#  $\mathbb{A}^{1}$-invariance of localizing invariants

Vladimir Sosnilo  M309
Universität Regensburg
Universitätsstraße 31
93053 Regensburg
Germany<EMAIL_ADDRESS>
###### Abstract.

Weibel proved that $p$-inverted K-theory is $\mathbb{A}^{1}$-invariant on
$\mathbb{F}_{p}$-schemes and K-theory with $\mathbb{Z}/p$-coefficients is
$\mathbb{A}^{1}$-invariant on $\mathbb{Z}[\frac{1}{p}]$-schemes. We extend
this result to all finitary localizing invariants of stable
$\infty$-categories. Along the way we study the Frobenius and Verschiebung
endofunctors defined by Tabuada and give an easy proof of Stienstra’s
projection formula.

###### Contents

  1. 1 Reminder on noncommutative motives and localizing invariants
  2. 2 Endomorphism categories
  3. 3 Nilpotent endomorphisms obstruct $\mathbb{A}^{1}$-invariance
  4. 4 Witt vectors
  5. 5 Torsion in Nil-theories

## Introduction

In the very foundation of motivic homotopy theory lies the idea of enforcing
the affine line to be contractible, hence imposing the
$\mathbb{A}^{1}$-invariance property111also called homotopy invariance on all
cohomology theories of one’s interest. Concretely, a cohomology theory222i.e.
a contravariant functor from the category of schemes $E$ is said to be
$\mathbb{A}^{1}$-invariant if the projection map induces an equivalence

$E(X)\simeq E(X\times\mathbb{A}^{1})$

for all schemes $X$. This is not as restrictive as it sounds at first, because
many important cohomology theories satisfy $\mathbb{A}^{1}$-property
automatically on regular schemes. For example, this is true for K-theory, Chow
groups, Morel-Levine’s algebraic cobordism in characteristic 0 (see [LM07]).

However, in general one is also interested in studying singular schemes. In
this context K-theory is known not to be $\mathbb{A}^{1}$-invariant—moreover,
conjecturally $\mathbb{A}^{1}$-invariance of K-theory on a scheme detects its
regularity ([Vor79, Conjecture], [GH10]). As another example, the derived
algebraic cobordism of Annala is not $\mathbb{A}^{1}$-invariant (see [Ann22,
Theorem 236]). This motivates the quest for non-$\mathbb{A}^{1}$-invariant
motivic homotopy theory.

###  0.1. Motivic theories

One approach is to replace $\mathbb{A}^{1}$-invariance with the property of
satisfying the projective bundle formula. This is done via pbf-local sheaves
with transfers and pbf-local $\mathbb{P}^{1}$-spectra in the recent works of
Annala and Iwasa (see [AI22] and [AI22]). Another more naïve approach is to
study localizing invariants (aka noncommutative motives) whose definition does
not refer to any geometric properties at all but is nevertheless sensible
enough to reflect many of them. Recall that a localizing invariant is a
functor $E$ from the $\infty$-category of small idempotent compelte stable
$\infty$-categories $\mathrm{Cat}^{\mathrm{perf}}_{\infty}$ to spectra
$\mathrm{Spt}$ satisfying:

  * •

for any Verdier sequence $\mathcal{A}\to\mathcal{B}\to\mathcal{C}$

$E(\mathcal{A})\to E(\mathcal{B})\to E(\mathcal{C})$

is a fiber sequence.

We say that a localizing invariant $E$ is finitary if it preserves filtered
colimits333In [BGT13] this property was included in the definition of a
localizing invariant.. Any localizing invariant gives rise to a pbf-local
$\mathbb{P}^{1}$-spectrum. Given a commutative ring $k$ we will also consider
localizing invariants of $k$-linear stable $\infty$-categories. We call them
$k$-localizing invariants.

Many “motivic” properties are shared by pbf-local $\mathbb{P}^{1}$-spectra and
localizing invariants. To list a few, the Nisnevich descent [BKRS22, Corollary
4.1.2], the pro-cdh-descent for noetherian schemes [LT19, Theorem A.8],
[BKRS22, Theorem C] hold for all localizing invariants. In the setting of pbf-
local $\mathbb{P}^{1}$-spectra the motives of Grassmanians
$\mathrm{Gr}_{n}(S)$ and classifying stacks of vector bundles
$\mathcal{V}ect_{n,S}$ can be computed as in the $\mathbb{A}^{1}$-invariant
theory [AI22, Corollary 4.2.5]. Annala and Iwasa also prove that K-theory can
be obtained via a direct analogue of Snaith’s construction [AI22, Theorem
0.1.1] (the $\mathbb{A}^{1}$-invariant analogue was previously proved in
[GS09]).

###  0.2. Back to $\mathbb{A}^{1}$-invariance

In many examples we have some control over the failure of
$\mathbb{A}^{1}$-invariance. In particular, this is illustrated by the
following result of Weibel:

######  Theorem 0.3 ([Wei81, Corollary 3.3]).

$K(-)[\frac{1}{p}]$ is $\mathbb{A}^{1}$-invariant on $\mathbb{F}_{p}$-schemes.
Moreover, $K(-)/l$ is $\mathbb{A}^{1}$-invariant on
$\mathbb{Z}[\frac{1}{l}]$-schemes.

Recently, Elmanto and Morrow have constructed a non-$\mathbb{A}^{1}$-invariant
version of motivic cohomology over schemes of characteristic $p$ (see [EM22]).
It is a well-behaved extension of the usual usual (meaning the one defined in
[Spi18]) $\mathbb{A}^{1}$-invariant motivic cohomology to non-regular schemes.
The two theories coincide after inverting $p$, so the direct analogue of the
above theorem holds. It follows from [Ann21, Theorem 4.22] that the
homological version of the derived algebraic cobordism is also
$\mathbb{A}^{1}$-invariant away from the characteristic. So, it is not
unreasonable to expect a similar theorem to hold even for a larger class of
non-$\mathbb{A}^{1}$-invariant motivic theories. We do not know whether this
is true for pbf-local $\mathbb{P}^{1}$-spectra of Annala-Iwasa under some
assumptions, or for the derived algebraic cobordism. However, we prove that
this is always true for all finitary localizing invariants:

######  Theorem 0.4 (Theorem 5.5).

Let $k$ be a connective ring spectrum with $\pi_{0}(k)$ of characteristic $p$
and $E$ a $k$-localizing invariant which is finitary. Then $E(-)[\frac{1}{p}]$
is $\mathbb{A}^{1}$-invariant.

We also generalize the other part of Weibel’s result:

######  Theorem 0.5 (Theorem 5.9).

Let $k$ be a connective $\mathbb{S}[\frac{1}{l}]$-algebra spectrum and $E$ a
finitary $k$-localizing invariant. Then $E(-)/l$ is
$\mathbb{A}^{1}$-invariant.

In particular, this implies that Theorem 0.3 holds for all spectral schemes.
In the context of dg-categories for $E$ being the functor of nonconnective
K-theory these theorems have been proved in [Tab17]. We note that that the
assumption of being finitary is necessary: for example, it follows from
[Elm21, Theorem 3.1.1] that $TC\otimes\mathbb{Q}(-)$ is not
$\mathbb{A}^{1}$-invariant.

###  0.6. Outline of the proof

We follow the same steps as the original Weibel’s proof [Wei81]. Given a
$k$-localizing invariant define

$NE(\mathcal{C}):=\operatorname{Fib}(E(\mathcal{C}\otimes_{k}\mathbf{Perf}_{k[t]})\to
E(\mathcal{C}))$

Similar to how it is done in [Tab17], we establish an action of the ring of
rational Witt vectors $\mathbb{W}_{0}(k)$ on the homotopy groups of
$NE(\mathcal{C})$. The main ingredients going into defining this action are
the Almkvist’s isomorphism [Alm74]

$\mathbb{Z}\oplus\mathbb{W}_{0}(k)\simeq
K_{0}(\mathcal{E}nd(\mathbf{Perf}_{k}))$

and an explicit isomorphism

$\pi_{i}E(\mathcal{C})\oplus\pi_{i+1}NE(\mathcal{C})\simeq\pi_{i}E(\mathcal{N}il(\mathcal{C})).$

Here $\mathcal{E}nd(\mathcal{C})$ is the stable $\infty$-category of
endomorphisms of $\mathcal{C}$ and $\mathcal{N}il(\mathcal{C})$ is the
subcategory of nilpotent endomorphisms. Having these, the action is induced by
the tensor product functor

$\mathcal{E}nd(\mathbf{Perf}_{k})\times\mathcal{N}il(\mathcal{C})\to\mathcal{N}il(\mathcal{C}).$

Our main results then follow from a thorough analysis of the action, granted
by the following steps:

  1. (1)

There are Frobenius and Verschiebung endofunctors
$\mathcal{F}_{n},\mathcal{V}_{n}$ on $\mathcal{N}il(\mathcal{C})$ as well as
$\mathcal{E}nd(\mathcal{C})$ (see [BGT15, Section 5]). We provide a different
way of defining them, which allows to show that they are nicely compatible
with the ring structure of $\mathbb{W}_{0}(k)$ and satisfy the projection
formula with respect to the action of $\mathbb{W}_{0}(k)$ on $NE(\mathcal{C})$
(see Lemma 4.12). For $\mathcal{C}=\mathbf{Perf}_{k}$ they induce classical
Frobenius and Verschiebung operations on
$K_{0}(\mathcal{E}nd(\mathbf{Perf}_{k}))\simeq\mathbb{Z}\oplus\mathbb{W}(\pi_{0}(k))$.
This part allows to represent the action of $p\in\mathbb{W}_{0}(k)$ on
$NE(\mathcal{C})$ by an explicit endofunctor of $\mathcal{N}il(\mathcal{C})$.

  2. (2)

The action of $\mathbb{W}_{0}(k)$ on the (discrete) abelian group
$\pi_{i}E(\mathcal{N}il(\mathcal{C}))$ is continuous. For $E=K$ this follows
from the construction of K-theory. In general, we introduce a filtration

$\cdots\to\mathcal{N}il^{\leqslant n}(\mathcal{C})\to\mathcal{N}il^{\leqslant
n+1}(\mathcal{C})\to\cdots$

of $\mathcal{N}il(\mathcal{C})$ such that

$\mathop{\mathchoice{\vtop{\halign{#\cr\hfil$\displaystyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\textstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptscriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
<EMAIL_ADDRESS>n}(\mathcal{C})\simeq\mathcal{N}il(\mathcal{C})$

and so that the action is continuous on the images of all
$\pi_{i}E(\mathcal{N}il^{\leqslant n}(\mathcal{C}))$ by construction (see
(5.1) and (5.3.1)). This allows to extend the action of $\mathbb{W}_{0}(k)$ to
an action of the full ring of Witt vectors $\mathbb{W}(k)$ and deduce the main
results from its properties.

The constructions in each of the steps are interesting on their own and appear
to be completely new.

###  0.7. Structure of the paper

In Section 1 we remind the reader all they need to know about localizing
invariants. In Section 2 we introduce the $\infty$-categories of endomorphisms
and of nilpotent endomorphisms, crucial for the main result. In Section 3 we
study nil-theories $NE(-)$ that measure the failure of
$\mathbb{A}^{1}$-invariance of $E$. We show that $NE(\mathcal{C})$ can be
recovered from the values of $E$ on the $\infty$-category of nilpotent
endomorphisms in $\mathcal{C}$ for any localizing invariant $E$. In Section 4
we remind the remind classical results about the ring of big Witt vectors. We
then study the Frobenius and Verschiebung endofunctors of the
$\infty$-categories of endomorphisms and relate them to the classical
Frobenius and Verschiebung operations on the ring of Witt vectors. Finally, in
Section 5 we prove the main result.

