\Omega _{[0,4]}\mathcal{N})\to K_0(\Omega \mathcal{N})$ is a surjection.
Theorem 3.2.
For every $n\in \mathbb{N}$ the canonical map $K_0(\Omega _{[0,7]}^n\mathcal{N}) \to K_0(\Omega ^n\mathcal{N}) \cong K_n(\mathcal{N})$ admits a natural section.
As before, $\mathcal{N}$ denotes an exact category. The basic approach is the same as that of Harris Reference Har15, Section 2.2 in showing that the canonical map from Bass’ $K_1$ to $K_0(\Omega \mathcal{N})$ 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 Reference Hel65, Lemma 2.1. We include a proof following Reference Tho97, Lemma 2.4 for the reader’s convenience.
Definition 3.3.
Let $J,K \in \mathcal{N}$.
(1)
We call $J$ and $K$extension-equivalent if there are objects $A, B \in \mathcal{N}$ such that there exist exact sequences$$\begin{equation*} \vcenter{\img[][85pt][9pt][{\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \begin{tikzcd} A\ar[r, rightarrowtail] & J\ar[r, twoheadrightarrow] & B \end{tikzcd}}]{Images/img1150e3588a7d389293a897f65f9fe373.svg}} \quad \text{and}\quad \vcenter{\img[][90pt][9pt][{\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \begin{tikzcd} A\ar[r, rightarrowtail] & K\ar[r, twoheadrightarrow] & B. \end{tikzcd}}]{Images/imgdd6eb175b4faca1214199aab43b3e779.svg}} \end{equation*}$$
(2)
We call $J$ and $K$stably extension-equivalent if there exists an object $S \in \mathcal{N}$ such that $J \oplus S$ and $K \oplus S$ 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.
Lemma 3.4 (Heller).
Let $\mathcal{N}$ be an exact category and let $J, J', K, K' \in \mathcal{N}$.
Then $[J] - [J'] = [K] - [K'] \in K_0(\mathcal{N})$ if and only if $J \oplus K'$ and $K \oplus J'$ are stably extension-equivalent.
Proof.
Define a relation on pairs of objects in $\mathcal{N}$ by setting $(J,J') \sim (K,K')$ if and only if $J \oplus K'$ and $K \oplus J'$ are stably extension-equivalent.
We claim that $\sim$ is an equivalence relation. Reflexivity and symmetry are obvious. To see transitivity, suppose that $(J,J') \sim (K,K') \sim (L,L')$, i.e., there exist $A$,$B$,$C$,$D$,$S$,$T \in \mathcal{N}$ such that there are exact sequences
$$\begin{equation*} \vcenter{\img[][127pt][10pt][{\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \begin{tikzcd} A\ar[r, rightarrowtail] & J \oplus K' \oplus S\ar[r, twoheadrightarrow] & B \end{tikzcd}}]{Images/img8750e183bc76da89e5526c01a938b8ee.svg}} \quad \text{and}\quad \vcenter{\img[][127pt][10pt][{\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \begin{tikzcd} A\ar[r, rightarrowtail] & K \oplus J' \oplus S\ar[r, twoheadrightarrow] & B \end{tikzcd}}]{Images/imge24fbbd0f39fcf2171c56cead7672301.svg}} \end{equation*}$$
as well as
$$\begin{equation*} \vcenter{\img[][128pt][10pt][{\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \begin{tikzcd} C\ar[r, rightarrowtail] & K \oplus L' \oplus T\ar[r, twoheadrightarrow] & D \end{tikzcd}}]{Images/img0945bcf7ddbd8a0933ad3b8e04e4a5ec.svg}} \quad \text{and}\quad \vcenter{\img[][131pt][10pt][{\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \begin{tikzcd} C\ar[r, rightarrowtail] & L \oplus K' \oplus T\ar[r, twoheadrightarrow] & D. \end{tikzcd}}]{Images/imgd267f5c6fb57d8689fef96dd60ee6725.svg}} \end{equation*}$$
Then the sequences formed by taking direct sums
$$\begin{equation*} \vcenter{\img[][230pt][10pt][{\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \begin{tikzcd} A \oplus C\ar[r, rightarrowtail] & J \oplus K' \oplus S \oplus K \oplus L' \oplus T\ar[r, twoheadrightarrow] & B \oplus D, \end{tikzcd}}]{Images/img52cb0be374652561ec4c1dacefd86c38.svg}} \end{equation*}$$
$$\begin{equation*} \vcenter{\img[][231pt][10pt][{\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \begin{tikzcd} A \oplus C\ar[r, rightarrowtail] & K \oplus J' \oplus S \oplus L \oplus K' \oplus T\ar[r, twoheadrightarrow] & B \oplus D, \end{tikzcd}}]{Images/img3b9947a5ad185f9db91c1fa344177eb1.svg}} \end{equation*}$$ are exact, too. Rewriting
$$\begin{equation*} J \oplus K' \oplus S \oplus K \oplus L' \oplus T \cong J \oplus L' \oplus K \oplus K' \oplus S \oplus T \end{equation*}$$
and
$$\begin{equation*} K \oplus J' \oplus S \oplus L \oplus K' \oplus T \cong L \oplus J' \oplus K \oplus K' \oplus S \oplus T \end{equation*}$$
proves transitivity, so $\sim$ is an equivalence relation. Denote by $k(\mathcal{N})$ the set of equivalence classes in $ob\mathcal{N}\times ob\mathcal{N}$ with respect to $\sim$. We write $[J,J']$ for the class of $(J,J')$ in $k(\mathcal{N})$.
