# Additivity of derived limits in the Cohen model

Nathaniel Bannister

###### Abstract.

We generalize the work of Jeffrey Bergfalk, Michael Hrušák, and Chris Lambie-
Hanson [5] to show that, in the model constructed by adding sufficiently many
Cohen reals, derived limits are additive on a large class of systems. In the
process, we isolate a partition principle responsible for the vanishing of
derived limits on collections of Cohen reals as well as reframe the
propagating trivializations result of [5] as a theorem of ZFC. The results
also build on the work of the author, Jeffrey Bergfalk, and Justin Moore [2]
to show additvity results for strong homology.

The author would like to thank James Cummings and Jeffrey Bergfalk for
insightful discussions and comments regarding this work. The author would also
like to thank Matteo Casarosa for interest in the project and encouragement to
complete this project.

##  1\. Introduction

In recent years, the set-theoretic content of derived limits and strong
homology have gained attention ([4], [5], [3], [2], [1]). Interesting
questions have been raised on the additivity of strong homology, including
consistency strength and the minimum compatible value of the continuum. We
extend the results of [5] to show that, in the model presented there, the
derived limits of the systems presented in [2] are additive; the results of
[2] will then apply to show that strong homology is additive and has compact
supports on the class of locally compact separable metric spaces. In the
process, we identify a partition principle responsible for the additivity of
higher derived limits on large sets of Cohen reals, similar to the principle
isolated in [1] governing additivity of derived limits after iterated Hechler
forcing of weakly compact length. We moreover show that the additivity of
derived limits follows from the triviality of certain coherent families which
allow the propogation of triviality used in [5]. The techniques shown here
also apply to wider systems, for instance the systems $\mathbf{A}_{\kappa}$ of
[3]. A topological interpretation in terms of strong homology of these wider
results is the following:

######  Theorem 1.

The following is forced by $\operatorname{Add}(\omega,\beth_{\omega}(\kappa))$
to add $\beth_{\omega}(\kappa)$ many Cohen reals: whenever $\langle X_{i}\mid
i<\kappa\rangle$ are compact metric spaces, the natural map

$\bigoplus_{i\in\kappa}\overline{H}_{*}(X_{i})\to\overline{H}_{*}\left(\coprod
X_{i}\right)$

is an isomorphism.

In the special case where $\kappa=\omega$, the results of [2] result in
additivity results on a larger class of spaces.

The structure of the argument is as follows: in section 2, we present our
basic conventions and the relevant facts about higher dimensional
$\Delta$-systems required for the argument. In section 3, we define
$\Omega_{\kappa}$ systems, which generalize the system $\mathbf{A}_{\kappa}$
and for which the additivity of derived limits holds implications for the
additivity of strong homology. We generalize the notions of coherence and
triviality used in [5] to this class of systems and present the class of
_injective_ systems, which simultaneously carry all the properties used of
$\mathbf{A}$ in [5] and are rich enough that additivity holds consequences for
all $\Omega_{\kappa}$ systems. In the latter part of this section, proofs are
simplified greatly through a broader categorical perspective, particularly in
reference to sheafs and abelian categories; however, the statements of the
relevant results should largely remain readable even without this background.

Section 4 begins our analysis of how to generalize the results of [5] to the
context of injective systems. We start by generalizing of the argument for [5,
Lemma 4.7] to injective $\Omega_{\kappa}$ systems; although the result there
was stated specifically as a result about Cohen forcing, at its core is a
$\mathrm{ZFC}$ theorem which we state therein. In section 5, we present a
partition principle at the core of the additivity results in the Cohen model.
In section 6, we show how the partition principle implies that coherent
families are trivial on a sufficiently large set of reals for the propagation
results of section 4. Finally, we note a number of questions still remaining.

##  2\. Preliminaries

###  2.1. Notational conventions

If $X$ is a set and $\kappa$ is a cardinal, then $[X]^{\kappa}=\\{Y\subseteq
X\,\mid\,|Y|=\kappa\\}$, $[X]^{<\kappa}=\\{Y\subseteq X\mid|Y|<\kappa\\}$. If
$\kappa$ and $\lambda$ are cardinals, then we say that $\lambda$ is _$
<\kappa$-inaccessible_ if $\nu^{<\kappa}<\lambda$ for all $\nu<\lambda$. If
$X$ is a set of ordinals, then $\operatorname{otp}(X)$ denotes the order-type
of $X$. will often view finite sets of ordinals as finite increasing sequences
of ordinals, and vice versa. For example, if $a\in[\mathrm{On}]^{<\omega}$ and
$\ell<\operatorname{otp}(a)$, then $a(\ell)$ is the unique $\alpha\in a$ such
that $|a\cap\alpha|=\ell.$ If $\mathbf{m}\subseteq\operatorname{otp}(a)$, then
$a[\mathbf{m}]=\\{a(\ell)\mid\ell\in\mathbf{m}\\}.$ For any set $X$ of
ordinals and natural number $n$, the notation
$\left(\alpha_{0},\ldots,\alpha_{n-1}\right)\in[X]^{n}$ will denote the
conjunction of the statements
$\left\\{\alpha_{0},\ldots,\alpha_{n-1}\right\\}\in[X]^{n}$ and
$\alpha_{0}<\ldots<\alpha_{n-1}.$

The forcings appearing herein will all be of the form
$\mathbb{P}=\operatorname{Add}(\omega,\chi)$, where $\chi$ is an uncountable
cardinal and $\chi\geq\kappa$ for some given $\kappa$. We think of the
conditions of $\mathbb{P}$ as finite partial functions from $\chi\times\kappa$
to $\omega$, ordered by reverse inclusion. Forcing with $\mathbb{P}$ produces
a generic function $F:\chi\times\kappa\rightarrow\omega.$ For a fixed
$\alpha<\chi$ we will still refer to the function
$F(\alpha,\cdot):\kappa\rightarrow\omega$ the $\alpha^{\text{th }}$ Cohen
sequence added by $\mathbb{P}$, and we will typically denote this function by
$f_{\alpha};$ we denote the canonical $\mathbb{P}$-name in $V$ for
$f_{\alpha}$ by $\dot{f}_{\alpha\cdot}$. If $G$ is $\mathbb{P}$-generic over
$V$ and $W\subseteq\chi$ then $G_{W}$ denotes $\\{p\in
G\mid\operatorname{dom}(p)\subseteq W\times\kappa\\}.$ For any condition $p$
in $\mathbb{P}$ let $u(p)$ denote the set
$\\{\alpha<\chi\mid\operatorname{dom}(p)\cap(\\{\alpha\\}\times\kappa)\neq\emptyset\\}$
and let $\bar{p}$ denote the finite partial function from
$\operatorname{otp}(u(p))\times\kappa$ to $\omega$ defined as follows: for all
$i<\operatorname{otp}(u(p))$ and all $\alpha<\kappa$, define $(i,\alpha)$ to
be in the domain of $\bar{p}$ if and only if
$(u(p)(i),\alpha)\in\operatorname{dom}(p)$; if so, let
$\bar{p}(i,\alpha)=p(u(p)(i),\alpha).$ Intuitively, $\bar{p}$ is a ”collapsed”
version of $p.$ Notice that the set $\\{\bar{p}\mid p\in\mathbb{P}\\}$ is a
subset of the set of finite partial functions from $\omega\times\kappa$ to
$\omega$.

###  2.2. Higher Dimensional $\Delta$-systems

######  Definition 1.

Suppose that $a$ and $b$ are sets of ordinals.

  * •

We say that $a$ and $b$ are _aligned_ if
$\operatorname{otp}(a)=\operatorname{otp}(b)$ and if
$\operatorname{otp}(a\cap\gamma)=\operatorname{otp}(b\cap\gamma)$ for all
$\gamma\in a\cap b.$ In other words, if $\gamma$ is a common element of two
aligned sets $a$ and $b$, then it occupies the same relative position in both
$a$ and $b$.

  * •

If $a$ and $b$ are aligned then we let
$\mathbf{r}(a,b):=\\{i<\operatorname{otp}(a)\mid a(i)=b(i)\\}.$ Notice that,
in this case, $a\cap b=a[\mathbf{r}(a,b)]=b[\mathbf{r}(a,b)]$

######  Definition 2.

Suppose that $H$ is a set of ordinals, $n$ is a positive integer, and $u_{b}$
is a set of ordinals for all $b\in[H]^{n}.$ We call $\left\langle u_{b}\mid
b\in[H]^{n}\right\rangle$ a uniform $n$-dimensional $\Delta$-system if there
is an ordinal $\rho$ and, for each $\mathbf{m}\subseteq n$, a set
$\mathbf{r}_{\mathbf{m}}\subseteq\rho$ satisfying the following statements.