### Notations & Conventions

  * •

Throughout the paper we fix a connective commutative ring spectrum $k$.

  * •

All spectral $k$-schemes are assumed quasi-compact and quasi-separated. They
form a category $\mathrm{Sch}_{k}$.

  * •

For an $\infty$-category $\mathcal{C}$ we denote by $\mathcal{C}^{\simeq}$ its
maximal groupoid and by $\mathcal{C}^{\omega}$ the subcategory of compact
objects.

  * •

For $\infty$-categories $\mathcal{C},\mathcal{D}$,
$\mathrm{Fun}(\mathcal{C},\mathcal{D})$ is the $\infty$-category of functors
between them. If $\mathcal{C}$ and $\mathcal{D}$ admit finite colimits
$\mathrm{Fun}^{\mathrm{ex}}(\mathcal{C},\mathcal{D})$ is the subcategory of
those functors that preserve finite colimits. If $\mathcal{C}$,$\mathcal{D}$
are also $k$-linear for some commutative ring spectrum $k$, we denote by
$\mathrm{Fun}^{\mathrm{ex}}_{k}(\mathcal{C},\mathcal{D})$ the
$\infty$-category of $k$-linear functors that preserve finite colimits.

  * •

Given a small stable $\infty$-category $\mathcal{C}$ we use the following
notation for the colimit-completion:

$\widehat{\mathcal{C}}:=\mathrm{Fun}_{ex}(\mathcal{C}^{\mathrm{op}},\mathrm{Spt}).$

This is a stable (presentable) $\infty$-category and it admits a fully
faithful Yoneda embedding $\mathcal{C}\to\widehat{\mathcal{C}}$.

  * •

For a stable $\infty$-category $\mathcal{C}$ and objects $x,y\in\mathcal{C}$
we denote by

$\mathrm{maps}_{\mathcal{C}}(x,y)$

the mapping spectrum in $\mathcal{C}$, while we keeping the notation

$\operatorname{\operatorname{Maps}}_{\mathcal{C}}(x,y)$

for the mapping space.

  * •

Given an $\mathbb{E}_{1}$-ring $R$, $\mathbf{Mod}_{R}$ is the
$\infty$-category of right $R$-modules, $\mathbf{Perf}_{R}$ is the subcategory
of perfect complexes.

  * •

We denote the category of $k$-linear idempotent complete stable
$\infty$-categories by $\mathrm{Cat}^{\mathrm{perf}}_{\infty,k}$.

### Acknowledgements

The results of the paper emerged after several busy conversations with Ryomei
Iwasa, Marc Hoyois and Niklas Kipp. The author thanks them for sharing their
ideas, wishes and insights. Toni Annala pointed out to the author that the
Frobenius and Verschiebung endofunctors $\mathcal{F}_{n}$, $\mathcal{V}_{n}$
are induced by the pullback and pushforward functors along the maps
$k[t]\stackrel{{\scriptstyle t\mapsto t^{n}}}{{\longrightarrow}}k[t]$. The
author also wants to thank Denis-Charles Cisinski for a discussion about non-
finitary localizing invariants. The author gladly acknowledges that he is a
member of the SFB 1085 “Higher Invariants” funded by the Deutsche
Forschungsgesellschaft (DFG).

##  1\. Reminder on noncommutative motives and localizing invariants

In this section we recall some basic notions and facts from the theory of
localizing invariants. We only discuss those results that are relevant for
this paper. For a general theory of localizing invariants we refer to [BGT13].
We start from definitons:

######  Definition 1.1.

We say that a diagram in $\mathrm{Cat}^{\mathrm{perf}}_{\infty,k}$

$\mathcal{A}\stackrel{{\scriptstyle
f}}{{\to}}\mathcal{B}\stackrel{{\scriptstyle g}}{{\to}}\mathcal{C}$

is an exact sequence of $k$-linear idempotent complete stable
$\infty$-categories if $f$ is fully faithful, the composite $g\circ f$ is
trivial and the induced functor

$\mathcal{B}/\mathcal{A}\to\mathcal{C}$

is an equivalence up to idempotent completion.

######  Definition 1.2.

A functor

$E:\mathrm{Cat}^{\mathrm{perf}}_{\infty,k}\to\mathrm{Spt}$

is said to be a $k$-localizing invariant if for any exact sequence
$\mathcal{A}\to\mathcal{B}\to\mathcal{C}$ of $k$-linear small idempotent
complete stable $\infty$-categories the sequence

$E(\mathcal{A})\to E(\mathcal{B})\to E(\mathcal{C})$

is a fiber sequence.

#### 1.2.1.

Any $k$-localizing invariant gives rise to a cohomology theory on all
$k$-schemes via assigment

$X\mapsto E(\mathbf{Perf}_{X})$

for any $X\in\mathrm{Sch}_{k}.$ We abuse the notation and write

$E(X):=E(\mathbf{Perf}_{X}).$

We also write

$E(R):=E(\mathrm{Spec}(R))$

when $R$ is a $k$-algebra.

#### 1.2.2.

One of the most useful elementary properties of localizing invariants is the
following result, sometimes called the additivity theorem. For a localizing
invariant $E$ and a fiber sequence

$f\to g\to h$

in $\mathrm{Fun}^{\operatorname{ex}}_{k}(\mathcal{C},\mathcal{D})$ for some
$k$-linear idempotent complete stable $\infty$-categories
$\mathcal{C},\mathcal{D}$ on has an equivalence of maps of spectra

$E(g)\simeq E(f)+E(h):E(\mathcal{C})\to E(\mathcal{D}).$

######  Example 1.3.

K-theory, topological Hochschild homology, topological Cyclic homology, TR,
Blanc’s topological K-theory, Selmer K-theory are $k$-localizing invariants
for any $k$. Homotopy K-theory is a $k$-localizing invariant if $k$ is a
$\mathbb{Z}$-algebra.

######  Proposition 1.4.

For any scheme $X$ the pullback along the projection
$\mathbb{P}^{1}_{X}\stackrel{{\scriptstyle p}}{{\to}}X$ and the pushforward
along any section $X\stackrel{{\scriptstyle i}}{{\to}}\mathbb{P}^{1}_{X}$
induce an equivalence

$E(\mathbf{Perf}_{X})\oplus
E(\mathbf{Perf}_{X})\stackrel{{\scriptstyle\begin{pmatrix}i_{*}&p^{*}\end{pmatrix}}}{{\simeq}}E(\mathbf{Perf}_{\mathbb{P}^{1}_{X}})$

###### Proof.

Follows from [Kha18, Theorem B]. ∎

######  Example 1.5.

Motivic cohomology of a fixed weight $m$ does not extend to a $k$-localizing
invariant for any $k$ because there is no equivalence as in Proposition 1.4.

######  Proposition 1.6.

Let $E$ be a $k$-localizing invariant $E$. Then the square of spectra

${E(\mathbb{P}^{1}_{k})}$${E(\mathbb{A}^{1}_{k})}$${E(\mathbb{A}^{1}_{k})}$${E(\mathbb{G}_{m,k})}$

is cartesian.

###### Proof.

The diagram in question is obtained via applying $E$ to the Nisnevich square

${\mathbb{G}_{m,k}}$${\mathbb{A}^{1}_{k}}$${\mathbb{A}^{1}_{k}}$${\mathbb{P}^{1}_{k},}$

so the result follows from the étale excision ([BKRS22, Theorem 4.1.1], see
also [TT90, Proposition 3.19]). ∎

### 1.7.

Recall that $\mathrm{Cat}^{\mathrm{perf}}_{\infty,k}$ is symmetric monoidal in
such a way that $\mathcal{C}\otimes_{k}\mathcal{D}$ corepresents functors
$\mathcal{C}\times\mathcal{D}$ that are $k$-linear and exact in each variable.
One can also see that $-\otimes_{k}\mathcal{D}$ left adjoint to
$\mathrm{Fun}^{\mathrm{ex}}_{k}(\mathcal{D},-)$ (see [Lur17, Theorem 4.5.2.1]
and [Lur17, 4.8.1]). In fact, one can also explicitly compute
$\mathcal{C}\otimes_{k}\mathcal{D}$ as the compact objects in
$\mathrm{Fun}^{\mathrm{ex}}(\mathcal{C}^{\mathrm{op}},\widehat{\mathcal{D}})$
(see [HSS17, Corollary 4.11]).

######  Lemma 1.8.

For any $k$-linear idempotent complete stable $\infty$-category $\mathcal{C}$

$-\otimes_{k}\mathcal{C}:\mathrm{Cat}^{\mathrm{perf}}_{\infty,k}\to\mathrm{Cat}^{\mathrm{perf}}_{\infty,k}$

preserves exact sequences and colimits.

###### Proof.

See Lemma 9.35 of [BGT13]. ∎

####  1.8.1. Non-commutative motives

Tabuada, along with Bloomberg, Cisinski and Gepner has developed a theory of
non-commutative motives based on localizing invariants. The idea to view
localizing invariants as motives of small idempotent complete stable
$\infty$-categories comes quite naturally after one settles on an idea that
localizing invariants are a good categorical analogue of a cohomology theory.
Indeed, in a “category of motives” any suitable cohomology theory is expected
to be representable by a motive and every motive should represent a cohomology
theory.

However, one also expects there to be a “motive” functor

$\mathrm{Cat}^{\mathrm{perf}}_{\infty}\to\mathrm{Fun}^{\mathrm{loc}}(\mathrm{Cat}^{\mathrm{perf}}_{\infty},\mathrm{Spt})$

such that $E(\mathcal{C})\simeq\mathrm{maps}(M(\mathcal{C}),E)$ for any
$E\in\mathrm{Fun}^{\mathrm{loc}}(\mathrm{Cat}^{\mathrm{perf}}_{\infty},\mathrm{Spt})$.
This is not easy to construct because of set-theoretic difficulties, so
instead, one considers finitary localizing invariants.

######  Definition 1.9.

We say that a functor

$E:\mathrm{Cat}^{\mathrm{perf}}_{\infty,k}\to\mathrm{Spt}$

is finitary if it preserves filtered colimits.

######  Example 1.10.

K-theory, topological Hochschild homology, homotopy K-theory, Blanc’s
topological K-theory are finitary localizing invariants.

We denote the $\infty$-category of finitary $k$-localizing invariants
$\mathcal{M}_{\mathrm{loc},k}$.

######  Theorem 1.11 ([BGT13, Theorem 1.1]).

The embedding functor

$\mathcal{M}_{\mathrm{loc},k}\to\mathrm{Fun}(\mathrm{Cat}^{\mathrm{perf}}_{\infty,k},\mathrm{Spt})$

admits a left adjoint. In particular, there is a functor

$\mathcal{U}_{\mathrm{loc},k}:\mathrm{Cat}^{\mathrm{perf}}_{\infty,k}\to\mathcal{M}_{\mathrm{loc},k}$

such that $E(\mathcal{C})\simeq\mathrm{maps}(M(\mathcal{C}),E)$ for any
$E\in\mathcal{M}_{\mathrm{loc},k}$.

######  Remark 1.12.

Although all the constructions and the arguments in [BGT13] are done in the
non-linear case, i.e. the case of localizing invariants, they work verbatim
for $k$-localizing invariants. See [HSS17, 4,5] for details.

######  Remark 1.13.

By construction, the $\infty$-category $\mathcal{M}_{\mathrm{loc},k}$ is
presentable. In fact, it admits a presentable symmetric monoidal enhancement
such that

$\mathcal{U}_{\mathrm{loc},k}:\mathrm{Cat}^{\mathrm{perf}}_{\infty,k}\to\mathcal{M}_{\mathrm{loc},k}$

is symmetric monoidal.