Clearly, if $(J,J')$ and $(K,K')$ are pairs of objects such that $J \cong K$ and $J' \cong K'$, then $[J,J'] = [K,K']$. Hence, the direct sum operation in $\mathcal{N}$ induces the structure of a commutative monoid on $k(\mathcal{N})$ via
are exact, it follows that $[J \oplus L, 0] = [K,0]$. Hence, the map $ob\mathcal{N}\to k(\mathcal{N}), J \mapsto [J,0]$ induces a homomorphism $\varphi \colon K_0(\mathcal{N}) \to k(\mathcal{N})$.
Note that $\varphi$ sends the class $[J] - [J'] \in K_0(\mathcal{N})$ to $\varphi ([J] - [J']) = [J,J']$, so $\varphi$ is an epimorphism. Moreover, it is immediate from the definition of $\sim$ that the kernel of $\varphi$ is trivial. This proves that $\varphi$ is an isomorphism, and the claim of the lemma follows.
■
We can now prove Theorem 3.1. Let $\mathbb{P}\coloneq (P_*, d, d')$ be a binary acyclic complex supported on $[0,m]$ for some $m\in \mathbb{N}$. Choose factorizations $d_n \colon P_n \twoheadrightarrow J_{n-1} \rightarrowtail P_{n-1}$ and $d_n' \colon P_n \twoheadrightarrow K_{n-1} \rightarrowtail P_{n-1}$ for all $n$. Since $J_n$ and $K_n$ both fit into an exact sequence with $P_{n-1},\ldots , P_0$, they represent the same class in $K_0(\mathcal{N})$. Therefore, there exist $A_n,B_n,S_n\in \mathcal{N}$ and exact sequences
We will show that $[ \mathbb{P}]=[\mathbb{S}_2]-[\mathbb{P}']$. The lemma then follows by iterating this procedure. For this notice that the complex $\mathbb{S}_2$ for $\mathbb{P}'$ is precisely the complex $\mathbb{S}_3$ for $\mathbb{P}$. 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 $\mathbb{P}, \mathbb{P}', \mathbb{S}_2$, and a fourth one explained in the diagram
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 $\mathbb{S}_k'$ and $\mathbb{S}_{k+1}'$ the same binary acyclic complexes as $\mathbb{S}_k$ and $\mathbb{S}_{k+1}$, 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 $\mathbb{S}_l$ is independent of the choice of $A_k$,$B_k$, and $S_k$ for $l \neq k, k+1$.