  1. (1)

$\operatorname{otp}\left(u_{b}\right)=\rho$ for all $b\in[H]^{n}$

  2. (2)

For all $a,b\in[H]^{n}$ and $\mathbf{m}\subseteq n$, if $a$ and $b$ are
aligned with $\mathbf{r}(a,b)=\mathbf{m}$, then $u_{a}$ and $u_{b}$ are
aligned with $\mathbf{r}\left(u_{a},u_{b}\right)=\mathbf{r}_{\mathbf{m}}$

  3. (3)

For all $\mathbf{m}_{0},\mathbf{m}_{1}\subseteq n$, we have
$\mathbf{r}_{\mathbf{m}_{0}\cap\mathbf{m}_{1}}=\mathbf{r}_{\mathbf{m}_{0}}\cap\mathbf{r}_{\mathbf{m}_{1}}$.

######  Definition 3.

$\alpha$ is $k$-addable for $a\in[H]^{m}$ if $a\not\in a$ and
$|a\cap\alpha|=k$. This is equivalent to the combination of

  * •

$k\leq m$

  * •

If $k>0$ then $\alpha>a(k-1)$

  * •

If $k<m$ then $\alpha<a(k)$

######  Lemma 1.

Suppose that $1\leq n<\omega$ and $\langle u_{b}\mid b\in[H]^{n}\rangle$ is a
uniform $n$-dimensional $\Delta$-system as witnessed by $\rho$ and
$\langle\mathbf{r_{m}}\mid\mathbf{m}\subseteq n\rangle$ and assume that $H$
has no largest element. If there is an $\alpha$ which is $k$-addable for
$a\in[H]^{m}$ with $m<n$, set
$u_{a,k}=u_{b}[\mathbf{r}_{m+1\setminus\\{k\\}}]$ for $b$ such that
$b[m+1\setminus\\{k\\}]=a$. Let $u_{a}=u_{a,|a|}$. Then the following hold.

  1. (1)

These definitions are independent of our choice of $b$

  2. (2)

For each $a\in[H]^{m}$ for $m\leq n$ and $k\leq m$, the collection

$\\{u_{a\cup\\{\beta\\}}\mid\beta\text{ is }k-\text{addable for }a\\}$

is a $\Delta$ system with root $u_{a,k}$

###### Proof.

We first show 1. Fix $a\in[H]^{m}$ and suppose that
$a=b[m+1\setminus\\{k\\}]=b^{\prime}[m+1\setminus\\{k\\}]$. Since $H$ has no
largest element, there is a $b^{\prime\prime}$ such that
$b^{\prime\prime}[m+1]=b[m+1]$ and $b^{\prime\prime}(j)\not\in b,b^{\prime}$
for any $j>m+1$. Then $b$ and $b^{\prime\prime}$ are aligned with
$\mathbf{r}(b,b^{\prime\prime})=m+1$ and $b^{\prime}$ and $b^{\prime\prime}$
are aligned with $\mathbf{r}(b^{\prime},b^{\prime\prime})$ either $m+1$ if
$b(k)=b^{\prime}(k)$ or $m+1\setminus\\{k\\}$ otherwise. Regardless, since
$\mathbf{r}_{m+1\setminus k}\subseteq\mathbf{r}_{m+1}$, we see that
$b[\mathbf{r}_{m+1\setminus\\{k\\}}]=b^{\prime\prime}[\mathbf{r}_{m+1\setminus\\{k\\}}]=b^{\prime}[\mathbf{r}_{m+1\setminus\\{k\\}}]$.

We now show 2. If $\alpha\neq\alpha^{\prime}$ are $k$ addable for $a$, pick
$b\sqsupseteq a\cup\\{\alpha\\}$ and $b^{\prime}\sqsupseteq
a\cup\\{\alpha^{\prime}\\}$ such that $b\cap b^{\prime}=a$. Then $b$ and
$b^{\prime}$ are aligned with $\mathbf{r}(b,b^{\prime})=m+1\setminus\\{k\\}$,
so $u_{b}$ and $u_{b^{\prime}}$ are aligned with
$\mathrm{r}(u_{b},u_{b}^{\prime})=\mathbf{r}_{m+1\setminus\\{k\\}}$. Then

$\begin{aligned} u_{a\cup\\{\alpha\\}}\cap
u_{a\cup\\{\alpha^{\prime}\\}}&=u_{b}[\mathbf{r}_{m+1}]\cap
u_{b^{\prime}}[\mathbf{r}_{m+1}]\\\ &=u_{b}[\mathbf{r}_{m+1}]\cap
u_{b^{\prime}}[\mathbf{r}_{m+1}]\cap
u_{b}[\mathbf{r}_{m+1\setminus\\{k\\}}]\cap
u_{b^{\prime}}[\mathbf{r}_{m+1\setminus\\{k\\}}]\\\
&=u_{b}[\mathbf{r}_{{m+1}\setminus\\{k\\}}]\cap
u_{b^{\prime}}[\mathbf{r}_{m+1\setminus\\{k\\}}]\\\
&=u_{b}[\mathbf{r}_{m+1\setminus\\{k\\}}]\\\ &=u_{a,k}\end{aligned}.$

∎

######  Definition 4.

Suppose that $\lambda$ is an infinite regular cardinal. Recursively define
cardinals $\sigma(\lambda,n)$ for $1\leq n<\omega$ by letting
$\sigma(\lambda,1)=\lambda$ and, given $1\leq n<\omega$, letting
$\sigma(\lambda,n+1)=\left(2^{<\sigma(\lambda,n)}\right)^{+}$.

######  Fact 1 ([8], Theorem 2.10).

Suppose that

  * •

$1\leq n<\omega;$

  * •

$\kappa<\lambda$ are infinite cardinals, $\lambda$ is regular and
$<\kappa$-inaccessible, and $\mu=\sigma(\lambda,n)$;

  * •

$c:[\mu]^{n}\rightarrow 2^{<\kappa}$;

  * •

for all $b\in[\mu]^{n}$, we are given a set $u_{b}\in[\mathrm{On}]^{<\kappa}$.

Then there are an $H\in[\mu]^{\lambda}$ and $k<2^{<\kappa}$ such that

  1. (1)

$c(b)=k$ for all $b\in[H]^{n}$;

  2. (2)

$\left\langle u_{b}\mid b\in[H]^{n}\right\rangle$ is a uniform $n$-dimensional
$\Delta$-system.

##  3\. $\Omega_{\kappa}$ systems

We now introduce a generalization of $\Omega$-systems which allows for wider
systems. Allowing wider systems will allow for statements about derived limits
indexed over $\omega^{\kappa}$.

######  Definition 5.

Suppose $\kappa$ is a cardinal. An $\Omega_{\kappa}$-system $\mathcal{G}$ is
specified by an indexed collection
$\\{G_{\alpha,k}\mid\alpha<\kappa,k\in\omega\\}$ of abelian groups along with
compatible homomorphisms $p_{\alpha,j,k}:G_{\alpha,j}\to G_{\alpha,k}$ for
each $\alpha<\kappa$ and $j\geq k$. Such data gives rise to the following
additional objects:

  * •

For each $x\in\omega^{\kappa}$ define ${\displaystyle
G_{x}:=\bigoplus_{\alpha<\kappa}G_{\alpha,x(\alpha)}}$ and
${\displaystyle\overline{G}_{x}:=\prod_{\alpha<\kappa}G_{\alpha,x(\alpha)}}$.
We regard $G_{x}$ as a subset of $\overline{G}_{x}$.

  * •

For each $x\leq y\in\omega^{\kappa}$ a homomorphism
$p_{y,x}:\overline{G}_{y}\to\overline{G}_{x}$ defined by
$p_{y,x}:=\prod_{\alpha<\kappa}p_{\alpha,y(\alpha),x(\alpha)}$.

  * •

The systems $G$ and $\overline{G}$ indexed over $\omega^{\kappa}$ with
structure given by the first two points

  * •

For each $\alpha<\kappa$ an inverse system $\mathcal{G}_{\alpha}$ indexed over
$\omega$ with $(\mathcal{G}_{\alpha})_{k}=G_{\alpha,k}$ and structure maps
given by $p_{\alpha,j,k}$.

######  Example 1.

Suppose $G_{\alpha,k}=\mathbb{Z}^{k}$ for each $\alpha,k$ and $p_{\alpha,j,k}$
is the canonical projection map. The resulting system $G$ is denoted
$\mathbf{A}_{\kappa}$ in, for instance, [3]. We may think of
$\overline{G}_{x}$ as functions from the area under the graph of $x$ into
$\mathbb{Z}$ and $G_{x}$ as the subgroup of finitely supported functions.