###  1.14. Smooth & proper categories

Given $\mathcal{C}\in\mathrm{Cat}^{\mathrm{perf}}_{\infty,k}$ we can make
sense of modules

######  Definition 1.15.

Given stable $\infty$-categories $\mathcal{C}$, $\mathcal{D}$ a
$\mathcal{C}$-$\mathcal{D}$-bimodule is an exact functor

$\mathcal{C}^{op}\otimes_{k}\mathcal{D}\to\mathrm{Spt}.$

For any $\mathcal{C}$ there is a canonical
$\mathcal{C}$-$\mathcal{C}$-bimodule given by the mapping spectrum functor,
which we call canonical. A perfect $\mathcal{C}$-$\mathcal{D}$-bimodule is a
bimodule which is a compact object of the functor category
$\mathrm{Fun}^{\mathrm{ex}}(\mathcal{C}^{op}\otimes_{k}\mathcal{D},\mathrm{Spt})$.

######  Definition 1.16.

For $\mathcal{C}\in\mathrm{Cat}^{\mathrm{perf}}_{\infty,k}$ we say:

  * •

$\mathcal{C}$ is smooth if $\mathcal{C}$ is perfect as a
$\mathcal{C}$-$\mathcal{C}$-bimodule.

  * •

$\mathcal{C}$ is proper if the mapping spectra in $\mathcal{C}$ are perfect
$k$-modules.

######  Remark 1.17.

The notions of smoothness and properness are evidently self-dual.

######  Remark 1.18.

For a scheme $X\in\mathrm{Sch}_{k}$ the smoothness and properness of
$\mathbf{Perf}_{X}$ correspond precisely to smoothness and properness of $X$.

######  Remark 1.19.

For $\mathcal{C},\mathcal{D}\in\mathrm{Cat}^{\mathrm{perf}}_{\infty,k}$ such
that $\mathcal{C}$ is smooth and proper, we have

$\mathrm{maps}_{\mathcal{M}_{\mathrm{loc},k}}(\mathcal{U}_{\mathrm{loc}}(\mathcal{C}),\mathcal{U}_{\mathrm{loc}}(\mathcal{D}))\simeq
K(\mathcal{C}^{\mathrm{op}}\otimes\mathcal{D}).$

This is [HSS17, Theorem 5.25] (see also [BGT13, Theorem 9.36]).

######  Lemma 1.20.

For $\mathcal{C},\mathcal{D}\in\mathrm{Cat}^{\mathrm{perf}}_{\infty,k}$, such
that $\mathcal{C}$ is smooth, the functor

$\mathrm{Fun}^{\mathrm{ex}}(\mathcal{C}^{\mathrm{op}},\mathcal{D})\to\mathrm{Fun}^{\mathrm{ex}}(\mathcal{C}^{\mathrm{op}},\widehat{\mathcal{D}})$

factors through the subcategory $\mathcal{C}\otimes\mathcal{D}$ (see 1.7).

###### Proof.

We need to show that this functor lands in the subcategory of compact objects.
The functor is natural in $\mathcal{D}$, so it suffices to show that the image
of the identity functor along

$\mathrm{Fun}^{\mathrm{ex}}(\mathcal{C}^{\mathrm{op}},\mathcal{C}^{\mathrm{op}})\to\mathrm{Fun}^{\mathrm{ex}}(\mathcal{C}^{\mathrm{op}},\widehat{\mathcal{C}^{\mathrm{op}}})\simeq\mathrm{Fun}^{\mathrm{ex}}(\mathcal{C}^{op}\otimes_{k}\mathcal{C},\mathrm{Spt})$

is compact. The image is the canonical bimodule which is compact by smoothness
of $\mathcal{C}$. ∎

##  2\. Endomorphism categories

######  Definition 2.1.

Let $\mathcal{C}$ be a small $k$-linear idempotent complete stable
$\infty$-category. We introduce the following $\infty$-categories:

  1. (1)

The stable $\infty$-category of endomorphisms on objects of $\mathcal{C}$:

$\mathcal{E}nd(\mathcal{C}):=\mathrm{Fun}(B\mathbb{N},\mathcal{C});$

  2. (2)

The $\infty$-category of nilpotent endomorphisms: $\mathcal{N}il(\mathcal{C})$
is defined to be the subcategory of $\mathcal{E}nd(\mathcal{C})$ consisting of
endomorphisms $x\stackrel{{\scriptstyle p}}{{\to}}x$ such that the colimit in
$\widehat{\mathcal{C}}$

$X[p^{-1}]:=\mathop{\mathchoice{\vtop{\halign{#\cr\hfil$\displaystyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\textstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptscriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
<EMAIL_ADDRESS>X\stackrel{{\scriptstyle p}}{{\to}}X\stackrel{{\scriptstyle
p}}{{\to}}X\stackrel{{\scriptstyle p}}{{\to}}\cdots$

is trivial. It is a stable subcategory because the colimit functor

$\mathrm{Fun}(B\mathbb{N},\mathcal{C})\to\mathrm{Fun}(B\mathbb{N},\widehat{\mathcal{C}})\stackrel{{\scriptstyle\mathop{\mathchoice{\vtop{\halign{#\cr\hfil$\displaystyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\textstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptscriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
<EMAIL_ADDRESS>
is exact.

######  Lemma 2.2.

There is a canonical equivalence between $\mathcal{E}nd(\mathcal{C})$ and the
subcategory of

$\mathrm{Fun}_{k}^{\mathrm{ex}}(\mathcal{C}^{\mathrm{op}},\mathbf{Mod}_{k[t]})$

consisting of functors $F:\mathcal{C}^{\mathrm{op}}\to\mathbf{Mod}_{k[t]}$ for
which the composite

$\mathcal{C}^{\mathrm{op}}\to\mathbf{Mod}_{k[t]}\stackrel{{\scriptstyle\mathcal{U}}}{{\to}}\mathbf{Mod}_{k}$

is representable.

###### Proof.

First we note that $k[t]$ is the free $k$-algebra on the monoid $\mathbb{N}$,
so we may identify $\mathcal{E}nd(\mathcal{C})$ with
$\mathrm{Fun}^{\mathrm{ex}}_{k}(\mathbf{Perf}_{k[t]},\mathcal{C})$ (see also
[BGT15, Proposition 3.8]). Under this identification taking the underlying
object of an endomorphism corresponds to evaluating a functor at the
tautological module $k[t]$-module $k[t]$. Now consider the fully faithful
composite

$\mathrm{Fun}^{\mathrm{ex}}_{k}(\mathbf{Perf}_{k[t]},\mathcal{C})\to\mathrm{Fun}^{\mathrm{ex}}_{k}(\mathbf{Perf}_{k[t]},\widehat{\mathcal{C}})\simeq\mathrm{Fun}^{\mathrm{ex}}_{k}(\mathbf{Perf}_{k[t]}\times\mathcal{C}^{\mathrm{op}},\mathbf{Mod}_{k})\simeq\mathrm{Fun}^{\mathrm{ex}}_{k}(\mathcal{C}^{\mathrm{op}},\mathbf{Mod}_{k[t]}).$

It sends an exact functor $F:\mathbf{Perf}_{k[t]}\to\mathcal{C}$ to a functor

$\mathcal{C}^{\mathrm{op}}\to\mathbf{Mod}_{k[t]}$
$X\mapsto\mathrm{maps}_{\mathcal{C}}(X,F(k[t])).$

Here the action of $k[t]$ is induced by the action on $F(k[t])$. Forgetting
the action we get a functor represented by $F(k[t])$. On the other hand, for
any exact functor $\mathcal{C}^{\mathrm{op}}\stackrel{{\scriptstyle
G}}{{\to}}\mathbf{Mod}_{k[t]}$ for which the composite

$\mathcal{C}^{\mathrm{op}}\stackrel{{\scriptstyle
G}}{{\to}}\mathbf{Mod}_{k[t]}\stackrel{{\scriptstyle\mathcal{U}}}{{\to}}\mathbf{Mod}_{k}$

is representable by $Y$, the functor

$\mathbf{Perf}_{k[t]}\to\mathcal{C}$

sending $k[t]$ to $Y$ with the induced action of $k[t]$ is in the preimage of
$G$. ∎

The usage of the word nilpotent in the definition above is not accidental:

######  Lemma 2.3.

For $(P,\phi)\in\mathcal{N}il(\mathcal{C})\subset\mathcal{E}nd(\mathcal{C})$
there exists $n\in\mathbb{N}$ such that $\phi^{n}\simeq 0$.

###### Proof.

In view of the exact sequences

$\operatorname{Fib}(P\stackrel{{\scriptstyle\phi^{n}}}{{\to}}P)\to
P\stackrel{{\scriptstyle\phi}}{{\to}}P$

and the vanishing of $P[\phi^{-1}]$ we have

$P=\mathop{\mathchoice{\vtop{\halign{#\cr\hfil$\displaystyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\textstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptscriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}}\displaylimits\limits_{n}\operatorname{Fib}(P\stackrel{{\scriptstyle\phi^{n}}}{{\to}}P).$

Now by compactness of $\mathcal{C}\subset\widehat{\mathcal{C}}$, we see that
there exists an $n\in\mathbb{N}$ and a map
$P\to\operatorname{Fib}(P\stackrel{{\scriptstyle\phi^{n}}}{{\to}}P)$ making
the following commutative diagram commute:

${P}$${\operatorname{Fib}(P\stackrel{{\scriptstyle\phi^{n}}}{{\to}}P)}$${P}$${P.}$$\scriptstyle{\mathrm{id}}$$\scriptstyle{0}$$\scriptstyle{\phi^{n}}$

In particular $\phi^{n}\simeq 0$. ∎

######  Remark 2.4.

Since $\mathbf{Perf}_{k[t]}$ is smooth, by Lemma 1.20 we have a fully faithful
embedding

$\mathcal{E}nd(\mathcal{C})\to\mathcal{C}\otimes_{k}\mathbf{Perf}_{k[t]}.$

We also have an embedding

$\mathcal{E}nd(\mathcal{C})\to\mathrm{Fun}_{k}^{\mathrm{ex}}(\mathcal{C}^{\operatorname{op}},\mathbf{Mod}_{k[t]})$

###  2.5. Tensoring endomorphisms

The symmetric monoidal structure on $\mathbf{Perf}_{k}$ induces a sectionwise
symmetric monoidal structure on

$\mathrm{Fun}^{\mathrm{ex}}_{k}(\mathbf{Perf}_{k[t]},\mathbf{Mod}_{k})\simeq\mathbf{Mod}_{k[t]}$

and on the subcategory

$\mathcal{E}nd(\mathbf{Perf}_{k}).$

Now for $\mathcal{C}$ being a $k$-linear stable $\infty$-category
$\mathcal{E}nd(\mathcal{C})$ is $\mathcal{E}nd(\mathbf{Perf}_{k})$-tensored.
In particular, the biexact pairing

$\otimes:\mathcal{C}\times\mathbf{Perf}_{k}\to\mathcal{C}$

induces biexact pairings

$\otimes:\mathrm{Fun}^{\mathrm{ex}}_{k}(\mathcal{C}^{\mathrm{op}},\mathbf{Mod}_{k[t]})\times\mathbf{Mod}_{k[t]}\to\mathrm{Fun}^{\mathrm{ex}}_{k}(\mathcal{C}^{\mathrm{op}},\mathbf{Mod}_{k[t]})$

and

$\otimes:\mathcal{E}nd(\mathcal{C})\times\mathcal{E}nd(\mathbf{Perf}_{k})\to\mathcal{E}nd(\mathcal{C})$
$(P,\phi),(Q,\psi)\mapsto(P\otimes Q,\phi\otimes\psi).$

This also restricts to an action of $\mathcal{E}nd(\mathbf{Perf}_{k})$ on
$\mathcal{N}il(\mathcal{C})$.