The binary acyclic complexes $\mathbb{S}_k' \oplus (\mathbb{S}_{k+1}[2])$ and $\mathbb{S}_k \oplus (\mathbb{S}_{k+1}'[2])$ have isomorphic underlying graded objects. We regard both as binary acyclic complexes
$$\begin{equation*} \vcenter{\img[][389pt][16pt][{\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \begin{tikzcd} J_k \oplus K_k \oplus S_k \oplus K_k \oplus S_k' \oplus J_k\ar[r, shift left]\ar[r, shift right] &[-10pt] B_k \oplus B_k' \oplus P_k \oplus A_{k-1} \ar[r, shift left]\ar[r, shift right] &[-10pt] J_{k-1} \oplus K_{k-1} \oplus S_{k-1}\ar[r, shift left]\ar[r, shift right] &[-10pt] B_{k-1}, \end{tikzcd}}]{Images/img19be9c034b974c65cf220b520df9e56f.svg}}{} \end{equation*}$$Both the pair of chain complexes given by the top differentials and the pair of chain complexes given by the bottom differentials of $\mathbb{S}_k' \oplus (\mathbb{S}_{k+1}[2])$ and $\mathbb{S}_k \oplus (\mathbb{S}_{k+1}'[2])$ 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 is $\mathbb{S}'_k \oplus (\mathbb{S}_{k+1}[2])$, whose lower row is $\mathbb{S}_k \oplus (\mathbb{S}_{k+1}'[2])$. Here all unmarked downward arrows are the identity, and $\tau _K$ and $\tau _J$ denote the automorphisms switching the two copies of $K_k$ and $J_k$, respectively,
Applying Lemma 2.9, the difference between the classes of $\mathbb{S}_k\oplus (\mathbb{S}'_{k+1}[2])$ and $\mathbb{S}'_k\oplus (\mathbb{S}_{k+1}[2])$ is therefore the same as
Therefore, $\mathbb{S}'_k\oplus (\mathbb{S}_{k+1}[2])$ and $\mathbb{S}_k\oplus (\mathbb{S}'_{k+1}[2])$ represent the same class in $K_0(\Omega _{[0,7]}\mathcal{N})$ by Lemma 2.11. In combination with Lemma 2.7, this shows
An analogous argument works for $k=2$, so the class $\sum _{n=2}^{\infty }(-1)^n[\mathbb{S}_n]$ is independent of the choices we make.
Next, we show that the map is independent of the choice of the representative $\mathbb{P}$ of $[\mathbb{P}]$. First note that if both differentials of the double complex $\mathbb{P}$ agree, then $K_n$ and $J_n$ agree and we can choose the same extension for both. In this case, both differentials for all $\mathbb{S}_n$ agree, so $[\mathbb{S}_n] = 0$ for all $n$.
It remains to see that for a short exact sequence $\mathbb{P}'\rightarrowtail \mathbb{P}\twoheadrightarrow \mathbb{P}''$ we also get short exact sequences $\mathbb{S}'_n\rightarrowtail \mathbb{S}_n\twoheadrightarrow \mathbb{S}''_n$ for all $n \geq 2$. For every $n$, we have short exact sequences $J'_n\rightarrowtail J_n\twoheadrightarrow J_n''$ and $K'_n\rightarrowtail K_n\twoheadrightarrow K''_n$. 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 Reference Qui73, Theorem 2, we have
where $\mathcal{E}\mathcal{N}$ is the exact category of exact sequences in $\mathcal{N}$. Therefore, we find short exact sequences $A'_n \rightarrowtail A_n\twoheadrightarrow A_n''$,$B'_n \rightarrowtail B_n \twoheadrightarrow B_n''$, and $S'_n \rightarrowtail S_n \twoheadrightarrow S''_n$ 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 $\mathbb{S}'_n,\mathbb{S}_n$, and $\mathbb{S}''_n$, we get the desired short exact sequence $\mathbb{S}'_n\rightarrowtail \mathbb{S}_n\twoheadrightarrow \mathbb{S}''_n$.
The map $K_0(\Omega ^n_{[0,7]}\mathcal{N})\to K_0(\Omega \Omega _{[0,7]}^{n-1}\mathcal{N})$ is a retract of $K_0(\Omega _{[0,7]}(B^q_{[0,7]})^{n-1}\mathcal{N})\to K_0(\Omega (B^q_{[0,7]})^{n-1}\mathcal{N})$ which admits a natural section by the case $n=1$. Hence $K_0(\Omega ^n_{[0,7]}\mathcal{N})\to K_0(\Omega \Omega _{[0,7]}^{n-1}\mathcal{N})$ admits a natural section as well.
Since $\Omega _{[0,7]}$ and $\Omega$ commute, it suffices to show that $K_0(\Omega _{[0,7]}^{n-1}\Omega \mathcal{N})\to K_0(\Omega ^n\mathcal{N})$ admits a natural section. But this map is a retract of the map $K_0(\Omega _{[0,7]}^{n-1}B^q\mathcal{N})\to K_0(\Omega ^{n-1}B^q\mathcal{N})$, which admits a natural section by the induction assumption.
■
4. Algebraic $K$-theory of infinite product categories
The results of Section 3 allow us to show that the comparison map $\mathbb{K}(\prod _{i\in I}\mathcal{N}_i) \to \prod _{i\in I} \mathbb{K}(\mathcal{N}_i)$ of connective $K$-theory spectra is a $\pi _*$-isomorphism.
Theorem 4.1.
For every family $\{\mathcal{N}_i\}_{i\in I}$ of exact categories and every $n\in \mathbb{N}$ the natural map
Note that the natural map $K_0(\prod _{i\in I}\mathcal{N}_i) \to \prod _{i\in I} K_0(\mathcal{N}_i)$ is clearly surjective, and that injectivity is a consequence of Lemma 3.4.
