An analogue of the Milnor conjecture for the de Rham-Witt complex in
characteristic 2

Emanuele Dotto

††footnotetext: _2020 Mathematics Subject Classification:_ Primary 19D55,
11E81, 13F35; Secondary 55P91, 14F30, 19D45

###### Abstract

We describe the modulo $2$ de Rham-Witt complex of a field of characteristic
$2$, in terms of the powers of the augmentation ideal of the
$\operatorname{\mathbb{Z}}/2$-geometric fixed points of real topological
restriction homology $\operatorname{TRR}$. This is analogous to the conjecture
of Milnor, proved in [Kat82] for fields of characteristic $2$, which describes
the modulo $2$ Milnor K-theory in terms of the powers of the augmentation
ideal of the Witt group of symmetric forms. Our proof provides a somewhat
explicit description of these objects, as well as a calculation of the
homotopy groups of the geometric fixed points of $\operatorname{TRR}$ and of
real topological cyclic homology, for all fields.

## Introduction

Let $k$ be a field. Let us recall that Milnor conjectured, in [Mil71],
[Mil70], that a certain canonical map of graded rings

$\operatorname{K}^{M}_{\ast}(k)/2\longrightarrow I^{\ast}/I^{\ast+1},$

should be an isomorphism. Here $\operatorname{K}^{M}_{\ast}(k)$ is the Milnor
K-theory of $k$, and
$I:=\ker(rk\colon\operatorname{W}^{s}(k)\to\operatorname{\mathbb{Z}}/2)$ is
the augmentation ideal of the Witt group $\operatorname{W}^{s}(k)$ of
symmetric forms over $k$. This conjecture was proved in [Kat82] when $k$ has
characteristic $2$, and subsequently in [OVV07, Mor05] in characteristic
different from $2$.

The starting point of our paper is the following, somewhat overloaded
observation. On one side of this isomorphism, we have a “symbolic version”
$\operatorname{K}^{M}_{\ast}(k)$ of the algebraic K-theory spectrum
$\operatorname{K}(k)$ of $k$. On the other side, we have the Witt group
$\operatorname{W}^{s}(k)$, which is $\pi_{0}$ of the
$\operatorname{\mathbb{Z}}/2$-geometric fixed-points spectrum of a certain
canonical $\operatorname{\mathbb{Z}}/2$-equivariant refinement
$\operatorname{KR}(k)$ of $\operatorname{K}(k)$ (see [CDH+20a, CDH+20b]). One
may then wonder if a similar relationship holds for other functors closely
related to algebraic K-theory. One does not need to look far for another such
example, which is already provided in Kato’s proof of Milnor’s conjecture in
characteristic $2$: The de Rham complex $\Omega^{\ast}_{k}$ is a “symbolic
version” of the topological Hochschild homology spectrum
$\operatorname{THH}(k)$, and $\operatorname{THH}(k)$ also admits a canonical
$\operatorname{\mathbb{Z}}/2$-equivariant refinement $\operatorname{THR}(k)$.
The calculation of [DMPR21, Corollary 5.2] provides an isomorphism between
$\pi_{0}$ of the $\operatorname{\mathbb{Z}}/2$-geometric fixed points of this
spectrum and $(k\otimes_{S}k)/2$, where $S\leq k$ is the subfield generated by
the squares. The result analogous to Milnor’s conjecture is then an
isomorphism

$\Omega^{\ast}_{k}/2\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}J^{\ast}/J^{\ast+1},$

where $J$ is the kernel of the multiplication map
$\mu\colon(k\otimes_{S}k)/2\to k/2$. Let us point out that, if $2$ is a unit
in $k$, the source and target of this map are clearly zero, so that this
statement has content only when the characteristic of $k$ is $2$. It seems to
be a standard result that this map is an isomorphism, and it plays an
important role in the proof of [Kat82, Lemma 7(3)] (see [Ara20] for a proof,
which we recast in Lemma 3.6). The main goal of our paper is to establish an
analogous result for topological restriction homology, whose “symbolic
version” is the de Rham-Witt complex of Bloch, Deligne and Illusie [Ill79].

Let $\operatorname{W}_{\langle 2^{\bullet}\rangle}\Omega^{\ast}_{k}$ be the
$2$-typical de Rham-Witt complex of $k$. We will take the definition of
[Cos08], as the initial object in the category of $2$-typical Witt complexes
over $k$ (see also [HM04]). For all integer $n\geq 0$, let
$\operatorname{TR}^{n+1}(k;2)$ be the $2$-typical $(n+1)$-truncated
topological restriction homology of [BHM93] (see also [AN21]). Similarly to
the relation between Milnor K-theory and algebraic K-theory,
$\operatorname{W}_{\langle 2^{\bullet}\rangle}\Omega^{\ast}_{k}$ and the
homotopy groups of $\operatorname{TR}^{n+1}(k;2)$ agree in low degrees, and
the former should be consider the symbolic version of the latter (see [Hes04]
and [GH99, §4]). We recall that the spectrum $\operatorname{TR}^{n+1}(k;2)$ is
defined as the $C_{2^{n}}$-fixed points of a $C_{2^{n}}$-equivariant structure
on $\operatorname{THH}(k)$, where $C_{2^{n}}$ is the cyclic group of order
$2^{n}$. This admits a $\operatorname{\mathbb{Z}}/2$-equivariant refinement
$\operatorname{TRR}^{n+1}(k;2)$, constructed by extending the
$C_{2^{n}}$-equivariant structure on $\operatorname{THH}(k)$ to an equivariant
spectrum $\operatorname{THR}(k)$ for the dihedral group $D_{2^{n}}$ of order
$2^{n+1}$. The fixed-point spectrum

$\operatorname{TRR}^{n+1}(k;2):=\operatorname{THR}(k)^{C_{2^{n}}}$

then inherits the structure of a $\operatorname{\mathbb{Z}}/2$-spectrum, since
$\operatorname{\mathbb{Z}}/2$ is the Weyl group of $C_{2^{n}}$ in $D_{2^{n}}$.
This construction is carried out in [Høg16] and [DMP24, §1], and we review it
in §1. There is then a canonical ring homomorphism

$\operatorname{res}^{D_{2^{n}}}_{C_{2^{n}}}\colon\pi_{0}\operatorname{TRR}^{n+1}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}\longrightarrow\pi_{0}\operatorname{TR}^{n+1}(k;2)/2,$

where $(-)^{\phi\operatorname{\mathbb{Z}}/2}$ denotes the geometric fixed-
points functor, and we let $J_{\langle 2^{n}\rangle}$ be its kernel. There are
operators between these spectra

$\textstyle{\operatorname{TRR}^{n+1}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{R}$$\scriptstyle{F}$$\textstyle{\operatorname{TRR}^{n}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{V}$

which correspond to the usual respective maps $R$, $V$ and $F$ on
$\operatorname{TR}^{n+1}(k;2)$ under the restriction map above. There is also
a map

$\sigma\colon\operatorname{TRR}^{n+1}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}\longrightarrow\operatorname{TRR}^{n+1}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}$

of order $2$, which is induced by the action of the Weyl group of
$\operatorname{\mathbb{Z}}/2$ in the quotient $D_{2^{n+1}}/C_{2^{n}}$. It is
easy to see that all these maps restrict to maps between the kernels
$J_{\langle 2^{n}\rangle}$. The main result of the paper is the following
analogue of Milnor’s conjecture.

######  Theorem A.

Let $k$ be a field of characteristic $2$. The maps $R,F,V$ and $d:=1+\sigma$
endow the sequence $J_{\langle 2^{\bullet}\rangle}^{\ast}/J_{\langle
2^{\bullet}\rangle}^{\ast+1}$ with the structure of a $2$-typical Witt complex
over $k$, and the unique map of $2$-typical Witt complexes over $k$

$(\operatorname{W}_{\langle
2^{\bullet}\rangle}\Omega^{\ast}_{k})/2\longrightarrow J_{\langle
2^{\bullet}\rangle}^{\ast}/J_{\langle 2^{\bullet}\rangle}^{\ast+1}$

is a strict isomorphism.

Let us remark on some special cases of this theorem:

  1. i)

For $\ast=0$, the isomorphism of Theorem A identifies with the modulo $2$
reduction of the isomorphism $\operatorname{W}_{\langle
2^{n}\rangle}(k)\cong\pi_{0}\operatorname{TR}^{n+1}(k;2)$ of [HM97, Theorem
F], where $\operatorname{W}_{\langle 2^{n}\rangle}(k)$ is the ring of
$(n+1)$-truncated $2$-typical Witt vectors of $k$.

  2. ii)

For $\bullet=0$, the isomorphism of Theorem A is the isomorphism
$\Omega^{\ast}_{k}\cong J^{\ast}/J^{\ast+1}$ discussed above.

  3. iii)

If $k$ has characteristic different from $2$, then both
$(\operatorname{W}_{\langle 2^{\bullet}\rangle}\Omega^{\ast}_{k})/2$ and
$\operatorname{TRR}^{n+1}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}$ vanish, so
Theorem A in fact holds in all characteristics.

  4. iv)

If $k$ is perfect of characteristic $2$, then $(\operatorname{W}_{\langle
2^{\bullet}\rangle}\Omega^{\ast}_{k})/2=0$ for $\ast>0$, and

$(\operatorname{W}_{\langle
2^{\bullet}\rangle}\Omega^{0}_{k})/2=\operatorname{W}_{\langle
2^{\bullet}\rangle}(k)/2\cong k.$

Similarly, in the case of perfect fields,
$\pi_{0}\operatorname{TRR}^{n+1}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}\cong
k$ and $J_{\langle 2^{\bullet}\rangle}=0$ (see [DMP24, Theorem 4.7]). Thus,
Theorem A has content only for non-perfect fields of characteristic $2$.

We prove the theorem by first explicitly calculating the homotopy groups of
$\operatorname{TRR}^{n+1}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}$ in §2.1
(even though we really only need $\pi_{0}$), extending the calculations for
perfect fields of [DMP24, §4.2]. We then use our calculation to provide
generators for
$\pi_{0}\operatorname{TRR}^{n+1}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}$ and
$J_{\langle 2^{\bullet}\rangle}$, analogous to the canonical generators
$V^{n-i}\tau_{i}(a)$ of the Witt vectors (see Propositions 2.8 and 2.15). This
allows us to define a Witt-complex structure on $J_{\langle
2^{\bullet}\rangle}^{\ast}/J_{\langle 2^{\bullet}\rangle}^{\ast+1}$, and to
prove Theorem A by induction on $\bullet$, using the exact sequences of
[Cos08, Lemma 3.5], in §3.2.

The description of the homotopy groups of
$\operatorname{TRR}^{n+1}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}$ is in
Theorem 2.6, and it is proved using the pullbacks of [DMP24, Theorem 2.7]. It
is somewhat technical, and we won’t state it here, but it is completely
explicit. There is however a closely related calculation which is more
straightforward to state. Let $\operatorname{TCR}(k;2)$ be the real
topological cyclic homology spectrum of $k$, which we may define as the
equaliser

$\operatorname{TCR}(k;2):=eq\big{(}\lx@xy@svg{\hbox{\raise 0.0pt\hbox{\kern
25.34377pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{TRR}(k;2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
31.4271pt\raise 7.58333pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{\operatorname{id}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 49.34377pt\raise
2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
31.60696pt\raise-7.54443pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.39166pt\hbox{$\scriptstyle{F}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
49.34377pt\raise-2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
49.34377pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{TRR}(k;2)}$}}}}}}}\ignorespaces}}}}\ignorespaces\big{)},$

where $\operatorname{TRR}(k;2)$ is the limit of
$\operatorname{TRR}^{n+1}(k;2)$ over the maps $R$. Let us point out that, by
[DMP24, Theorem A], if $2$ is a unit in $k$, then
$\operatorname{TCR}(k;p)^{\phi\operatorname{\mathbb{Z}}/2}=0$ for every prime
$p$, so that we may assume that $k$ has characteristic $2$. Let $C_{2}$ act on
$k\otimes_{S}k$ by swapping the two tensor factors, where $S\leq k$ is the
subfield of squares. Let us denote by $w$ the generator of $C_{2}$. The
following is proved in Corollary 2.4.

######  Theorem B.

Let $k$ be a field of characteristic $2$. For every integer $l\geq 0$, there
is an exact sequence

$0\to\pi_{2l}\operatorname{TCR}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}\to(k\otimes_{S}k)^{C_{2}}\xrightarrow{\pi-\phi}(k\otimes_{S}k)^{C_{2}}/Im(1+w)\to\pi_{2l-1}\operatorname{TCR}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}\to
0,$

where $\pi$ is the quotient map, and $\phi$ is the ring homomorphism defined
by $\phi(a\otimes b)=ba^{2}\otimes b$.

The map $\phi$ in fact determines an isomorphism $\phi\colon
k\otimes_{S}k\to(k\otimes_{S}k)^{C_{2}}/Im(1+w)$, and is in a sense a
replacement of the Frobenius of $k$ when $k$ is not perfect (see Lemma 2.1).
It plays a crucial role in the calculations of §2.1 and in the description of
$J_{\langle 2^{\bullet}\rangle}$.

In [Kat82, Theorem (1)], Kato exhibits a closely related exact sequence,
involving the symmetric and quadratic Witt groups $\operatorname{W}^{s}(k)$
and $\operatorname{W}^{q}(k)$. Combined with Theorem B, it gives isomorphisms

$\pi_{2l}\operatorname{TCR}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}\cong\operatorname{W}^{s}(k)\
\ \ \ \ \ \mbox{and}\ \ \ \ \ \
\pi_{2l-1}\operatorname{TCR}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}\cong\operatorname{W}^{q}(k)$

for every $l\geq 0$. In fact, this identifies the homotopy groups of
$\operatorname{TCR}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}$ with the genuine
normal L-groups of $k$, as conjectured by Nikolaus, proved in great generality
in [HNS21], and verified in [DMP24] in the case of perfect fields (see Remark
2.5).

From Theorem B, we can also deduce a version of the Milnor conjecture for
$\operatorname{TC}$. Let us choose the respective equaliser and coequaliser

$\textstyle{\nu^{\ast}_{dRW/2}(k;2)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\operatorname{W}_{\langle
2^{\infty}\rangle}\Omega^{\ast}_{k})/2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{id}}$$\scriptstyle{F}$$\textstyle{(\operatorname{W}_{\langle
2^{\infty}\rangle}\Omega^{\ast}_{k})/2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\epsilon^{\ast}_{dRW/2}(k;2)}$

as possible symbolic versions of topological cyclic homology modulo $2$, where
$\operatorname{W}_{\langle 2^{\infty}\rangle}\Omega^{\ast}_{k}$ is the limit
over the map $R$ of $\operatorname{W}_{\langle
2^{\bullet}\rangle}\Omega^{\ast}_{k}$ (and we are intentionally quotienting
out $2$ before taking the equaliser). Now let

$K:=\ker\big{(}\operatorname{res}^{\operatorname{\mathbb{Z}}/2}_{e}\colon\pi_{0}\operatorname{TCR}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}\longrightarrow(\pi_{0}\operatorname{TC}(k;2))^{\operatorname{\mathbb{Z}}/2}/Im(1+w)\big{)}$

be the kernel of the restriction map, where $w$ is the involution on
$\pi_{0}\operatorname{TC}(k;2)$ induced from the
$\operatorname{\mathbb{Z}}/2$-action on $\operatorname{TCR}(k;2)$. Let us also
denote
$T_{-1}:=\pi_{-1}\operatorname{TCR}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}$,
which we consider as a
$\pi_{0}\operatorname{TCR}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}$-module. The
following is a $\operatorname{TC}$ analogue of [Kat82, Theorem (2)].

######  Corollary C.

For every field $k$ of characteristic $2$, there is an isomorphism of graded
rings

$\nu_{dRW/2}^{\ast}(k;2)\cong K^{\ast}/K^{\ast+1},$

and an isomorphism of graded $K^{\ast}/K^{\ast+1}$-modules

$\epsilon^{\ast}_{dRW/2}(k;2)\cong K^{\ast}T_{-1}/K^{\ast+1}T_{-1}.$

We prove this last result in §3.3. Our argument is fairly straightforward,
combining the Milnor conjecture at the prime $2$, Theorem B, and the
identification from [CMM21, Proposition 2.26] of $\nu_{dRW/2}^{\ast}(k;2)$ and
$\epsilon_{dRW/2}^{\ast}(k;2)$ with the respective equaliser and coequaliser

$\textstyle{\nu^{\ast}(k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Omega^{\ast}_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\scriptstyle{C^{-1}}$$\textstyle{\Omega^{\ast}_{k}/d(\Omega^{\ast-1}_{k})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\epsilon^{\ast}(k)}$

of the projection $\pi$ and the inverse Cartier operator $C^{-1}$.

### Acknowledgements

I genuinely thank Ib Madsen for encouraging me to look into a possible
relation between real THH and the Milnor conjecture, which eventually led to
the ideas of this paper. I also thank Irakli Patchkoria, Thomas Read, and
Damiano Testa for helpful conversations regarding some technical aspects of
the project.

The author is supported by EPSRC grant EP/W019620/1. For the purpose of open
access, the author has applied a Creative Commons Attribution (CC-BY) licence
to any Author Accepted Manuscript version arising from this submission.

###### Contents

  1. 1 Preliminaries on real topological Hochschild homology
  2. 2 Real TR and real TC of fields of characteristic $2$
    1. 2.1 The geometric fixed points of TRR and TCR for fields of characteristic $2$
    2. 2.2 The canonical generators of TRR
    3. 2.3 The fundamental ideal of TRR
  3. 3 Real TR and the de Rham-Witt complex
    1. 3.1 The Witt complex associated to TRR
    2. 3.2 The Milnor conjecture for the deRham-Witt complex
    3. 3.3 The Milnor conjecture and TCR

##  1 Preliminaries on real topological Hochschild homology

Here we recall the basic definitions surrounding real topological cyclic
homology. In order to streamline this section, we recast the definitions in
the special case where the input is a discrete commutative ring $A$ with the
trivial involution (which in the next sections of the paper will be a field
$k$ of characteristic $2$). We refer the details of these constructions to
[DMPR21] and [DMP24], and we will freely use the language of stable
equivariant homotopy theory.

Let $O(2)$ be the infinite dihedral group, that we identify with the semi-
direct product $\operatorname{\mathbb{Z}}/2\rtimes S^{1}$ by choosing the
reflection across the real axis as the generator for
$\operatorname{\mathbb{Z}}/2$. The real topological Hochschild homology of $A$
is a ring $O(2)$-spectrum $\operatorname{THR}(A)$, whose underlying ring
$S^{1}$-spectrum is the topological Hochschild homology spectrum
$\operatorname{THH}(A)$, originally defined in [Bök86] (see also [BHM93] and
[NS18]). It can be constructed, as an $O(2)$-equivariant ring orthogonal
spectrum, as the geometric realisation of the dihedral bar construction

$\operatorname{THR}(A):=|N^{di}\operatorname{H}A|=|[n]\mapsto(\operatorname{H}A)^{\otimes
n+1}|,$

where $\operatorname{H}A$ is (a flat model for) the Eilenberg-MacLane ring
orthogonal $\operatorname{\mathbb{Z}}/2$-spectrum of $A$, and $\otimes$
denotes the smash product of spectra (see [DMPR21]). The action of $O(2)$ is
defined from the structure of a dihedral object in the sense of [FL91, S 1.5,
Example 5] and [Lod87], where the cyclic group $C_{n+1}$ acts in simplicial
degree $n$ by rotating the $n+1$ smash factors, and the reflection acts in
degree $n$ by reversing the order of the last $n$ smash factors.