### 2.6.

Sending an object $P\in\mathcal{C}$ to the pair $(P,0)$ gives rise to an exact
functor

$\iota:\mathcal{C}\to\mathcal{E}nd(\mathcal{C})$

which factors through an exact functor

$\iota_{\operatorname{nil}}:\mathcal{C}\to\mathcal{N}il(\mathcal{C}).$

This functor admits a left inverse

$\mathcal{E}nd(\mathcal{C})\to\mathcal{C}$

given by sending an endomorphism to its underlying module. We denote the
respective cofibers of the maps

$\mathcal{U}_{\operatorname{loc},k}(\mathcal{C})\to\mathcal{U}_{\operatorname{loc},k}(\mathcal{E}nd(\mathcal{C}))$
$\mathcal{U}_{\operatorname{loc},k}(\mathcal{C})\to\mathcal{U}_{\operatorname{loc},k}(\mathcal{N}il(\mathcal{C}))$

in the $\infty$-category $\mathrm{NMot}_{k}$ by
$\mathcal{E}nd_{0}(\mathcal{C})$ and $\mathcal{N}il_{0}(\mathcal{C})$. We also
may think of them as the images of idempotents

$\pi:(P,\phi)\mapsto(P,\phi)-(P,0).$

Abusing the notation, we denote by $E(\mathcal{E}nd_{0}(\mathcal{C}))$ and
$E(\mathcal{N}il_{0}(\mathcal{C}))$ the images of the corresponding
idempotents on $E(\mathcal{E}nd(\mathcal{C}))$ and
$E(\mathcal{N}il(\mathcal{C}))$ for any $k$-localizing invariant $E$ (i.e. not
necessarily finitary). This is justified by the fact that for a finitary $E$
we have

$E(\mathcal{E}nd_{0}(\mathcal{C}))=\mathrm{maps}(\mathcal{E}nd_{0}(\mathcal{C}),E),$
$E(\mathcal{E}nd_{0}(\mathcal{C}))=\mathrm{maps}(\mathcal{N}il_{0}(\mathcal{C}),E).$

######  Remark 2.7.

The image of $K_{0}(\mathcal{C})$ in $K_{0}(\mathcal{E}nd(\mathcal{C}))$ gives
an ideal $\mathcal{I}_{\operatorname{triv}}$, so
$K_{0}(\mathcal{E}nd(\mathcal{C}))$ and
$K_{0}(\mathcal{E}nd_{0}(\mathcal{C}))$ are both rings.

##  3\. Nilpotent endomorphisms obstruct $\mathbb{A}^{1}$-invariance

Most of the paper is dedicated to studying the $\infty$-categories
$\mathcal{N}il(\mathcal{C})$ and $\mathcal{E}nd(\mathcal{C})$. This section
explains the main reason we need to do it: as Proposition 3.4 shows, the
$\infty$-category $\mathcal{N}il(\mathcal{C})$ captures the failure of
$\mathbb{A}^{1}$-invariance.

######  Definition 3.1.

Let $E$ be a functor

$\mathrm{Cat}^{\mathrm{perf}}_{\infty,k}\to\mathrm{Spt}$

For any $\mathcal{C}\in\mathrm{Cat}^{\mathrm{perf}}_{\infty,k}$ we define

$NE(\mathcal{C}):=\operatorname{Fib}(E(\mathcal{C}\otimes_{k}\mathbf{Perf}_{k[t]})\to
E(\mathcal{C}\otimes_{k}\mathbf{Perf}_{k})),$

where $k[t]\to k$ is the evaluation map.

#### 3.1.1.

$NE(-)$ promotes to a functor
$\mathrm{Cat}^{\mathrm{perf}}_{\infty,k}\to\mathrm{Spt}$. Equivalently
$NE(\mathcal{C})$ can be defined as the cofiber of the map induced by the
$k$-algebra structure:

$E(\mathcal{C}\otimes_{k}\mathbf{Perf}_{k})\to
E(\mathcal{C}\otimes_{k}\mathbf{Perf}_{k[t]}).$

######  Example 3.2.

The nonconnective K-theory spectrum gives rise to a functor
$\mathrm{Cat}^{\mathrm{perf}}_{\infty,k}\to\mathrm{Spt}$. We have
$NK(\mathbf{Perf}_{X})=0$ if $X$ is a regular noetherian scheme, however, in
general it is non-trivial.

######  Example 3.3.

Assume $k$ is $\mathbb{Z}$-linear. Then given any functor
$E:\mathrm{Cat}^{\mathrm{perf}}_{\infty,k}\to\mathrm{Spt}$ one can define a
functor

$EH(\mathcal{C}):=\mathop{\mathchoice{\vtop{\halign{#\cr\hfil$\displaystyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\textstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptscriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}}\displaylimits\limits_{\Delta^{op}}E(\mathcal{C}\otimes_{k}\mathbf{Perf}_{k[\Delta^{\bullet}]}).$

This satisfies $NEH(\mathcal{C})\simeq 0$ for all
$\mathcal{C}\in\mathrm{Cat}^{\mathrm{perf}}_{\infty,k}$. See [Tab15, 5.1] and
[LT19, 3.13].

######  Proposition 3.4.

For a $k$-linear idempotent complete stable $\infty$-category $\mathcal{C}$
the canonical embedding (see Remark 2.4) induces an equivalence

$\Psi:\mathcal{N}il(\mathcal{C})\to\mathrm{Ker}(\mathcal{C}\otimes_{k}\mathbf{Perf}_{k[t]}\to\mathcal{C}\otimes_{k}\mathbf{Perf}_{k[t,t^{-1}]}).$

Futhermore, the map $\iota_{\mathrm{nil}}$ corresponds to the functor

$\mathcal{C}\otimes_{k}ev_{*}:\mathcal{C}\otimes_{k}\mathbf{Perf}_{k}\to\mathcal{C}\otimes_{k}\mathbf{Perf}_{k[t]}$

under this equivalence. Here $ev_{*}$ is the restriction of scalars functor
along the evaluation map.

###### Proof.

$\Psi$ is fully faithful by construction, so we need to show that for any

$(P,\phi)\in(\mathrm{Fun}^{\mathrm{ex}}(B\mathbb{N},\widehat{\mathcal{C}}))^{\omega}\simeq\mathcal{C}\otimes_{k}\mathbf{Perf}_{k[t]}$

such that $P[\phi^{-1}]\simeq 0$ we have that $P$ belongs to
$\mathcal{C}=(\widehat{\mathcal{C}})^{\omega}$.

First assume that $\phi\simeq 0$. The functor

$\widehat{\mathcal{C}}\to\mathrm{Fun}^{\mathrm{ex}}(B\mathbb{N},\widehat{\mathcal{C}})$
$X\mapsto(X,0)$

is fully faithful and preserves colimits, so it also reflects compact objects.
In particular $P$ should be compact.

In general, by Lemma 2.3 there exists an $n\in\mathbb{N}$ such that
$\phi^{n}\simeq 0$. The pushforward along the map of rings $t\mapsto t^{p}$
yields a functor

$\mathbf{Perf}_{k[t]}\to\mathbf{Perf}_{k[t]}.$

This gives a functor

$\mathcal{C}\otimes_{k}\mathbf{Perf}_{k[t]}\to\mathcal{C}\otimes_{k}\mathbf{Perf}_{k[t]}$

which sends $(P,\phi)$ to $(P,\phi^{n})\simeq(P,0)$. By the above reasoning,
$P$ belongs to $\mathcal{C}$.

The second part of the claim follows from the construction. ∎

might be interesting but not used

######  Corollary 3.5.

There is an equivalence
$\mathcal{N}il(\mathcal{C})\simeq\mathcal{C}\otimes_{k}\mathcal{N}il(\mathbf{Perf}_{k})$.

###### Proof.

By Proposition 3.4 we have a Verdier sequence

$\mathcal{N}il(\mathbf{Perf}_{k})\to\mathbf{Perf}_{k[t]}\to\mathbf{Perf}_{k[t,t^{-1}]}$

which by Lemma 1.8 induces a Verdier sequence

$\mathcal{C}\otimes_{k}\mathcal{N}il(\mathbf{Perf}_{k})\to\mathcal{C}\otimes_{k}\mathbf{Perf}_{k[t]}\to\mathcal{C}\otimes_{k}\mathbf{Perf}_{k[t,t^{-1}]}.$

Applying Proposition 3.4 again we get the desired equivalence. ∎

######  Proposition 3.6.

For a $k$-localizing invariant $E$ we have a natural equivalence

$\Omega NE(\mathcal{C})\simeq E(\mathcal{N}il_{0}(\mathcal{C}))$

of functors $\mathrm{Cat}^{\mathrm{perf}}_{\infty,k}\to\mathrm{Spt}$.

###### Proof.

Denote the projection maps $\mathbb{P}^{1}_{k}\to\mathrm{Spec}(k)$ and
$\mathbb{A}^{1}_{k}\to\mathrm{Spec}(k)$ by $pr_{\mathbb{P}^{1}}$ and
$pr_{\mathbb{A}^{1}}$, respectively. We have maps given by embedding $0$ and
its complement into $\mathbb{P}^{1}_{k}$:

$\mathrm{Spec}(k)\stackrel{{\scriptstyle
i_{0}}}{{\to}}\mathbb{P}^{1}_{k}\stackrel{{\scriptstyle
j_{0}}}{{\leftarrow}}\mathbb{A}^{1}_{k}.$

We also have an embedding given the complement to the section at infinity:

$\mathbb{A}^{1}_{k}\stackrel{{\scriptstyle
j_{\infty}}}{{\to}}\mathbb{P}^{1}_{k}.$

Consider the following diagram:

${E(\mathcal{C}\otimes_{k}\mathbf{Perf}_{\mathbb{P}_{k}^{1},\\{0\\}})}$${E(\mathcal{C}\otimes_{k}\mathbf{Perf}_{\mathbb{P}_{k}^{1}})}$${E(\mathcal{C}\otimes_{k}\mathbf{Perf}_{\mathbb{A}_{k}^{1}})}$${E(\mathcal{N}il(\mathcal{C}))}$${E(\mathcal{C}\otimes_{k}\mathbf{Perf}_{\mathbb{A}_{k}^{1}})}$${E(\mathcal{C}\otimes_{k}\mathbf{Perf}_{\mathbb{G}_{m,k}}).}$$\scriptstyle{j}$$\scriptstyle{j_{\infty}^{*}}$$\scriptstyle{j_{0}^{*}}$

By Proposition 3.4 the rows in diagram are fiber sequences. By Lemma 1.8
$E(\mathcal{C}\otimes_{k}-)$ is a $k$-localizing invariant, so the right
square is a pullback square by Proposition 1.6. This implies that the left
vertical map $j$ is an equivalence. It suffices to construct a horizontal map
in the commutative diagram

${E(\mathcal{C})}$${E(\mathcal{C}\otimes_{k}\mathbf{Perf}_{\mathbb{P}_{k}^{1},\\{0\\}})}$${E(\mathcal{N}il(\mathcal{C}))}$$\scriptstyle{\iota_{nil}}$$\scriptstyle{j}$

whose cofiber is $\Omega NE(\mathcal{C})$. It follows from the “furthermore”
part of Proposition 3.4 that the direct image functor444note that $i_{0,*}$ is
well defined because $i_{0}$ is a regular embedding
$i_{0,*}:\mathcal{C}\otimes_{k}\mathbf{Perf}_{k}\to\mathcal{C}\otimes_{k}\mathbf{Perf}_{\mathbb{P}^{1}_{k}}$
makes the diagram commutative. So we only need to show that its cofiber $C$ is
naturally equivalent to $\Omega NE(\mathcal{C})$.