Recall that $K_n(\mathcal{N})$ is naturally isomorphic to $K_0(\Omega ^n\mathcal{N})$. Consider the following diagram, where the vertical maps are the sections from Theorem 3.2 followed by the canonical homomorphisms:
Since the natural functors $C^q_{[0,7]}(\prod _{i\in I}\mathcal{N}_i) \to \prod _{i\in I} C^q_{[0,7]}(\mathcal{N}_i)$ and $B^q_{[0,7]}(\prod _{i\in I}\mathcal{N}_i) \to \prod _{i\in I} B^q_{[0,7]}(\mathcal{N}_i)$ are isomorphisms, the middle horizontal map is an isomorphism. A diagram chase implies that the natural map $K_n(\prod _{i\in I}\mathcal{N}_i) \to \prod _{i\in I} K_n(\mathcal{N}_i)$ 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$\mathbb{K}^{-\infty }$ of an exact category is Schlichting’s delooping Reference Sch06, Section 12.
The argument to extend Theorem 4.1 to non-connective algebraic $K$-theory is based on a localization sequence of Schlichting Reference Sch04. To state it, we need to recall the following definition.
Let $\mathcal{N}$ be an exact category, and let $\mathcal{A}\subseteq \mathcal{N}$ be an extension closed full subcategory.
(1)
An admissible epimorphism $N \twoheadrightarrow A$ with $N \in \mathcal{N}$ and $A \in \mathcal{A}$ is special if there exists an admissible monomorphism $B \rightarrowtail N$ with $B \in \mathcal{A}$ such that the composition $B \to A$ is an admissible epimorphism.
(2)
The inclusion $\mathcal{A}\subseteq \mathcal{N}$ is called left $s$-filtering if the following holds:
(a)
The subcategory $\mathcal{A}$ is closed under admissible subobjects and admissible quotients in $\mathcal{N}$.
(b)
Every admissible epimorphism $N \twoheadrightarrow A$ from an object $N \in \mathcal{N}$ to an object $A \in \mathcal{A}$ is special.
(c)
For every morphism $f \colon A \to N$ with $A \in \mathcal{A}$ and $N \in \mathcal{N}$ there exists an object $B \in \mathcal{A}$, a morphism $f' \colon A \to B$, and an admissible monomorphism $i \colon B \rightarrowtail N$ such that $if' = f$.
Let $\mathcal{A}\subseteq \mathcal{N}$ be a left $s$-filtering subcategory. A weak isomorphism in $\mathcal{N}$ is a morphism which can be written as the composition of admissible monomorphisms with cokernel in $\mathcal{A}$ and admissible epimorphisms with kernel in $\mathcal{A}$. Let $\Sigma$ denote the collection of weak isomorphisms in $\mathcal{N}$. The set $\Sigma$ satisfies a calculus of left fractions Reference Sch04, Lemma 1.13, so one can form the localization $\mathcal{N}[\Sigma ^{-1}]$. The localization inherits an exact structure from $\mathcal{N}$ by declaring a sequence to be exact if it is isomorphic to the image of an exact sequence under the localization functor $\mathcal{N}\to \mathcal{N}[\Sigma ^{-1}]$Reference Sch04, Proposition 1.16. The resulting exact category is denoted $\mathcal{N}/\mathcal{A}$.
Let $\mathcal{A}$ be an idempotent complete, left $s$-filtering subcategory of the exact category $\mathcal{N}$. Then the sequence $\mathcal{A}\to \mathcal{N}\to \mathcal{N}/\mathcal{A}$ of exact functors induces a homotopy fiber sequence of spectra
Finally, recall the countable envelope$\mathcal{F}\mathcal{N}$ of an idempotent complete exact category $\mathcal{N}$Reference Sch04, Section 3 (and the references therein). The concrete definition need not concern us here. It suffices to know that $\mathcal{F}\mathcal{N}$ is an exact category which contains $\mathcal{N}$ as a left $s$-filtering subcategory, and that $\mathbb{K}^{-\infty }(\mathcal{F}\mathcal{N})$ is contractible Reference Sch04, Lemma 3.2; the latter claim holds because $\mathcal{F}\mathcal{N}$ admits countable coproducts. Moreover, $\mathcal{F}\mathcal{N}$ depends functorially on $\mathcal{N}$. Denote by $\mathcal{S}\mathcal{N}$ the quotient category $\mathcal{F}\mathcal{N}/\mathcal{N}$. The category $\mathcal{S}\mathcal{N}$ is called the suspension of $\mathcal{N}$. Write $\mathcal{S}^n\mathcal{N}$ for the $n$-fold suspension of $\mathcal{N}$. From Theorem 4.3, it follows directly that $\Omega ^n\mathbb{K}^{-\infty }(\mathcal{S}^n\mathcal{N})$ is naturally equivalent to $\mathbb{K}^{-\infty }(\mathcal{N})$. In particular, we have $K_{-n}(\mathcal{N}) \cong K_0(\operatorname {Idem}(\mathcal{S}^n\mathcal{N}))$ for all $n > 0$, where $\operatorname {Idem}(-)$ denotes the idempotent completion functor.