Now let $p$ be a prime, $n\geq 0$ an integer, and
$D_{p^{n}}=\operatorname{\mathbb{Z}}/2\rtimes C_{p^{n}}$ the finite dihedral
subgroup of $O(2)$ of order $2p^{n}$. Since the Weyl group of $C_{p^{n}}$
inside $D_{p^{n}}$ is $\operatorname{\mathbb{Z}}/2$, the (genuine) fixed-
points ring spectrum $\operatorname{THR}(A)^{C_{p^{n}}}$ is canonically a ring
$\operatorname{\mathbb{Z}}/2$-spectrum. The inclusion of subgroups
$C_{p^{n-1}}\leq C_{p^{n}}$ induces a restriction map $F$, also called
Frobenius, and a transfer map $V$, also called Verschiebung, which are maps of
$\operatorname{\mathbb{Z}}/2$-spectra

$\textstyle{\operatorname{THR}(A)^{C_{p^{n}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\textstyle{\operatorname{THR}(A)^{C_{p^{n-1}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces.}$$\scriptstyle{V}$

There is a further map $R$ of $\operatorname{\mathbb{Z}}/2$-spectra, sometimes
called restriction or truncation

$\operatorname{THR}(A)^{C_{p^{n}}}\stackrel{{\scriptstyle
R}}{{\longrightarrow}}\operatorname{THR}(A)^{C_{p^{n-1}}},$

which is defined from the real cyclotomic structure of $\operatorname{THR}(A)$
(see [DPJM22, Definition 3.9]). The maps $R$ and $F$ are moreover maps of ring
spectra (see [DPJM22, Remark 3.10]). On underlying spectra, these are the maps
$F,V$ and $R$ of $\operatorname{THH}(A)$, which after applying $\pi_{0}$
correspond to the operators on the ring of Witt vectors with the same name,
see [HM97, Theorem 3.3].

######  Definition 1.1.

Let $A$ be a commutative ring, and $p$ a prime. The $p$-typical truncated real
topological restriction homology, real topological restriction homology, and
real topological cyclic homology of $A$ are the ring
$\operatorname{\mathbb{Z}}/2$-spectra defined respectively as:

$\displaystyle\operatorname{TRR}^{n+1}(A;p):=\operatorname{THR}(A)^{C_{p^{n}}},$
$\displaystyle\operatorname{TRR}(A;p):=\lim\big{(}\dots\xrightarrow{R}\operatorname{TRR}^{n+1}(A;p)\xrightarrow{R}\operatorname{TRR}^{n}(A;p)\xrightarrow{R}\dots\xrightarrow{R}\operatorname{TRR}^{1}(A;p)=\operatorname{THR}(A)\big{)},$
$\displaystyle\operatorname{TCR}(A;p):=eq\big{(}\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
26.34897pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{TRR}(A;p)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
32.4323pt\raise 7.58333pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.43056pt\hbox{$\scriptstyle{\operatorname{id}}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 50.34897pt\raise
2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
32.61217pt\raise-7.54443pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-2.39166pt\hbox{$\scriptstyle{F}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
50.34897pt\raise-2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
50.34897pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{TRR}(A;p)}$}}}}}}}\ignorespaces}}}}\ignorespaces\big{)},$

where the map $F$ in the equaliser is induced by the Frobenius maps above,
since $R$ and $F$ commute.

The $\operatorname{\mathbb{Z}}/2$-geometric fixed points of these spectra are
characterised in [DMP24], as we now recall. These results will be used in §2.1
below, and we encourage the reader, at least for the purpose of the present
paper, to take them as definitions of these objects.

In [DMP24, Lemma 1.2], we give a canonical equivalence of ring spectra

$\operatorname{THR}(A)^{\phi\operatorname{\mathbb{Z}}/2}=\operatorname{TRR}^{1}(A;p)^{\phi\operatorname{\mathbb{Z}}/2}\simeq(\operatorname{H}\underline{A})^{\phi\operatorname{\mathbb{Z}}/2}\otimes_{\operatorname{H}A}(\operatorname{H}\underline{A})^{\phi\operatorname{\mathbb{Z}}/2},$

where $\operatorname{H}\underline{A}$ is the Eilenberg MacLane spectrum of the
$\operatorname{\mathbb{Z}}/2$ Mackey functor (or Tambara functor) with
constant value $A$ and transfer map $2$. Its geometric fixed-points spectrum
is then regarded as an $\operatorname{H}A$-module via the map of ring spectra

$\operatorname{H}A\simeq(N^{\operatorname{\mathbb{Z}}/2}_{e}\operatorname{H}A)^{\phi\operatorname{\mathbb{Z}}/2}\xrightarrow{\epsilon^{\phi\operatorname{\mathbb{Z}}/2}}(\operatorname{H}\underline{A})^{\phi\operatorname{\mathbb{Z}}/2},$

where $N^{\operatorname{\mathbb{Z}}/2}_{e}\operatorname{H}A$ is the Hill-
Hopkins-Ravenel norm construction of the ring spectrum $\operatorname{H}A$ of
[HHR16] and [Sto11], and $\epsilon$ is the counit of the free-forgetful
adjunction between commutative ring $\operatorname{\mathbb{Z}}/2$-spectra and
commutative ring spectra. We will call this the Frobenius module structure of
$(\operatorname{H}\underline{A})^{\phi\operatorname{\mathbb{Z}}/2}$, and refer
to [DMPR21, §2.5] for the details of its construction. The Weyl group of
$\operatorname{\mathbb{Z}}/2$ in $D_{2}=\operatorname{\mathbb{Z}}/2\times
C_{2}$ is $C_{2}$, and therefore
$\operatorname{THR}(A)^{\phi\operatorname{\mathbb{Z}}/2}$ is canonically a
ring $C_{2}$-spectrum. In [DMP24, Lemma 1.2], we lift the equivalence above to
an equivalence of ring $C_{2}$-spectra

$\operatorname{THR}(A)^{\phi\operatorname{\mathbb{Z}}/2}\simeq\operatorname{H}\underline{A}\otimes_{N^{C_{2}}_{e}\operatorname{H}A}N^{C_{2}}_{e}((\operatorname{H}\underline{A})^{\phi\operatorname{\mathbb{Z}}/2}),$

where the right factor is a module by applying the norm to the map
$\operatorname{H}A\to(\operatorname{H}\underline{A})^{\phi\operatorname{\mathbb{Z}}/2}$,
and the left factor is now regarded as a $C_{2}$-spectrum.

This $C_{2}$-equivariant homotopy type will help us characterise the
$\operatorname{\mathbb{Z}}/2$-geometric fixed points of
$\operatorname{TRR}^{n+1}(A;p)$, inductively on $n$. For every $n\geq 1$, the
$\operatorname{\mathbb{Z}}/2$-geometric fixed points of
$\operatorname{TRR}^{n+1}(A;p)$ is equivalent to the product of $(n+1)$-copies
of $\operatorname{THR}(A)^{\phi\operatorname{\mathbb{Z}}/2}$ if $p$ is odd,
see [DMP24, Theorem 2.1]. For $p=2$, they are given by a pullback of ring
spectra

$\textstyle{\operatorname{TRR}^{n+1}(A;2)^{\phi\operatorname{\mathbb{Z}}/2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{R}$$\scriptstyle{cF^{n-1}\times
cF^{n-1}\sigma_{n+1}}$$\textstyle{\operatorname{TRR}^{n}(A;2)^{\phi\operatorname{\mathbb{Z}}/2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(F^{n-1},\sigma_{1}F^{n-1}\sigma_{n})}$$\textstyle{(\operatorname{THR}(A)^{\phi\operatorname{\mathbb{Z}}/2})^{C_{2}}\times(\operatorname{THR}(A)^{\phi\operatorname{\mathbb{Z}}/2})^{C_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r\times\sigma_{1}r}$$\textstyle{\operatorname{THR}(A)^{\phi\operatorname{\mathbb{Z}}/2}\times\operatorname{THR}(A)^{\phi\operatorname{\mathbb{Z}}/2},}$

see [DMP24, Theorem 2.7]. Here $\sigma_{n}$ is the generator of the Weyl group
of $\operatorname{\mathbb{Z}}/2$ inside the quotient $D_{2^{n}}/C_{2^{n-1}}$,
which is also of order $2$. The map

$c\colon(\operatorname{THR}(A)^{C_{2}})^{\phi\operatorname{\mathbb{Z}}/2}\longrightarrow(\operatorname{THR}(A)^{\phi\operatorname{\mathbb{Z}}/2})^{C_{2}}$

is a certain canonical map, and $r$ is the canonical map to the
$C_{2}$-geometric fixed points followed by the equivalence given by the
cyclotomic structure (see above [DMP24, Theorem 2.7] for the definitions).

In [DMP24, Theorem A], we also characterise the real topological cyclic
homology of $A$, by providing an equivalence of ring spectra

$\operatorname{TCR}(A;2)^{\phi\operatorname{\mathbb{Z}}/2}\simeq\big{(}\lx@xy@svg{\hbox{\raise
0.0pt\hbox{\kern
34.75043pt\hbox{\ignorespaces\ignorespaces\ignorespaces\hbox{\vtop{\kern
0.0pt\offinterlineskip\halign{\entry@#!@&&\entry<EMAIL_ADDRESS>0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern 3.0pt\raise
0.0pt\hbox{$\textstyle{(\operatorname{THR}(A)^{\phi\operatorname{\mathbb{Z}}/2})^{C_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$}}}}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
42.07416pt\raise 6.65971pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.50694pt\hbox{$\scriptstyle{r}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern 58.75043pt\raise
2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces{}{\hbox{\lx@xy@droprule}}\ignorespaces\ignorespaces\ignorespaces{\hbox{\kern
41.66014pt\raise-8.26387pt\hbox{{}\hbox{\kern 0.0pt\raise
0.0pt\hbox{\hbox{\kern 3.0pt\hbox{\hbox{\kern
0.0pt\raise-1.75pt\hbox{$\scriptstyle{f}$}}}\kern
3.0pt}}}}}}\ignorespaces{\hbox{\kern
58.75043pt\raise-2.15277pt\hbox{\hbox{\kern 0.0pt\raise
0.0pt\hbox{\lx@xy@tip{1}\lx@xy@tip{-1}}}}}}{\hbox{\lx@xy@droprule}}{\hbox{\lx@xy@droprule}}{\hbox{\kern
58.75043pt\raise 0.0pt\hbox{\hbox{\kern 0.0pt\raise 0.0pt\hbox{\hbox{\kern
3.0pt\raise
0.0pt\hbox{$\textstyle{\operatorname{THR}(A)^{\phi\operatorname{\mathbb{Z}}/2}}$}}}}}}}\ignorespaces}}}}\ignorespaces\big{)},$

where $f$ is the forgetful map.

Finally, we will need to briefly use the existence of norm maps on
$\operatorname{THR}(A)$ in order calculate a certain restriction map, in
Propositions 2.11 and 2.13. To establish their existence, we simply observe
that the dihedral bar construction employed above to define
$\operatorname{THR}$ has a canonical symmetric monoidal structure, and
therefore $\operatorname{THR}(A)$ is a strictly commutative $O(2)$-equivariant
ring spectrum (provided we choose a strictly commutative and flat model for
the Eilenberg-MacLane ring $C_{2}$-spectrum $\operatorname{H}\underline{A}$,
which we can achieve by a cofibrant replacement in the flat model structure of
[Sto11, BDS16]). Thus, we obtain non-additive norm maps

$N_{H}^{G}\colon\pi_{0}\operatorname{THR}(A)^{H}\longrightarrow\pi_{0}\operatorname{THR}(A)^{G}$

for every pair of finite subgroups $H\leq G\leq O(2)$, which, when composed
with a restriction map, satisfy the multiplicative double-coset formula.

##  2 Real TR and real TC of fields of characteristic $2$

###  2.1 The geometric fixed points of TRR and TCR for fields of
characteristic $2$

Let $k$ be a field of characteristic $2$, and $S\leq k$ the subfield of
squares. We regard $k$ as an $S$-vector space, and endow the abelian group
$k\otimes_{S}k$ with the involution $w$ which flips the two tensor factors.

The homotopy groups of
$\operatorname{TRR}(k)^{\phi\operatorname{\mathbb{Z}}/2}$ have been computed
in [DMP24, Theorem 4.7, Corollary 4.8] when the field $k$ is perfect, as a sum
of copies of $k$. In this section we give an analogous description of these
homotopy groups for a general field of characteristic $2$ (and an analogous
proof), where some of the copies of $k$ appearing in the calculation for
perfect fields are replaced by expressions involving $k\otimes_{S}k$ (which is
isomorphic to $k$ if $k$ is perfect). This is Theorem 2.6 below, and its
statement and proof will be the content of §2.1.

The key algebraic input for extending the calculation to non-perfect fields
lies in the following Lemma, which we will use several times throughout the
paper. For every elementary tensor $a\otimes b\in k\otimes_{S}k$, let us
define

$\phi(a\otimes b):=ba^{2}\otimes b\in(k\otimes_{S}k)^{C_{2}},$

where the $C_{2}$-invariants on the right are with respect to the involution
$w$. We note that this map does not obviously extend to $k\otimes_{S}k$, as it
is unclear how to define it on a sum of elementary tensors. It will serve as a
replacement of the Frobenius of $k$, and will be related to the cyclotomic
structure of $\operatorname{THR}(k)$ by Proposition 2.3, and to the fibre
sequence of [Kat82, Theorem (1)] describing the Witt groups of $k$ in Remark
2.5.

######  Lemma 2.1.

The assignment $\phi$ induces a well-defined additive isomorphism
$k\otimes_{S}k\xrightarrow{\cong}(k\otimes_{S}k)^{C_{2}}/Im(1+w)$. This
isomorphism moreover fits into a commutative diagram

$\textstyle{k\otimes_{S}k\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\scriptstyle{\cong}$$\scriptstyle{\phi}$$\textstyle{(k\otimes_{S}k)^{C_{2}}/Im(1+w)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu}$$\textstyle{k\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(-)^{2}}$$\textstyle{k}$

where the map $\mu$ is the multiplication map, which is an isomorphism if and
only if $k$ is perfect.

###### Proof.

It is easy to see that $\phi$ extends to a well-defined additive map after we
quotient the image of $1+w$ in the target. To see that it is an isomorphism,
choose a basis $k\cong\oplus_{X}S$ of $k$ as an $S$-vector space. This induces
an isomorphism of $C_{2}$-equivariant abelian groups

$k\otimes_{S}k\cong\bigoplus_{X\times X}S$

where the involution on the right-hand side sends a basis element $(x,y)$ of
$X\times X$ to $(y,x)$. Under this isomorphism, the map $\phi$ corresponds to
the map

$\bigoplus_{X\times
X}S\cong\bigoplus_{X}(\bigoplus_{X}S)\cong\bigoplus_{X}k\cong(\bigoplus_{X\times
X}S)^{C_{2}}/Im(1+w)$

where the second isomorphism is the sum over $X$ of the isomorphism
$k\cong\oplus_{X}S$, and the last isomorphism sends the summand $x$ to the
summand $(x,x)$ via the square map $(-)^{2}\colon k\xrightarrow{\cong}S$. ∎

We calculate the homotopy groups of
$\operatorname{TRR}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}$ using the iterated
pullback description of [DMP24, Theorem 2.7], reviewed in §1. This description
relies on the $C_{2}$-equivariant homotopy type of
$\operatorname{THR}(k)^{{\phi\operatorname{\mathbb{Z}}/2}}$, which we
calculate in the following proposition using Lemma 2.1. We denote by
$\operatorname{H}(k\otimes_{S}k,w)$ the $C_{2}$-equivariant Eilenberg-MacLane
spectrum of the abelian group $k\otimes_{S}k$ with $C_{2}$-action $w$.

######  Proposition 2.2.

Let $k$ be a field of characteristic $2$. Then there is a natural equivalence
of $C_{2}$-equivariant spectra

${\operatorname{THR}(k)}^{{\phi\operatorname{\mathbb{Z}}/2}}\simeq\bigoplus_{n\geq
0}\Sigma^{n\rho}\operatorname{H}(k\otimes_{S}k,w)\oplus\bigoplus_{\begin{smallmatrix}(n,m)\\\
0\leq
n<m\end{smallmatrix}}\Sigma^{n+m}{C_{2}}_{+}\otimes\operatorname{H}(k\otimes_{S}k),$

where $\rho$ is the regular representation of $C_{2}$. It follows that there
is a natural equivalence of spectra

$({\operatorname{THR}(k)}^{{\phi\operatorname{\mathbb{Z}}/2}})^{C_{2}}\simeq(\bigoplus_{n\geq
0}\big{(}(\bigoplus_{0\leq
j<n}\Sigma^{n+j}\operatorname{H}(k\otimes_{S}k))\oplus\Sigma^{2n}\operatorname{H}(k\otimes_{S}k)^{C_{2}}\big{)})\oplus(\bigoplus_{\begin{smallmatrix}(n,m)\\\
0\leq n<m\end{smallmatrix}}\Sigma^{n+m}\operatorname{H}(k\otimes_{S}k)).$

###### Proof.

Let $\operatorname{H}\underline{k}$ be the Eilenberg MacLane $C_{2}$-spectrum
of the ring with trivial involution $k$. The geometric fixed points of
$\operatorname{H}\underline{k}$ equipped with the Frobenius module structure
of §1 decompose, as a $k$-module, as a direct sum

$\operatorname{H}\underline{k}^{{\phi\operatorname{\mathbb{Z}}/2}}\simeq\bigoplus_{n\geq
0}\Sigma^{n}\operatorname{H}(\varphi^{\ast}k),$

where $\varphi=(-)^{2}\colon k\to k$ denotes the Frobenius homomorphism of
$k$. By [DMP24, Lemma 4.3], there is an equivalence of $C_{2}$-spectra

${\operatorname{THR}(k)}^{{\phi\operatorname{\mathbb{Z}}/2}}\simeq\bigoplus_{n\geq
0}\Sigma^{n\rho}\operatorname{H}\underline{k}\otimes_{N_{e}^{C_{2}}Hk}N_{e}^{C_{2}}\operatorname{H}(\varphi^{\ast}k)\oplus\bigoplus_{\begin{smallmatrix}(n,m)\\\
0\leq
n<m\end{smallmatrix}}\Sigma^{n+m}{C_{2}}_{+}\otimes\operatorname{H}(\varphi^{\ast}k\otimes_{k}\varphi^{\ast}k),$

which is moreover natural in $k$ since the decomposition of
$\operatorname{H}\underline{k}^{{\phi\operatorname{\mathbb{Z}}/2}}$ above is.
Clearly $\varphi^{\ast}k\otimes_{k}\varphi^{\ast}k=k\otimes_{S}k$, and
therefore to obtain the first decomposition of the Lemma it is sufficient to
show that the canonical map

$\operatorname{H}\underline{k}\otimes_{N_{e}^{C_{2}}Hk}N_{e}^{C_{2}}\operatorname{H}(\varphi^{\ast}k)\longrightarrow\operatorname{H}(k\otimes_{k\otimes
k}(\varphi^{\ast}k\otimes\varphi^{\ast}k))\cong\operatorname{H}(k\otimes_{S}k,w)$

is an equivalence, where the middle term is $\pi_{0}$ of the underlying
spectrum of the left term, with the induced involution.

Let us choose a basis of the $k$-vector space $\varphi^{\ast}k$, that is we
write $\varphi^{\ast}k$ as a direct sum

$\varphi^{\ast}k\cong\bigoplus_{X}k$

over some set $X$. Since the norm commutes with direct sums we obtain an
equivalence of $C_{2}$-spectra

$\displaystyle\operatorname{H}\underline{k}\otimes_{N_{e}^{C_{2}}\operatorname{H}k}N_{e}^{C_{2}}\operatorname{H}(\varphi^{\ast}k)$
$\displaystyle\simeq\operatorname{H}\underline{k}\otimes_{N_{e}^{C_{2}}\operatorname{H}k}N_{e}^{C_{2}}\operatorname{H}(\bigoplus_{X}k)$
$\displaystyle\simeq\bigoplus_{X\times
X}\operatorname{H}\underline{k}\otimes_{N_{e}^{C_{2}}\operatorname{H}k}N_{e}^{C_{2}}\operatorname{H}k\simeq\bigoplus_{X\times
X}\operatorname{H}\underline{k},$

where the last term is the indexed sum of $\operatorname{H}\underline{k}$ with
the involution on $X\times X$ that swaps the product factors. Under this
equivalence, the canonical map above corresponds to the equivalence

$\bigoplus_{X\times
X}\operatorname{H}\underline{k}\cong\operatorname{H}((\bigoplus_{X}k)\otimes_{k}(\bigoplus_{X}k))\cong\operatorname{H}((\varphi^{\ast}k)\otimes_{k}(\varphi^{\ast}k))=\operatorname{H}(k\otimes_{S}k)$

where the middle isomorphism is the tensor product of two copies of the choice
of basis above.