The projective bundle formula of Proposition 1.4 yields that the map

$E(\mathcal{C}\otimes_{k}\mathbf{Perf}_{k})\oplus
E(\mathcal{C}\otimes_{k}\mathbf{Perf}_{k})\stackrel{{\scriptstyle\begin{pmatrix}i_{0,*}&pr^{*}_{\mathbb{P}^{1}}\end{pmatrix}}}{{\to}}E(\mathcal{C}\otimes_{k}\mathbf{Perf}_{\mathbb{P}^{1}_{k}})$

is an equivalence. To finish the proof, observe that the following diagram is
commutative

${E(\mathcal{C}\otimes_{k}\mathbf{Perf}_{k})}$${E(\mathcal{C}\otimes_{k}\mathbf{Perf}_{k})\oplus
E(\mathcal{C}\otimes_{k}\mathbf{Perf}_{k})}$${E(\mathcal{C}\otimes_{k}\mathbf{Perf}_{k})}$${E(\mathcal{C}\otimes_{k}\mathbf{Perf}_{\mathbb{P}^{1}_{k},\\{0\\}})}$${E(\mathcal{C}\otimes_{k}\mathbf{Perf}_{\mathbb{P}^{1}_{k}})}$${E(\mathcal{C}\otimes_{k}\mathbf{Perf}_{\mathbb{A}^{1}_{k}})}$${C}$${0}$${NE(\mathcal{C})}$$\scriptstyle{\begin{pmatrix}1\\\
0\end{pmatrix}}$$\scriptstyle{i_{0,*}}$$\scriptstyle{\begin{pmatrix}0&1\end{pmatrix}}$$\scriptstyle{\begin{pmatrix}i_{0,*}&pr_{\mathbb{P}^{1}}^{*}\end{pmatrix}}$$\scriptstyle{pr^{*}_{\mathbb{A}^{1}}}$$\scriptstyle{j_{0}^{*}}$

and all its rows and columns are fiber sequences. ∎

######  Corollary 3.7.

Let $E$ be any $k$-localizing invariant and $\mathcal{C}$ be a $k$-linear
stable $\infty$-category. Then $\pi_{i}NE(\mathcal{C})$ admits an action of
the ring $K_{0}(\mathcal{E}nd_{0}(\mathbf{Perf}_{k}))$ for any
$i\in\mathbb{Z}$.

###### Proof.

By Proposition 3.6 $\pi_{i}NE(\mathcal{C})\simeq
E_{i-1}(\mathcal{N}il_{0}(\mathcal{C}))$. Thus it suffices to construct the
action of $K_{0}(\mathcal{E}nd_{0}(\mathbf{Perf}_{k}))$ on
$\pi_{i}E(\mathcal{N}il_{0}(\mathcal{C}))$. The action of
$\mathcal{E}nd(\mathbf{Perf}_{k})$ on $\mathcal{N}il(\mathcal{C})$ (see 2.5)

$\mathcal{E}nd(\mathbf{Perf}_{k})\otimes\mathcal{N}il(\mathcal{C})\to\mathcal{N}il(\mathcal{C})$

by adjunction yields a functor

$\mathcal{E}nd(\mathbf{Perf}_{k})^{\simeq}\to\operatorname{Fun}_{ex}(\mathcal{N}il(\mathcal{C}),\mathcal{N}il(\mathcal{C}))^{\simeq}=\mathrm{End}_{\mathrm{Cat}^{\mathrm{perf}}_{\infty,k}}(\mathcal{N}il(\mathcal{C}))\to\mathrm{End}_{\mathbb{Z}}(\pi_{i}E(\mathcal{N}il(\mathcal{C}))).$

By additivity theorem this descents into a map

$K_{0}(\mathcal{E}nd(\mathbf{Perf}_{k}))\to\mathrm{End}_{\mathbb{Z}}(\pi_{i}E(\mathcal{N}il(\mathcal{C})))$

which is a map of rings by construction. This action commutes with the
idempotent given by sending $(P,\phi)\in\mathcal{N}il(\mathcal{C})$ to $(P,0)$
(see 2.6), so we in fact have a map of rings

$K_{0}(\mathcal{E}nd(k))\to\mathrm{End}_{\mathbb{Z}}(\pi_{i}E(\mathcal{N}il_{0}(\mathcal{C}))).$

Finally, the ideal $\mathcal{I}_{\operatorname{triv}}$ is sent to $0$, so we
obtain the desired action

$K_{0}(\mathcal{E}nd_{0}(k))\to\mathrm{End}_{\mathbb{Z}}(\pi_{i}E(\mathcal{N}il_{0}(\mathcal{C}))).$

∎

##  4\. Witt vectors

Assume $k$ is a discrete ring. One can check that the map

$\mathcal{E}nd(k)\to(1+tk[\\![t]\\!])^{\times}$

given by sending a pair $(P,\phi)\in\mathcal{E}nd(k)$ to its characteristic
polynomial $\mathrm{det}(1-t\phi)$ is a group homomorphism. Here
$\mathcal{E}nd(k)$ denotes the exact category of endomorphisms on finitely
generated projective $k$-modules. In fact, the codomain of the map can be
endowed with a ring structure in such a way that this map is also a ring
homomorphism. This ring is called the ring of Witt vectors and is denoted by
$\mathbb{W}(k)$. A trivial endomorphism has trivial characteristic polynomial,
so we end up having a ring homomorphism

$K_{0}(\mathcal{E}nd_{0}(k))\to\mathbb{W}(k).$

The image of this map consists of fractions of the form

$\frac{1+a_{1}t+\cdots a_{n}t^{n}}{1+b_{1}t+\cdots b_{l}t^{l}}.$

We call the subring spanned by the image the ring of rational Witt vectors and
denote by $\mathbb{W}_{0}(k)$.

If $k$ is an arbitrary connective ring spectrum and
$\mathcal{E}nd(\mathbf{Perf}_{k})$ is considered instead of
$\mathcal{E}nd(k)$, the same story goes through, except that now the
characteristic polynomial lands in power series over $\pi_{0}(k)$. The
following is, in essence, a classical result of Almkvist [Alm74]. As stated,
however, it follows from [BGT15, Theorem 1.12] (see also [BGT15, Theorem
1.9]):

######  Theorem 4.1 (Almkvist).

The characteristic polynomial map

$K_{0}(\mathcal{E}nd_{0}(\mathbf{Perf}_{k}))\to\mathbb{W}(\pi_{0}(k))$

is an isomorphism onto its image $\mathbb{W}_{0}(\pi_{0}(k))$. Moreover, it
becomes an isomorphism after completing with respect to the topology defined
by preimages of the ideals $(1+t^{N}\pi_{0}(k)[\\![t]\\!])$.

######  Example 4.2.

$\mathbb{W}(\mathbb{F}_{p})$ is isomorphic to the countable product of copies
of $\mathbb{Z}_{p}$ (see [Hes08, Proposition 10]).

Later we will make use of the following fact:

######  Lemma 4.3.

Assume $R$ is a discrete $\mathbb{Z}[\frac{1}{l}]$-algebra for
$l\in\mathbb{Z}$. Then $l$ is also invertible in $\mathbb{W}(R)$.

###### Proof.

Define $a_{i}\in\pi_{0}(k)$ inductively:

$a_{0}=1,a_{1}=-\frac{1}{l},$
$a_{n}=-\frac{1}{l}(\sum\limits_{i_{1}+\cdots+i_{l}=n,i_{k}<n}a_{i_{1}}\ldots
a_{i_{l}}).$

Now it is tedious to check that the power series

$\mathbf{a}=\sum\limits_{i\in\mathbb{N}}a_{i}t^{i}$

belongs to $\mathbb{W}(\pi_{0}(k))$ and satisfies

$\mathbf{a}^{l}=(1-t).$

∎

###  4.4. Verschiebung and Frobenius

Any element $x$ of the ring $\mathbb{W}(\pi_{0}(k))$ can be represented as an
infinite product

$\prod\limits_{i\in\mathbb{N}}(1-a_{i}t).$

We define

$V_{n}(1-at)=1-at^{n}$ $F_{n}(1-at)=1-a^{n}t.$

By additivity and continuity this extends to natural additive Verschiebung and
Frobenius operations

$V_{n},F_{n}:\mathbb{W}(\pi_{0}(k))\to\mathbb{W}(\pi_{0}(k))$

for all $n\in\mathbb{N}$ (cf. [Gra78]).

These operations have been studied thoroughly over the past century and they
have been shown to satisfy many interesting properties (see [Hes08], [Sti82]).
These operations restrict to the subring $\mathbb{W}_{0}(\pi_{0}(k))$ and we
use the same notation for the restricted operations. For our purposes the most
important property is the following:

######  Lemma 4.5.

Assume $R$ is a discrete $\mathbb{F}_{p}$-algebra. Then the equality

$V_{p^{i}}F_{p^{l}}(x)=p^{l}x$

holds for all $x\in\mathbb{W}(R)$.

###### Proof.

By continuity and additivity it suffices to show the equality for $x=(1+at)$.
In this case this is obvious from the definitions and the fact that taking the
$p^{l}$-th power on $R$ is additive. ∎

######  Remark 4.6.

One has a nice description of the topology on $\mathbb{W}(R)$ from Theorem 4.1
using the Verschiebung operations. Recall that any element $x\in\mathbb{W}(R)$
can be uniquely represented as a product

$\prod\limits_{i\in\mathbb{N}}(1-a_{i}t^{i})=\prod\limits_{i\in\mathbb{N}}V_{i}(1-a_{i}t).$

If $x\in 1+t^{N}R[\\![t]\\!]$ then $a_{i}=0$ for $i<N$. Hence $x$ is a
(possibly infinite) sum of $V_{i}(x_{i})$ for $x_{i}\in\mathbb{W}(R)$ and
$i\geqslant N$. Conversely, any such sum belongs to the ideal, so we have:

$1+t^{N}R[\\![t]\\!]=\\{\prod\limits_{i\geqslant
N}V_{i}(x_{i})|x_{i}\in\mathbb{W}(R)\\}.$

###  4.7. Categorical Frobenius & Verschiebung

A crucial observation about the operations $V_{n}$ and $F_{n}$ is that they
are induced by explicit endofunctors of $\mathcal{E}nd(k)$.

#### 4.7.1.

For a map of commutative ring spectra $f:R\to S$ one has an adjunction

${\mathbf{Mod}_{S}}$${\mathbf{Mod}_{R}}$$\scriptstyle{f_{*}}$$\scriptstyle{f^{!}}$

The functor $f^{!}$ is called the coinduction functor. It can be described
explicitly as follows:

$f^{!}(M):=\mathrm{maps}_{R}(S,M)$

If $S$ is perfect over $R$ (for example, it is finite and flat), both $f_{*}$
and $f^{!}$ restrict to the subcategories of perfect modules and we get an
adjunction

${\mathrm{Perf}_{S}}$${\mathrm{Perf}_{R}.}$$\scriptstyle{f_{*}}$$\scriptstyle{f^{!}}$

######  Remark 4.8.

If $S$ if perfect over $R$, then the functor $f^{!}$ commutes with colimits.
This follows from the equivalences

$f^{!}\simeq\mathrm{maps}_{R}(S,P)\simeq S^{\vee}\otimes_{R}P$

that make $f^{!}$ a left adjoint.