Let $\{ \mathcal{N}_i \}_{i \in I}$ be a family of exact categories.
Since the natural map $\mathbb{K}^{-\infty }(\mathcal{N}) \xrightarrow {\sim } \mathbb{K}^{-\infty }(\operatorname {Idem}(\mathcal{N}))$ is an equivalence for every exact category and $\operatorname {Idem}(\prod _{i \in I} \mathcal{N}_i) \cong \prod _{i \in I} \operatorname {Idem}(\mathcal{N}_i)$, we may assume that $\mathcal{N}_i$ is idempotent complete for all $i \in I$.
Consider the left $s$-filtering inclusion $\prod _{i \in I} \mathcal{N}_i \subseteq \mathcal{F}(\prod _{i \in I} \mathcal{N}_i)$. The various projection functors $\prod _{i \in I} \mathcal{N}_i \to \mathcal{N}_j$ induce an exact functor $\mathcal{F}(\prod _{i \in I} \mathcal{N}_i) \to \prod _{i \in I} \mathcal{F}\mathcal{N}_i$. Moreover, the inclusion $\prod _{i \in I} \mathcal{N}_i \subset \prod _{i \in I} \mathcal{F}\mathcal{N}_i$ is left $s$-filtering since it is left $s$-filtering on each factor. Since $\mathcal{S}\mathcal{N}_i$ is obtained from $\mathcal{F}\mathcal{N}_i$ by a calculus of left fractions, we can identify $\prod _{i \in I} \mathcal{F}\mathcal{N}_i / \prod _{i \in I} \mathcal{N}_i \cong \prod _{i \in I} \mathcal{S}\mathcal{N}_i$ (for an explicit description of the morphism sets making this obvious, cf. Reference GZ67, Chapter I, Section 2.2). Therefore, we have by Theorem 4.3 a map of homotopy fiber sequences of spectra:
Since both $\mathcal{F}(\prod _{i \in I} \mathcal{N}_i)$ and $\prod _{i \in I} \mathcal{F}\mathcal{N}_i$ admit countable coproducts, the $K$-theory of both vanishes and the middle vertical arrow is a $\pi _*$-isomorphism. Hence, the right vertical map is a $\pi _*$-isomorphism. By induction, it follows that the canonical map
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.
5. The relation to Nenashev’s $K_1$
The abelian group $K_0(\Omega \mathcal{N})$ is not the first algebraic description of $K_1$ of an exact category. Nenashev gave the following description of $K_1(\mathcal{N})$.
Definition 5.1.
Define $K_1^N(\mathcal{N})$ as the abelian group generated by binary acyclic complexes of length two
The main result of Reference Nen98 states that $K_1^N(\mathcal{N})$ is isomorphic to $K_1(\mathcal{N})$. By Lemma 2.9, regarding a binary acyclic complex of length two as a class in $K_0(\Omega \mathcal{N})$ defines a natural homomorphism
as already remarked in the introduction. In this section, we prove Theorem 1.1. Before doing so, we give the following corollary.
Corollary 5.2.
For all $n\geq 1$, the homomorphism $K_0(\Omega _{[0,2]}^n\mathcal{N}) \to K_0(\Omega ^n\mathcal{N})$ is a surjection and the homomorphism $K_0(\Omega _{[0,4]}^n\mathcal{N}) \to K_0(\Omega ^n\mathcal{N})$ admits a natural section.
Proof.
By Theorem 1.1, $\Phi$ is an isomorphism. Since $K_0(\Omega _{[0,2]}\mathcal{N}) \to K_1^N(\mathcal{N})$ is a surjection, so is $K_0(\Omega _{[0,2]}\mathcal{N})\to K_0(\Omega \mathcal{N})$. By Lemma 2.9, $\Phi$ factors as
This exhibits $K_0(\Omega \mathcal{N})$ as a natural retract of $K_0(\Omega _{[0,4]}\mathcal{N})$. 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 $K_1^N(\mathcal{N})$, which we will need later in the argument.