Now let us identify the $C_{2}$-fixed points of
${\operatorname{THR}(k)}^{{\phi\operatorname{\mathbb{Z}}/2}}$. Notice that
$\operatorname{H}(k\otimes_{S}k,w)$ is a module over
$\operatorname{H}\underline{k}$ (via the ring map $k\to k\otimes_{S}k$ that
sends $a$ to $a^{2}\otimes 1$) and therefore its $C_{2}$-fixed-points spectrum
is an $\operatorname{H}k$-module, and therefore it decomposes canonically as a
wedge of Eilenberg-MacLane spectra. Its homotopy groups are isomorphic to the
Bredon homology groups

$\pi_{i}^{C_{2}}(\Sigma^{n\rho}\operatorname{H}(k\otimes_{S}k,w))=H^{C_{2}}_{i}(S^{n\rho};(k\otimes_{S}k,w)),$

which in turn are the homology groups of the chain complex

$0\xleftarrow{}(k\otimes_{S}k)^{C_{2}}\xleftarrow{1+w}k\otimes_{S}k\xleftarrow{1+w}k\otimes_{S}k\xleftarrow{1+w}\dots\xleftarrow{1+w}k\otimes_{S}k\xleftarrow{}0$

where the first non-zero group on the left is in degree $n$ and the last non-
zero group on the right is in degree $2n$ (notice that all the signs on the
arrows are $+$ since $k$ has characteristic $2$). It follows that all the
groups below $n$ and above $2n$ vanish, that

$\pi_{2n}^{C_{2}}(\Sigma^{n\rho}\operatorname{H}(k\otimes_{S}k,w))\cong(k\otimes_{S}k)^{C_{2}}$

and that

$\pi_{i}^{C_{2}}(\Sigma^{n\rho}\operatorname{H}(k\otimes_{S}k,w))\cong(k\otimes_{S}k)^{C_{2}}/Im(1+w)\xleftarrow{\cong}k\otimes_{S}k$

for every $n\leq i<2n$, where the left-pointing isomorphism is the map $\phi$
from Lemma 2.1. ∎

In order to calculate the homotopy groups of
$\operatorname{TRR}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}$ and
$\operatorname{TCR}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}$, we also need to
determine the maps $r$ and $f$ (see §1), under the equivalences of Proposition
2.2. In the following proposition, the summands are arranged exactly as in
Proposition 2.2. In particular, the summands indexed on $(n,m)$ with $n<m$ in
the source, and those indexed on $(n,m)$ with $n\neq m$ in the target,
correspond to the induced summands.

######  Proposition 2.3.

For any perfect field $k$ of characteristic $2$, the maps
$r,f\colon{({\operatorname{THR}(k)}^{{\phi\operatorname{\mathbb{Z}}/2}})}^{C_{2}}\to{\operatorname{THR}(k)}^{{\phi\operatorname{\mathbb{Z}}/2}}$
induce on $\pi_{\ast}$ the maps

$r,f\colon(\bigoplus_{\begin{smallmatrix}(n,m)\\\ n+m=\ast\\\ n>m\geq
0\end{smallmatrix}}k\otimes_{S}k)\oplus(\bigoplus_{\begin{smallmatrix}(n,n)\\\
2n=\ast\\\ n\geq
0\end{smallmatrix}}(k\otimes_{S}k)^{C_{2}})\oplus(\bigoplus_{\begin{smallmatrix}(n,m)\\\
n+m=\ast\\\ 0\leq
n<m\end{smallmatrix}}k\otimes_{S}k)\longrightarrow\bigoplus_{\begin{smallmatrix}(n,m)\\\
n+m=\ast\\\ n,m\geq 0\end{smallmatrix}}k\otimes_{S}k,$

where $r$ kills the $(n,m)$-summands with $n<m$, maps the $(n,m)$-summands
with $n>m$ to the $(n,m)$-summand via the identity, and maps the
$(n,n)$-summand to the $(n,n)$-summand via the composite

$(k\otimes_{S}k)^{C_{2}}\stackrel{{\scriptstyle\pi}}{{\longrightarrow}}(k\otimes_{S}k)^{C_{2}}/Im(1+w)\xrightarrow{\phi^{-1}}k\otimes_{S}k$

of the quotient map and the isomorphism of Lemma 2.1. The map $f$ kills the
$(n,m)$-summands with $n>m$, is the identity on the summand $(n,n)$, and
embeds diagonally the $(n,m)$-summands with $n<m$ into the sum of the summands
$(n,m)$ and $(m,n)$.

###### Proof.

By [DMP24, Lemma 4.3], the map $r$ vanishes on the summands $(n,m)$ with
$n<m$. By the same Lemma, under the identification of Proposition 2.2, it is
given on the other summands, for a fixed $n\geq 0$, by the outer composite of
the maps in the diagram

$\textstyle{(\Sigma^{n\rho}\operatorname{H}(k\otimes_{S}k,w))^{C_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$Prop
2.2$\textstyle{\Sigma^{n}\operatorname{H}(k\otimes_{S}k,w)^{\phi
C_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\Sigma^{n}(\operatorname{H}\underline{k}\otimes_{N_{e}^{C_{2}}\operatorname{H}k}N_{e}^{C_{2}}\operatorname{H}(\varphi^{\ast}k))^{\phi
C_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\displaystyle(\bigoplus_{0\leq
j<n}\Sigma^{n+j}\operatorname{H}(k\otimes_{S}k))\oplus\Sigma^{2n}\operatorname{H}(k\otimes_{S}k)^{C_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\displaystyle\bigoplus_{0\leq
j}\Sigma^{n+j}\operatorname{H}\big{(}\frac{(k\otimes_{S}k)^{C_{2}}}{Im(1+w)}\big{)}}$$\textstyle{\Sigma^{n}(\operatorname{H}\underline{k})^{\phi
C_{2}}\otimes_{(N_{e}^{C_{2}}\operatorname{H}k)^{\phi
C_{2}}}(N_{e}^{C_{2}}\operatorname{H}(\varphi^{\ast}k))^{\phi
C_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\Sigma^{n}(\operatorname{H}\underline{k})^{\phi
C_{2}}\otimes_{\operatorname{H}k}\operatorname{H}(\varphi^{\ast}k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\scriptstyle{\simeq}$$\textstyle{\displaystyle\bigoplus_{j\geq
0}\Sigma^{n+j}\operatorname{H}(k\otimes_{S}k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{=}$$\textstyle{\displaystyle\bigoplus_{j\geq
0}\Sigma^{n+j}(\operatorname{H}\varphi^{\ast}k)\otimes_{\operatorname{H}k}(\operatorname{H}\varphi^{\ast}k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{\displaystyle\Sigma^{n}(\bigoplus_{j\geq
0}\Sigma^{j}\operatorname{H}(\varphi^{\ast}k))\otimes_{\operatorname{H}k}(\operatorname{H}\varphi^{\ast}k).}$

Here, the left map on the top row is the canonical map, and the right map on
the top row is the equivalence of the proof of Proposition 2.2. In the right
column, the top vertical map is the monoidality of the geometric fixed points,
the second map is the diagonal equivalence, and the third one is the splitting
induced by the Frobenius module structure. The two bottom horizontal maps are
the canonical equivalences.

Let us now consider the top left square. Its right vertical equivalence is
given by splitting $\operatorname{H}(k\otimes_{S}k,w)^{\phi C_{2}}$ as the sum
of its homotopy groups using the $\operatorname{H}k$-module induced by the map
$k\to k\otimes_{S}k$ as we did in Proposition 2.2 for
$\operatorname{H}(k\otimes_{S}k,w)^{C_{2}}$, and then by identifying these
homotopy groups with the homology of the chain complex

$0\xleftarrow{}(k\otimes_{S}k)^{C_{2}}\xleftarrow{1+w}k\otimes_{S}k\xleftarrow{1+w}k\otimes_{S}k\xleftarrow{1+w}\cdots\
,$

where the first non-zero group on the left is in degree zero. The horizontal
map on the second row sends the summand $j<n$ to the summand $j$ via $\phi$,
and it maps the last summand to the summand $j=2n$ via the projection map
(here $\phi$ appears because we used it to identify the homotopy groups of the
source of the map in the proof of Proposition 2.2). The square commutes by the
naturality of the canonical map from fixed points to geometric fixed points.

Thus, the identification of the map $r$ follows once we prove that the
equivalence from the bottom left corner of the diagram to the second entry of
the second row is the map $\phi$ on homotopy groups. Let
$a,b\colon\mathbb{S}\to\operatorname{H}k$, so that the suspension of $a\otimes
b\colon\mathbb{S}\to\operatorname{H}(k\otimes_{S}k)$ is a generator of a
homotopy group of the bottom left entry of the diagram. The composite of the
equivalences up to the top right corner of the diagram sends $a\otimes b$ to
the element of the homotopy group represented by $a\otimes N_{e}^{C_{2}}(b)$.
The remaining two equivalences send this to the element represented by $b\cdot
a\otimes b$, where the multiplication is with respect to the $k$-module action
on $\varphi^{\ast}k$, and this is precisely $ba^{2}\otimes b=\phi(a\otimes
b)$.

The identification of $f$ is simpler: by [DMP24, Lemma 4.3], it is the
diagonal on the summands $(n,m)$ with $n<m$. The identification on the other
summands follows from the fact that the restriction map

$\operatorname{res}^{C_{2}}_{e}\colon
H^{C_{2}}_{*}(S^{n\rho};(k\otimes_{S}k,w))\to H_{*}(S^{2n};k\otimes_{S}k)$

is the inclusion of fixed points in degree $*=2n$, and zero otherwise. ∎

######  Corollary 2.4.

For every field $k$ of characteristic $2$, and every integer $l\geq 0$, there
is an exact sequence

$0\to\pi_{2l}\operatorname{TCR}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}\to(k\otimes_{S}k)^{C_{2}}\xrightarrow{\pi-\phi}(k\otimes_{S}k)^{C_{2}}/Im(1+w)\to\pi_{2l-1}\operatorname{TCR}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}\to
0$

where $\pi$ quotients the image of $1+w$, and $\phi$ is the isomorphism of
Lemma 2.1 restricted to the fixed points. By Kato’s calculation [Kat82,
Theorem (1)], this identifies
$\pi_{2l}\operatorname{TCR}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}$ with the
symmetric Witt group of $k$, and
$\pi_{2l-1}\operatorname{TCR}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}$ with the
quadratic Witt group of $k$.

###### Proof.

By Proposition 2.3, the map $r-f$ is an isomorphism in $\pi_{\ast}$ when
restricted and corestricted to the summands $(n,m)$ with $n\neq m$. It is
therefore an isomorphism in odd degrees, and its long exact sequence
decomposes into exact sequences

$0\to\pi_{2l}\operatorname{TCR}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}\to(k\otimes_{S}k)^{C_{2}}\oplus\bigoplus_{\begin{smallmatrix}(n,m)\\\
n+m=2l\\\ n,m\geq 0\\\ n\neq
m\end{smallmatrix}}k\otimes_{S}k\xrightarrow{r-f}\bigoplus_{\begin{smallmatrix}(n,m)\\\
n+m=2l\\\ n,m\geq
0\end{smallmatrix}}k\otimes_{S}k\to\pi_{2l-1}\operatorname{TCR}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}\to
0$

for every $l\geq 0$. Again by Proposition 2.3, the kernel and cokernel of
$r-f$ are the same as those of $\iota-\phi^{-1}\pi$. These are respectively
equivalent to the kernel and cokernel of $\pi-\phi$, by applying the
isomorphism $\phi$ of Lemma 2.1 to the target.

In [Kat82], Kato exhibits an exact sequence

$0\to\operatorname{W}^{s}(k)\to
k\otimes_{S}k\xrightarrow{\pi-\phi}(k\otimes_{S}k)/Im(1+w)\to\operatorname{W}^{q}(k)\to
0$

where $\operatorname{W}^{s}(k)$ and $\operatorname{W}^{q}(k)$ are respectively
the symmetric and quadratic Witt groups of $k$. It is easy to see that the
kernel and cokernel of $\pi-\phi$ agree with those above, by restricting and
corestricting the maps to the fixed points. ∎

######  Remark 2.5.

Corollary 2.4 in particular shows that the homotopy groups of the spectrum
$\operatorname{TCR}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}$ agree with the
homotopy groups of the cofibre $\operatorname{L}^{n}(k)$ of the canonical map

$\operatorname{L}^{q}(k)\longrightarrow\operatorname{L}(\operatorname{Mod}^{\omega}_{A},\text{\Qoppa}^{gs}_{k})$

induced by the symmetrisation map from the quadratic to the genuine Poincaré
structure, as defined in [CDH+23, CDH+20a, CDH+20b]. This confirms a
conjecture of Nikolaus, proved in [HNS21], in the case of fields. This is
because the even and odd homotopy groups of $\operatorname{L}^{n}(k)$, in
degrees greater or equal to $-1$, are respectively the Witt groups of
symmetric and quadratic forms of $k$, as explained in [DMP24, Remark 4.6].

Let us denote by
$\pi\colon(k\otimes_{S}k)^{C_{2}}\to(k\otimes_{S}k)^{C_{2}}/Im(1+w)$ the
projection map, so that for every $x\in(k\otimes_{S}k)^{C_{2}}$ we can
consider the element $(\phi^{-1}\pi)(x)$ of $k\otimes_{S}k$. For every $n\geq
0$, we define a subgroup of $(k\otimes_{S}k)^{C_{2}}$ by

$\displaystyle\phi^{n}\big{(}(k\otimes_{S}k)^{C_{2}}\big{)}:=\\{$
$\displaystyle x\in(k\otimes_{S}k)^{C_{2}}\ |\ \ \
(\phi^{-1}\pi)(x)\in(k\otimes_{S}k)^{C_{2}},(\phi^{-1}\pi)^{2}(x)\in(k\otimes_{S}k)^{C_{2}},$
$\displaystyle(\phi^{-1}\pi)^{3}(x)\in(k\otimes_{S}k)^{C_{2}},\ldots,(\phi^{-1}\pi)^{n}(x)\in(k\otimes_{S}k)^{C_{2}}\\}$

where by convention
$\phi^{0}((k\otimes_{S}k)^{C_{2}})=(k\otimes_{S}k)^{C_{2}}$. Thus, by
construction, there is a well-defined map

$\phi^{-1}\pi\colon\phi^{n}\big{(}(k\otimes_{S}k)^{C_{2}}\big{)}\longrightarrow\phi^{n-1}\big{(}(k\otimes_{S}k)^{C_{2}}\big{)},$

for every $n\geq 1$, and a map
$(\phi^{-1}\pi)^{n+1}\colon\phi^{n}\big{(}(k\otimes_{S}k)^{C_{2}}\big{)}\to
k\otimes_{S}k$. Let us consider the pullback

$\textstyle{\phi^{n}\big{(}(k\otimes_{S}k)^{C_{2}}\big{)}\times_{k\otimes_{S}k}\phi^{n}\big{(}(k\otimes_{S}k)^{C_{2}}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\phi^{n}\big{(}(k\otimes_{S}k)^{C_{2}}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{w(\phi^{-1}\pi)^{n+1}}$$\textstyle{\phi^{n}\big{(}(k\otimes_{S}k)^{C_{2}}\big{)}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\phi^{-1}\pi)^{n+1}}$$\textstyle{k\otimes_{S}k}$

where we keep in mind that one of the two maps which we pull back is composed
with the involution $w$ of $k\otimes_{S}k$.

######  Theorem 2.6.

Let $k$ be a field of characteristic $2$. For any $l\geq 1$, there is an
isomorphism

$\pi_{\ast}{\operatorname{TRR}^{l+1}(k;2)}^{{\phi\operatorname{\mathbb{Z}}/2}}\cong\left\\{\begin{array}[]{ll}\displaystyle\big{(}\bigoplus_{\begin{smallmatrix}(n,m)\\\
n+m=\ast\\\ n,m\geq 0\\\ n\neq
m\end{smallmatrix}}k\otimes_{S}k\big{)}\oplus\big{(}\phi^{l-1}\big{(}(k\otimes_{S}k)^{C_{2}}\big{)}\times_{k\otimes_{S}k}\phi^{l-1}\big{(}(k\otimes_{S}k)^{C_{2}}\big{)}\big{)}&\ast\mbox{even}\\\
\displaystyle\bigoplus_{\begin{smallmatrix}(n,m)\\\ n+m=\ast\\\ n,m\geq 0\\\
n\neq m\end{smallmatrix}}k\otimes_{S}k&\ast\mbox{odd}\end{array}\right.$

In particular, in degree zero, we obtain a ring isomorphism

$\pi_{0}{\operatorname{TRR}^{l+1}(k;2)}^{{\phi\operatorname{\mathbb{Z}}/2}}\cong\phi^{l-1}\big{(}(k\otimes_{S}k)^{C_{2}}\big{)}\times_{k\otimes_{S}k}\phi^{l-1}\big{(}(k\otimes_{S}k)^{C_{2}}\big{)}.$

The maps
$R,F\colon{\operatorname{TRR}^{l+1}(k;2)}^{{\phi\operatorname{\mathbb{Z}}/2}}\to{\operatorname{TRR}^{l}(k;2)}^{{\phi\operatorname{\mathbb{Z}}/2}}$
and the Weyl action are described on homotopy groups as follows.

The map $R$ kills the $(n,m)$-summands with $n\neq m$, and in even degrees it
sends an element $(x,y)$ of the right-hand pullback to
$(\phi^{-1}\pi(x),\phi^{-1}\pi(y))$.

The map $F$ kills the $(n,m)$-summands with $m<n$, embeds the $(n,m)$-summands
with $n<m$ diagonally into the sum of the $(n,m)$ and $(m,n)$-summands, and in
even degrees it sends an element $(x,y)$ of the right-hand pullback to
$(x,x)$.

The Weyl action swaps the $(n,m)$-summand and the $(m,n)$-summand for all
$n\neq m$, and in even degrees takes an element $(x,y)$ in the pullback to
$(y,x)$.

###### Proof.

By [DMP24, Theorem 2.7] and §1, for every $l\geq 1$, there is a pullback
square of ring spectra

$\textstyle{\operatorname{TRR}^{l+1}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{R}$$\scriptstyle{(cF^{l-1},cF^{l-1}\sigma_{l+1})}$$\textstyle{\operatorname{TRR}^{l}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(F^{l-1},\sigma_{1}F^{l-1}\sigma_{l})}$$\textstyle{(\operatorname{THR}(k)^{\phi\operatorname{\mathbb{Z}}/2})^{C_{2}}\times(\operatorname{THR}(k)^{\phi\operatorname{\mathbb{Z}}/2})^{C_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{r\times\sigma_{1}r}$$\textstyle{\operatorname{THR}(k)^{\phi\operatorname{\mathbb{Z}}/2}\times\operatorname{THR}(k)^{\phi\operatorname{\mathbb{Z}}/2}}$

where $\sigma_{l}$ denotes the action of the generator of the Weyl group of
$\operatorname{\mathbb{Z}}/2$ in $D_{2^{l}}/C_{2^{l-1}}$, which is of order
$2$. We prove, by induction on $l$, that the connecting homomorphism in the
Meyer-Vietoris long exact sequence of this pullback square vanish, and
therefore that the square gives a pullback square of homotopy groups. One can
see, again by induction, that these pullbacks of homotopy groups indeed match
the description of the homotopy groups of the Theorem. However, we will need
to prove the vanishing of the connecting maps and the explicit description of
the pullback in the same induction step.