Our main point of interest is the canonical map

$\bigoplus_{\alpha<\kappa}\lim{}^{n}\mathcal{G}_{\alpha}\to\lim{}^{n}G;$

for instance by the following theorem.

######  Theorem 2 ([2], Theorem 6).

The following are equivalent:

  * •

Strong homology is additive and compactly supported on the class of locally
compact separable metric spaces

  * •

For every $\Omega_{\omega}$ system $\mathcal{G}$ such that each $G_{\alpha,k}$
is finitely generated and every $n$, the canonical map

$\bigoplus_{k}\lim{}^{n}\mathcal{G}_{k}\to\lim{}^{n}G$

is an isomorphism

We will show the following:

######  Theorem 3.

Suppose that $\chi\geq\beth_{\omega}(\kappa)$. Then
$\operatorname{Add}(\omega,\chi)$ forces that whenever $\mathcal{G}$ is an
$\Omega_{\kappa}$ system, the canonical map

$\bigoplus_{\alpha}\lim{}^{n}\mathcal{G}_{\alpha}\to\lim{}^{n}G$

is an isomorphism.

The proof of [2, Theorem 6] generalizes to the present context to show the
result on additivity of strong homology mentioned in the introduction. For
$\kappa>\omega$, it is unclear whether there are any implications for
additivity of strong homology on any large classes of spaces beyond those
mentioned in Theorem 2.

The key ingredient of Theorem 3 will be to establish connections between the
coherence and triviality of certain families with the additivity of derived
limits.

###  3.1. Coherence and Triviality

We now describe the general machinery by which we will prove additivity of
derived limits. We will generally make use of the following cochain complex,
whose cohomology is gives the derived limits of an inverse system.

######  Definition 6.

Suppose that $X\subseteq\omega^{\kappa}$ and $\mathbf{G}$ is an inverse system
of abelian groups indexed over $\omega^{\kappa}$. The _alternating cochain
complex_ $K^{\bullet}(\mathbf{G}\upharpoonright X)$ is given by

$K^{n}(\mathbf{G}\upharpoonright
X)=\left\\{y\in\prod_{x_{0},...,x_{n}}\mathbf{G}_{\bigwedge
x_{i}}\,\middle|\,\text{for any }\sigma\in
S_{n},\,y(x_{0},...,x_{n})=\mathrm{sgn}(\sigma)y(x_{\sigma(0)},...,x_{\sigma(n)})\right\\}$
with $d\colon K^{n}\to K^{n+1}$ given by

$dy(x_{0},...,x_{n+1})=\sum_{i}(-1)^{i}p_{\bigwedge_{j\in
n+2\setminus\\{i\\}}x_{j},\bigwedge
x_{j}}y(x_{0},...,\widehat{x_{i}},...,x_{n+1}).$

Note that we do not require that $\bigwedge x_{i}$ to be a member of $X$ in
the above definition. It is readily verified that $d^{n+1}d^{n}=0$ for any
$n<\omega$; our main object of interest will be the cohomology groups
$H^{n}(K^{\bullet}(\mathbf{G}))$.

In general, $H^{n}(K^{\bullet}(\mathbf{G}\upharpoonright X))$ _does not_
correspond to $\lim^{n}(\mathbf{G}\upharpoonright X)$, the derived limits of
the subsystem indexed by those elements of $\omega^{\kappa}$ which are in $X$.
For an extreme example, if $X$ consists of two incompatible points $x,y$ then
$H^{n}(K^{\bullet}(\mathbf{G}\upharpoonright X))$ can be identified with all
elements of $G_{x}\times G_{y}$ which restrict to the same element of
$G_{x\wedge y}$ whereas $\lim^{n}(\mathbf{G}\upharpoonright X)$ is all of
$G_{x}\times G_{y}$. Nevertheless.
$H^{n}(K^{\bullet}(\mathbf{G}\upharpoonright X))$ does have a characterization
in terms of derived limits.

######  Proposition 1.

$H^{n}(K^{\bullet}(\mathbf{G}\upharpoonright
X))\cong\lim^{n}(\mathbf{G}\upharpoonright Y)$ where $Y$ is the downwards
closure of $X$.

###### Proof.

An observation in [7] shows that $\lim^{n}(\mathbf{G}\upharpoonright Y)$ is
naturally isomorhic to the sheaf cohomology $H^{n}(Y,\mathcal{F})$, where the
topology on $Y$ is generated by the sets $U_{x}=\\{y\mid y\leq x\\}$ and
$\mathcal{F}$ is the sheaf $U\mapsto\lim(\mathbf{G}\upharpoonright U)$. We now
consider the cover $\mathcal{U}=\\{U_{x}\mid x\in X\\}$; observe that these
sets cover $Y$ and that $U_{x_{1}}\cap\ldots\cap
U_{x_{n}}=U_{x_{1}\wedge\ldots\wedge x_{n}}$. Since
$U_{x_{1}\wedge\ldots\wedge x_{n}}$ has a maximal element,
$\lim^{s}(\mathbf{G}\upharpoonright U_{x_{1}\wedge\ldots\wedge x_{n}})=0$ for
all $s>0$. A theorem of Leray allows us to conclude that the Čech cohomology
of $\mathcal{U}$ is $\lim^{n}(\mathbf{G}\upharpoonright Y)$ (see, for
instance, [6, Exercise III.4.11]). Since Čech cohomology can be computed using
alternating chains by [10, I.20 Proposition 2], the proposition follows. ∎

From this point on, we will sometimes use $\lim^{n}(\mathbf{G}\upharpoonright
X)$ to refer to $H^{n}(K^{\bullet}(\mathbf{G}\upharpoonright X))$.

######  Definition 7.

Fix $X\subseteq\omega^{\kappa}$ and a positive integer $n$. $\Phi\in
K^{n-1}(\overline{G}\upharpoonright X)$ is _$n$ -coherent_ if $d\Phi\in
K^{n}(G\upharpoonright X)$. Equivalently, $\Phi$ represents a cycle of
$K^{n-1}(\overline{G}/G\upharpoonright X)$.

######  Example 2.

Suppose $\mathcal{G}$ is as in Example 1. An $n$-coherent family corresponds
to, for each $\mathbf{x}\in X^{n}$, functions $f_{\mathbf{x}}$ from the area
under the graph of $\bigwedge\mathbf{x}$ into $\mathbb{Z}$ such that for
$\mathbf{y}\in X^{n+1}$,
$\sum_{i}f_{x_{0},\ldots,\widehat{x_{i}},\ldots,x^{n}}$ is finitely supported
on $\bigwedge y$. This is the notion of $n$-coherent families used in, for
instance, [3].

The long exact sequence

${0}$${\lim G\upharpoonright X}$${\lim\overline{G}\upharpoonright
X}$${\lim\overline{G}/G\upharpoonright X}$${\lim{}^{1}G\upharpoonright
X}$${\lim{}^{1}\overline{G}\upharpoonright X}$${\cdots}$

will play a crucial role in connecting coherent families with the derived
limits of $G$.

######  Definition 8.

Fix $X\subseteq\omega^{\kappa}$ and a positive integer $n$. If $n=1$, an
$n$-coherent family $\Phi\in K^{0}(\overline{G}\upharpoonright X)$ is _type
$I$ trivial_ if there exists a $\psi\in\lim\overline{G}$ such that for each
$f\in X$ there is a finite $S$ such that
$\Phi_{f}\upharpoonright\bigoplus_{i\in\kappa\setminus
S}G_{i,f(i)}=p_{f}\psi\upharpoonright\bigoplus_{i\in\kappa\setminus
S}G_{i,f(i)}$. In this case, we say that $\Phi_{f}=^{*}\psi$.

If $n>1$, an $n$-coherent family $\Phi\in K^{n-1}(\overline{G}\upharpoonright
X)$ is _type $I$ trivial_ if there exists a $\Psi\in K^{n-2}(\overline{G})$
such that for each $f\in X$ there is a finite $S$ such that
$\Phi_{f}\upharpoonright\bigoplus_{i\in\kappa\setminus
S}G_{i,f(i)}=d\Psi_{f}\upharpoonright\bigoplus_{i\in\kappa\setminus
S}G_{i,f(i)}$. In this case, we say that $\Phi_{f}=^{*}d\Psi$.

Such a $\psi$ or $\Psi$ is referred to a _type $I$ trivialization_.