Lemma 5.3.
For any binary acyclic complex of length two, we have
Let $\mathbb{P}\coloneq (P_*, d, d')$ be a binary acyclic complex. In a first step we will not shorten $\mathbb{P}$ but produce a complex $\widehat{\mathbb{P}}$ representing the same class in $K_0(\Omega \mathcal{N})$, which we will then be able to shorten.
Choose factorizations $d_2 \colon P_2 \twoheadrightarrow J \rightarrowtail P_1$ and $d_2' \colon P_2 \twoheadrightarrow K \rightarrowtail P_{1}$. Since $J$ and $K$ both are the kernel of an admissible epimorphism $P_1 \twoheadrightarrow P_0$, they represent the same class in $K_0(\mathcal{N})$. Therefore, there exist by Lemma 3.4$A,B,S\in \mathcal{N}$, and exact sequences
$$\begin{equation*} \vcenter{\img[][104pt][9pt][{\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \begin{tikzcd} A\ar[r, rightarrowtail] & J \oplus S\ar[r, twoheadrightarrow] & B \end{tikzcd}}]{Images/imgaf596e112fea759255df1ca967133163.svg}} \quad \text{and}\quad \vcenter{\img[][108pt][9pt][{\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \begin{tikzcd} A\ar[r, rightarrowtail] & K \oplus S\ar[r, twoheadrightarrow] & B. \end{tikzcd}}]{Images/img61fd88474c5d37b662b4726204197f62.svg}} \end{equation*}$$
Let $\mathbb{S}$ denote the binary acyclic complex
Note that $[\Delta _M]=0\in K_0(\Omega \mathcal{N})$.
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 $\mathbb{P}\oplus \Delta _B, \mathbb{P}'$, and $\mathbb{S}$
We can build the following binary acyclic double complex involving $\widehat{\mathbb{P}}\oplus \Delta _B, \mathbb{P}'\oplus \Delta _J[1]\oplus \Delta _K[1]$, and a third non-trivial complex.
The non-trivial vertical complex differs from $\Delta _J[1]\oplus \mathbb{S}\oplus \Delta _K[1]$ by identifying $J\oplus P_1\oplus K\oplus B \cong J\oplus B\oplus P_1 \oplus K$ and using $\tau _K$ and $\tau _J$ in the bottom differential. Since $K$ and $J$ represent the same class in $K_0$, we have
Let us fix the following notation: If $M$ is an object containing $N$ as a direct summand, denote by $e_N$ the obvious idempotent $M \to M$ whose image is $N$.
Consider the following double complex involving $\widehat{\mathbb{P}}$ and $\mathbb{Q}[1]$. Note that only the rows are acyclic. This suffices to see that the total complex shifted down by one represents the same class as $[\mathbb{Q}]+[\widehat{\mathbb{P}}]$
Assume that $\mathbb{P}$ was supported on $[0,m]$, then $\mathbb{T}$ admits a projection onto $\Delta _{P_m}[m\!-\!1]$. The kernel of this projection admits a projection to $\Delta _{P_{m-1}}[m-2]$ and so on until we take the kernel of the projection to $\Delta _{P_2}[1]$. The remaining acyclic binary complex $\mathbb{T}'$ is
It follows that $[\mathbb{P}]=[\mathbb{T}']-[\mathbb{Q}]$. Since $\mathbb{T}'$ is supported on $[0,2]$ and $\mathbb{Q}$ has length one shorter than $\mathbb{P}$, iterating this argument already shows that $K_0(\Omega _{[0,2]}\mathcal{N})\to K_0(\Omega \mathcal{N})$ is surjective.
Remark 5.5.