For $l=1$, the pullback above describing
${\operatorname{TRR}^{2}(k;2)}^{{\phi\operatorname{\mathbb{Z}}/2}}$ is
equivalent to the pullback

$(\operatorname{THR}(k)^{\phi\operatorname{\mathbb{Z}}/2})^{C_{2}}{\times_{\operatorname{THR}(k)^{\phi\operatorname{\mathbb{Z}}/2}}}(\operatorname{THR}(k)^{\phi\operatorname{\mathbb{Z}}/2})^{C_{2}}$

along the maps $r$ and $\sigma_{1}r$ (since the right vertical map in the
square above is the diagonal for $l=1$). By the characterisation of $r$ of
Proposition 2.3, the map $r-\sigma_{1}r$ in the corresponding Mayer-Vietoris
sequence is surjective in every degree. Therefore, there is a pullback

$\pi_{\ast}{\operatorname{TRR}^{2}(k;2)}^{{\phi\operatorname{\mathbb{Z}}/2}}\cong\big{(}\big{(}\bigoplus_{\begin{smallmatrix}(n,m)\\\
n,m\geq 0,n+m=\ast\\\ (n\neq
m)\end{smallmatrix}}k\otimes_{S}k\big{)}\oplus(k\otimes_{S}k)^{C_{2}})\big{)}\
{}_{r}\times_{\sigma r}\big{(}\big{(}\bigoplus_{\begin{smallmatrix}(n,m)\\\
n,m\geq 0,n+m=\ast\\\ (n\neq
m)\end{smallmatrix}}k\otimes_{S}k\big{)}\oplus(k\otimes_{S}k)^{C_{2}})\big{)}$

in even degrees, and an analogous pullback without the summands
$(k\otimes_{S}k)^{C_{2}}$ in odd degrees. By the description of $r$ from
Proposition 2.3, this is isomorphic to the pullback of the statement of the
Theorem. The characterisation of the maps $R,F$ and of the Weyl action follows
by the description of the corresponding maps of [DMP24, Theorem 2.7].

Now let $l\geq 2$, and suppose that the decomposition above holds for
$\pi_{\ast}{\operatorname{TRR}^{h}(k;2)}^{{\phi\operatorname{\mathbb{Z}}/2}}$
for all $h\leq l$, and that the maps
$R,F\colon{\operatorname{TRR}^{h}(k;2)}^{{\phi\operatorname{\mathbb{Z}}/2}}\to{\operatorname{TRR}^{h-1}(k;2)}^{{\phi\operatorname{\mathbb{Z}}/2}}$
and $\sigma_{h}$ are given in homotopy groups by the formulas of the Theorem.
We will show that the same holds for $l+1$. The Mayer-Vietoris sequence of the
pullback square above is then (we recall that $\sigma_{1}F=F$)

$\textstyle{\dots{}_{\
}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\textstyle{\pi_{\ast}{\operatorname{TRR}^{l+1}(k;2)}^{{\phi\operatorname{\mathbb{Z}}/2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(cF^{l-1},cF^{l-1}\sigma_{l+1},R)}$$\textstyle{\displaystyle\big{[}\big{(}\bigoplus_{\begin{smallmatrix}(n,m)\\\
n+m=\ast\\\ n,m\geq 0\\\ n\neq
m\end{smallmatrix}}k\otimes_{S}k\big{)}\oplus(k\otimes_{S}k)^{C_{2}}\big{]}\oplus}$$\textstyle{\displaystyle\big{[}\big{(}\bigoplus_{\begin{smallmatrix}(n,m)\\\
n+m=\ast\\\ n,m\geq 0\\\ n\neq
m\end{smallmatrix}}k\otimes_{S}k\big{)}\oplus(k\otimes_{S}k)^{C_{2}}\big{]}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\oplus}$$\scriptstyle{r\oplus\sigma_{1}r-(F^{l-1},F^{l-1}\sigma_{l})}$$\textstyle{\displaystyle\big{[}\big{(}\bigoplus_{\begin{smallmatrix}(n,m)\\\
n+m=\ast\\\ n,m\geq 0\\\ n\neq
m\end{smallmatrix}}k\otimes_{S}k\big{)}\oplus\big{(}\phi^{l-1}((k\otimes_{S}k)^{C_{2}})\underset{k\otimes_{S}k}{\times}\phi^{l-1}((k\otimes_{S}k)^{C_{2}})\big{)}\big{]}}$$\textstyle{\displaystyle(\bigoplus_{\begin{smallmatrix}(n,m)\\\
n+m=\ast\end{smallmatrix}}k\otimes_{S}k)\oplus(\bigoplus_{\begin{smallmatrix}(n,m)\\\
n+m=\ast\end{smallmatrix}}k\otimes_{S}k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\partial}$$\textstyle{\dots}$

for $\ast$ even, and a similar expression without the fixed points terms for
$\ast$ odd. An argument completely analogous that of the proof of [DMP24,
Theorem 4.7] shows that the bottom vertical map is surjective, and identifies
its kernel with the formula of the Theorem. The description of the maps $R$
and $F$ also follow by a similar argument. ∎

######  Remark 2.7.

From the proof of Theorem 2.6 we see that the isomorphism for the $0$-th
homotopy group is explicitly given by the map

$(F^{l},F^{l}\sigma)\colon\pi_{0}{\operatorname{TRR}^{l+1}(k;2)}^{{\phi\operatorname{\mathbb{Z}}/2}}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\phi^{l-1}\big{(}(k\otimes_{S}k)^{C_{2}}\big{)}\times_{k\otimes_{S}k}\phi^{l-1}\big{(}(k\otimes_{S}k)^{C_{2}}\big{)}$

where we implicitly identify the target
$\pi_{0}{\operatorname{THR}(k;2)}^{{\phi\operatorname{\mathbb{Z}}/2}}$ of
$F^{l}$ with $k\otimes_{S}k$, and $\sigma:=\sigma_{l}$ denotes the Weyl action
on the source of this map. We can see this directly as follows. Let us express
the source of this map as the iterated pullback

$\pi_{0}{\operatorname{TRR}^{l+1}(k;2)}^{{\phi\operatorname{\mathbb{Z}}/2}}\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}(k\otimes_{S}k)^{C_{2}}{}_{r}\\!\times_{f}\dots{}_{r}\\!\times_{f}(k\otimes_{S}k)^{C_{2}}{}_{r}\\!\times_{wr}(k\otimes_{S}k)^{C_{2}}{}_{f}\\!\times_{wr}\dots{}_{f}\\!\times_{wr}(k\otimes_{S}k)^{C_{2}}$

as in [DMP24, Remark 2.8], where the pullback has $2l$ factors, and the
isomorphism is given by the map
$(F^{l},F^{l-1}R,F^{l-2}R^{2},\dots,FR^{l-1},F\sigma
R^{l-1},\dots,F^{l-2}\sigma R^{2},F^{l-1}\sigma R,F^{l}\sigma)$. We still have
a pullback after applying $\pi_{0}$ because the connecting maps of the Meyer-
Vietoris sequences vanish as seen in the proof of 2.6. By Proposition 2.3,
$r=\phi^{-1}\pi$ and $f$ is the fixed points inclusion
$(k\otimes_{S}k)^{C_{2}}\to k\otimes_{S}k$. Since $f$ is injective, the
projection onto the first and last factors defines an isomorphism between this
pullback and
$\phi^{l-1}\big{(}(k\otimes_{S}k)^{C_{2}}\big{)}\times_{k\otimes_{S}k}\phi^{l-1}\big{(}(k\otimes_{S}k)^{C_{2}}\big{)}$,
and the composite map is indeed $(F^{l},F^{l}\sigma)$.

###  2.2 The canonical generators of TRR

We recall, that for every commutative ring $R$, the ring of $2$-typical
$(n+1)$-truncated Witt vectors $\operatorname{W}_{\langle 2^{n}\rangle}(R)$ is
the set $R^{\times n+1}$ equipped with the unique functorial ring structure
which makes the Witt polynomials into ring homomorphisms (see e.g. [Hes15,
§1]). Additively, it is generated by the elements

$V^{n-i}\tau_{i}(a)=(0,\dots,0,a,0,\dots,0),$

where the entry $a$ is in the $(n-i+1)$-st component, $a$ ranges through the
elements of $R$ and $i=0,\dots,n$.

The goal of this section is to define canonical generators for the pullback of
Theorem 2.6, analogous to the generators $V^{n-i}\tau_{i}(a)$ of the
$(n+1)$-truncated Witt vectors, thus providing generators for
$\pi_{0}{\operatorname{TRR}^{n+1}(k;2)}^{\phi\operatorname{\mathbb{Z}}/2}$
analogous to those of $\operatorname{W}_{\langle 2^{n}\rangle}(k)$.

Recall that, for every elementary tensor $a\otimes b\in k\otimes_{S}k$, we
have defined

$\phi(a\otimes b):=ba^{2}\otimes b\in(k\otimes_{S}k)^{C_{2}}$

(see Lemma 2.1). Similarly, for any elementary tensor $a\otimes b\in
k\otimes_{S}k$ and $n\geq 0$, let us iterate this construction and define

$\phi^{n}(a\otimes b):=b^{2^{n-1}}a^{2^{n}}\otimes
b\in\phi^{-n+1}\big{(}(k\otimes_{S}k)^{C_{2}}\big{)}.$

Let us also define $\tau_{0}(a\otimes b):=a\otimes b\in k\otimes_{S}k$, and
for every $0\leq i<n$, the following elements of the pullback
$\phi^{n-1}((k\otimes_{S}k)^{C_{2}})\underset{k\otimes_{S}k}{\times}\phi^{n-1}((k\otimes_{S}k)^{C_{2}})$:

$\displaystyle\tau_{n}(a\otimes b)$ $\displaystyle:=(\phi^{n}(a\otimes
b),\phi^{n}(b\otimes a))$ $\displaystyle V^{n-i}\tau_{i}(a\otimes b)$
$\displaystyle:=(\phi^{i}(a\otimes b)+\phi^{i}(b\otimes a),0)$
$\displaystyle\sigma V^{n-i}\tau_{i}(a\otimes b)$
$\displaystyle:=(0,\phi^{i}(a\otimes b)+\phi^{i}(b\otimes a)).$

Here we recall that the pullback is taken with respect to the maps
$(\phi^{-1}\pi)^{n}$ and $w(\phi^{-1}\pi)^{n}$, see the diagram above Theorem
2.6, and $\sigma$ is the Weyl action which switches the two pullback
components.

######  Proposition 2.8.

Let $k$ be a field of characteristic $2$. For every $n\geq 0$, the subgroup
$\phi^{n}\big{(}(k\otimes_{S}k)^{C_{2}}\big{)}$ of $k\otimes_{S}k$ is
generated by elements of the form $\phi^{n+1}(a\otimes b)$ and
$\phi^{i}(a\otimes b)+\phi^{i}(b\otimes a)$, for $0\leq i\leq n$ and $a\otimes
b\in k\otimes_{S}k$.

It follows that, for every $n\geq 1$,

$\pi_{0}\operatorname{TRR}^{n+1}(k;2)^{{\phi\operatorname{\mathbb{Z}}/2}}\cong\phi^{n-1}((k\otimes_{S}k)^{C_{2}})\underset{k\otimes_{S}k}{\times}\phi^{n-1}((k\otimes_{S}k)^{C_{2}})$

is generated by the elements $\tau_{n}(a\otimes b)$, $V^{n-i}\tau_{i}(a\otimes
b)$ and $\sigma V^{n-i}\tau_{i}(a\otimes b)$, for $0\leq i\leq n-1$ and
$a\otimes b\in k\otimes_{S}k$.

###### Proof.

Let us first show that the proposed generators belong to
$\phi^{n}\big{(}(k\otimes_{S}k)^{C_{2}}\big{)}$. For the first, we see that
for every $1\leq j\leq n$ we have that

$(\phi^{-1}\pi)^{j}(\phi^{n+1}(a\otimes b))=\phi^{n+1-j}(a\otimes b),$

which belongs to $(k\otimes_{S}k)^{C_{2}}$ since $n+1-j\geq 1$. On the other
hand, for all $0\leq i\leq n-1$, we have that

$(\phi^{-1}\pi)^{j}(\phi^{i}(a\otimes b)+\phi^{i}(b\otimes
a))=\phi^{i-j}(a\otimes b)+\phi^{i-j}(b\otimes a)$

if $0\leq j\leq i$, which is a fixed point, and for $i<j\leq n$ this is

$(\phi^{-1}\pi)^{j}(\phi^{i}(a\otimes b)+\phi^{i}(b\otimes
a))=(\phi^{-1}\pi)^{j-i}(a\otimes b+b\otimes a)=0$

since $\pi$ quotients off the image of $1+w$.

The proof that these elements generate
$\phi^{n}\big{(}(k\otimes_{S}k)^{C_{2}}\big{)}$ is by induction on $n$. For
$n=0$, consider the exact sequence

$k\otimes_{S}k\xrightarrow{1+w}(k\otimes_{S}k)^{C_{2}}\longrightarrow(k\otimes_{S}k)^{C_{2}}/Im(1+w)\to
0.$

By Lemma 2.1, the right term is generated by the equivalence classes of the
elements of the form $\phi(a\otimes b)$, and the image of $1+w$ is generated
by the elements of the form $a\otimes b+b\otimes a$, which proves the claim.

Now suppose that the claim holds for $n-1$, and consider the exact sequence

$k\otimes_{S}k\xrightarrow{1+w}\phi^{n}((k\otimes_{S}k)^{C_{2}})\longrightarrow\phi^{n}((k\otimes_{S}k)^{C_{2}})/Im(1+w)\to
0.$

By an argument analogous to the proof of Lemma 2.1, $\phi$ defines an
isomorphism between $\phi^{n-1}((k\otimes_{S}k)^{C_{2}})$ and
$\phi^{n}((k\otimes_{S}k)^{C_{2}})/Im(1+w)$. Thus, by the inductive
assumption, the classes of $\phi^{n+1}(a\otimes b)$ and $\phi^{i}(a\otimes
b)+\phi^{i}(b\otimes a)$, for $1\leq i\leq n$ and $a\otimes b\in
k\otimes_{S}k$, generate the quotient. The image of $1+w$ is generated by the
elements of the form $a\otimes b+b\otimes a$, which concludes the induction.

The proof for
$\pi_{0}\operatorname{TRR}^{n+1}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}$ is
completely analogous, by induction on the exact sequences

$\textstyle{(k\otimes_{S}k)\oplus(k\otimes_{S}k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(1+w,0)+(0,1+w)}$$\textstyle{\phi^{n-1}((k\otimes_{S}k)^{C_{2}})\underset{k\otimes_{S}k}{\times}\phi^{n-1}((k\otimes_{S}k)^{C_{2}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\phi^{-1}\pi,\phi^{-1}\pi)}$$\textstyle{\phi^{n-2}((k\otimes_{S}k)^{C_{2}})\underset{k\otimes_{S}k}{\times}\phi^{n-2}((k\otimes_{S}k)^{C_{2}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

∎

Next, we want to understand the effect of the transfer and norm maps of
$\operatorname{TRR}(k)$ under the isomorphism of 2.6, and their relation to
the generators of Proposition 2.8. For every $0\leq h<l$, let

$\operatorname{tran}^{D_{2^{l}}}_{D_{2^{h}}}\colon\pi_{0}\operatorname{TRR}^{h+1}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}\to\pi_{0}\operatorname{TRR}^{l+1}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}$

be the transfer map associated to the subgroup inclusion $D_{2^{h}}\leq
D_{2^{l}}$.

######  Proposition 2.9.

For every $0<h<l$, the map $\operatorname{tran}^{D_{2^{l}}}_{D_{2^{h}}}$
corresponds, under the isomorphism of Theorem 2.6, to the group homomorphism

$V^{l-h}\colon\phi^{h-1}((k\otimes_{S}k)^{C_{2}})\underset{k\otimes_{S}k}{\times}\phi^{h-1}((k\otimes_{S}k)^{C_{2}})\longrightarrow\phi^{l-1}((k\otimes_{S}k)^{C_{2}})\underset{k\otimes_{S}k}{\times}\phi^{l-1}((k\otimes_{S}k)^{C_{2}})$

which sends $(x,y)$ to $(x+y,0)$. For $h=0$, it corresponds to the group
homomorphism

$V^{l}\colon
k\otimes_{S}k\longrightarrow\phi^{l-1}((k\otimes_{S}k)^{C_{2}})\underset{k\otimes_{S}k}{\times}\phi^{l-1}((k\otimes_{S}k)^{C_{2}})$

which sends $a\otimes b$ to $(a\otimes b+b\otimes a,0)$.

###### Proof.

Let us first suppose $h>0$. We need to show that the unique map in the bottom
row of the commutative square

$\textstyle{\pi_{0}\operatorname{TRR}^{h+1}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{tran}^{D_{2^{l}}}_{D_{2^{h}}}}$$\scriptstyle{(F^{h},F^{h}\sigma)}$$\scriptstyle{\cong}$$\textstyle{\pi_{0}\operatorname{TRR}^{l+1}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(F^{l},F^{l}\sigma)}$$\scriptstyle{\cong}$$\textstyle{\phi^{h-1}((k\otimes_{S}k)^{C_{2}})\underset{k\otimes_{S}k}{\times}\phi^{h-1}((k\otimes_{S}k)^{C_{2}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\phi^{l-1}((k\otimes_{S}k)^{C_{2}})\underset{k\otimes_{S}k}{\times}\phi^{l-1}((k\otimes_{S}k)^{C_{2}})}$

agrees with $V^{l-h}$, where the vertical maps are the isomorphisms of Theorem
2.6 and Remark 2.7. By the double coset formula of the $D_{2^{l}}$-Mackey
functor $\underline{\pi}_{0}\operatorname{THR}(k)$, the upper composite has
first component

$\displaystyle F^{l}\operatorname{tran}^{D_{2^{l}}}_{D_{2^{h}}}$
$\displaystyle=\operatorname{res}^{D_{2^{l}}}_{\operatorname{\mathbb{Z}}/2}\operatorname{tran}^{D_{2^{l}}}_{D_{2^{h}}}=\sum_{g\in\operatorname{\mathbb{Z}}/2/D_{2^{l}}/D_{2^{h}}}\operatorname{tran}^{\operatorname{\mathbb{Z}}/2}_{{}^{g}D_{2^{h}}\cap\operatorname{\mathbb{Z}}/2}c_{g}\operatorname{res}^{D_{2^{h}}}_{D_{2^{h}}\cap\operatorname{\mathbb{Z}}/2^{g}}.$

The double coset $\operatorname{\mathbb{Z}}/2/D_{2^{l}}/D_{2^{h}}$ is the
quotient of the cyclic group $C_{2^{l-h}}$ by the involution which acts by
inversion. It therefore consists of two fixed points (the unit and the
rotation $g_{0}$ of order $2$ in $D_{l}$) which conjugate
$\operatorname{\mathbb{Z}}/2$ to itself, and $(2^{l-h}-2)/2$ points whose
corresponding intersection $D_{2^{h}}\cap\operatorname{\mathbb{Z}}/2^{g}$ is
trivial. Thus

$\displaystyle F^{l}\operatorname{tran}^{D_{2^{l}}}_{D_{2^{h}}}$
$\displaystyle=\operatorname{res}^{D_{2^{h}}}_{\operatorname{\mathbb{Z}}/2}+c_{g_{0}}\operatorname{res}^{D_{2^{h}}}_{\operatorname{\mathbb{Z}}/2}+\sum_{1,g_{0}\neq
g\in\operatorname{\mathbb{Z}}/2/D_{2^{l}}/D_{2^{h}}}\operatorname{tran}^{\operatorname{\mathbb{Z}}/2}_{e}c_{g}\operatorname{res}^{D_{2^{h}}}_{e}$
$\displaystyle=\operatorname{res}^{D_{2^{h}}}_{\operatorname{\mathbb{Z}}/2}+\operatorname{res}^{D_{2^{h}}}_{\operatorname{\mathbb{Z}}/2}c_{g_{0}}=F^{h}+F^{h}\sigma$

where the transfer $\operatorname{tran}^{\operatorname{\mathbb{Z}}/2}_{e}$ is
zero since $k$ has characteristic $2$ (see [DMPR21, Theorem 5.1]), $\sigma$ is
the action of the Weyl group of $D_{h}$ in $D_{l}$, and the equality is
regarded as elements of $k\otimes_{S}k$. The map $F^{h}$ is determined in
Theorem 2.6: it sends an element in the upper left corner of the square,
corresponding to $(x,y)$ in the bottom left corner, to $x$. Thus the unique
bottom horizontal map in the square above sends $(x,y)$ to the pair with first
component $x+y$.