######  Construction 4.9.

Fix a positive natural number $l$. Consider the $k$-linear map of algebras
$p_{l}:k[t]\to k[t]$ that sends $t$ to $t^{l}$. The functors $p_{l,*}$ and
$p_{l}^{!}$ induce functors

$\mathrm{Fun}^{\mathrm{ex}}_{k}(\mathcal{C}^{\mathrm{op}},\mathbf{Mod}_{k[t]})\to\mathrm{Fun}^{\mathrm{ex}}_{k}(\mathcal{C}^{\mathrm{op}},\mathbf{Mod}_{k[t]}).$

By Lemma 2.2 we may identify $\mathcal{E}nd(\mathcal{C})$ as a subcategory
$\mathrm{Fun}^{\mathrm{ex}}_{k}(\mathcal{C}^{\mathrm{op}},\mathbf{Mod}_{k[t]})$
of functors for which the composite

$\mathcal{C}^{\mathrm{op}}\to\mathbf{Mod}_{k[t]}\stackrel{{\scriptstyle\mathcal{U}}}{{\to}}\mathbf{Mod}_{k}$

is representable. Now $p_{l,*}$ and $p_{l}^{!}$ preserve this subcategory
because $p_{l}$ is finite and free. We denote the restriction of $p_{l,*}$ to
$\mathcal{E}nd(\mathcal{C})$ by $\mathcal{F}_{l}$ and the restriction of
$p_{l}^{!}$ by $\mathcal{V}_{l}$. We call these functors the categorical
Frobenius and Verschiebung.

######  Remark 4.10.

Unravelling the definitions we can describe the functors from Construction 4.9
explicitly as follows. Given any $(P,\phi)\in\mathcal{E}nd(\mathcal{C})$ we
have

  * •

$\mathcal{F}_{l}(P,\phi)\simeq(P,\phi^{l})$

  * •

$\mathcal{V}_{l}(P,\phi)\simeq\mathrm{maps}_{k[t]}(k[t^{\frac{1}{l}}],P)\simeq(P^{l},\begin{pmatrix}0&\cdots&0&(-1)^{l+1}\phi\\\
1&\cdots&0&0\\\ \vdots&\ddots&\vdots&\vdots\\\ 0&\cdots&1&0\end{pmatrix}).$

######  Proposition 4.11.

For $\mathcal{C}=\mathbf{Perf}_{k}$ the induced maps

$K_{0}(\mathcal{E}nd_{0}(\mathcal{C}))\to
K_{0}(\mathcal{E}nd(\mathcal{C}))\stackrel{{\scriptstyle
K_{0}(\mathcal{V}_{n}),K_{0}(\mathcal{F}_{n})}}{{\longrightarrow}}K_{0}(\mathcal{E}nd(\mathcal{C}))\to
K_{0}(\mathcal{E}nd_{0}(\mathcal{C}))$

are equal to the Verschiebung and Frobenius maps under the isomorphism from
Theorem 4.1.

###### Proof.

By Remark 4.10 the functors $\mathcal{F}_{n}$ and $\mathcal{V}_{n}$ are the
same as those defined in Section 5 of [BGT15]. The analogous functors on the
exact category of endomorphisms of finitely generated projective modules over
$\pi_{0}(k)$ were also defined by Grayson in [Gra78] and they were shown to
induce the Verschiebung and Frobenius maps on the corresponding summand of
$K_{0}$. By Theorem 5.7 of [BGT15] the operations are compatible with the
operations defined by Grayson. ∎

#### 4.11.1.

Recall that

$\mathrm{Fun}^{\mathrm{ex}}_{k}(\mathbf{Perf}_{k[t]},\mathbf{Mod}_{k})\simeq\mathbf{Mod}_{k[t]}$

admits a sectionwise symmetric monoidal structure (see (2.5)). By
construction, the functor $p_{l,*}$ is symmetric monoidal. The usual abstract
nonsense implies that the adjoint functor $p_{*}^{!}$ is lax symmetric
monoidal, i.e. there is a natural transformation

$\mu_{X,Y}:p_{l}^{!}(X)\otimes p_{l}^{!}(Y)\to p_{l}^{!}(X\otimes Y).$

Now let $\mathcal{C}$ be a $k$-linear idempotent complete stable
$\infty$-category.

######  Lemma 4.12.

There are natural equivalences for $X\in\mathcal{E}nd(\mathcal{C})$ and
$Y\in\mathcal{E}nd(\mathbf{Perf}_{k})$

  1. (1)

$\mathcal{F}_{l}(X\otimes Y)\simeq\mathcal{F}_{l}(X)\otimes\mathcal{F}_{l}(Y)$

  2. (2)

$X\otimes V_{l}(Y)\simeq\mathcal{V}_{l}(F_{l}(X)\otimes Y)$

###### Proof.

The first equivalence follows from commutativity of the diagram

${\mathrm{Fun}^{\mathrm{ex}}(\mathcal{C}^{\mathrm{op}},\mathbf{Mod}_{k[t]})\times\mathbf{Mod}_{k[t]}}$${\mathrm{Fun}^{\mathrm{ex}}(\mathcal{C}^{\mathrm{op}},\mathbf{Mod}_{k[t]})}$${\mathrm{Fun}^{\mathrm{ex}}(\mathcal{C}^{\mathrm{op}},\mathbf{Mod}_{k[t]})\times\mathbf{Mod}_{k[t]}}$${\mathrm{Fun}^{\mathrm{ex}}(\mathcal{C}^{\mathrm{op}},\mathbf{Mod}_{k[t]})}$$\scriptstyle{\otimes}$$\scriptstyle{p_{l,*}\times
p_{l,*}}$$\scriptstyle{p_{l,*}}$$\scriptstyle{\otimes}$

which is simply a manifestation of monoidality of $p_{l,*}$. To construct the
second equivalence consider the natural transformation

$\xi_{X,Y}:X\otimes
p_{l}^{!}(Y)\stackrel{{\scriptstyle\eta_{X}\otimes\mathrm{id}_{p_{l}^{!}(Y)}}}{{\to}}p_{l}^{!}(p_{*}(X))\otimes
p_{l}^{!}(Y)\stackrel{{\scriptstyle\mu_{p_{*}(X),Y}}}{{\to}}p_{l}^{!}(p_{l,*}(X)\otimes
Y)$

of functors

$\mathbf{Mod}_{k[t]}\times\mathbf{Mod}_{k[t]}\longrightarrow\mathbf{Mod}_{k[t]}.$

Notice that both functors commute with filtered colimits in both variables
(see Remark 4.8). The natural transformation for $X=Y=(k,\mathrm{id})$ is
equivalent to the composite

$\mathrm{maps}_{k[t]}(k[t^{\frac{1}{l}}],k)\stackrel{{\scriptstyle
can\otimes\mathrm{id}}}{{\to}}\mathrm{maps}_{k[t]}(k[t^{\frac{1}{l}}],k)\otimes\mathrm{maps}_{k[t]}(k[t^{\frac{1}{l}}],k)\stackrel{{\scriptstyle\times}}{{\to}}\mathrm{maps}_{k[t]}(k[t^{\frac{1}{l}}],k),$

where $can$ is induced by the ring map $k[t^{\frac{1}{l}}]\to k$ that sends
the generator to the unit in $k$ and the second map is the pointwise product
in $k$. We observe that this composite is the identity. Now, the
$\infty$-category $\mathbf{Mod}_{k[t]}$ is generated under colimits and shifts
by the object $(k,\mathrm{id})$, so this natural transformation is an
equivalence in general. Now if $X$ is a functor from
$\mathrm{Fun}^{\mathrm{ex}}(\mathcal{C},\mathbf{Mod}_{k[t]})$, we get a
natural equivalence

$X(C)\otimes p^{!}_{l}(Y)\simeq p^{!}_{l}(p_{l,*}(X(C))\otimes Y)$

of functors

$\mathrm{Fun}^{\mathrm{ex}}(\mathcal{C}^{\mathrm{op}},\mathbf{Mod}_{k[t]})\times\mathbf{Mod}_{k[t]}\longrightarrow\mathbf{Mod}_{k[t]}.$

By construction, the restriction of the equivalence to the subcategories of
endomorphisms gives

$X\otimes V_{l}(Y)\simeq\mathcal{V}_{l}(F_{l}(X)\otimes Y).$

∎

######  Remark 4.13.

A quite involved proof of the relation Lemma 4.12(2) in K-theory appears in
[Sti82] and later in [Tab17]. We are able to significantly simplify the proof
by employing the adjunction between $p_{l,*}$ and $p^{!}_{l}$ and also obtain
an equivalence on the level of $\infty$-categories and not just an equality of
K-theory classes.

######  Corollary 4.14.

Assume $k$ is an $\mathbb{F}_{p}$-algebra. Let $\mathcal{C}$ be a small
$k$-linear idempotent complete stable $\infty$-category. Then multiplication
by $p^{l}\in K_{0}(\mathcal{E}nd_{0}(\mathbf{Perf}_{k}))$ on
$K_{0}(\mathcal{E}nd_{0}(\mathcal{C}))$ and
$K_{0}(\mathcal{N}il_{0}(\mathcal{C}))$ are equal to the composites

$K_{0}(\mathcal{E}nd_{0}(\mathcal{C}))\to
K_{0}(\mathcal{E}nd(\mathcal{C}))\stackrel{{\scriptstyle
K_{0}(\mathcal{V}_{p^{l}}\circ\mathcal{F}_{p^{l}})}}{{\to}}K_{0}(\mathcal{E}nd(\mathcal{C}))\to
K_{0}(\mathcal{E}nd_{0}(\mathcal{C}))$

and

$K_{0}(\mathcal{N}il_{0}(\mathcal{C}))\to
K_{0}(\mathcal{N}il(\mathcal{C}))\stackrel{{\scriptstyle
K_{0}(\mathcal{V}_{p^{l}}\circ\mathcal{F}_{p^{l}})}}{{\to}}K_{0}(\mathcal{N}il(\mathcal{C}))\to
K_{0}(\mathcal{N}il_{0}(\mathcal{C}))$

for any $l\in\mathbb{N}$.

###### Proof.

If $\mathcal{C}=\mathbf{Perf}_{k}$, this follows from Proposition 4.11 and
Lemma 4.5. In general, there is an equivalence

$X\otimes V_{p^{l}}(Y)\simeq\mathcal{V}_{p^{l}}(F_{p^{l}}(X)\otimes Y)$

for $X\in\mathcal{E}nd(\mathcal{C})$ and
$Y\in\mathcal{E}nd(\mathbf{Perf}_{k})$ by Lemma 4.12. In particular, for
$Y=(k,id)$ the following equalities hold in $K_{0}$ of either
$\mathcal{N}il(\mathcal{C})$ or $\mathcal{E}nd(\mathcal{C})$

$p^{l}[X]=[X][\mathcal{V}_{p^{l}}\circ\mathcal{F}_{p^{l}}(k,id)]=[X][\mathcal{V}_{p^{l}}(k,id)]=[\mathcal{V}_{p^{l}}\circ\mathcal{F}_{p^{l}}(X)][(k,id)]=[\mathcal{V}_{p^{l}}\circ\mathcal{F}_{p^{l}}(X)]$

that prove the claim. ∎

##  5\. Torsion in Nil-theories

This section is dedicated to proving the main results (Theorems 0.4 and 0.5).
So far we never had to require our localizing invariants to be finitary. The
reason we need it is to show that the action of $\mathbb{W}_{0}(\pi_{0}(k))$
constructed in Corollary 3.7 is continuous. This allows to extend the action
to the action of the ring of Witt vectors $\mathbb{W}(\pi_{0}(k))$. The ring
$\mathbb{W}(\pi_{0}(k))$ has many nice properties which the ring
$\mathbb{W}_{0}(\pi_{0}(k))$ does not (for example, an integer $l$ is
invertible in in $\mathbb{W}(\pi_{0}(k))$ whenever it is invertible in $k$).
The $\mathbb{W}(\pi_{0}(k))$-modules $NE(\mathcal{C})$ inherit these
properties allowing to prove the main theorems.