The idea to use the complexes $\widehat{\mathbb{P}},\mathbb{T}$, and $\mathbb{T}'$ is from the aforementioned, unpublished result of Grayson. He uses a different argument to show that $[\widehat{\mathbb{P}}]-[\mathbb{P}]$ is contained in the image of $\Phi$. 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 $\mathbb{T}'$. Let $\mathbb{T}_{triv}'$ denote the binary acyclic complex whose underlying graded object is that of $\mathbb{T}'$, but with both differentials equal to the top differential of $\mathbb{T}'$. Then the following diagram, where the upper row is $\mathbb{T}'$ and the second row is $\mathbb{T}'_{triv}$, commutes:
Both differentials of $\mathbb{T}'_{triv}$ agree and thus it represents the trivial class. Since $\tau _K$ and $\tau _J$ are of order two, we conclude from Lemma 2.9 that
Since $J\rightarrowtail P_1\twoheadrightarrow P_0$ is exact, this is the same as $[\vcenter{\img[][79pt][22pt][{\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \begin{tikzcd} J\oplus J\ar[r, shift left, "\id"]\ar[r, shift right, "\tau_{J}"']&J\oplus J \end{tikzcd}}]{Images/img620a8357d1389d69d2a8ab1a8d59b623.svg}}]$. This shows that
We are now going to iterate this argument. Choose factorizations $d_n \colon P_n \twoheadrightarrow J_{n-1} \rightarrowtail P_{n-1}$ and $d_n' \colon P_n \twoheadrightarrow K_{n-1} \rightarrowtail P_{n-1}$ for all $n \geq 2$ such that $J_n \rightarrowtail P_n \twoheadrightarrow J_{n-1}$ and $K_n \rightarrowtail P_n \twoheadrightarrow K_{n-1}$ are exact for all $n$. Set $J_0 \coloneq P_0$ and $K_0 \coloneq P_0$. For any natural number $k$, fix the following auxiliary notation:
Note that $\mathbb{P}_1$ is precisely the complex $\mathbb{Q}$ appearing in Equation 5.6. Moreover, if $k$ is sufficiently large so that $P_n \cong 0$ for all $n > k$, then $\mathbb{P}_{k+1}$ is obtained from $\mathbb{P}_k$ by interchanging the top and bottom differential.
For every $k$ let $\mathbb{Q}_k$ denote the complex obtained from $\mathbb{P}_k$ by the same procedure as $\mathbb{Q}$ is obtained from $\mathbb{P}$.
Suppose now that $k$ is odd. Substituting appropriately in Equation 5.6, we obtain the equation
$$\begin{equation*} Y \coloneq K_{k+1} \oplus \bigoplus _{n=1}^k(J_n \oplus K_n) \end{equation*}$$
is the kernel of the first bottom differential of $\mathbb{P}_k$. 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 $\mathbb{Q}_k$, we see that, up to automorphisms flipping the two copies of $Z$ in the three lowest degrees of $\mathbb{Q}_k$,$\mathbb{Q}_k$ coincides with the sum of $\mathbb{P}_{k+1}$ with some complexes in the image of the diagonal functor $\Delta$. Since, by Equation 5.7, $[ \vcenter{\img[][86pt][22pt][{\renewcommand{\arraystretch}{1} \setlength{\unitlength}{1.0pt} \begin{tikzcd} Z\oplus Z\ar[r, shift left, "\id"]\ar[r, shift right, "\tau_{Z}"']&Z\oplus Z \end{tikzcd}}]{Images/img8017ad54447469f580e22c76e46de458.svg}} ] = 0$, we see that $[\mathbb{Q}_k] = [\mathbb{P}_{k+1}]$. Hence,
$$\begin{equation*} k(\mathbb{P}) \coloneq \min \{ n \in \mathbb{N}\mid P_{n'} \cong 0\ \text{for all}\ n' > n \}. \end{equation*}$$
We have to show that this is a well-defined homomorphism. By our definition of $k(\mathbb{P})$, the complex $\mathbb{P}_{k(\mathbb{P})}$ has length two.
Let $\mathbb{P}' \rightarrowtail \mathbb{P}\twoheadrightarrow \mathbb{P}''$ be an exact sequence of binary acyclic complexes. Evidently, $x(\mathbb{P}',k(\mathbb{P})) + x(\mathbb{P}'',k(\mathbb{P})) = x(\mathbb{P},k(\mathbb{P}))$. Note that $k(\mathbb{P}'), k(\mathbb{P}'') \leq k(\mathbb{P})$ and at least one of $k(\mathbb{P}')$ and $k(\mathbb{P}'')$ equals $k(\mathbb{P})$. If $k(\mathbb{P}') = k(\mathbb{P}) = k(\mathbb{P}'')$, we already have $\Psi ([\mathbb{P}']) + \Psi ([\mathbb{P}'']) = \Psi ([\mathbb{P}])$. Suppose $k(\mathbb{P}') < k(\mathbb{P})$. Then $\mathbb{P}_{k(\mathbb{P})}$ arises from $\mathbb{P}_{k(\mathbb{P}')}$ by interchanging the role of top and bottom differential $k(\mathbb{P}) - k(\mathbb{P}')$ times. Since interchanging the top and bottom differential results only in a change of sign (Lemma 5.3) and $J'_n \cong 0$ for $n > k(\mathbb{P}')$, we have $x(\mathbb{P}',k(\mathbb{P}')) = x(\mathbb{P}',k(\mathbb{P}))$. The case $k(\mathbb{P}'') < k(\mathbb{P})$ is analogous.