Note that when $n>1$, this is equivalent to $\Phi$ representing $0$ in

$\lim{}^{n-1}(\overline{G}/G\upharpoonright X)$

and when $n=1$, this is equivalent to $[\Phi]$ being in the image of the map

$\lim{}^{0}(\overline{G}\upharpoonright
X)\to\lim{}^{0}(\overline{G}/G\upharpoonright X).$

By subtracting $\Phi$ and the projections of $\psi$ in the $n=1$ case, we see
that the $n=1$ case is also equivalent to the existence of a
$\Psi=\langle\psi_{f}\,\mid\,f\in X\rangle$ such that $d\Psi=0$ and
$\Psi_{\vec{f}}=^{*}\Phi_{\vec{f}}$. For general $\Omega_{\kappa}$-systems, we
cannot hope for this to occur for every $n$-coherent family; however, type $I$
triviality is the sort of triviality which we can extend from small subsets to
larger sets. Another sort of triviality which can hold, and was used in [2],
is the following.

######  Definition 9.

Fix $X\subseteq\omega^{\kappa}$ and a positive integer $n$. An $n$-coherent
family $\Phi\in K^{n-1}(\overline{G}\upharpoonright X)$ is _type $II$ trivial_
if there exists a $\Psi\in K^{n-1}(G\upharpoonright X)$ such that
$d\Phi=d\Psi$. Such a $\Psi$ is referred to as a _type $II$ trivialization_.

Chasing through the proof of the snake lemma, this is equivalent to the image
of $\Phi$ under the connecting homomorphism to $\lim^{n}G\upharpoonright X$ is
$0$. Our overall plan, and what is done in [5], is to obtain a type $II$
trivialization over a large set and then propagate the trivialization to
obtain that the entire system is type $I$ trivial. Although this is not
possible for all $\Omega_{\kappa}$ systems, there is a sufficiently rich class
on which we can, namely the _injective_ systems.

###  3.2. Injective Systems

Several particular features of the system $\mathbf{A}$ are used heavily in [5]
which do not apply to arbitrary $\Omega$ systems. It turns out, however, that
most of The collection of $\Omega_{\kappa}$ systems forms a category, with a
morphism $\varphi$ between two systems $\mathcal{G}$ and $\mathcal{H}$
consisting of group homomorphisms $\varphi_{\alpha,k}\colon G_{\alpha,k}\to
H_{\alpha,k}$ commuting with the structure maps of $\mathcal{G}$ and
$\mathcal{H}$. We note here that the category is abelian and that $\varphi$ is
monic if and only if each $\varphi_{\alpha,k}$ is injective (see, for
instance, [9, Theorem 11.2]).

######  Definition 10.

An $\Omega_{\kappa}$ system $\mathcal{I}$ is injective if whenever
$\mathcal{G},\mathcal{H}$ are $\Omega_{\kappa}$ systems and
$\varphi\colon\mathcal{G}\to\mathcal{I}$ and
$\psi\colon\mathcal{G}\to\mathcal{H}$ are maps of $\Omega_{\kappa}$ systems
such that each $\psi_{\alpha,k}$ is injective, there is a
$\overline{\varphi}\colon\mathcal{H}\to\mathcal{I}$ such that
$\overline{\varphi}\psi=\varphi$.

Note that there is a sensible notion of injective objects for
$\mathrm{Ab}^{\Lambda}$ for any inverse system $\Lambda$, which we will make
use of in Lemma 7. See, for instance, [9, Chapter 11] for more details on the
general theory of such objects.

The class of injective systems admits the following concrete characterization:

######  Lemma 2.

An $\Omega_{\kappa}$ system $\mathcal{I}$ is injective if and only if

  * •

Each $p_{\alpha,k}$ is surjective;

  * •

Each $I_{\alpha,k}$ is a divisible group;

  * •

For each $x\in I_{\alpha,n+1}$, $y\in I_{\alpha,n}$, and $k\in\mathbb{Z}$ such
that $ky=p_{\alpha,k}(x)$, there is a $z\in I_{n+1}$ such that $kz=x$ and
$p_{n}(z)=y$.

###### Proof.

We first show these conditions imply injectivity. Suppose
$\varphi\colon\mathcal{G}\to\mathcal{I}$ and
$\psi\colon\mathcal{G}\to\mathcal{H}$ are as in the definition of injective.
Consider the poset of all pairs $\langle\mathcal{K},\theta\rangle$ where
$\mathcal{K}$ is a subsystem of $\mathcal{H}$ containing the image of
$\mathcal{G}$ and $\theta\colon\mathcal{K}\to\mathcal{I}$ satisfies
$\theta\psi=\varphi$, ordered by
$\langle\mathcal{K},\theta\rangle\leq\langle\mathcal{K}^{\prime},\theta^{\prime}\rangle$
if and only if $\mathcal{K}$ is a subsystem of $\mathcal{K}^{\prime}$ and
$\theta^{\prime}\upharpoonright\mathcal{K}=\theta$. The hypothesis of Zorn’s
lemma are readily satisfied, so let $(\mathcal{M},\psi)$ be a maximal element.
We claim that $\mathcal{M}=\mathcal{H}$; note that this is sufficient. Suppose
not at let $x,\alpha,k$ be such that

  * •

$x\in H_{\alpha,k}\setminus M_{\alpha,k}$;

  * •

$k$ is minimal such that for some $\alpha$, $M_{\alpha,k}\neq H_{\alpha,k}$.

Let $M^{\prime}$ be the subsystem of $H$ with

$M^{\prime}_{\beta,n}=\left\\{\begin{array}[]{cc}\langle
M_{\alpha,k},x\rangle&n=k,\beta=\alpha\\\
M_{\beta,n}&\text{otherwise}\end{array}\right.;$

we claim that $\psi$ can be extended to $M^{\prime}$, contradicting the
maximality of $M$. First, suppose $\langle x\rangle\cap M_{\alpha,k}=\langle
0\rangle$. If $k=0$ then mapping $x$ to $0$ defines an extension of $\psi$ to
$M^{\prime}$. If $x$ is not in the torsion subgroup of $H_{\alpha,k}$, then we
may extend $\psi$ arbitrarily to include $x$ such that $\psi
p_{\alpha,k-1}^{H}(x)=p_{\alpha,k-1}\psi(x)$; note this is possible by
surjectivity of $p_{\alpha,k-1}$. Otherwise, let $\ell\in\mathbb{Z}_{+}$ be
minimal such that $\ell x=0$; note that $\ell\psi p_{\alpha,k-1}^{H}(x)=0$
since $\psi p_{\alpha,k-1}$ is a homomorphism. Then by hypothesis, there is a
$y\in I_{\alpha,k}$ such that $\ell y=0$ and $p_{\alpha,k-1}(y)=\ell\psi
p_{\alpha,k-1}^{H}(x)$; we may then extend $\psi$ to $M^{\prime}$ by setting
$\psi_{\alpha,k}(x)=y$.

Otherwise, there is some minimal $\ell\in\mathbb{Z}$ such that $\langle
x\rangle\cap M_{\alpha,k}=\\{x\ell i\mid i\in\mathbb{Z}\\}$. If $k=0$ then we
may use that $I_{\alpha,0}$ is divisible to find some $y\in I_{\alpha,0}$ such
that $\ell y=\psi(\ell x)$. But then $\psi$ together with $x\mapsto y$
determines a mapping on a larger subsystem of $\mathcal{H}$, contradicting
maximality of $\mathcal{M}$. If $k>0$, then we may find some $y\in
I_{\alpha,k}$ such that $\ell y=\psi(\ell x)$ and
$p_{\alpha,k-1}(y)=\psi(p_{\alpha,k-1}^{\mathcal{H}}(x))$. But then once again
$\psi$ together with $x\mapsto y$ determines a mapping on a larger subsystem
of $\mathcal{H}$, contradicting maximality of $\mathcal{M}$.

Conversely, if $x$ is not in the image of $p_{\alpha,n}$, consider the system
$\mathcal{G}$ with

$G_{\beta,k}=\left\\{\begin{array}[]{cc}I_{\alpha,k}\oplus\mathbb{Z}&k>n,\beta=\alpha\\\
I_{\beta,k}&\text{otherwise}\end{array}\right.$

with structure maps $q_{\beta,k}$ given by $p_{\alpha,k}\oplus 1$ if $k>n$ and
$\beta=\alpha$, $p_{\alpha,n}$ the unique map which restricts to $p_{n}$ on
$I_{k+1}$ and maps $(0,1)$ to $x$, and $p_{\beta,k}$ otherwise. Then the
identity map from $\mathcal{I}$ to $\mathcal{I}$ does not extend to a map from
$\mathcal{G}$ to $\mathcal{I}$, so $\mathcal{I}$ is not injective.

If some $x\in I_{\alpha,n}$ is not divisible by $k$, take the system $G$ with

$G_{\beta,k}=\left\\{\begin{array}[]{cc}I_{\alpha,k}\oplus\mathbb{Z}/\langle(p_{n,k}(X),k)\rangle&k\leq
n,\beta=\alpha\\\ I_{\beta,k}k&\text{otherwise}\end{array}\right.;$

there is no map from $G$ to $I$ extending the identity on $I$.