Now let $\operatorname{\mathbb{Z}}/2^{\prime}$ be the subgroup of $D_{l}$
generated by a reflection non-conjugate to $\operatorname{\mathbb{Z}}/2$.
Similarly to the calculation above, the second component of the top composite
is

$\displaystyle F^{l}\sigma\operatorname{tran}^{D_{2^{l}}}_{D_{2^{h}}}$
$\displaystyle=\operatorname{res}^{D_{2^{l}}}_{\operatorname{\mathbb{Z}}/2^{\prime}}\operatorname{tran}^{D_{2^{l}}}_{D_{2^{h}}}=\sum_{g\in\operatorname{\mathbb{Z}}/2^{\prime}/D_{2^{l}}/D_{2^{h}}}\operatorname{tran}^{\operatorname{\mathbb{Z}}/2}_{{}^{g}D_{2^{h}}\cap\operatorname{\mathbb{Z}}/2}c_{g}\operatorname{res}^{D_{2^{h}}}_{D_{2^{h}}\cap\operatorname{\mathbb{Z}}/2^{g}}.$

Now the double coset
$\operatorname{\mathbb{Z}}/2^{\prime}/D_{2^{l}}/D_{2^{h}}$ is the quotient of
the cyclic group $C_{2^{l-h}}$ by the free involution, and none of the
conjugates of $\operatorname{\mathbb{Z}}/2$ is contained in $D_{2^{h}}$. Thus

$\displaystyle F^{l}\sigma\operatorname{tran}^{D_{2^{l}}}_{D_{2^{h}}}$
$\displaystyle=\sum_{g\in\operatorname{\mathbb{Z}}/2^{\prime}/D_{2^{l}}/D_{2^{h}}}\operatorname{tran}^{\operatorname{\mathbb{Z}}/2}_{e}c_{g}\operatorname{res}^{D_{2^{h}}}_{e}=0,$

again since $\operatorname{tran}^{\operatorname{\mathbb{Z}}/2}_{e}=0$. Thus
the second component of the bottom horizontal map is null as claimed.

The proof of the case $h=0$ is similar, by calculating the upper composite of
the diagram

$\textstyle{\pi_{0}\operatorname{THR}(k)^{\phi\operatorname{\mathbb{Z}}/2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{tran}^{D_{2^{l}}}_{\operatorname{\mathbb{Z}}/2}}$$\scriptstyle{\cong}$$\textstyle{\pi_{0}\operatorname{TRR}^{l+1}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(F^{l},F^{l}\sigma)}$$\scriptstyle{\cong}$$\textstyle{k\otimes_{S}k\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\phi^{l-1}((k\otimes_{S}k)^{C_{2}})\underset{k\otimes_{S}k}{\times}\phi^{l-1}((k\otimes_{S}k)^{C_{2}})}$

where the left vertical map is the isomorphism of [DMPR21, Theorem 5.1]. ∎

######  Remark 2.10.

The notation used in Proposition 2.9 for the transfer map is consistent with
our notation for the generators of Proposition 2.8, since

$V^{n-i}\tau_{i}(a\otimes b)=V^{n-i}(\phi^{i}(a\otimes b),\phi^{i}(b\otimes
a))=(\phi^{i}(a\otimes b)+\phi^{i}(b\otimes a),0).$

The generators $\tau_{n}(a\otimes b)$ also have a somewhat topological
interpretation, as we now explain. As seen at the end of §1, the there is a
non-additive norm map

$N_{\operatorname{\mathbb{Z}}/2}^{D_{2^{n}}}\colon\pi_{0}\operatorname{THR}(k)^{\operatorname{\mathbb{Z}}/2}=\pi_{0}\operatorname{TRR}^{1}(k;2)^{\operatorname{\mathbb{Z}}/2}\longrightarrow\pi_{0}\operatorname{TRR}^{n+1}(k;2)^{\operatorname{\mathbb{Z}}/2}=\pi_{0}\operatorname{THR}(k)^{D_{2^{n}}}.$

Moreover, since $k$ is of characteristic $2$, the canonical map
$\pi_{0}\operatorname{THR}(k)^{\operatorname{\mathbb{Z}}/2}\to\pi_{0}\operatorname{THR}(k)^{\phi\operatorname{\mathbb{Z}}/2}$
is an isomorphism (since the transfer from the trivial subgroup to
$\operatorname{\mathbb{Z}}/2$ is zero by [DMPR21, Theorem 5.1]), and therefore
by post-composing with the canonical projection we also obtain a non-additive
map

$\pi_{0}\operatorname{THR}(k)^{\phi\operatorname{\mathbb{Z}}/2}\cong\pi_{0}\operatorname{THR}(k)^{\operatorname{\mathbb{Z}}/2}\xrightarrow{N_{\operatorname{\mathbb{Z}}/2}^{D_{2^{n}}}}\pi_{0}\operatorname{TRR}^{n+1}(k;2)^{\operatorname{\mathbb{Z}}/2}\longrightarrow\pi_{0}\operatorname{TRR}^{n+1}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}$

on geometric fixed points, which we still denote by
$N_{\operatorname{\mathbb{Z}}/2}^{D_{2^{n}}}$.

######  Proposition 2.11.

Under the isomorphism of Theorem 2.6, the map
$N_{\operatorname{\mathbb{Z}}/2}^{D_{2^{n}}}$ corresponds to the map
$N^{n}\colon
k\otimes_{S}k\to\phi^{n-1}((k\otimes_{S}k)^{C_{2}})\times_{k\otimes_{S}k}\phi^{n-1}((k\otimes_{S}k)^{C_{2}})$
that sends an elementary tensor $a\otimes b$ to

$N^{n}(a\otimes b)=\tau_{n}(ab\otimes 1)=(\phi_{n}(ab\otimes
1),\phi_{n}(1\otimes ab)).$

In particular, we find that

$\tau_{n}(a\otimes b)=N^{n}(a\otimes 1)\cdot\sigma N^{n}(b\otimes 1).$

###### Proof.

The identification of $N^{n}(a\otimes b)$ is similar to the proof of
Proposition 2.9. It is sufficient to show that the unique map in the bottom
row of the commutative square

$\textstyle{\pi_{0}\operatorname{TRR}^{1}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{N^{D_{2^{n}}}_{\operatorname{\mathbb{Z}}/2}}$$\scriptstyle{\cong}$$\textstyle{\pi_{0}\operatorname{TRR}^{n+1}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(F^{n},F^{n}\sigma)}$$\scriptstyle{\cong}$$\textstyle{k\otimes_{S}k\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\phi^{n-1}((k\otimes_{S}k)^{C_{2}})\underset{k\otimes_{S}k}{\times}\phi^{n-1}((k\otimes_{S}k)^{C_{2}})}$

agrees with $N^{n}$ on the elementary tensors $a\otimes b$. Indeed, the value
on a general element in the tensor product is determined by iterations of the
relation

$N(x+y)=N(x)+N(y)+V(xy).$

By the multiplicative double coset formula of the $D_{2^{l}}$-Tambara functor
$\underline{\pi}_{0}\operatorname{THR}(k)$, the upper composite has first
component

$\displaystyle F^{n}N^{D_{2^{n}}}_{\operatorname{\mathbb{Z}}/2}$
$\displaystyle=\operatorname{res}^{D_{2^{n}}}_{\operatorname{\mathbb{Z}}/2}N^{D_{2^{n}}}_{\operatorname{\mathbb{Z}}/2}=\prod_{g\in\operatorname{\mathbb{Z}}/2/D_{2^{n}}/\operatorname{\mathbb{Z}}/2}N^{\operatorname{\mathbb{Z}}/2}_{{}^{g}\operatorname{\mathbb{Z}}/2\cap\operatorname{\mathbb{Z}}/2}c_{g}\operatorname{res}^{\operatorname{\mathbb{Z}}/2}_{\operatorname{\mathbb{Z}}/2\cap\operatorname{\mathbb{Z}}/2^{g}}.$

The double coset
$\operatorname{\mathbb{Z}}/2/D_{2^{n}}/\operatorname{\mathbb{Z}}/2$ is the
quotient of the cyclic group $C_{2^{n}}$ by the involution which acts by
inversion, and consists of two fixed points (the unit and the rotation $g_{0}$
of order $2$ in $D_{n}$) which conjugate $\operatorname{\mathbb{Z}}/2$ to
itself, and $(2^{n}-2)/2$ points whose corresponding intersection
$\operatorname{\mathbb{Z}}/2\cap\operatorname{\mathbb{Z}}/2^{g}$ is trivial.
Moreover, since the cyclic group acts trivially on
$\pi_{0}\operatorname{THH}(k)=k$, the conjugation $c_{g}$ is trivial except
for $g=g_{0}$. Thus

$\displaystyle F^{n}N^{D_{2^{n}}}_{\operatorname{\mathbb{Z}}/2}$
$\displaystyle=(\operatorname{id})\cdot(c_{g_{0}})\cdot(N^{\operatorname{\mathbb{Z}}/2}_{e}\operatorname{res}^{\operatorname{\mathbb{Z}}/2}_{e})^{2^{n-1}-1}.$

Since the restriction map
$\operatorname{res}^{\operatorname{\mathbb{Z}}/2}_{e}$ corresponds to the
multiplication map $\mu\colon k\otimes_{S}k\to k$ and
$N^{\operatorname{\mathbb{Z}}/2}_{e}$ to the map $k\to k\otimes_{S}k$ which
sends $a$ to $a^{2}\otimes 1$ (see [DMPR21, Corollary 5.2]), this sends
$a\otimes b$ to

$a\otimes b\cdot b\otimes a\cdot((ab)^{2}\otimes
1)^{2^{n-1}-1}=(ab)^{2^{n}-1}\otimes ab=\phi_{n}(1\otimes ab).$

Similarly, by letting $\operatorname{\mathbb{Z}}/2^{\prime}$ be the subgroup
of $D_{n}$ generated by a reflection non-conjugate to
$\operatorname{\mathbb{Z}}/2$, the second component of the upper composite in
the square above is

$\displaystyle F^{n}\sigma N^{D_{2^{n}}}_{\operatorname{\mathbb{Z}}/2}$
$\displaystyle=\operatorname{res}^{D_{2^{n}}}_{\operatorname{\mathbb{Z}}/2^{\prime}}N^{D_{2^{n}}}_{\operatorname{\mathbb{Z}}/2}=\prod_{g\in\operatorname{\mathbb{Z}}/2^{\prime}/D_{2^{l}}/\operatorname{\mathbb{Z}}/2}N^{\operatorname{\mathbb{Z}}/2^{\prime}}_{{}^{g}\operatorname{\mathbb{Z}}/2\cap\operatorname{\mathbb{Z}}/2}c_{g}\operatorname{res}^{\operatorname{\mathbb{Z}}/2}_{\operatorname{\mathbb{Z}}/2\cap\operatorname{\mathbb{Z}}/2^{\prime
g}}.$

Now the double coset
$\operatorname{\mathbb{Z}}/2^{\prime}/D_{2^{n}}/\operatorname{\mathbb{Z}}/2$
is the quotient of the cyclic group $C_{2^{n}}$ by the free involution, and
since $\operatorname{\mathbb{Z}}/2$ and $\operatorname{\mathbb{Z}}/2^{\prime}$
are not conjugate

$\displaystyle F^{n}\sigma N^{D_{2^{n}}}_{\operatorname{\mathbb{Z}}/2}$
$\displaystyle=(N^{\operatorname{\mathbb{Z}}/2^{\prime}}_{e}\operatorname{res}^{\operatorname{\mathbb{Z}}/2}_{e})^{2^{n-1}}.$

Thus the second component of the bottom horizontal map of the square sends
$a\otimes b$ to

$((ab)^{2})^{2^{n-1}}\otimes 1=\phi_{n}(ab\otimes 1).$

This identifies the map $N^{n}$ as claimed. Finally, observe that

$\displaystyle N^{n}(a\otimes 1)\cdot\sigma N^{n}(b\otimes 1)$
$\displaystyle=(a^{2^{n}}\otimes 1,a^{2^{n}-1}\otimes
a)\cdot(b^{2^{n}-1}\otimes b,b^{2^{n}}\otimes 1)$
$\displaystyle=(a^{2^{n}}b^{2^{n}-1}\otimes b,a^{2^{n}-1}b^{2^{n}}\otimes
a)=\tau_{n}(a\otimes b).$

∎

######  Remark 2.12.

For the usual Witt vectors, the elements $\tau_{n}(a)=(a,0,\dots,0)$ assemble
into a (non-additive) multiplicative map $\tau_{n}\colon R\to W_{\langle
2^{n}\rangle}(R)$, which is a section for the truncation map $R$. We do not
think that this is the case for
$\pi_{0}\operatorname{TRR}^{n+1}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}$,
since there seem to be no way of extending $\tau_{n}(a\otimes b)$ to a sum of
elementary tensors. Even without a canonical splitting for the truncation map
$R\colon\pi_{0}\operatorname{TRR}^{n+1}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}\to\pi_{0}\operatorname{TRR}^{1}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}$
at hand, having a set of generators for
$\pi_{0}\operatorname{TRR}^{n+1}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}$
defined from the $\tau_{i}(a\otimes b)$ will suffice for our purposes.

###  2.3 The fundamental ideal of TRR

The components of the geometric fixed points of any connective
$\operatorname{\mathbb{Z}}/2$-spectrum $X$ admit a restriction map, defined as
the canonical map of cokernels

$\textstyle{\pi_{0}(X_{h\operatorname{\mathbb{Z}}/2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{\operatorname{tran}_{e}^{\operatorname{\mathbb{Z}}/2}}$$\textstyle{\pi_{0}(X^{\operatorname{\mathbb{Z}}/2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{res}^{\operatorname{\mathbb{Z}}/2}_{e}}$$\textstyle{\pi_{0}(X^{\phi\operatorname{\mathbb{Z}}/2})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{(\pi_{0}X)_{\operatorname{\mathbb{Z}}/2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1+w}$$\textstyle{(\pi_{0}X)^{\operatorname{\mathbb{Z}}/2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(\pi_{0}X)^{\operatorname{\mathbb{Z}}/2}/Im(1+w)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where $w$ is the action of the generator of $\operatorname{\mathbb{Z}}/2$ on
$\pi_{0}X$. This map is moreover a monoidal natural transformation. By
applying this construction to the $\operatorname{\mathbb{Z}}/2$-spectrum
$\operatorname{TRR}^{n+1}(k;2)$, we obtain a ring homomorphism which we denote
by

$\operatorname{res}^{D_{2^{n}}}_{C_{2^{n}}}\colon\pi_{0}\operatorname{TRR}^{n+1}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}\longrightarrow(\pi_{0}\operatorname{TR}^{n+1}(k;2))^{\operatorname{\mathbb{Z}}/2}/(1+w)\cong\operatorname{W}_{\langle
2^{n}\rangle}(k)/2.$

Here $\operatorname{W}_{\langle 2^{n}\rangle}(k)$ is the ring of
$(n+1)$-truncated $2$-typical Witt vectors of $k$, and the isomorphism is from
[HM97, Theorem F]. Here we use that the involution $w$ induced by the
$\operatorname{\mathbb{Z}}/2$-action of $\operatorname{TRR}^{n+1}(k;2)$ is the
identity (see the proof of [DPJM22, Theorem 3.7], the first paragraph of page
522 holds also for $p=2$).

The goal of this section is to describe explicitly the map
$\operatorname{res}^{D_{2^{n}}}_{C_{2^{n}}}$ under the isomorphism of Theorem
2.6, and provide generators for its kernel. We recall that for every
commutative ring $R$, as a set, $\operatorname{W}_{\langle
2^{n}\rangle}(R)=R^{\times n+1}$, with the unique functorial ring structure
which makes the Witt polynomials into ring homomorphisms. For any
$\operatorname{\mathbb{F}}_{2}$-algebra $R$, we moreover have that as a set

$\operatorname{W}_{\langle 2^{n}\rangle}(R)=R\times(R/R^{2})^{\times n}.$

We denote by $V^{n-i}\tau_{i}(a)=(0,\dots,0,a,0,\dots,0)$ the canonical
additive generators of $\operatorname{W}_{\langle 2^{n}\rangle}(R)$, for $a\in
R$.

######  Proposition 2.13.

Let $k$ be a field of characteristic $2$, and $n\geq 1$. Under the
isomorphisms of Theorem 2.6 and [HM97, Theorem F], the restriction map
corresponds to the unique ring homomorphism

$\operatorname{res}^{D_{2^{n}}}_{C_{2^{n}}}\colon\phi^{n-1}((k\otimes_{S}k)^{C_{2}})\underset{k\otimes_{S}k}{\times}\phi^{n-1}((k\otimes_{S}k)^{C_{2}})\longrightarrow\operatorname{W}_{\langle
2^{n}\rangle}(k)/2$

which sends the respective generators of Proposition 2.8 to

$\begin{array}[]{ll}\operatorname{res}^{D_{2^{n}}}_{C_{2^{n}}}\tau_{n}(a\otimes
b)&=(ab,0,\dots,0)=\tau_{n}(ab)\\\
\operatorname{res}^{D_{2^{n}}}_{C_{2^{n}}}V^{n-i}\tau_{i}(a\otimes
b)&=(0,\dots,0,[ab],0,\dots,0)=[V^{n-i}\tau_{i}(ab)]\\\
\operatorname{res}^{D_{2^{n}}}_{C_{2^{n}}}\sigma V^{n-i}\tau_{i}(a\otimes
b)&=(0,\dots,0,[ab],0,\dots,0)=[V^{n-i}\tau_{i}(ab)]\end{array}$

for all $0\leq i\leq n-1$, where the mod $k^{2}$ reduction $[ab]$ of $ab$ sits
in the $(n-i+1)$-st component.

###### Proof.

Since $\underline{\pi_{0}}\operatorname{THR}(k)$ is a $D_{2^{n}}$-Tambara
functor, the restriction $\operatorname{res}^{D_{2^{n}}}_{C_{2^{n}}}$ is a
ring homomorphism, and by the double-coset formulas it commutes with norms and
transfers, and with the Weyl action. The operators $V^{n-i}$ and $\tau_{i}$
are described in terms of norms and transfers by Propositions 2.9 and 2.11,
and by [HM97, Theorem 3.3] for the usual Witt vectors. It therefore follows
that

$\operatorname{res}^{D_{2^{n}}}_{C_{2^{n}}}\sigma V^{n-i}\tau_{i}(a\otimes
b)=[wV^{n-i}\tau_{i}\operatorname{res}^{\operatorname{\mathbb{Z}}/2}_{e}(a\otimes
b)]=[V^{n-i}\tau_{i}(ab)],$

where $\operatorname{res}^{\operatorname{\mathbb{Z}}/2}_{e}$ is the
multiplication map of $k\otimes_{S}k$ by [DMPR21, Theorem 5.1]. The proof for
the other generators is similar. ∎

######  Definition 2.14.

The fundamental ideal $J_{\langle 2^{n}\rangle}$ of
$\pi_{0}\operatorname{TRR}^{n+1}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}$ is
the kernel of the ring homomorphism

$J_{\langle
2^{n}\rangle}:=\ker\big{(}\operatorname{res}^{D_{2^{n}}}_{C_{2^{n}}}\colon\phi^{n-1}((k\otimes_{S}k)^{C_{2}})\underset{k\otimes_{S}k}{\times}\phi^{n-1}((k\otimes_{S}k)^{C_{2}})\longrightarrow\operatorname{W}_{\langle
2^{n}\rangle}(k)/2\big{)}$

from Proposition 2.13, for $n\geq 1$, and for $n=0$ it is the kernel of the
multiplication map

$J_{\langle
1\rangle}:=\ker\big{(}\operatorname{res}^{\operatorname{\mathbb{Z}}/2}_{e}\colon
k\otimes_{S}k\longrightarrow k=\operatorname{W}_{\langle
1\rangle}(k)/2\big{)}.$

######  Proposition 2.15.

For every $n\geq 0$, $J_{\langle 2^{n}\rangle}$ is the subgroup of
$\pi_{0}\operatorname{TRR}^{n+1}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}$
generated by the elements

$\displaystyle\tau_{n}(a\otimes b)+\tau_{n}(ab\otimes 1),$ $\displaystyle
V^{n-i}\tau_{i}(a\otimes b)+\sigma V^{n-i}\tau_{i}(a\otimes b),$
$\displaystyle V^{n-i}\tau_{i}(a\otimes b)+V^{n-i}\tau_{i}(ab\otimes 1),$

for all $0\leq i\leq n-1$ and $a\otimes b\in k\otimes_{S}k$.

###### Proof.

The proof is by induction on $n$. For $n=0$, this is the claim that the kernel
of the multiplication map

$\mu\colon k\otimes_{S}k\longrightarrow k$

is generated by $a\otimes b+ab\otimes 1$ for $a\otimes b\in k\otimes_{S}k$,
which is clear.