### 5.1.

Now we introduce a filtration on the $\infty$-category
$\mathcal{N}il(\mathcal{C})$. We start from the case
$\mathcal{C}=\mathbf{Perf}_{k}$. Define $\mathcal{N}il^{\leqslant
n}(\mathbf{Perf}_{k})$ to be the stable $\infty$-category of those
$k[t]/(t^{n})$-modules that are perfect over $k$. Restricting scalars gives
rise to canonical functors

$\tau_{n,m}:\mathcal{N}il^{\leqslant
n}(\mathbf{Perf}_{k})\to\mathcal{N}il^{\leqslant m}(\mathbf{Perf}_{k})\text{
for any }n\leqslant m$ $\tau_{n}:\mathcal{N}il^{\leqslant
n}(\mathbf{Perf}_{k})\to\mathcal{N}il(\mathbf{Perf}_{k})\text{ for any }n.$

######  Lemma 5.2.

Let $\\{\mathcal{C}_{i}\\}_{i\in I}$ be a filtered diagram of
$\infty$-categories and $\mathcal{C}$ be an $\infty$-category. A map
$\mathop{\mathchoice{\vtop{\halign{#\cr\hfil$\displaystyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\textstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptscriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
<EMAIL_ADDRESS>is an equivalence if any object of $\mathcal{C}$ is an image of an object of
$\mathcal{C}_{i}$ for some $i$ and for all $x,y\in\mathcal{C}_{n}$ the maps

$\mathop{\mathchoice{\vtop{\halign{#\cr\hfil$\displaystyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\textstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptscriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}}\displaylimits_{I_{n/}}\operatorname{\operatorname{Maps}}_{\mathcal{C}_{i}}(x,y)\to\mathcal{C}$

are equivalences where $I_{n/}$ is the category of objects of $I$ under $n$.

###### Proof.

This can be seen via expressing $\infty$-categories as complete Segal spaces
inside the functor category $\mathrm{Fun}(\Delta^{\mathrm{op}},\mathrm{Spc})$,
where filtered colimits are computed sectionwise. See also the short note
[Roz12]. ∎

######  Lemma 5.3.

The canonical map

$\mathop{\mathchoice{\vtop{\halign{#\cr\hfil$\displaystyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\textstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptscriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
<EMAIL_ADDRESS>n}(k)\to\mathcal{N}il(k)$

is an equivalence.

###### Proof.

Any object of $\mathcal{N}il(k)$ is the image of $\mathcal{N}il^{\leqslant
k}(k)$ for big enough $k$. So it suffices to show that for any
$U,V\in\mathcal{N}il^{\leqslant n}(k)$ the map

$\mathop{\mathchoice{\vtop{\halign{#\cr\hfil$\displaystyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\textstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptscriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
<EMAIL_ADDRESS>m}(k)}(\tau_{n,m}U,\tau_{n,m}V)\to\operatorname{\operatorname{Maps}}_{\mathcal{N}il(k)}(\tau_{n}U,\tau_{n}V)$

is an equivalence. Translating into the world of modules we need to show that
the map

$\mathop{\mathchoice{\vtop{\halign{#\cr\hfil$\displaystyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\textstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptscriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}}\displaylimits\limits_{m}\operatorname{\operatorname{Maps}}_{k[x]/(x^{m})}(U,V)\to\operatorname{\operatorname{Maps}}_{k[x]}(U,V)$

is an equivalence. By adjunction it is the same as the map

$\mathop{\mathchoice{\vtop{\halign{#\cr\hfil$\displaystyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\textstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptscriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}}\displaylimits\limits_{m}\operatorname{\operatorname{Maps}}_{k[x]/(x^{n})}(U\otimes_{k[x]/(x^{m})}k[x]/(x^{n}),V)\to\operatorname{\operatorname{Maps}}_{k[x]}(U,V).$

It suffices to show that the map of pro-objects

$\\{U\\}_{m}\to\\{U\otimes_{k[x]/(x^{m})}k[x]/(x^{n})\\}_{m},$

is a weak equivalence (here the left-hand side is the constant pro-system). To
compute the right hand-side we use the following free resolution for
$k[x]/(x^{n})$:

$k[x]/(x^{n})\leftarrow k[x]/(x^{m})\stackrel{{\scriptstyle
x^{m-n}}}{{\leftarrow}}k[x]/(x^{m})\stackrel{{\scriptstyle
x^{n}}}{{\leftarrow}}k[x]/(x^{m})\stackrel{{\scriptstyle
x^{m-n}}}{{\leftarrow}}k[x]/(x^{m})\stackrel{{\scriptstyle
x^{n}}}{{\leftarrow}}\cdots$

For $l>m+n$ we have a map of complexes

${k[x]/(x^{n})}$${k[x]/(x^{l})}$${k[x]/(x^{l})}$${k[x]/(x^{l})}$${k[x]/(x^{l})}$${\cdots}$${k[x]/(x^{n})}$${k[x]/(x^{m})}$${k[x]/(x^{m})}$${k[x]/(x^{m})}$${k[x]/(x^{m})}$${\cdots}$$\scriptstyle{=}$$\scriptstyle{pr_{l,m}}$$\scriptstyle{pr_{l,n}}$$\scriptstyle{0}$$\scriptstyle{x^{l-n}}$$\scriptstyle{0}$$\scriptstyle{x^{l}}$$\scriptstyle{0}$$\scriptstyle{x^{l-n}}$$\scriptstyle{pr_{m,n}}$$\scriptstyle{x^{m-n}}$$\scriptstyle{x^{n}}$$\scriptstyle{x^{m-n}}$

Now we can compute the map $U\otimes_{k[x]/(x^{l})}k[x]/(x^{n})\to
U\otimes_{k[x]/(x^{m})}k[x]/(x^{n})$ as the totalization of the map of
complexes

${U}$${U}$${U}$${U}$${\cdots}$${U}$${U}$${U}$${U}$${\cdots}$$\scriptstyle{\mathrm{id}}$$\scriptstyle{0}$$\scriptstyle{x^{l-n}}$$\scriptstyle{0}$$\scriptstyle{x^{l}}$$\scriptstyle{0}$$\scriptstyle{x^{l-n}}$$\scriptstyle{x^{m-n}}$$\scriptstyle{x^{n}}$$\scriptstyle{x^{m-n}}$

Taking the brutal truncation of these complexes for all $m$ we get a fiber
sequence of pro-systems:

$\\{U\\}_{m}\to\\{U\otimes_{k[x]/(x^{m})}k[x]/(x^{n})\\}_{m}\to\\{(U\otimes_{k[x]/(x^{m})}k[x]/(x^{n}))_{\geqslant
1}\\}_{m}.$

From the computation above we see that the morphisms in the pro-system on the
right are eventually zero, so it has trivial limit. In particular, the map in
question is a pro-equivalence. ∎

#### 5.3.1.

For $\mathcal{C}$ a $k$-linear idempotent complete stable $\infty$-category we
define

$\mathcal{N}il^{\leqslant
n}(\mathcal{C}):=\mathcal{C}\otimes_{k}\mathcal{N}il^{\leqslant
n}(\mathbf{Perf}_{k}).$

By Corollary 3.5 we have an equivalence

$\mathcal{N}il(\mathcal{C})\simeq\mathcal{C}\otimes_{k}\mathcal{N}il(\mathbf{Perf}_{k})$

so we have a filtration of $\mathcal{N}il(\mathcal{C})$ by
$\mathcal{N}il^{\leqslant n}(\mathcal{C})$.

######  Corollary 5.4.

In the context above we have

$\mathop{\mathchoice{\vtop{\halign{#\cr\hfil$\displaystyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\textstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptscriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
<EMAIL_ADDRESS>n}(\mathcal{C})\simeq\mathcal{N}il(\mathcal{C})$

###### Proof.

Follows from Lemma 1.8 and Lemma 5.3. ∎

Now we are ready to state and prove our main result

######  Theorem 5.5.

Assume that $\pi_{0}(k)$ is an $\mathbb{F}_{p}$-algebra and $E$ is a finitary
$k$-localizing invariant. Then for any small $k$-linear idempotent complete
stable $\infty$-category $\mathcal{C}$ we have

$NE(\mathcal{C})[\frac{1}{p}]\simeq 0.$

###### Proof.

It suffices to show that every element of
$\pi_{i+1}NE(\mathcal{C})\simeq\pi_{i}E(\mathcal{N}il_{0}(\mathcal{C}))$ is
$p$-torsion (the equivalence is the content of Proposition 3.6). In other
words, by Corollary 4.14 it suffices to show that for every
$x\in\pi_{i}E(\mathcal{N}il_{0}(\mathcal{C}))$ there exists $l\in\mathbb{N}$
such that $(\mathcal{V}_{p^{l}}\circ\mathcal{F}_{p^{l}})(x)$ is in the image
of the map

$\pi_{i}E(\iota_{\operatorname{nil}}):\pi_{i}E(\mathcal{C})\to\pi_{i}E(\mathcal{N}il(\mathcal{C})).$

Since $E$ is finitary and by Corollary 5.4 we have

$\pi_{i}E(\mathcal{N}il(\mathcal{C}))\simeq\mathop{\mathchoice{\vtop{\halign{#\cr\hfil$\displaystyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\textstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptscriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}}\displaylimits_{l}\pi_{i}E(\mathcal{N}il^{\leqslant
p^{l}}(\mathcal{C})).$

Hence any element $x\in\pi_{i}E(\mathcal{N}il(\mathcal{C}))$ is the image of
an element $\overline{x}$ of $\pi_{i}E(\mathcal{N}il^{\leqslant
p^{l}}(\mathcal{C}))$. Restricting scalars along maps in the commutative
diagram of ring spectra

${k[t]}$${k[t]}$${k}$${k[t]/(t^{p^{l}})}$$\scriptstyle{p_{l}}$$\scriptstyle{t\mapsto
0}$

we get a commutative diagram

${\mathcal{N}il^{\leqslant
p^{l}}(\mathcal{C})}$${\mathcal{N}il(\mathcal{C})}$${\mathcal{C}}$${\mathcal{N}il(\mathcal{C}).}$$\scriptstyle{\tau_{p^{l}}}$$\scriptstyle{\mathcal{U}}$$\scriptstyle{\mathcal{F}_{p^{l}}}$$\scriptstyle{\iota_{\mathrm{nil}}}$

Now

$\mathcal{F}_{p^{l}}(x)=\mathcal{F}_{p^{l}}(\tau_{p^{l}}(\overline{x}))=\iota_{\mathrm{nil}}(\mathcal{U}(\overline{x})).$

It suffices to show that $\mathcal{V}_{p^{l}}$ preserves the image of
$\iota_{\mathrm{nil}}$ which follows from the following lemma. ∎

######  Lemma 5.6.

There is an equivalence of maps

$\mathcal{V}_{i}\circ\iota_{\mathrm{nil}}\simeq
i\cdot\iota_{\mathrm{nil}}:E(\mathcal{C})\to E(\mathcal{N}il(\mathcal{C})).$