If the third condition fails for some $\alpha,x,y,k$ then take $G$ with

$G_{\beta,k}=\left\\{\begin{array}[]{cc}I_{\alpha,n}\oplus\mathbb{Z}/\langle(x,-k)\rangle&k=n,\beta=\alpha\\\
I_{\beta,k}\text{otherwise}\end{array}\right.$

with map in dimension $p$ induced by $p_{n}$ on $I_{n}$ and $(0,1)\mapsto y$;
there is no map from $G$ to $I$ extending the identity on $I$. ∎

We now record some facts about injective objects within $\Omega_{\kappa}$
systems. Noting that an $\Omega_{\kappa}$ system consists of simply $\kappa$
many inverse sequences of abelian groups, we will state several of the results
in terms objects of $\mathrm{Ab}^{\omega}$.

######  Lemma 3.

The category of $\Omega_{\kappa}$ systems has enough injective objects. That
is, whenever $\mathcal{G}$ is an $\Omega_{\kappa}$ system there is an
injective system $\mathcal{I}$ and a monic
$\varphi\colon\mathcal{G}\to\mathcal{I}$.

See, for instance, [9, Theorem 11.18] for a proof in a more general context.
We now establish an analogue of “extension by $0$” used in [5].

######  Corollary 1.

Suppose $\mathcal{I}$ is an injective $\Omega_{\kappa}$ system. There is a
family $\\{i_{fg}\colon\overline{I}_{f}\to\overline{I}_{g}\mid f\leq g\\}$
such that

  * •

$p_{gf}i_{fg}=\mathrm{id}_{\overline{I_{f}}}$;

  * •

if $x\in I_{f}$ then $i_{fg}(z)\in I_{g}$.

###### Proof.

For each $\alpha,n$, since $\ker(p_{\alpha,n})$ is a divisible abelian group
by Lemma 2, it is injective, so the short exact sequence

${0}$${\ker(p_{\alpha,n})}$${I_{\alpha,n+1}}$${I_{\alpha,n}}$${0}$$\scriptstyle{p_{n}}$

splits. In particular, there is an $i_{\alpha,n}\colon I_{\alpha,n}\to
I_{\alpha,n+1}$ such that
$p_{\alpha,n}i_{\alpha,n}=\mathrm{id}_{I_{\alpha,n}}$. Composing yields that
for each $\alpha$ and each $n\leq m$ there are compatible
$i_{\alpha,n,m}\colon I_{\alpha,n}\to I_{\alpha,m}$ such that
$p_{\alpha,m,n}i_{\alpha,n,m}=\mathrm{id}_{I_{\alpha,n}}$. We may then set
$i_{fg}=\prod_{\alpha}i_{\alpha,f(\alpha),g(\alpha)}$. ∎

We now turn to establishing that the two forms of triviality are, in fact,
equivalent for injective systems. This will allow us to emulate the
“propagating trivializations” step of [5].

######  Lemma 4.

Suppose that $I\in\mathrm{Ab}^{\omega}$ is injective and
$X\subseteq\omega^{\kappa}$. Then the system $I_{\alpha}$ indexed over $X$
defined by $I_{\alpha}(x)=I_{x(\alpha)}$ is injective in $\mathrm{Ab}^{X}$.

###### Proof.

Let $\pi_{\alpha}^{*}\colon\mathrm{Ab}^{\omega}\to\mathrm{Ab}^{X}$ be the
functor with $\pi_{\alpha}(G)(x)=G_{x(\alpha)}$; then
$\pi_{\alpha}^{*}(I)=I_{\alpha}$. $\pi_{\alpha}^{*}$ has a left adjoint
$(\pi_{\alpha})_{!}$ which is exact: for $G\in\mathrm{Ab}^{X}$, we take

$(\pi_{\alpha})_{!}(G)(k)=\varinjlim_{\begin{subarray}{c}x\in X\\\
x(\alpha)\geq k\end{subarray}}G_{x}.$

Therefore, since both $\mathrm{Ab}^{\omega}$ and $\mathrm{Ab}^{X}$ are abelian
categories, $\pi_{\alpha}$ preserves injective objects. ∎

######  Corollary 2.

Suppose that $\mathcal{I}=\langle I_{\alpha,k},p_{\alpha,j,k}\rangle$ is an
injective object in the category of $\Omega_{\kappa}$-systems. Given
$X\subseteq\omega^{\kappa}$, let $\overline{I}\upharpoonright X$ be the
inverse system indexed over $X$ induced by $\overline{I}$. Then

$\lim{}^{n}(\overline{I}\upharpoonright X)=0$

for all $X\subseteq\omega^{\kappa}$ and $n>0$.

###### Proof.

Note that for each $\alpha$, $\mathcal{I}_{\alpha}$ is injective in
$\mathrm{Ab}^{\omega}$. Therefore, each
$\pi_{\alpha}^{*}(\mathcal{I}_{\alpha})$ is injective in $\mathrm{Ab}^{X}$ by
Lemma 4. In particular, noting that $\overline{I}\upharpoonright
X=\prod_{\alpha}\pi_{\alpha}^{*}(\mathcal{I}_{\alpha})$ and that a product of
injective objects is injective (in any category), $\overline{I}\upharpoonright
X$ is an injective object, so derived functors for any left exact functor (for
instance $\lim$) will vanish. ∎

The next lemma shows that two types of triviality are equivalent for injective
systems.

######  Lemma 5.

Suppose that $\mathcal{I}$ is an injective $\Omega_{\kappa}$ system. For every
$X\subseteq\omega^{\kappa}$, positive integer $n$, and every $n$-coherent
$\Phi\in K^{n-1}(\overline{I}\upharpoonright X)$, $\Phi$ is type $I$ trivial
if and only if $\Phi$ is type $II$ trivial.

###### Proof.

Let $Y$ be the downwards closure of $X$. By Corollary 2,
$\lim^{n}\overline{I}\upharpoonright Y=0$ for every $n>0$. Using the long
exact sequence of derived limits corresponding to

${0}$${{I}\upharpoonright Y}$${\overline{{I}}\upharpoonright
I}$${\overline{I}/I\upharpoonright Y}$${0,}$

and Proposition 1, if $n>1$ then an $n$-coherent $\Phi\in
K^{n-1}(\overline{I}\upharpoonright X)$ represents $0$ in
$\lim^{n-1}(\overline{I}/I\upharpoonright Y)$ if and only if $[\Phi]$ is in
the kernel of the connecting homomorphism from
$\lim^{n-1}(\overline{I}/I\upharpoonright Y)$ to $\lim^{n}(I\upharpoonright
Y)$. But these are precisely the meanings of type $I$ and type $II$
triviality. The fact type $I$ and type $II$ triviality are always equivalent
for $\Omega$ systems if $n=1$ completes the proof. ∎

We now connect coherent families being trivial for injective systems with
additivity for all systems.

######  Lemma 6.

Suppose that for every injective $\Omega_{\kappa}$-system and each $n$, every
corresponding $n$-coherent family is $n$-trivial. Then for each
$\Omega_{\kappa}$ system $\mathcal{G}$, the induced map
$\bigoplus_{\alpha}\lim^{n}\mathcal{G}_{\alpha}\to\lim^{n}G$ is an
isomorphism.

###### Proof.

Consider the functor $F\colon\Omega_{\kappa}$-systems$\to$Ab given by
$\mathcal{G}\mapsto\bigoplus_{\alpha}\lim\mathcal{G}_{\alpha}$. The
$\delta$-functors

$\mathcal{G}\mapsto\bigoplus_{\alpha}\lim{}^{n}\mathcal{G}_{\alpha}$

and

$\mathcal{G}\mapsto\lim{}^{n}G$

both coincide in dimension $0$ with $F$ and (using the hypothesis for the
second), map injective objects to $0$ in all positive dimensions. Therefore,
they coincide and are the derived limits of $F$, yielding the desired
isomorphism. ∎

The remainder of our work will be showing that coherent families are trivial
for injective systems.

##  4\. Propagating Trivializations for Injective Systems

In this section, we demonstrate how to extend trivializations of an injective
system on a set containing sufficiently many Cohen sequences to a
trivialization on the whole domain.

######  Definition 11.

Fix an injective $\Omega_{\kappa}$ system $\mathcal{I}$, an
$X\subseteq\omega^{\kappa}$, a function $g\in\omega^{\kappa}$, and a positive
integer $n$. We say a family
$\Phi=\left\langle\phi_{\vec{f}}\in\overline{I}_{\bigwedge\vec{f}\wedge
g}\right\rangle$ is _$n$ -coherent below $g$_ if it satisfies Definition 7,
with $K^{n}$ replaced with the complex where each $G_{\bigwedge\vec{f}}$ is
replaced with $G_{\bigwedge\vec{f}\wedge g}$. Such a family is $n$-trivial if
there is a $\psi$ or $\Psi$ as in Definition 8, the only difference is that in
the case $n=1$, $\psi\in\overline{I}_{g}$ and in the case $n>1$,
$\psi_{\vec{f}}\in I_{\bigwedge\vec{f}\wedge g}$.