Now suppose the claim holds for $n$, and consider the commutative diagram with
exact rows

$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{V^{n+1}+\sigma
V^{n+1}}$$\textstyle{J_{\langle
2^{n+1}\rangle}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{R}$$\textstyle{J_{\langle
2^{n}\rangle}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(k\otimes_{S}k)\oplus(k\otimes_{S}k)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{V^{n+1}+\sigma
V^{n+1}}$$\scriptstyle{\mu+\mu}$$\textstyle{\pi_{0}\operatorname{TRR}^{n+2}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{R}$$\scriptstyle{\operatorname{res}^{D_{2^{n+1}}}_{C_{2^{n+1}}}}$$\textstyle{\pi_{0}\operatorname{TRR}^{n+1}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\operatorname{res}^{D_{2^{n}}}_{C_{2^{n}}}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{k/k^{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{V^{n+1}}$$\textstyle{\operatorname{W}_{\langle
2^{n+1}\rangle}(k)/2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{R}$$\textstyle{\operatorname{W}_{\langle
2^{n}\rangle}(k)/2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where the vertical maps from the top row to the middle row are kernel
inclusions. The middle row is exact by [DMP24, Proof of Theorem 4.9], or by
the explicit calculation of Theorem 2.6 and Proposition 2.9. Thus, if we find
a set of generators for $K$ and show that the map $R$ in the first row is
surjective, then $J_{\langle 2^{n+1}\rangle}$ is generated by the image by
$V^{n+1}+\sigma V^{n+1}$ of the generators of $K$, and by a choice of lifts of
the generators of $J_{\langle 2^{n}\rangle}$ given by the inductive
assumption. The kernel $K$ consists of those elements $(x,y)$ such that
$\mu(x)+\mu(y)$ is a square in $k$. Since every square $c^{2}$ in $k$ is hit
by $c\otimes c$ under $\mu$, $K$ is the subgroup of elements of the form
$(x,y)$ where

$x=y+c\otimes c+z$

for some $c\in k$ and $z\in\ker(\mu)$. Since the kernel of $\mu$ is generated
by elements of the form $a\otimes b+ab\otimes 1$, we conclude that $K$ is
generated by elements of the form $(a\otimes b,a\otimes b)$, and elements of
the form $(a\otimes b+ab\otimes 1+c\otimes c,0)$. The images of these
generators by $V^{n+1}+\sigma V^{n+1}$ are respectively of the form

$V^{n+1}(a\otimes b)+\sigma V^{n+1}(a\otimes b)$

and

$V^{n+1}(a\otimes b+ab\otimes 1+c\otimes c)+\sigma V^{n+1}(0)=V^{n+1}(a\otimes
b)+V^{n+1}(ab\otimes 1)+V^{n+1}(c\otimes c),$

where $V^{n+1}(c\otimes c)=0$ by Proposition 2.9. It therefore remains to show
that by applying $R$ to the elements $\tau_{n+1}(a\otimes
b)+\tau_{n+1}(ab\otimes 1)$ , $V^{n+1-i}\tau_{i}(a\otimes b)+\sigma
V^{n+1-i}\tau_{i}(a\otimes b)$ and $V^{n+1-i}\tau_{i}(a\otimes
b)+V^{n+1-i}\tau_{i}(ab\otimes 1)$, for $1\leq i\leq n$, we hit all the
generators of $J_{\langle 2^{n}\rangle}$ given by the inductive assumption.
This is the case since $R\tau_{i+1}=\tau_{i}$, $RV^{n+1-i}=V^{n-i}R$, and
$R\sigma=\sigma R$, by Theorem 2.6 and Proposition 2.9. ∎

######  Corollary 2.16.

For every $n\geq 0$, the ideal $J_{\langle 2^{n}\rangle}$ is generated, as a
$\pi_{0}\operatorname{TRR}^{n+1}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}$-module,
by the elements of the form

$\displaystyle\tau_{n}(1\otimes c)+\tau_{n}(c\otimes 1)=V^{0}\tau_{n}(c\otimes
1)+\sigma V^{0}\tau_{n}(c\otimes 1),$ $\displaystyle V^{n-i}\tau_{i}(a\otimes
b)+\sigma V^{n-i}\tau_{i}(a\otimes b),$

for all $0\leq i\leq n-1$, $a\otimes b\in k\otimes_{S}k$ and $c\in k$. In
particular, $J_{\langle 2^{n}\rangle}$ is generated, as a
$\pi_{0}\operatorname{TRR}^{n+1}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}$-module,
by fixed points for the involution $\sigma$.

###### Proof.

By Proposition 2.15, the corollary follows from the identities

$\displaystyle\tau_{n}(a\otimes b)+\tau_{n}(ab\otimes 1)$
$\displaystyle=\tau_{n}(a\otimes 1)\cdot(\tau_{n}(1\otimes
b)+\tau_{n}(b\otimes 1)),$ $\displaystyle V^{n-i}\tau_{i}(a\otimes
b)+V^{n-i}\tau_{i}(ab\otimes 1)$ $\displaystyle=(V^{n-i}\tau_{i}(a\otimes
1)+\sigma V^{n-i}\tau_{i}(b\otimes 1))\cdot(V^{n-i}\tau_{i}(b\otimes 1)+\sigma
V^{n-i}\tau_{i}(b\otimes 1)),$

which can be easily verified from the definitions. ∎

##  3 Real TR and the de Rham-Witt complex

###  3.1 The Witt complex associated to TRR

Since the operators $F,V,\sigma$ and $R$ of Theorem 2.6 and Proposition 2.9
commute with the restriction map to the Witt vectors modulo $2$, they induce
maps on the fundamental ideals

$F,R\colon J_{\langle 2^{n+1}\rangle}\to J_{\langle 2^{n}\rangle}\ \ \ \ \ \
,\ \ \ \ \ \ V\colon J_{\langle 2^{n}\rangle}\to J_{\langle 2^{n+1}\rangle}\ \
\ \ \ \ and\ \ \ \ \ \ \ \sigma\colon J_{\langle 2^{n}\rangle}\to J_{\langle
2^{n}\rangle}$

for all $n\geq 0$.

######  Proposition 3.1.

For every integer $q\geq 2$, the maps $F,R,V,\sigma$ above restrict to maps

$F,R\colon J_{\langle 2^{n+1}\rangle}^{q}\to J_{\langle 2^{n}\rangle}^{q}\ \ \
\ \ \ ,\ \ \ \ \ \ V\colon J_{\langle 2^{n}\rangle}^{q}\to J_{\langle
2^{n+1}\rangle}^{q}\ \ \ \ \ \ and\ \ \ \ \ \ \ \sigma\colon J_{\langle
2^{n}\rangle}^{q}\to J_{\langle 2^{n}\rangle}^{q}.$

Moreover, $1+\sigma$ induces a well-defined map

$1+\sigma\colon J_{\langle 2^{n}\rangle}^{q}\to J_{\langle
2^{n}\rangle}^{q+1},$

which satisfies $(1+\sigma)^{2}=0$.

###### Proof.

The claim about $R,F$ and $\sigma$ are clear since these maps are
multiplicative. For the map $V$, we employ Corollary 2.16. First suppose that
$n\geq 1$, so that the power $J_{\langle 2^{n}\rangle}^{q}$ is additively
generated by elements of the form

$(x,y)\cdot(x_{1},x_{1})\cdot\dots\cdot(x_{q},x_{q})=(xx_{1}\dots
x_{q},yx_{1}\dots x_{q}),$

where $(x,y)$ is a generator of
$\pi_{0}\operatorname{TRR}^{n+1}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}$ from
Proposition 2.8, and $(x_{l},x_{l})$ is a generator of $J_{\langle
2^{n}\rangle}$ from Corollary 2.16, which is diagonal since they are invariant
by the Weyl action $\sigma$. Since $V$ is additive, it is sufficient to show
that $V$ sends these elements to $J_{\langle 2^{n+1}\rangle}^{q}$. Now by
Proposition 2.9, we have that

$\displaystyle V(xx_{1}\dots x_{q},yx_{1}\dots x_{q})$
$\displaystyle=((x+y)x_{1}\dots x_{q},0)$
$\displaystyle=(x+y,x+y)\cdot(x_{1},x_{1})\cdot\dots\cdot(x_{q-2},x_{q-2})\cdot(x_{q-1},x_{q-1})\cdot(x_{q},0).$

The first factor is

$(x+y,x+y)=V(x,y)+\sigma V(x,y),\ \ \ \ \ $

which belongs to $J_{\langle 2^{n+1}\rangle}$ since it is sent to zero by the
restriction map by Proposition 2.13. By Corollary 2.16, each of the factors
$(x_{1},x_{1}),\dots,(x_{q-1},x_{q-1})$ is of the form

$V^{n-i}\tau_{i}(a\otimes b)+\sigma V^{n-i}\tau_{i}(a\otimes b)$

for some $0\leq i\leq n$, which as an element of
$\pi_{0}\operatorname{TRR}^{n+2}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}$ is of
the form

$V^{n+1-i}\tau_{i}(a\otimes b)+\sigma V^{n+1-i}\tau_{i}(a\otimes b)$

for $0\leq i\leq n$, and therefore belongs to $J_{\langle 2^{n+1}\rangle}$.
Thus, it suffices to show that $(x_{q-1},x_{q-1})\cdot(x_{q},0)$ is also in
$J_{\langle 2^{n+1}\rangle}$. But since $(x_{q},x_{q})$ is of the form
$V^{n-i}\tau_{i}(a\otimes b)+\sigma V^{n-i}\tau_{i}(a\otimes b)$, we have that
$(x_{q},0)$ is a well-defined element of
$\pi_{0}\operatorname{TRR}^{n+2}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}$, and
since $J_{\langle 2^{n+1}\rangle}$ is an ideal,
$(x_{q-1},x_{q-1})\cdot(x_{q},0)$ indeed belongs to $J_{\langle
2^{n+1}\rangle}$.

If $n=0$, the ideal $J_{\langle 1\rangle}^{q}$ of $k\otimes_{S}k$ is
additively generated by elements of the form $xx_{1}\dots x_{q}$ with $x\in
k\otimes_{S}k$ and $x_{1},\dots,x_{q}$ fixed by the involution $w$. Then by
Proposition 2.9

$V(xx_{1}\dots x_{q})=(xx_{1}\dots x_{q}+w(xx_{1}\dots
x_{q}),0)=((x+w(x))x_{1}\dots x_{q},0),$

and one can repeat the argument used in the case $n\geq 1$.

Finally, let us show that $1+\sigma$ sends $J_{\langle 2^{n}\rangle}^{q}$ to
$J_{\langle 2^{n}\rangle}^{q+1}$. By Corollary 2.16, every element of
$J_{\langle 2^{n}\rangle}^{q}$ is a sum of elements of the form $z\cdot
g_{1}\cdot\dots\cdot g_{q}$ with
$z\in\pi_{0}\operatorname{TRR}^{n+1}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}$
and each $g_{i}\in J_{\langle 2^{n}\rangle}$ fixed by $\sigma$. Since
$1+\sigma$ is additive we only need to show that these elements are sent to
$J_{\langle 2^{n}\rangle}^{q+1}$. Since $\sigma$ is multiplicative, we have
that

$(1+\sigma)(z\cdot g_{1}\cdot\dots\cdot g_{q})=z\cdot g_{1}\cdot\dots\cdot
g_{q}+\sigma(z)\cdot\sigma(g_{1})\cdot\dots\cdot\sigma(g_{q})=(z+\sigma(z))\cdot
g_{1}\cdot\dots\cdot g_{q}.$

It therefore suffices to show that $z+\sigma(z)$ belongs to $J_{\langle
2^{n}\rangle}$, which is the case since the restriction map to the Witt
vectors modulo $2$ is invariant under the action of $\sigma$. Clearly, since
$\sigma^{2}$ is the identity, we have that $(1+\sigma)^{2}=0$. ∎

For every $n\geq 0$, let us denote by $J_{\langle
2^{n}\rangle}^{\ast}/J_{\langle 2^{n}\rangle}^{\ast+1}$ the graded ring
defined by the quotients $J_{\langle 2^{n}\rangle}^{q}/J_{\langle
2^{n}\rangle}^{q+1}$ for $q\geq 0$, and by the multiplication of $J_{\langle
2^{n}\rangle}$. We will show that the sequence of graded rings $J_{\langle
2^{n}\rangle}^{\ast}/J_{\langle 2^{n}\rangle}^{\ast+1}$ where $n\geq 0$,
equipped with the operators $R,F,V$ and $d:=(1+\sigma)$, define the structure
of a $2$-typical Witt complex. We recall its definition, from [Cos08], in the
special case where the base ring has characteristic $2$. In this case item
$v)$ simplifies since $d\log[-1]=0$, and the definition agrees to the one for
odd primes from [HM04].

######  Definition 3.2 ([Cos08]).

A $2$-typical Witt complex over an $\operatorname{\mathbb{F}}_{2}$-algebra $A$
consists of:

  1. i)

a graded-commutative pro-graded ring $\\{E^{\ast}_{n},R\colon
E^{\ast}_{n+1}\to E^{\ast}_{n}\\}_{n\geq 0}$,

  2. ii)

A strict map of pro-rings $\lambda\colon\operatorname{W}_{\langle
2^{\bullet}\rangle}(A)\to E^{0}_{\bullet}$ from the pro-ring of $2$-typical
Witt vectors of $A$,

  3. iii)

a strict map of pro-graded rings

$F\colon E^{\ast}_{\bullet+1}\longrightarrow E^{\ast}_{\bullet}$

such that $\lambda F=F\lambda$,

  4. iv)

a strict map of pro-graded $E^{\ast}_{\bullet}$-modules

$V\colon F^{\ast}E^{\ast}_{\bullet}\longrightarrow E^{\ast}_{\bullet+1}$

such that $\lambda V=V\lambda$ and $FV=2$. The linearity of $V$ means that
$V(x)y=V(xF(y))$ for all $x\in E^{\ast}_{n}$ and $y\in E^{\ast}_{n+1}$,

  5. v)

a strict map of pro-graded abelian groups $d\colon E^{\ast}_{\bullet}\to
E^{\ast+1}_{\bullet}$, which is a derivation, in the sense that

$d(xy)=d(x)y+(-1)^{|x|}xd(y)$

for all $x,y\in E^{\ast}_{n}$, and which satisfies the relations

$\displaystyle FdV$ $\displaystyle=d$ $\displaystyle dd$ $\displaystyle=0$
$\displaystyle Fd\lambda\tau_{n}$
$\displaystyle=(\lambda\tau_{n-1})\cdot(d\lambda\tau_{n-1}),$

where $\tau_{n}\colon A\to\operatorname{W}_{\langle 2^{n}\rangle}(A)$ is the
Teichmüller map sending $a$ to $(a,0,\dots,0)$.

Before showing that the graded ring defined by the ideals $J_{\langle
2^{n}\rangle}$ admits the structure of a Witt-complex, let us point out that
since the map $\operatorname{res}^{D_{2^{n}}}_{C_{2^{n}}}$ is surjective by
Propositions 2.13, it induces an isomorphism

$J_{\langle 2^{n}\rangle}^{0}/J_{\langle
2^{n}\rangle}=\pi_{0}\operatorname{TRR}^{n+1}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}/\ker(\operatorname{res}^{D_{2^{n}}}_{C_{2^{n}}})\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\pi_{0}\operatorname{TR}^{n+1}(k;2)/2\cong\operatorname{W}_{\langle
2^{n}\rangle}(k)/2,$

where the last isomorphism is from [HM97, Theorem F].

######  Proposition 3.3.

The sequence of graded rings $\\{J_{\langle 2^{n}\rangle}^{\ast}/J_{\langle
2^{n}\rangle}^{\ast+1}\\}_{n\geq 0}$ equipped with the operators $R,F,V$ and
$d:=(1+\sigma)$ from Proposition 3.1, and the quotient maps

$\lambda\colon\operatorname{W}_{\langle
2^{n}\rangle}(k)\longrightarrow\operatorname{W}_{\langle
2^{n}\rangle}(k)/2\cong J_{\langle 2^{n}\rangle}^{0}/J_{\langle
2^{n}\rangle},$

defines a $2$-typical Witt complex over the field $k$ of characteristic $2$.

###### Proof.

First of all, the maps $R,F,V$ and $d:=(1+\sigma)$ are well-defined on the
quotients of the powers of the ideals by Proposition 3.1. Axioms $i)$-$iv)$ of
Definition 3.2 follow immediately from either the fact that $F,V$ and
$\operatorname{res}^{D_{2^{n}}}_{C_{2^{n}}}$ are induced from the maps of a
Mackey functor, or from their explicit formulas from Theorem 2.6 and
Propositions 2.9 and 2.13. This is except from the identity $FV=2$ (which in
our case is zero), since by these arguments we only know that $FV=1+\sigma$.
However, for every $x\in J_{\langle 2^{n}\rangle}^{q}$, we have that

$FV(x)=1+\sigma(x)$

belongs to $J_{\langle 2^{n}\rangle}^{q+1}$ by Proposition 3.1, and it is
therefore indeed zero in $J_{\langle 2^{n}\rangle}^{q}/J_{\langle
2^{n}\rangle}^{q+1}$.

Let us show axiom $v)$. To see that $d$ satisfies the Leibniz rule, let $x\in
J_{\langle 2^{n}\rangle}^{q}/J_{\langle 2^{n}\rangle}^{q+1}$ and $y\in
J_{\langle 2^{n}\rangle}^{q^{\prime}}/J_{\langle
2^{n}\rangle}^{q^{\prime}+1}$, and let us calculate

$\displaystyle d(xy)+d(x)y+xd(y)$
$\displaystyle=xy+\sigma(x)\sigma(y)+(x+\sigma(x))y+x(y+\sigma(y))$
$\displaystyle=xy+\sigma(x)\sigma(y)+\sigma(x)y+x\sigma(y)=(x+\sigma(x))(y+\sigma(y))$
$\displaystyle=d(x)d(y).$

Since $d(x)$ belongs to $J_{\langle 2^{n}\rangle}^{q+1}$ and $d(y)$ to
$J_{\langle 2^{n}\rangle}^{q^{\prime}+1}$ by Proposition 3.1, we have that
$d(x)d(y)$ belongs to $J_{\langle 2^{n}\rangle}^{q+q^{\prime}+2}$, and
therefore it vanishes in $J_{\langle 2^{n}\rangle}^{q+q^{\prime}+1}/J_{\langle
2^{n}\rangle}^{q+q^{\prime}+2}$.

Let us now verify the last three identities involving $d$ in axiom $v)$. For
the first one, let $x\in J_{\langle 2^{n}\rangle}^{q}/J_{\langle
2^{n}\rangle}^{q+1}$. Then

$FdV(x)=FV(x)+F\sigma V(x)=FV(x)=(1+\sigma)(x)=d(x)$

in $J_{\langle 2^{n}\rangle}^{q+1}/J_{\langle 2^{n}\rangle}^{q+2}$, where
$F\sigma V(x)=0$ by the double coset formula (or by direct calculation). For
the second identity, we have that

$d^{2}=(1+\sigma)^{2}=1+2\sigma+\sigma^{2}=2+2\sigma=0$

since $\sigma$ has order $2$. Finally, for the third one, let $a\in
k=\operatorname{W}_{\langle 1\rangle}(k)$. On the one hand, by Proposition
2.13,

$\displaystyle Fd\lambda\tau_{n}(a)$ $\displaystyle=Fd\tau_{n}(a\otimes
1)=F(\tau_{n}(a\otimes 1)+\sigma\tau_{n}(a\otimes 1))$
$\displaystyle=\tau_{n}(a\otimes 1)+\sigma\tau_{n}(a\otimes
1)=(a^{2^{n}}\otimes 1+a^{2^{n}-1}\otimes a,a^{2^{n}-1}\otimes
a+a^{2^{n}}\otimes 1)$

where the third equality holds by the formula for $F$ of Theorem 2.6. On the
other hand

$\displaystyle(\lambda\tau_{n-1}(a))\cdot(d\lambda\tau_{n-1}(a))=\tau_{n-1}(a\otimes
1)\cdot(\tau_{n-1}(a\otimes 1)+\sigma\tau_{n-1}(a\otimes 1))$
$\displaystyle=(a^{2^{n-1}}\otimes 1,a^{2^{n-1}-1}\otimes
a)\cdot(a^{2^{n-1}}\otimes 1+a^{2^{n-1}-1}\otimes a,a^{2^{n-1}-1}\otimes
a+a^{2^{n-1}}\otimes 1)$ $\displaystyle=(a^{2^{n}}\otimes 1+a^{2^{n}-1}\otimes
a,a^{2^{n}-2}\otimes a^{2}+a^{2^{n}-1}\otimes a),$

and these are equal since we are tensoring over $S$.