Suppose now that $\mathbb{P}$ lies in the image of the diagonal functor $\Delta \colon C^q\mathcal{N}\to B^q\mathcal{N}$. Then we may choose $J_n = K_n$ for all $n$. In this case, the top and bottom differential of $\mathbb{P}_{k(\mathbb{P})}$ 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 $\mathbb{P}_{k(\mathbb{P})}$, applying the Nenashev relation we see that
Consequently, $\Psi ([\mathbb{P}]) = 0$ by Lemma 2.11. This shows that $\Psi$ is a well-defined homomorphism $K_0(\Omega \mathcal{N}) \to K_1^N(\mathcal{N})$.
Our previous discussion implies that $\Phi \circ \Psi = \operatorname {id}_{K_0(\Omega \mathcal{N})}$. What is left to do is to show that $\Psi \circ \Phi = \operatorname {id}_{K_1^N(\mathcal{N})}$. To do so, it suffices to establish equation Equation 5.6 in $K_1^N(\mathcal{N})$ for all binary acyclic complexes of length two. Let
Consider the following binary acyclic double complex where the upper row is $\mathbb{P}_1$, the lower row is $\mathbb{P}$ with switched differentials plus $\Delta _{P_2}\oplus \Delta _{P_2}\oplus \Delta _{P_2}[1]$, and the middle vertical map $\tau _{P_2}$ denotes the flip of the second and third summand:
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.
■
Acknowledgments
The authors are indebted to Daniel Grayson for sharing with us his proof of surjectivity of $\Phi$. We thank Robin Loose for helpful discussions and Bernhard Köck for useful comments on a previous version.
is a $\pi _*$-isomorphism. Here $\mathbb{K}^{-\infty }$ denotes non-connective algebraic $K$-theory.
Theorem 1.3.
The canonical map $K_0(\Omega ^n_{[0,2]}\mathcal{N})\to K_0(\Omega ^n\mathcal{N})$ is a surjection.
Theorem 1.4.
For every $n\in \mathbb{N}$ the canonical map $K_0(\Omega _{[0,3]}^n\mathcal{N}) \to K_0(\Omega ^n\mathcal{N}) \cong K_n(\mathcal{N})$ admits a natural section.
in $K_0(\Omega _{\operatorname {supp}\mathbb{T}}\mathcal{N})$.
Remark 2.10.
Specifying a binary double complex involves a sizeable amount of data. In order to write down such complexes without occupying too much space, we will follow Nenashev’s convention and depict binary double complexes by diagrams of the form
where it is understood that the left vertical morphisms commute with the top horizontal morphisms (corresponding to $d^h$ and $d^v$), and that the right vertical morphisms commute with the bottom horizontal morphisms (corresponding to $d^{\prime ,h}$ and $d^{\prime ,v}$).
The canonical map $K_0(\Omega _{[0,4]}\mathcal{N})\to K_0(\Omega \mathcal{N})$ is a surjection.
Theorem 3.2.
For every $n\in \mathbb{N}$ the canonical map $K_0(\Omega _{[0,7]}^n\mathcal{N}) \to K_0(\Omega ^n\mathcal{N}) \cong K_n(\mathcal{N})$ admits a natural section.
Lemma 3.4 (Heller).
Let $\mathcal{N}$ be an exact category and let $J, J', K, K' \in \mathcal{N}$.
Then $[J] - [J'] = [K] - [K'] \in K_0(\mathcal{N})$ if and only if $J \oplus K'$ and $K \oplus J'$ are stably extension-equivalent.
Let $\mathcal{A}$ be an idempotent complete, left $s$-filtering subcategory of the exact category $\mathcal{N}$. Then the sequence $\mathcal{A}\to \mathcal{N}\to \mathcal{N}/\mathcal{A}$ of exact functors induces a homotopy fiber sequence of spectra
The idea to use the complexes $\widehat{\mathbb{P}},\mathbb{T}$, and $\mathbb{T}'$ is from the aforementioned, unpublished result of Grayson. He uses a different argument to show that $[\widehat{\mathbb{P}}]-[\mathbb{P}]$ is contained in the image of $\Phi$. Grayson’s argument avoids the use of Heller’s lemma, but the computation of the appearing correction terms is more complicated.
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Both authors were funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy - GZ 2047/1, Projekt-ID 390685813.
The second author was furthermore supported by the Max Planck Society and Wolfgang Lück’s ERC Advanced Grant “KL2MG-interactions” (no. 662400).
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author={Kasprowski, Daniel},
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