The following proposition is crucial for the arguments of this section and is
one of the main reasons we use injective objects.

######  Proposition 2.

Suppose that $\mathcal{I}$ is an injective $\Omega_{\kappa}$ system,
$X\subseteq\omega^{\kappa}$, $g\in\omega^{\kappa}$, $n$ is a positive integer,
and every $n$-coherent family corresponding to $\mathcal{I}$ indexed by
$X^{n}$ is trivial. Then every $n$-coherent family below $g$ corresponding to
$\mathcal{I}$ indexed by $X^{n}$ is trivial below $f$.

###### Proof.

Fix $\\{i_{h,k}\colon I_{h}\to I_{k}\mid h\leq k\\}$ as in Corollary 1. Let
$\Phi=\langle\varphi_{\vec{f}}\in I_{\bigwedge\vec{f}\wedge g}\mid\vec{f}\in
X^{n}\rangle$ be $n$-coherent below $g$. Let $\Phi^{*}=\langle
i_{\bigwedge\vec{f}\wedge g,\bigwedge\vec{f}}(\varphi_{\vec{f}})\rangle$;
since the $i_{x,y}$ are compatible and injective $\Phi^{*}$ is $n$-coherent.
By assumption, $\Phi^{*}$ is trivial, as witnessed by a single $\psi^{*}$ if
$n=1$ or a family $\Psi^{*}$ if $n>1$. Then applying
$p_{\bigwedge\vec{f},\bigwedge\vec{f}\wedge g}$ pointwise, we obtain a
trivialization of $\Phi$. ∎

Our main lemma for this section is the following. We note that, from what we
have proven above about injective systems, the proof follows mutatis mutandis
from the proof of [5, Lemma 4.2]; we repeat the argument for completeness.

######  Lemma 7.

Suppose that

  * •

$\mathcal{I}$ is an injective $\Omega_{\kappa}$ system;

  * •

$A\subseteq X\subseteq\omega^{\kappa}$;

  * •

$\Phi=\langle\varphi_{\vec{f}}\,\mid\,\vec{f}\in X^{n}\rangle$ is an
$n$-coherent family corresponding to $\mathcal{I}$;

  * •

$\Phi\upharpoonright A$ is $n$-trivial;

  * •

$H^{j}(K^{\bullet}(\mathcal{I}\upharpoonright A))=0$ for any $1\leq j\leq
n-1$;

  * •

for any infinite $E\subseteq\kappa$ and $f\colon E\to\omega$ there is an $x\in
A$ such that $\\{\alpha\in E\mid f(\alpha)\leq x(\alpha)\\}$ is infinite.

Then $\Phi$ is $n$-trivial.

###### Proof.

We first show the $n=1$ case. Since $\Phi\upharpoonright A$ is trivial, there
is a $\Psi\in\lim\overline{I}$ trivializing $\Phi$; we claim that $\Psi$ also
trivializes $\Phi$. Otherwise, there is an $f\in X$ such that $E=\\{\alpha\mid
p_{\alpha,\omega,f(\alpha)}\Psi(\alpha)\neq\Phi_{f}(\alpha)\\}$ is infinite.
By hypothesis on $A$, we may find some $g\in A$ such that $\\{\alpha\in E\mid
f(\alpha)\leq g(\alpha)\\}$ is infinite. But this is absurd since, once
projected onto $f\wedge g$, $\Psi=^{*}\Phi_{g}=^{*}\Phi_{f}$.

We now present the general argument. Let
$\mathscr{T}_{1}=\langle\tau_{\vec{\alpha}}\mid\vec{\alpha}\in[A]^{n-1}\rangle$
be a (type $I$) trivialization of $\Phi\upharpoonright A$. For each $f\in X$,
let $C_{n-1}^{f}$ be the $n-1$-coherent family below $f$ indexed by $A^{n-1}$
defined by

$\zeta_{\vec{\alpha}}^{f}=\varphi_{\vec{\alpha}f}+(-1)^{n}p_{\bigwedge\vec{\alpha},\bigwedge\vec{\alpha}\wedge
f}\tau_{\vec{\alpha}}.$

After adding appropriate bonding maps, the following computation verifies that
$C_{n-1}$ is coherent:

$\displaystyle\sum_{i=0}^{n-1}(-1)^{i}\zeta^{f}_{\vec{\alpha}^{i}}$
$\displaystyle=\sum_{i=0}^{n-1}(-1)^{i}\varphi_{\vec{\alpha}^{i}f}+(-1)^{n}\sum_{i=0}^{n-1}(-1)\tau_{\vec{\alpha}^{i}}$
$\displaystyle=^{*}\sum_{i=0}^{n-1}(-1)^{i}\varphi_{\vec{\alpha}^{i}f}+(-1)^{n}\varphi_{\vec{\alpha}}$
$\displaystyle=^{*}0$

where the first $=^{*}$ is the fact that $\mathscr{T}_{1}$ trivializes
$\Phi\upharpoonright A$ and the second uses that $\Phi$ is $n$-coherent. We
now define by mutual recursion, for $2\leq k\leq n$,

  * •

$\mathscr{T}_{k}\langle\tau_{\vec{\alpha}}^{\vec{f}}\,\mid\,\vec{f}\in
X^{k-1},\vec{\alpha}\in A^{n-k}\rangle$ trivializing $C_{n-k+1}$

  * •

Families
$C_{n-k}^{\vec{f}}=\langle\zeta_{\vec{\alpha}}^{\vec{f}}\,\mid\,\vec{\alpha}\in
A^{n-k}\rangle$ which are $n-k$ coherent below $\bigwedge\vec{f}$ defined by

$\zeta_{\vec{\alpha}}^{\vec{f}}=\varphi_{\vec{\alpha}\vec{f}}+(-1)^{n-k+1}\sum_{i=0}^{k-1}(-1)^{i}\tau_{\vec{\alpha}}^{\vec{f}^{i}}.$

The hypothesis of the lemma plus Proposition 2 guarantees that we can pass
from this coherent family to a trivialization. Therefore, it suffices to show
that

  1. (1)

$C_{n-k}$ are $n-k$ coherent

  2. (2)

$\mathscr{T}_{n}$ does indeed trivialize $\Phi$.

The first can be verified by the series of steps (after applying appropriate
maps $p$ to restrict to $\bigwedge\vec{f}\wedge\bigwedge\vec{\alpha}$),

$\displaystyle\sum_{i=0}^{n-k}(-1)^{i}\zeta_{\vec{\alpha}^{i}}^{\vec{f}}$
$\displaystyle=\sum_{i=0}^{n-k}(-1)^{i}\varphi_{\vec{\alpha}^{i}\vec{f}}+(-1)^{n-k+1}\sum_{i=0}^{n-k}(-1)^{i}\sum_{j=0}^{k-1}(-1)^{j}\tau_{\vec{\alpha}^{i}}^{\vec{f}^{j}}$
$\displaystyle=^{*}\sum_{i=0}^{n-k}(-1)^{i}\varphi_{\vec{\alpha}^{i}\vec{f}}+(-1)^{n-k+1}\sum_{j=0}^{k-1}(-1)^{j}\zeta_{\vec{\alpha}}^{\vec{f}}$
$\displaystyle=^{*}\sum_{i=0}^{n-k}(-1)^{i}\varphi_{\vec{\alpha}^{i}\vec{f}}+(-1)^{n-k+1}\left(\sum_{j=0}^{k-1}(-1)^{j}\left(\varphi_{\vec{\alpha}\vec{f}^{j}}+(-1)^{n-k}\sum_{\ell=0}^{k-2}(-1)^{\ell}\tau_{\vec{\alpha}}^{\left(\vec{f}^{j}\right)^{\ell}}\right)\right)$
$\displaystyle=^{*}\sum_{i=0}^{n}(-1)^{i}\varphi_{(\vec{\alpha}\vec{f})^{i}}+(-1)^{n-k+1}\left(\sum_{j=0}^{k-1}(-1)^{j+n-k}\sum_{\ell=0}^{k-2}(-1)^{\ell}\tau_{\vec{\alpha}}^{\left(\vec{f}^{j}\right)^{\ell}}\right)$
$\displaystyle=^{*}-\left(\sum_{j=0}^{k-1}(-1)^{j}\sum_{\ell=0}^{k-2}(-1)^{\ell}\tau_{\vec{\alpha}}^{\left(\vec{f}^{j}\right)^{\ell}}\right)$
$\displaystyle=^{*}-\left(\sum_{j\leq\ell\leq
k-2}(-1)^{j+\ell}\tau_{\vec{\alpha}}^{\left(\vec{f}^{j}\right)^{\ell}}+\sum_{\ell<j\leq
k-1}(-1)^{j+\ell}\tau_{\vec{\alpha}}^{\left(\vec{f}^{\jmath}\right)^{\ell}}\right)$
$\displaystyle=^{*}-\left(\sum_{j\leq\ell\leq
k-2}(-1)^{j+\ell}\tau_{\vec{\alpha}}^{\left(\vec{f}^{2+1}\right)^{j}}+\sum_{\ell<j\leq
k-1}(-1)^{j+\ell}\tau_{\vec{\alpha}}^{\left(\vec{f}^{j}\right)^{\ell}}\right)$
$\displaystyle=^{*}-\left(\sum_{\ell<j\leq
k-1}(-1)^{j+\ell+1}\tau_{\vec{\alpha}}^{\left(\vec{f}^{j}\right)^{\ell}}+\sum_{\ell<j\leq
k-1}(-1)^{j+\ell}\tau_{\vec{\alpha}}^{\left(\vec{f}^{\prime}\right)^{\ell}}\right)$
$\displaystyle=0.$