∎

###  3.2 The Milnor conjecture for the deRham-Witt complex

Let us endow the sequence $J_{\langle 2^{\bullet}\rangle}^{\ast}/J_{\langle
2^{\bullet}\rangle}^{\ast+1}$ with the structure of a $2$-typical Witt complex
of Proposition 3.3. We recall that, by definition, the $2$-typical deRham-Witt
complex $\operatorname{W}_{\langle 2^{\bullet}\rangle}\Omega^{\ast}_{k}$ of
$k$ is the initial object in the category of $2$-typical Witt complexes over
$k$ (see [Cos08] and [HM04]). Thus, there is a unique map of $2$-typical Witt
complexes

$\operatorname{W}_{\langle 2^{\bullet}\rangle}\Omega^{\ast}_{k}\longrightarrow
J_{\langle 2^{\bullet}\rangle}^{\ast}/J_{\langle
2^{\bullet}\rangle}^{\ast+1}.$

Let us denote by $\operatorname{W}_{\langle
2^{\bullet}\rangle}\Omega^{\ast}_{k}/2$ the degreewise cokernel of the
multiplication by $2$ map. Since all the maps defining the structure of a Witt
complex are additive, this is again a Witt-complex, where the map

$\operatorname{W}_{\langle
2^{\bullet}\rangle}(k)\longrightarrow\operatorname{W}_{\langle
2^{\bullet}\rangle}\Omega^{0}_{k}/2=\operatorname{W}_{\langle
2^{\bullet}\rangle}(k)/2$

is the quotient map. Since $2$ vanishes in $J_{\langle
2^{\bullet}\rangle}^{\ast}/J_{\langle 2^{\bullet}\rangle}^{\ast+1}$, the
unique map above descends to a unique map of Witt-complexes

$u\colon\operatorname{W}_{\langle
2^{\bullet}\rangle}\Omega^{\ast}_{k}/2\longrightarrow J_{\langle
2^{\bullet}\rangle}^{\ast}/J_{\langle 2^{\bullet}\rangle}^{\ast+1}.$

######  Theorem 3.4.

The unique morphism of Witt-complexes $u\colon\operatorname{W}_{\langle
2^{\bullet}\rangle}\Omega^{\ast}_{k}/2\to J_{\langle
2^{\bullet}\rangle}^{\ast}/J_{\langle 2^{\bullet}\rangle}^{\ast+1}$ is an
isomorphism.

######  Remark 3.5.

Let us discuss a few special cases of this theorem. For $\ast=0$, the unique
map $u$ is by construction the isomorphism

$\lambda\colon\operatorname{W}_{\langle
2^{\bullet}\rangle}\Omega^{0}_{k}/2=\operatorname{W}_{\langle
2^{\bullet}\rangle}(k)/2\stackrel{{\scriptstyle\cong}}{{\longrightarrow}}\pi_{0}\operatorname{TR}(k;2)/2\cong\pi_{0}\operatorname{TRR}^{\bullet+1}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}/J_{\langle
2^{\bullet}\rangle}$

where the arrow is the isomorphism of [HM97, Theorem F].

On the other hand, for $\bullet=0$, the map $u$ is the unique map of
commutative differential graded algebras

$\operatorname{W}_{\langle
1\rangle}\Omega^{\ast}_{k}/2=\Omega^{\ast}_{k}\longrightarrow
J^{\ast}_{\langle 1\rangle}/J^{\ast+1}_{\langle 1\rangle}.$

This is well-known to be an isomorphism, as claimed in [Kat82] (see e.g.
[Ara20] for a proof, which is also recasted in Lemma 3.6 below). In
particular, for $\ast=1$, this is equivalent to the fact that, since $k$ has
characteristic $2$, a $\operatorname{\mathbb{Z}}$-linear derivation out of $k$
is automatically $S$-linear (where we recall that $S\leq k$ is the subfield of
squares).

The rest of the section is dedicated to the proof of Theorem 3.4. The proof is
by induction on $n$, by means of the exact sequences

$\textstyle{\Omega^{q}_{k}\oplus\Omega^{q-1}_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{V^{n}+dV^{n}}$$\textstyle{\operatorname{W}_{\langle
2^{n}\rangle}\Omega^{q}_{k}/2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{R}$$\textstyle{\operatorname{W}_{\langle
2^{n-1}\rangle}\Omega^{q}_{k}/2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

from [Cos08, Lemma 3.5], where $n,q\geq 1$.

The base case for the induction $n=1$ seems to be well-known to the experts,
and is used without proof in [Kat82]. We recall the argument from [Ara20, Fact
1] for the reader’s convenience, and to introduce some notation that we will
use in the proof of the induction step.

######  Lemma 3.6 ([Ara20]).

The unique map $u\colon\Omega^{\ast}_{k}\longrightarrow J_{\langle
1\rangle}^{\ast}/J_{\langle 1\rangle}^{\ast+1}$ of commutative differential
graded algebras is an isomorphism.

###### Proof.

For every $a\in k$, let us denote $\Delta(a):=1\otimes a+a\otimes 1\in
k\otimes_{S}k$. We note that the map $u$ is necessarily given by the formula

$u(ada_{1}\dots da_{q}):=a\Delta(a_{1})\cdot\dots\cdot\Delta(a_{q}).$

In order to show that this is an isomorphism, we choose suitable bases of the
source and target as $k$-vector spaces. Let $\\{x_{i}\\}_{i\in I}$ be a
$2$-basis of $k$. We recall that this is a set of elements of $k$ whose
differentials $\\{dx_{i}\\}_{i\in I}$ form a basis of the $k$-vector space
$\Omega^{1}_{k}$ or, equivalently, such that the elements

$x^{\xi}:=\prod_{i\in\xi}x_{i}$

form a basis of $k$ as an $S$-vector space, where $\xi$ ranges through the
finite subsets of $I$ (see [Gro67, Chapter 0, §21.4]). Here we use the
convention that $x^{\emptyset}=1$, and we will write $\xi\subset^{f}\\!I$ if
$\xi$ is a finite subset of $I$. It is easy to see that the set $\\{1\otimes
x^{\xi}\\}_{\xi\subset^{f}I}$ is a basis of $k\otimes_{S}k$ as a $k$-vector
space, where $k$ acts by multiplication on the left tensor factor. Now let us
denote

$\Delta(x)^{\xi}:=\prod_{i\in\xi}\Delta(x_{i})$

for every $\xi\subset^{f}\\!I$ and $a\in k$ (with the convention that
$\Delta(x)^{\emptyset}=1\otimes 1$). These elements satisfy the identities

$\displaystyle\Delta(x)^{\xi}$
$\displaystyle=\sum_{\nu\subset\xi}x^{\xi\setminus\nu}\cdot(1\otimes x^{\nu})$
(1) $\displaystyle 1\otimes x^{\xi}$
$\displaystyle=\sum_{\nu\subset\xi}x^{\xi\setminus\nu}\cdot\Delta(x)^{\nu}$
(2)

for every finite subset $\xi$ of $I$. It follows that
$\\{\Delta(x)^{\xi}\\}_{\xi\subset^{f}I}$ is also a basis of $k\otimes_{S}k$.
Since the multiplication map sends $\Delta(x)^{\xi}$ to a non-zero element of
$k$ if and only if $\xi=\emptyset$, it follows that
$\\{\Delta(x)^{\xi}\\}_{\emptyset\neq\xi\subset^{f}I}$ is a basis for
$J_{\langle 1\rangle}$ as a $k$-vector space with respect to multiplication on
the left factor. It then readily follows that the elements $\Delta(x)^{\xi}$
with $|\xi|\geq q$ form a basis of $J_{\langle 1\rangle}^{q}$, and that the
elements $\Delta(x)^{\xi}$ with $|\xi|=q$ form a basis of $J_{\langle
1\rangle}^{q}/J_{\langle 1\rangle}^{q+1}$. Since for every $\xi\subset I$ with
$|\xi|=q$, the map $u$ sends a basis element
$(dx)^{\xi}:=\prod_{i\in\xi}dx_{i}$ of $\Omega^{q}_{k}$ to $\Delta(x)^{\xi}$,
the claim follows. ∎

The induction step for proving Theorem 3.4 will rely on the following two key
technical Lemmas. Let us choose a $2$-basis $\\{x_{i}\\}_{i\in I}$ of $k$ as
in the proof of Lemma 3.6.

######  Lemma 3.7.

Let $q\geq 1$, and let $\mu_{\nu}\in k$ for every subset $\nu\subset I$ with
$|\nu|=q-1$. Suppose that

$\sum_{\begin{smallmatrix}\nu\subset I\\\
|\nu|=q-1\end{smallmatrix}}\Delta(\mu_{\nu})\Delta(x)^{\nu}\in J_{\langle
1\rangle}^{q+1}.$

Then $\sum_{\nu\subset I,|\nu|=q-1}V(\mu_{\nu}(dx)^{\nu})$ is divisible by $2$
in $\operatorname{W}_{\langle 2\rangle}\Omega^{q-1}_{k}$.

###### Proof.

Let us write $\mu_{\nu}\in k$ uniquely as a linear combination
$\mu_{\nu}=\sum_{\delta\subset^{f}I}s_{\nu,\delta}^{2}x^{\delta}$ with respect
to the basis $\\{x^{\delta}\\}_{\delta\subset^{f}I}$ of $k$ as an $S$-vector
space. Let us notice that, since we are tensoring over $S$, the map
$\Delta\colon k\to k\otimes_{S}k$ is $S$-linear. Then by applying formula (2):

$\displaystyle\Delta(\mu_{\nu})$
$\displaystyle=\sum_{\delta\subset^{f}I}s_{\nu,\delta}^{2}\Delta(x^{\delta})=\sum_{\delta\subset^{f}I}s_{\nu,\delta}^{2}(1\otimes
x^{\delta}+x^{\delta}\cdot(1\otimes 1))$
$\displaystyle=\sum_{\delta\subset^{f}I}s_{\nu,\delta}^{2}\sum_{\gamma\subset\delta}x^{\delta\setminus\gamma}\Delta(x)^{\gamma}+\sum_{\delta\subset^{f}I}s_{\nu,\delta}^{2}x^{\delta}\cdot\Delta(x)^{\emptyset}=\sum_{\emptyset\neq\gamma\subset\delta\subset^{f}I}s_{\nu,\delta}^{2}x^{\delta\setminus\gamma}\Delta(x)^{\gamma}.$

It follows that

$\displaystyle\sum_{\begin{smallmatrix}\nu\subset I\\\
|\nu|=q-1\end{smallmatrix}}\Delta(\mu_{\nu})\Delta(x)^{\nu}$
$\displaystyle=\sum_{\begin{smallmatrix}\nu\subset I\\\
|\nu|=q-1\end{smallmatrix}}(\sum_{\emptyset\neq\gamma\subset\delta^{f}\subset
I}s_{\nu,\delta}^{2}x^{\delta\setminus\gamma}\Delta(x)^{\gamma})\Delta(x)^{\nu}=\sum_{\begin{smallmatrix}\nu\subset
I\\\
|\nu|=q-1\end{smallmatrix}}\sum_{\begin{smallmatrix}\emptyset\neq\gamma\subset\delta\subset^{f}I\\\
\gamma\cap\nu=\emptyset\end{smallmatrix}}s_{\nu,\delta}^{2}x^{\delta\setminus\gamma}\Delta(x)^{\gamma\amalg\nu},$

where in the last sum $\gamma$ and $\nu$ are disjoint, since
$\Delta(a)\Delta(a)=0$ in $k\otimes_{S}k$ for all $a\in k$. Since, by
assumption, this element belongs to $J_{\langle 1\rangle}^{q+1}$, and the
$\Delta(x)^{\xi}$ with $|\xi|=q+1$ are a basis for $J_{\langle
1\rangle}^{q+1}$, we must have that, for every $\nu\subset I$ with $|\nu|=q-1$
and every $j\in I\setminus\nu$ (corresponding to the terms $\gamma=\\{j\\}$ in
the sum above),

$\sum_{\begin{smallmatrix}\nu\subset I\\\
|\nu|=q-1\end{smallmatrix}}\sum_{\delta\subset^{f}I}\sum_{j\in\delta\setminus\nu}s_{\nu,\delta}^{2}x^{\delta\setminus
j}\Delta(x)^{j\amalg\nu}=0.$

Since the $\Delta(x)^{\xi}$ are linearly independent, we have that, for every
subset $\xi\subset I$ with $|\xi|=q$ (corresponding to the terms
$\epsilon=j\amalg\nu$), the coefficient of $\Delta(x)^{\xi}$ must vanish, i.e.
that

$0=\sum_{j\in\xi}\sum_{\begin{smallmatrix}\delta\subset^{f}I\\\
j\in\delta\end{smallmatrix}}s_{\xi\setminus j,\delta}^{2}x^{\delta\setminus
j}=\sum_{\alpha\subset^{f}I}\sum_{j\in\xi\setminus\alpha}s_{\xi\setminus
j,\alpha\amalg j}^{2}x^{\alpha}.$

Since the $x^{\alpha}$ for a basis of $k$ as an $S$-vector space, we find that
for every finite $\alpha\subset I$ and $\xi\subset I$ with $|\xi|=q$ the
corresponding coefficient must vanish:

$\sum_{j\in\xi\setminus\alpha}s_{\xi\setminus j,\alpha\amalg j}^{2}=0.$ (3)

We now show that these relations among the coefficients $s_{\nu,\delta}$ imply
that $V(\sum_{\nu\subset I,|\nu|=q-1}\mu_{\nu}(dx)^{\nu})$ is divisible by $2$
in $\operatorname{W}_{\langle 2\rangle}\Omega^{q-1}_{k}$. We do this by
showing that the sum $\sum_{\nu\subset I,|\nu|=q-1}\mu_{\nu}(dx)^{\nu}$ is in
the image of the Frobenius map. The claim will then follow since, by the
linearity of $V$ of axiom $iv)$ of Definition 3.2,

$V(F(z))=V(1)\cdot z$

for every $z\in\operatorname{W}_{\langle 2\rangle}\Omega^{q-1}_{k}$, and
$V(1)=2\in\operatorname{W}_{\langle 2\rangle}(k)$ since $k$ has characteristic
$2$.

By rearranging the terms and grouping pairs $(\nu,\delta)$ with the same
intersection $\beta$, we can write

$\displaystyle\begin{split}\sum_{\begin{smallmatrix}\nu\subset I\\\
|\nu|=q-1\end{smallmatrix}}\mu_{\nu}(dx)^{\nu}&=\sum_{\begin{smallmatrix}\nu\subset
I\\\
|\nu|=q-1\end{smallmatrix}}\sum_{\delta\subset^{f}I}s^{2}_{\nu,\delta}x^{\delta}(dx)^{\nu}=\sum_{\beta\subset^{f}I}\sum_{\begin{smallmatrix}\nu\subset
I\\\ |\nu|=q-1\end{smallmatrix}}\sum_{\begin{smallmatrix}\delta\subset^{f}I\\\
\delta\cap\nu=\beta\end{smallmatrix}}s^{2}_{\nu,\delta}x^{\delta}(dx)^{\nu}\\\
&=\sum_{\beta\subset^{f}I}x^{\beta}(dx)^{\beta}\sum_{\begin{smallmatrix}\nu\subset
I\\\ |\nu|=q-1\end{smallmatrix}}\sum_{\begin{smallmatrix}\delta\subset^{f}I\\\
\delta\cap\nu=\beta\end{smallmatrix}}s^{2}_{\nu,\delta}x^{\delta\setminus\beta}(dx)^{\nu\setminus\beta}.\end{split}$
(4)

Each term $x^{\beta}(dx)^{\beta}$ is in the image of the Frobenius, since by
axiom $v)$ of Definition 3.2 (we recall that in the de Rham-Witt complex the
map $\lambda$ is the identity)

$\displaystyle x^{\beta}(dx)^{\beta}$
$\displaystyle=\prod_{i\in\beta}x_{i}dx_{i}=\prod_{i\in\beta}\tau_{0}(x_{i})d\tau_{0}(x_{i})=\prod_{i\in\beta}F(d\tau_{1}(x_{i}))=F(\prod_{i\in\beta}d\tau_{1}(x_{i})).$

It is therefore sufficient to show that for every fixed $\beta\subset I$, the
double sum in equation (4) is in the image of the Frobenius. Let us now group
those terms by the union $\lambda$ of $\nu$ and $\delta$, and write

$\displaystyle\sum_{\begin{smallmatrix}\nu\subset I\\\
|\nu|=q-1\end{smallmatrix}}\sum_{\begin{smallmatrix}\delta\subset^{f}I\\\
\delta\cap\nu=\beta\end{smallmatrix}}s^{2}_{\nu,\delta}x^{\delta\setminus\beta}(dx)^{\nu\setminus\beta}$
$\displaystyle=\sum_{\lambda\subset^{f}I}\sum_{\begin{smallmatrix}\nu\subset
I\\\ |\nu|=q-1\end{smallmatrix}}\sum_{\begin{smallmatrix}\delta\subset^{f}I\\\
\delta\cap\nu=\beta\\\
\delta\cup\nu=\lambda\end{smallmatrix}}s^{2}_{\nu,\delta}x^{\delta\setminus\beta}(dx)^{\nu\setminus\beta}.$
(5)

We now show that for every fixed $\beta,\lambda\subset^{f}I$, the inner double
sum in (5) is in the image of the Frobenius. Notice that, after fixing $\beta$
and $\lambda$, the subset $\delta$ is determined by $\nu$, and let us write
$\delta_{\nu}:=(\lambda\setminus\nu)\amalg\beta$. That is, we show that

$\displaystyle\sum_{\begin{smallmatrix}\nu\subset I\\\ |\nu|=q-1\\\
\beta\subset\nu\subset\lambda\end{smallmatrix}}s^{2}_{\nu,\delta_{\nu}}x^{\delta_{\nu}\setminus\beta}(dx)^{\nu\setminus\beta}=\sum_{\begin{smallmatrix}\nu\subset
I\\\ |\nu|=q-1\\\
\beta\subset\nu\subset\lambda\end{smallmatrix}}s^{2}_{\nu,\delta_{\nu}}x^{\lambda\setminus\nu}(dx)^{\nu\setminus\beta}$
(6)

is in the image of the Frobenius. Let us first treat the case where
$\beta=\lambda$ (with $q-1$ elements, otherwise the sum is trivially zero). In
this case the sum is just $s^{2}_{\beta,\beta}$, and every square of $k$ is in
the image of the Frobenius since $s^{2}=F(\tau_{1}(s))$. Thus, suppose that
$\beta$ is a proper subset of $\lambda$, and choose an element
$j_{0}\in\lambda\setminus\beta$. We claim that (6) is equal to

$\sum_{\begin{smallmatrix}\nu\subset I\\\ |\nu|=q-1\\\
\beta\subset\nu\subset\lambda\\\
j_{0}\in\nu\end{smallmatrix}}s^{2}_{\nu,\delta_{\nu}}d(x^{(\lambda\setminus\nu)\amalg
j_{0}})(dx)^{(\nu\setminus\beta)\setminus j_{0}}.$

This will conclude the proof, since any square is in the image of the
Frobenius by the argument above, and so is each differential by the relation
$d=FdV$ of axiom $v)$ (observe that $\lambda\setminus\nu$ and
$\nu\setminus\beta$ are disjoint, with union $\lambda\setminus\beta$, so that
no summands contains the square of a differential). To see that the last claim
holds, let us notice that by iterating the Leibniz rule