###### Proof.

The base change theorem tells us that for all $i\in\mathbb{N}\setminus\\{0\\}$
we have commutative diagrams

${\mathbf{Mod}_{k}}$${\mathbf{Mod}_{k[t]/(t^{i})}}$${\mathbf{Mod}_{k[t]}}$${\mathbf{Mod}_{k[t]}}$$\scriptstyle{u_{i}^{!}}$$\scriptstyle{a_{0,*}}$$\scriptstyle{a_{i,*}}$$\scriptstyle{p^{!}_{i}}$

induced by the pushout diagrams of ring spectra

${k[t]}$${k[t]}$${k}$${k[t]/(t^{i})}$$\scriptstyle{a_{0}}$$\scriptstyle{p_{i}}$$\scriptstyle{a_{i}}$$\scriptstyle{u_{i}}$

where the vertical maps are the obvious projections. By construction the
composite $p^{!}_{i}a_{0,*}$ induces

$\mathcal{V}_{i}\circ\iota_{\mathrm{nil}}:\mathcal{N}il(\mathcal{C})\to\mathcal{N}il(\mathcal{C})$

The composite functors $a_{i}^{!}u_{i,*}$ fit into exact sequences

${a_{0,*}(-)}$${a_{i}^{!}u_{i,*}(-)}$${a_{i-1}^{!}u_{i-1,*}(-).}$${a_{0,*}(-)}$${a_{0,*}(-)\otimes_{k}k[t]/(t^{i})}$${a_{0,*}(-)\otimes_{k}k[t]/(t^{i-1})}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$

which induce exact sequences of functors

$\mathcal{C}\to\mathcal{N}il(\mathcal{C})$
$\iota_{\mathrm{nil}}\to\mathcal{V}_{i}\circ\iota_{\mathrm{nil}}\to\mathcal{V}_{i-1}\circ\iota_{\mathrm{nil}}.$

Applying additivity inductively we see that

$\mathcal{V}_{i}\circ\iota_{\mathrm{nil}}:E(\mathcal{C})\to
E(\mathcal{N}il(\mathcal{C})).$

is equivalent to $i$ copies of $\iota_{\mathrm{nil}}$. ∎

### 5.7.

Fix $E$ a finitary $k$-localizing invariant and $\mathcal{C}$ a $k$-linear
idempotent complete stable $\infty$-category. Arguing as in the proof of
Theorem 5.5 one can see that $V_{n}(x)$ acts trivially on the image
$\mathbf{N}_{n}$ of $\pi_{i}E(\mathcal{N}il^{\leqslant n}(\mathcal{C}))$ in
$\pi_{i}E(\mathcal{N}il_{0}(\mathcal{C}))$ for any
$x\in\mathbb{W}_{0}(\pi_{0}(k))$. Indeed, for $y$ in the image we have an
equality in $\pi_{i}E(\mathcal{N}il_{0}(\mathcal{C}))$ (Lemma 4.12 and Lemma
5.6):

$y\cdot V_{n}(x)=V_{n}(F_{n}(y)\cdot x)=V_{n}(0\cdot x)=0.$

Now, any element of $\mathbb{W}(\pi_{0}(k))$ can be written as a product

$\prod\limits_{i\in\mathbb{N}}V_{i}(x_{i})$

for some $x_{i}\in\mathbb{W}_{0}(\pi_{0}(k))$ (see Remark 4.6), hence we can
make sense of the action

$\mathbb{W}(\pi_{0}(k))\times\mathbf{N}_{n}\to\mathbf{N}_{n}$
$(\prod\limits_{i\in\mathbb{N}}V_{i}(x_{i}),y)\mapsto\sum\limits_{i\in\mathbb{N}}V_{i}(x_{i})\cdot
y=\sum\limits_{0\leqslant i\leqslant n}V_{i}(x_{i})\cdot y,$

where $x_{i}\in\mathbb{W}_{0}(\pi_{0}(k))$. This action is continuous and
extends the action of
$\mathbb{W}_{0}(\pi_{0}(k))\subset\mathbb{W}(\pi_{0}(k))$, so summarizing we
have:

######  Lemma 5.8.

The action of Corollary 3.7

$\mathbb{W}_{0}(\pi_{0}(k))\times\pi_{i}(E(\mathcal{N}il_{0}(\mathcal{C})))\to\pi_{i}(E(\mathcal{N}il_{0}(\mathcal{C})))$

is continuous.

###### Proof.

We need to show that for any $x\in\pi_{i}(E(\mathcal{N}il_{0}(\mathcal{C})))$
there exists $k$ such that $1+t^{k}R[\\![t]\\!]\subset\mathbb{W}(\pi_{0}(k))$
acts trivially on $x$. We have

$\pi_{i}(E(\mathcal{N}il_{0}(\mathcal{C})))\simeq\mathop{\mathchoice{\vtop{\halign{#\cr\hfil$\displaystyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\textstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
1.5pt\cr\rightarrowfill@\textstyle\cr\nointerlineskip\kern-1.0pt\cr\cr}}}{\vtop{\halign{#\cr\hfil$\scriptscriptstyle\operator@font
colim$\hfil\cr\nointerlineskip\kern
<EMAIL_ADDRESS>
in the category of $\mathbb{W}_{0}(\pi_{0}(k))$-modules. Hence there exists
$n$ such that $x\in\mathbf{N}_{n}$. As explained above, the action on
$\mathbf{N}_{n}$ extends to a continuous action of $\mathbb{W}(\pi_{0}(k))$
with $1+t^{n}R[\\![t]\\!]\subset\mathbb{W}(\pi_{0}(k))$ acting trivially. ∎

Now we can prove our second main result.

######  Theorem 5.9.

Assume $l$ is invertible in $\pi_{0}(k)$ for some $l\in\mathbb{Z}$. Let $E$ a
finitary $k$-localizing invariant and $\mathcal{C}$ a $k$-linear idempotent
complete stable $\infty$-category. Then

  1. (1)

$l$ is invertible in $\pi_{i}NE(\mathcal{C})$;

  2. (2)

$E/l$ is $\mathbb{A}^{1}$-invariant.

###### Proof.

By Lemma 5.8
$\pi_{i}NE(\mathcal{C})\simeq\pi_{i-1}E(\mathcal{N}il_{0}(\mathcal{C}))$ are
continuous $\mathbb{W}(\pi_{0}(k))$-modules. Now the first claim follows from
Lemma 4.3. The second claim follows from the equivalence

$NE/l(\mathcal{C})\simeq N(E/l)(\mathcal{C}).$

∎

## References

  * [AI22] T. Annala and R. Iwasa, _Motivic spectra and universality of $K$-theory_, 2022
  * [Alm74] G. Almkvist, _The Grothendieck ring of the category of endomorphisms_ , Journal of Algebra 28 (1974), no. 3, pp. 375–388, https://www.sciencedirect.com/science/article/pii/0021869374900477
  * [Ann21] T. Annala, _Algebraic Spivak’s theorem and applications_ , 2021
  * [Ann22] T. Annala, _Derived Algebraic Cobordism_ , 2022
  * [BGT13] A. J. Blumberg, D. Gepner, and G. Tabuada, _A universal characterization of higher algebraic $K$-theory_, Geom. Topol. 17 (2013), no. 2, pp. 733–838
  * [BGT15] A. Blumberg, D. Gepner, and G. Tabuada, _K-theory of endomorphisms via noncommutative motives_ , Transactions of the American Mathematical Society 368 (2015), no. 2, p. 1435–1465, http://dx.doi.org/10.1090/TRAN/6507
  * [BKRS22] T. Bachmann, A. A. Khan, C. Ravi, and V. Sosnilo, _Categorical Milnor squares and K-theory of algebraic stacks_ , Selecta Mathematica 28 (2022), no. 5, http://dx.doi.org/10.1007/s00029-022-00796-w
  * [Elm21] E. Elmanto, _THH and TC are (very) far from being homotopy functors_ , Journal of Pure and Applied Algebra 225 (2021), no. 8, p. 106640, http://dx.doi.org/10.1016/j.jpaa.2020.106640
  * [EM22] E. Elmanto and M. Morrow, _Motivic cohomology of schemes_ , in preparation
  * [GH10] T. Geisser and L. Hesselholt, _On a conjecture of Vorst_ , Mathematische Zeitschrift 270 (2010), no. 1–2, p. 445–452, http://dx.doi.org/10.1007/s00209-010-0806-2
  * [Gra78] D. R. Grayson, _Grothendieck rings and Witt vectors_ , Communications in Algebra 6 (1978), no. 3, pp. 249–255, preprint https://doi.org/10.1080/00927877808822245
  * [GS09] D. Gepner and V. P. Snaith, _On the Motivic Spectra Representing Algebraic Cobordism and Algebraic K-Theory_ , Documenta Math. (2009), no. 14, pp. 359–396
  * [Hes08] L. Hesselholt, _Lecture notes on Witt vectors_ , 2008
  * [HSS17] M. Hoyois, S. Scherotzke, and N. Sibilla, _Higher traces, noncommutative motives, and the categorified Chern character_ , Adv. Math. 309 (2017), pp. 97–154, https://doi.org/10.1016/j.aim.2017.01.008
  * [Kha18] A. A. Khan, _Algebraic $K$-theory of quasi-smooth blow-ups and cdh descent_, 2018, to appear in Annales Henri Lebesgue
  * [LM07] M. Levine and F. Morel, _Algebraic Cobordism_ , Springer, 2007
  * [LT19] Land and Tamme, _On the $K$-theory of pullbacks_, Annals of Mathematics 190 (2019), no. 3, p. 877, http://dx.doi.org/10.4007/annals.2019.190.3.4
  * [Lur17] J. Lurie, _Higher Algebra_ , September 2017, http://www.math.harvard.edu/~lurie/papers/HA.pdf
  * [Roz12] N. Rozenblyum, _Filtered colimits of $\infty$-categories_, 2012, https://people.math.harvard.edu/~gaitsgde/GL/colimits.pdf
  * [Spi18] M. Spitzweck, _A commutative $\mathbf{P}^{1}$-spectrum representing motivic cohomology over Dedekind domains_, Mém. Soc. Math. Fr. 157 (2018), preprint arXiv:1207.4078
  * [Sti82] J. Stientstra, _Operations in the higher K-theory of endomorphisms_ , no. 2, pp. 59–115, https://books.google.de/books?id=Deb0fn093m4C
  * [Tab15] G. Tabuada, _$\mathbf{A}^{1}$ -homotopy theory of noncommutative motives_, Journal of Noncommutative Geometry 9 (2015), no. 3, p. 851–875
  * [Tab17] by same author, _$\mathbf{A}^{1}$ -homotopy invariance of algebraic K-theory with coefficients and du Val singularities_, Annals of K-Theory 2 (2017), no. 1, p. 1–25, http://dx.doi.org/10.2140/AKT.2017.2.1
  * [TT90] R. W. Thomason and T. Trobaugh, _Higher algebraic K-theory of schemes and of derived categories_ , The Grothendieck Festschrift III, Progress in Mathematics, vol. 88, Birkhäuser, 1990, pp. 247–435
  * [Vor79] T. Vorst, _Localization of the K-Theory of Polynomial Extensions._ , Mathematische Annalen 244 (1979), pp. 33–54, http://eudml.org/doc/163289
  * [Wei81] C. A. Weibel, _Mayer-Vietoris sequences and module structures on $NK_{\ast}$_, pp. 466–493, https://doi.org/10.1007/BFb0089534