The fact that the functions $\tau_{\vec{\alpha}^{i}}^{\vec{f}}$ trivialize the
functions $\zeta_{\vec{\alpha}}^{\vec{f}^{j}}$ underlies the passage from the
first line to the second; replace $\tau_{\vec{\alpha}}^{\vec{f}}$ with its
definition at (5) to pass from the second line to the third. Nothing more than
a regrouping underlies the passage from the third line to the fourth,
whereupon the first sum vanishes by the $n$-coherence of $\Phi.$ Simple
bookkeeping converts the fifth line into the sixth, and the fact that
$\left(\vec{f}^{j}\right)^{\ell}=\left(\vec{f}^{\ell+1}\right)^{j}$ for all
$j\leq\ell\leq k-2$ converts the sixth line into the seventh. $\mathrm{A}$
renaming of variables in the first sum then yields the eighth line, whose
terms all plainly cancel.

To verify that the $\mathscr{T}_{n}$ trivialize $\Phi$, note that for any
$g\in A$ and $\vec{f}\in X^{n}$, once projected onto $\vec{f}\wedge g$, we
have

$\begin{aligned}
d\mathscr{T}_{n}(\vec{f})&=\sum_{i=0}^{n-1}(-1)^{i}\mathscr{T}_{n}(\vec{f}^{i})\\\
&=^{*}\sum_{i=0}^{n-1}(-1)^{i}\zeta^{\vec{f}^{i}}_{g}\\\
&=^{*}\sum_{i=0}^{n-1}(-1)^{i}\left[\varphi_{g\vec{f}}+(-1)^{n}\sum_{j=0}^{n-2}(-1)^{j}(\tau_{g}^{(\vec{f}^{i})^{j}})\right]\\\
&=^{*}\sum_{i=0}^{n-1}(-1)^{i}\varphi_{g\vec{f}}\\\
&=^{*}\varphi_{\vec{f}}\end{aligned},$

where the first equality is definition of $d$, the second uses that
$\tau^{\vec{f}^{i}}$ trivializes $\zeta_{g}^{\vec{f}^{i}}$, the third is
expanding the definition of $\zeta$, the fourth is cancelling identical terms
with opposite signs, and the last equality is the coherence of $\varphi$. Now,
if $d\mathscr{T}_{n}(\vec{f})$ and $\Phi(\vec{f})$ disagreed infinitely often,
let $E=\\{\alpha\mid
d\mathscr{T}_{n}(\vec{f})(\alpha)\neq\Phi(\vec{f})(\alpha)\\}$. By hypothesis,
we may find an $g\in A$ such that $\\{\alpha\in
E\mid\bigwedge\vec{f}(\alpha)\leq g(\alpha)\\}$ is infinite. But then
$d\mathscr{T}_{n}(\vec{f})$ disagrees with $\varphi_{\vec{f}}$, even after
projecting onto $\vec{f}\wedge g$, contrary to the above calculation. ∎

The relevance of Lemma 7 to Cohen forcing is the following:

######  Proposition 3.

Any set of Cohen sequences added by $\operatorname{Add}(\omega,\chi)$ of
cardinality at least $\kappa^{+}$ satisfies the last clause of the lemma.

###### Proof.

Work in $V[G]$. Suppose $A\subseteq\chi$ has cardinality at least $\kappa^{+}$
and $E\subseteq\kappa$ is infinite. By the countable chain condition, there is
a $W\subseteq\chi$ such that $|W|\leq\kappa$ such that $E\in V[G_{W}]$. By
mutual genericity, the sequence indexed by any $\beta\in A\setminus W$ will
do. ∎

The plan for the remainder of the argument will consist of defining a sequence
$\langle\lambda_{n}\mid n<\omega\rangle$ of cardinals such that any
$n$-coherent family indexed by a set containing at least $\lambda_{n}$ many of
the Cohen sequences is trivial once restricted to a set containing at least
$\lambda_{n-1}$ many Cohen sequences. Lemma 7 will then allow us to
inductively conclude that every $n$ coherent family indexed by a set
containing at least $\lambda_{n}$ many Cohen sequences is trivial. The first
step factors through a partition principle, analogous to the one defined in
[1].

##  5\. A Local Partition Result in Cohen Extensions

We now extract a partition result in Cohen extensions which is essentially
proven in section 6 of [5] and will allow us to create type $II$
trivializations of families to sets with sufficiently many Cohen reals. We
first make a preliminary definition.

######  Definition 12.

A string on $[X]^{\leq n}$ is a sequence $\langle a_{i}\mid 1\leq i\leq
m\rangle$ of finite subsets of $X$ such that

  * •

for each $i$, $a_{i}\subsetneq a_{i+1}$

  * •

for $i<m$, $|a_{i+1}|=|a_{i}|+1$

  * •

$|a_{m}|=n$

We denote the maximal strings on $X$ as $X^{{}^{[\\![n]\\!]}}$

######  Lemma 8.

Let $\kappa$ be a cardinal and let $\lambda$ be regular and $<\kappa^{+}$
inaccessible. Fix $n\geq 2$, let $\mu=\sigma(\lambda^{+},2n-1)$, let
$\chi\geq\mu$ be a cardinal, and let
$\mathbb{P}=\operatorname{Add}(\omega,\chi)$. The following holds in
$V^{\mathbb{P}}:$ Suppose that $X\subseteq\omega^{\kappa}$ consists of $\mu$
many Cohen sequences added by $\mathbb{P}$ and $c\colon[X]^{n}\to\kappa$. Then
there is an $A\subseteq X$ of cardinality $\lambda$, finite sets
$\\{S_{a}\subseteq\kappa\,\mid\,a\in[A]^{\leq n}\\}$, functions
$\\{F_{\alpha}\colon[A]^{\leq n}\setminus\varnothing\to X\,\mid\,\alpha\not\in
S_{\varnothing}\\}$, and ordinals $\\{\beta_{a}\,\mid\,a\in[A]^{\leq n}\\}$
such that for each $\alpha$

  * •

$F_{\alpha}(\\{a\\})=a$

  * •

For $\tau\subsetneq\sigma$,
$F_{\alpha}(\tau)(\alpha)<F_{\alpha}(\sigma)(\alpha)$

  * •

For any string $\langle a_{i}\mid 1\leq i\leq m\rangle$ on $[A]^{\leq n}$ and
any $\alpha\not\in S_{a_{1}}$,

$c\left(a_{1}\cup\bigcup_{2\leq i\leq
m}F_{\alpha}(a_{i})\right)=\beta_{a_{1}}$

######  Remark 1.

It is perhaps worth comparing the principle mentioned above to the one used in
[1]. Although we now have a sequence of $F_{\alpha}$ (rather than a single
$F$), the conclusion is stronger in the sense that we may additionally freeze
out several initial entries. Because of this, it is difficult to extend the
partition result to all of $\omega^{\kappa}$, as was done in [1], leading to
our need to propagate trivializations.

###### Proof.

Fix a condition $p$ such that

  * •

$p\Vdash|\\{\alpha<\chi\mid\dot{f}_{\alpha}\in\dot{X}\\}|\geq\lambda_{n}$ and

  * •

$p\Vdash\dot{c}\colon X\to\kappa$.