$d(x^{\xi})=\sum_{j\in\xi}x^{\xi\setminus j}dx_{j}$

for every finite subset $\xi\subset I$, and therefore

$\displaystyle\sum_{\begin{smallmatrix}\nu\subset I\\\ |\nu|=q-1\\\
\beta\subset\nu\subset\lambda\\\
j_{0}\in\nu\end{smallmatrix}}s^{2}_{\nu,\delta_{\nu}}d(x^{(\lambda\setminus\nu)\amalg
j_{0}})(dx)^{(\nu\setminus\beta)\setminus
j_{0}}=\sum_{\begin{smallmatrix}\nu\subset I\\\ |\nu|=q-1\\\
\beta\subset\nu\subset\lambda\\\
j_{0}\in\nu\end{smallmatrix}}\sum_{j\in(\lambda\setminus\nu)\amalg
j_{0}}s^{2}_{\nu,\delta_{\nu}}x^{((\lambda\setminus\nu)\amalg j_{0})\setminus
j}(dx_{j})(dx)^{(\nu\setminus\beta)\setminus j_{0}}$
$\displaystyle=\sum_{\begin{smallmatrix}\nu\subset I\\\ |\nu|=q-1\\\
\beta\subset\nu\subset\lambda\\\
j_{0}\in\nu\end{smallmatrix}}\big{(}s^{2}_{\nu,\delta_{\nu}}x^{\lambda\setminus\nu}(dx)^{\nu\setminus\beta}+\sum_{j\in\lambda\setminus\nu}s^{2}_{\nu,\delta_{\nu}}x^{((\lambda\setminus\nu)\amalg
j_{0})\setminus j}(dx)^{(\nu\setminus\beta)\setminus j_{0}\amalg j}\big{)}$
$\displaystyle=\big{(}\sum_{\begin{smallmatrix}\nu\subset I\\\ |\nu|=q-1\\\
\beta\subset\nu\subset\lambda\\\
j_{0}\in\nu\end{smallmatrix}}s^{2}_{\nu,\delta_{\nu}}x^{\lambda\setminus\nu}(dx)^{\nu\setminus\beta}\big{)}+\big{(}\sum_{\begin{smallmatrix}\nu\subset
I\\\ |\nu|=q-1\\\ \beta\subset\nu\subset\lambda\\\
j_{0}\in\nu\end{smallmatrix}}\sum_{j\in\lambda\setminus\nu}s^{2}_{\nu,\delta_{\nu}}x^{((\lambda\setminus\nu)\amalg
j_{0})\setminus j}(dx)^{(\nu\setminus\beta)\setminus j_{0}\amalg j}\big{)}.$

By setting $\zeta=(\nu\setminus j_{0})\amalg j$ in the second sum, and
observing that $\delta_{\nu}=(\delta_{\zeta}\setminus j_{0})\amalg j$, we find
that this is equal to

$\displaystyle\big{(}\sum_{\begin{smallmatrix}\nu\subset I\\\ |\nu|=q-1\\\
\beta\subset\nu\subset\lambda\\\
j_{0}\in\nu\end{smallmatrix}}s^{2}_{\nu,\delta_{\nu}}x^{\lambda\setminus\nu}(dx)^{\nu\setminus\beta}\big{)}+\big{(}\sum_{\begin{smallmatrix}\zeta\subset
I\\\ |\zeta|=q-1\\\ \beta\subset\zeta\subset\lambda\\\
j_{0}\notin\zeta\end{smallmatrix}}\sum_{j\in\zeta\setminus\beta}s^{2}_{(\zeta\amalg
j_{0})\setminus j,(\delta_{\zeta}\setminus j_{0})\amalg
j}x^{((\lambda\setminus(\zeta\amalg j_{0}\setminus j))\amalg j_{0})\setminus
j}(dx)^{((\zeta\amalg j_{0}\setminus j)\setminus\beta)\setminus j_{0}\amalg
j}\big{)}$ (7) $\displaystyle=\big{(}\sum_{\begin{smallmatrix}\nu\subset I\\\
|\nu|=q-1\\\ \beta\subset\nu\subset\lambda\\\
j_{0}\in\nu\end{smallmatrix}}s^{2}_{\nu,\delta_{\nu}}x^{\lambda\setminus\nu}(dx)^{\nu\setminus\beta}\big{)}+\big{(}\sum_{\begin{smallmatrix}\zeta\subset
I\\\ |\zeta|=q-1\\\ \beta\subset\nu\subset\lambda\\\
j_{0}\notin\zeta\end{smallmatrix}}\big{(}\sum_{j\in\zeta\setminus\beta}s^{2}_{(\zeta\amalg
j_{0})\setminus j,(\delta_{\zeta}\setminus j_{0})\amalg
j}\big{)}x^{\lambda\setminus\zeta}(dx)^{\zeta\setminus\beta}\big{)}.$ (8)

Finally, by applying the relation (3) for $\xi=\zeta\amalg j_{0}$ and
$\alpha=\delta_{\zeta}\setminus j_{0}$, so that
$\xi\setminus\alpha=(\zeta\setminus\beta)\amalg j_{0}$, we find that

$\sum_{j\in\zeta\setminus\beta}s^{2}_{(\zeta\amalg j_{0})\setminus
j,(\delta_{\zeta}\setminus j_{0})\amalg j}=s^{2}_{\zeta,\delta_{\zeta}},$

which identifies (7) and (6). ∎

Let us remark that, if $(x,y)$ is one of the generators of $J_{\langle
2^{n}\rangle}$ of Proposition 2.15, then $x$ and $y$, when regarded as
elements of $k\otimes_{S}k$, belong to $J_{\langle 1\rangle}$. Thus, if an
element $(z,w)$ of
$\pi_{0}\operatorname{TRR}^{n+1}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}$
belongs to $J_{\langle 2^{n}\rangle}^{q+1}$ for some $q\geq 0$, then $z$ and
$w$ belong to $J_{\langle 1\rangle}^{q+1}$. The following lemma strengthen
this property when the second component $w$ is zero.

######  Lemma 3.8.

Let $z\in(k\otimes_{S}k)^{C_{2}}$ be such that $(z,0)$ belongs to $J_{\langle
2^{n}\rangle}^{q+1}$ for some $q\geq 0$. Then $z$ belongs to $J_{\langle
1\rangle}^{q+2}$.

###### Proof.

By the characterisation of $J_{\langle 2^{n}\rangle}$ of Proposition 2.15, we
see that $(x,0)$ can be expressed as a sum of elements $u_{1}\dots u_{q+1}$,
with $u_{j}\in J_{\langle 2^{n}\rangle}$, of three types:

  1. i)

At least one of the $u_{j}$ is of the form $V^{n-i}\tau_{i}(a\otimes
b)+V^{n-i}\tau_{i}(ab\otimes 1)$ for $a,b\in k$ and $0\leq i\leq n-1$. We can
then write this generator in components as

$u_{1}\dots u_{q+1}=(w_{1}+w_{1}^{\prime},0)u_{2}\dots u_{q+1}$

where $w_{1}$ is the first component of $V^{n-i}\tau_{i}(a\otimes b)$ and
$w_{1}^{\prime}$ the first component of $V^{n-i}\tau_{i}(ab\otimes 1)$.

  2. ii)

All of the $u_{j}$ are of the form $V^{n-i}\tau_{i}(a\otimes b)+\sigma
V^{n-i}\tau_{i}(a\otimes b)$ for $a,b\in k$ and $0\leq i\leq n-1$. Then each
factor is diagonal, that is $u_{j}=(v_{j},v_{j})$ where $v_{j}$ is in
$J_{\langle 1\rangle}$, and

$u_{1}\dots u_{q+1}=(v_{1}\dots v_{q+1},v_{1}\dots v_{q+1}).$

  3. iii)

It is not of the first two types. In this case, at least one of the $u_{j}$ is
of the form $\tau_{n}(a\otimes b)+\tau_{n}(ab\otimes 1)$ and the other factors
are diagonal. We can then write such a generator as

$\displaystyle u_{1}\dots u_{q+1}$
$\displaystyle=(t_{1},t_{1}^{\prime})\dots(t_{l},t_{l}^{\prime})(s_{l+1},s_{l+1})\dots(s_{q+1},s_{q+1})$
$\displaystyle=(t_{1}\dots t_{l}s_{l+1}\dots s_{q+1},t_{1}^{\prime}\dots
t_{l}^{\prime}s_{l+1}\dots s_{q+1})$

for some $1\leq l\leq q+1$, where $t_{j}=\phi_{n}(a_{j}\otimes
b_{j})+\phi_{n}(a_{j}b_{j}\otimes 1)$, $t_{j}^{\prime}=\phi_{n}(b_{j}\otimes
a_{j})+\phi_{n}(1\otimes a_{j}b_{j})$, and $s_{j}$ is in $J_{\langle
1\rangle}$.

Thus, let us write $(z,0)$ as a sum of these types of generators, where we
omit the indexing from the sums to make this expression more digestible:

$(z,0)=\sum(w_{1}+w_{1}^{\prime},0)u_{2}\dots u_{q+1}+\sum(v_{1}\dots
v_{q+1},v_{1}\dots v_{q+1})+\sum(t_{1}\dots t_{l}s_{l+1}\dots
s_{q+1},t_{1}^{\prime}\dots t_{l}^{\prime}s_{l+1}\dots s_{q+1})$

Since the second component of $(z,0)$ is null, we must have that

$\sum v_{1}\dots v_{q+1}=\sum t_{1}^{\prime}\dots t_{l}^{\prime}s_{l+1}\dots
s_{q+1}.$

By replacing the left-hand side in the first component above we have that $z$
is, by denoting $r_{j}$ the first component of $u_{j}$, of the form

$\displaystyle z$ $\displaystyle=\sum(w_{1}+w_{1}^{\prime})r_{2}\dots
r_{q+1}+\sum t_{1}^{\prime}\dots t_{l}^{\prime}s_{l+1}\dots s_{q+1}+\sum
t_{1}\dots t_{l}s_{l+1}\dots s_{q+1}$
$\displaystyle=\sum(w_{1}+w_{1}^{\prime})r_{2}\dots r_{q+1}+\sum(t_{1}\dots
t_{l}+t_{1}^{\prime}\dots t_{l}^{\prime})s_{l+1}\dots s_{q+1}.$

Thus, since the $r_{j}$ and $s_{j}$ belong to $J_{\langle 1\rangle}$, to
conclude the proof it is sufficient to show that $w_{1}+w_{1}^{\prime}$
belongs to $J_{\langle 1\rangle}^{2}$ and that $(t_{1}\dots
t_{l}+t_{1}^{\prime}\dots t_{l}^{\prime})$ belongs to $J_{\langle
1\rangle}^{l+1}$. For the first case, we have that

$\displaystyle w_{1}+w_{1}^{\prime}$ $\displaystyle=\phi_{i}(a\otimes
b)+\phi_{i}(b\otimes a)+\phi_{i}(ab\otimes 1)+\phi_{i}(1\otimes ab)$
$\displaystyle=a^{2^{i}-1}b^{2^{i}}\otimes a+b^{2^{i}-1}a^{2^{i}}\otimes
b+(ab)^{2^{i}-1}\otimes ab+(ab)^{2^{i}}\otimes 1$
$\displaystyle=(ab)^{2^{i}-1}(b\otimes a+a\otimes b+1\otimes ab+ab\otimes 1)$
$\displaystyle=(ab)^{2^{i}-1}(1\otimes a+a\otimes 1)(1\otimes b+b\otimes 1)$

which indeed belongs to $J_{\langle 1\rangle}^{2}$. For the second case, we
have that

$\displaystyle(t_{1}\dots t_{l}+t_{1}^{\prime}\dots t_{l}^{\prime})$
$\displaystyle=\prod_{j=1}^{l}(\phi_{n}(a_{j}\otimes
b_{j})+\phi_{n}(a_{j}b_{j}\otimes 1))+\prod_{j=1}^{l}(\phi_{n}(b_{j}\otimes
a_{j})+\phi_{n}(1\otimes a_{j}b_{j}))$
$\displaystyle=\prod_{j=1}^{l}(a_{j}^{2^{n}-1}b_{j}^{2^{n}}\otimes
a_{j}+(a_{j}b_{j})^{2^{n}-1}\otimes
a_{j}b_{j})+\prod_{j=1}^{l}(b_{j}^{2^{n}-1}a_{j}^{2^{n}}\otimes
b_{j}+(a_{j}b_{j})^{2^{n}}\otimes 1)$
$\displaystyle=\prod_{j=1}^{l}\big{(}((a_{j}b_{j})^{2^{n}-1}\otimes
a_{j})\cdot(b_{j}\otimes 1+1\otimes
b_{j})\big{)}+\prod_{j=1}^{l}\big{(}((a_{j})^{2^{n}}(b_{j})^{2^{n}-1}\otimes
1)\cdot(b_{j}\otimes 1+1\otimes b_{j})\big{)}$
$\displaystyle=\big{(}\prod_{j=1}^{l}((a_{j}b_{j})^{2^{n}-1}\otimes
a_{j})+\prod_{j=1}^{l}((a_{j})^{2^{n}}(b_{j})^{2^{n}-1}\otimes
1)\big{)}\cdot\prod_{j=1}^{l}(b_{j}\otimes 1+1\otimes b_{j})$
$\displaystyle=\big{(}\prod_{j=1}^{l}(a_{j}b_{j})^{2^{n}-1}(1\otimes
a_{j})+\prod_{j=1}^{l}(a_{j}b_{j})^{2^{n}-1}(a_{j}\otimes
1)\big{)}\cdot\prod_{j=1}^{l}(b_{j}\otimes 1+1\otimes b_{j})$
$\displaystyle=(\prod_{j=1}^{l}(a_{j}b_{j})^{2^{n}-1})\big{(}(1\otimes(\prod_{j=1}^{l}a_{j})+(\prod_{j=1}^{l}a_{j})\otimes
1)\big{)}\cdot\prod_{j=1}^{l}(b_{j}\otimes 1+1\otimes b_{j})$

which belongs to $J_{\langle 1\rangle}^{l+1}$. ∎

###### Proof of Theorem 3.4.

By Remark 3.5, in degree $q=0$ the map $u_{n}$ is an isomorphism for all
$n\geq 0$. Thus, let $q\geq 1$. By Lemma 3.6, the map
$u_{0}\colon\Omega^{q}_{k}\longrightarrow J_{\langle 1\rangle}^{q}/J_{\langle
1\rangle}^{q+1}$ is an isomorphism for every $q\geq 1$. Thus, let $n\geq 1$,
assume that $u_{n-1}$ is an isomorphism, and let us show that $u_{n}$ is an
isomorphism. Since $u_{n}$ is a map of Witt complexes, it clearly hits all the
generators of $J_{\langle 2^{n}\rangle}^{q}$ from Proposition 2.15, and it is
therefore surjective. To see that it is injective, consider the commutative
diagram with exact rows

$\textstyle{\Omega^{q}_{k}\oplus\Omega^{q-1}_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{V^{n}+dV^{n}}$$\textstyle{(\operatorname{W}_{\langle
2^{n}\rangle}\Omega^{q}_{k})/2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{R}$$\scriptstyle{u_{n}}$$\textstyle{(\operatorname{W}_{\langle
2^{n-1}\rangle}\Omega^{q}_{k})/2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{u_{n-1}}$$\scriptstyle{\cong}$$\textstyle{0}$$\textstyle{J_{\langle
2^{n}\rangle}^{q}/J_{\langle
2^{n}\rangle}^{q+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{R}$$\textstyle{J_{\langle
2^{n-1}\rangle}^{q}/J_{\langle
2^{n-1}\rangle}^{q+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

where the top row is exact by [Cos08, Lemma 3.5]. It then suffices to show
that $u_{n}$ is injective when restricted to the image of the top left
horizontal map $V^{n}+dV^{n}$. Thus, let $a\in\Omega^{q}_{k}$ and
$b\in\Omega^{q-1}_{k}$, and suppose that $u_{n}(V^{n}(a)+dV^{n}(b))$ can be
represented by an element of $J_{\langle 2^{n}\rangle}^{q+1}$. We need to show
that $c:=V^{n}(a)+dV^{n}(b)$ is divisible by $2$ in $\operatorname{W}_{\langle
2^{n}\rangle}\Omega^{q}_{k}$. Let us explicitly calculate
$u_{n}(V^{n}(a)+dV^{n}(b))$. Given a $2$-basis $\\{x_{i}\\}_{i\in I}$ of $k$,
let us write

$a=\sum_{\begin{smallmatrix}\xi\subset I\\\
|\xi|=q\end{smallmatrix}}\lambda_{\xi}(dx)^{\xi}\ \ \ \ \ \ \ \mbox{and}\ \ \
\ \ \ \ \ b=\sum_{\begin{smallmatrix}\nu\subset I\\\
|\nu|=q-1\end{smallmatrix}}\mu_{\nu}(dx)^{\nu}$

where $\lambda_{\xi},\mu_{\nu}\in k$ and $(dx)^{\xi}=\prod_{i\in\xi}dx_{i}$.
Since $u$ is a map of Witt complexes, we must have that

$u_{n}(V^{n}(a)+dV^{n}(b))=V^{n}(u_{0}(a))+dV^{n}(u_{0}(b))=\sum_{\begin{smallmatrix}\xi\subset
I\\\
|\xi|=q\end{smallmatrix}}V^{n}(\lambda_{\xi}\Delta(x)^{\xi})+\sum_{\begin{smallmatrix}\nu\subset
I\\\ |\nu|=q-1\end{smallmatrix}}dV^{n}(\mu_{\nu}\Delta(x)^{\nu}).$

Recall from Proposition 2.9 that for every $x\otimes y\in k\otimes_{S}k$ we
have that $V^{n}(x\otimes y)$ is represented by $(\operatorname{tran}(x\otimes
y),0)$ in
$\pi_{0}\operatorname{TRR}^{n+1}(k;2)^{\phi\operatorname{\mathbb{Z}}/2}$,
where $\operatorname{tran}(x\otimes y)=x\otimes y+y\otimes x$. Thus,

$\displaystyle u_{n}(V^{n}(a)+dV^{n}(b))$
$\displaystyle=\sum_{\begin{smallmatrix}\xi\subset I\\\
|\xi|=q\end{smallmatrix}}(\operatorname{tran}(\lambda_{\xi}\Delta(x)^{\xi}),0)+\sum_{\begin{smallmatrix}\nu\subset
I\\\
|\nu|=q-1\end{smallmatrix}}(\operatorname{tran}(\mu_{\nu}\Delta(x)^{\nu}),\operatorname{tran}(\mu_{\nu}\Delta(x)^{\nu})).$

By choosing $i\in\xi$, we can write

$\displaystyle\operatorname{tran}(\lambda_{\xi}\Delta(x)^{\xi})$
$\displaystyle=\operatorname{tran}(\lambda_{\xi}\Delta(x_{i})\Delta(x)^{\xi\setminus
i})=\operatorname{tran}(\lambda_{\xi}\Delta(x_{i}))\Delta(x)^{\xi\setminus
i}=\operatorname{tran}(\lambda_{\xi}\otimes x_{i}+\lambda_{\xi}x_{i}\otimes
1)\Delta(x)^{\xi\setminus i}$ $\displaystyle=(\lambda_{\xi}\otimes
x_{i}+\lambda_{\xi}x_{i}\otimes
1+x_{i}\otimes\lambda_{\xi}+1\otimes\lambda_{\xi}x_{i})\Delta(x)^{\xi\setminus
i}=\Delta(\lambda_{\xi})\Delta(x_{i})\Delta(x)^{\xi\setminus i}$
$\displaystyle=\Delta(\lambda_{\xi})\Delta(x)^{\xi},$

and similarly
$\operatorname{tran}(\mu_{\nu}\Delta(x)^{\nu})=\Delta(\mu_{\nu})\Delta(x)^{\nu}$
(which is obvious in the case where $q=1$ and $\nu=\emptyset$). Thus we find
that

$\displaystyle u_{n}(V^{n}(a)+dV^{n}(b))$
$\displaystyle=\sum_{\begin{smallmatrix}\xi\subset I\\\
|\xi|=q\end{smallmatrix}}(\Delta(\lambda_{\xi})\Delta(x)^{\xi},0)+\sum_{\begin{smallmatrix}\nu\subset
I\\\
|\nu|=q-1\end{smallmatrix}}(\Delta(\mu_{\nu})\Delta(x)^{\nu},\Delta(\mu_{\nu})\Delta(x)^{\nu})$
$\displaystyle=\big{(}\sum_{\begin{smallmatrix}\xi\subset I\\\