Let $Y$ be the set of all $\alpha<\chi$ for which there is a condition
$p_{\alpha}\leq p$ such that $p_{\alpha}\Vdash\dot{f}_{\alpha}\in\dot{X}$;
observe that $|Y|\geq\mu$ by assumption. For each $\alpha\in Y$, fix such a
condition $p_{\alpha}$. Since $\mathbb{P}$ has $\mu$ as a precalibre, there
exists a centered $Y^{\prime}\subseteq Y$ of cardinality $\mu$. For each
$a\in[Y^{\prime}]^{n}$, let $q_{a,\ell}$ enumerate a maximal antichain of
conditions below $\bigcup_{\alpha\in a}p_{\alpha}$ deciding the value of
$\dot{c}(a)$ to be some $\beta_{a,\ell}$. Let

$u(a,\ell)=\\{\gamma<\chi\mid\operatorname{dom}(q_{a,\ell}\cap(\\{\gamma\\}\times\kappa)\neq\varnothing)\\}.$

For each $b\in[Y^{\prime}]^{2n-1}$ let
$v_{b}=\bigcup\\{u(a,\ell)\,\mid\,a\in[b]^{n},\ell<\omega\\}$. Define
$F\colon[Y^{\prime}]^{2n-1}\to H(\kappa^{+})$ as follows. First, for
$b\in[Y^{\prime}]^{2n-1}$ and $\mathbf{m}\in[2n-1]^{n}$ and $\ell<\omega$, let

$w_{\mathbf{m},\ell}^{b}=\\{\eta<\operatorname{otp}(v_{b})\,\mid\,v_{b}(\eta)\in
u(b[\mathbf{m}],\ell)\\}.$

Then set

$F(b)=\langle\langle\overline{q}_{b[\mathbf{m}],\ell},w_{\mathbf{m},\ell}^{b},\beta_{b[\mathbf{m}],\ell}\rangle\,\mid\,\mathbf{m\in[2n-1]^{n},\ell<\omega}\rangle.$

By Fact 1, there exists $H\in[Y^{\prime}]^{\lambda^{+}}$ such that

  * •

$F$ is constant on $H^{2n-1}$, taking value
$\langle\langle\overline{q}_{\mathbf{m},\ell},w_{\mathbf{m},\ell},\beta_{\mathbf{m},\ell}\,\mid\,\mathbf{m}\in[2n-1]^{n},\ell<\omega\rangle$;

  * •

$\langle v_{b}\,\mid\,b\in[H]^{2n-1}\rangle$ is a uniform $2n-1$ dimensional
$\Delta$ system as witnessed by $\rho$ and
$\langle\mathbf{r_{m}}\,\mid\,\mathbf{m}\subseteq 2n-1\rangle$.

By taking an initial segment if necessary, we may assume that $H$ has order
type $\lambda^{+}$. Let $\langle
v_{a,k}\,\mid\,a\in[H]^{<2n-1},k\leq|a|\rangle$ be given by Lemma 1 whenever
possible.

######  Claim 1.

Let $a\in[H]^{n-1}$, $k\leq n-1$, and $\ell<\omega$ and suppose that
$\alpha,\alpha^{\prime}$ are both $k$-addable for $a$. Then
$q_{a\cup\\{\alpha\\},\ell}\upharpoonright(v_{a,k}\times\kappa)=q_{a\cup\\{\alpha^{\prime}\\},\ell}\upharpoonright(v_{a,k}\times\kappa)$.

###### Proof.

We can assume that $\alpha\neq\alpha^{\prime}$. Fix $c\in[H]^{n-1}$ with $\min
c>\max(a\cup\\{\alpha,\alpha^{\prime}\\})$ and let $b=a\cup\\{\alpha\\}\cup c$
and $b^{\prime}=a\cup\\{\alpha^{\prime}\\}\cup c$. Then $b$ and $b^{\prime}$
are aligned with $\mathbf{r}(b,b^{\prime})=2n-1\setminus\\{k\\}$, so $v_{b}$
and $v_{b^{\prime}}$ are aligned with
$\mathbf{r}(v_{b},v_{b^{\prime}})=\mathbf{r}_{2n-1\setminus\\{k\\}}$.
Moreover,
$v_{a,k}=v_{b}[\mathbf{r}_{n\setminus\\{k\\}}]=v_{b^{\prime}}[\mathbf{r}_{n\setminus\\{k\\}}]$.
By homogeneity of $H$, observe that $u(a\cup\\{\alpha\\},\ell)$ and
$u(a\cup\\{\alpha^{\prime}\\},\ell)$ occupy the same relative positions in
$v_{b}$ and $v_{b^{\prime}}$ respectively. In particular,
$u(a\cup\\{\alpha\\},\ell)$ and $u(a\cup\\{\alpha^{\prime}\\},\ell)$ are
aligned with $u(a\cup\\{\alpha\\},\ell)\cap
u(a\cup\\{\alpha^{\prime}\\},\ell)=v_{a,k}\cap
u(a\cup\\{\alpha\\},\ell)=v_{a,k}\cap u(a\cup\\{\alpha^{\prime}\\},\ell)$.
Moreover, by homogeneity of $H$,
$\overline{q}_{a\cup\\{\alpha\\},\ell}=\overline{q}_{a\cup\\{\alpha^{\prime}\\},\ell}=\overline{q}_{n,\ell}$.
Thus, if $(\beta,\gamma)\in v_{a,k}\times\kappa$ is in the domain of
$q_{a\cup\\{\alpha\\},\ell}$ with $\beta=u(a\cup\\{\alpha\\},\ell)(i)$ then it
is also in the domain of $q_{a\cup\\{\alpha^{\prime}\\},\ell}$ and the value
of each is $\overline{q}_{n,\ell}(i,\gamma)$ and the same holds in reverse. ∎

We now note that the values of $\overline{q}_{\mathbf{m},\ell}$ and
$\beta_{\mathbf{m},\ell}$ are independent of $\mathbf{m}\in[2n-1]^{n}$. To see
this, given $b\in[H]^{2n-1}$ and $\mathbf{m}\subseteq n$, there is a
$b^{*}\in[H]^{2n-1}$ such that $b[\mathbf{m}]=b^{*}[n]$. Then

$\overline{q}_{\mathbf{m},\ell}=\overline{q}_{b\mathbf{m},\ell}=\overline{q}_{b^{*}[n],\ell}=\overline{q}_{n,\ell}$

and similarly for $\beta_{\mathbf{m},\ell}$. Let $\overline{q}_{\ell}$ and
$\beta_{\ell}$ be the constant values of $\overline{q}_{\mathbf{m},\ell}$ and
$\beta_{\mathbf{m},\ell}$ respectively. Now, let $H_{0},...,H_{n-1}$ be
disjoint subsets of $H$ such that $\operatorname{otp}(H_{k})=\lambda$ and
$H_{k}<H_{k^{\prime}}$ for all $k<k^{\prime}<n$. For each $k<n$ let
$\delta_{k}=\min(H_{k})$. For $a\in[H_{0}]^{\leq n-1}$, let
$d_{a}=\\{\delta_{i}\mid|a|\leq i<n\\}$ and let
$d^{+}=\\{\delta_{k}\,\mid\,1\leq k<n\\}$. Let $q=q_{d_{\varnothing},0}$ and
let $\dot{A}$ be a $\mathbb{P}$-name for $\\{\alpha\in
H_{0}\,\mid\,d_{\\{\alpha\\}\in\dot{G}}\\}$. We claim that $q$ forces that
there are functions $\dot{F}_{j}$ as in the lemma on the set
$\\{\dot{f}_{\alpha}\mid\alpha\in\dot{A}\\}$. We first claim that $\dot{A}$ is
forced to have cardinality $\lambda$. By regularity of $\lambda$, it suffices
to show the following.

######  Claim 2.

$q$ forces that $\dot{A}$ is unbounded in $H_{0}$.

###### Proof.

Fix an arbitrary $\gamma\in H_{0}$ and an arbitrary $r\leq q$. By choice of
$v_{d^{+},0}$, $\\{v_{d_{\alpha}}\,\mid\,\alpha\in H_{0}\setminus\gamma\\}$ is
an infinite 1-dimensional $\Delta$-system with root $v_{d^{+},0}$. Therefore,
there exists an $\delta\in H_{0}\setminus\gamma$ such that
$v_{d_{\alpha}}\setminus v_{d^{+},0}$ is disjoint from
$\\{\xi\mid\exists\zeta(\xi,\zeta\in\operatorname{dom}(r))\\}$. By Claim 1,
$q_{d_{\\{\alpha\\}},0}\upharpoonright(v_{d^{+},0}\times\kappa)=q_{d_{\varnothing},0}\upharpoonright(v_{d^{+},0}\times\kappa)$.
Since $r$ extends $q_{d_{\varnothing},0}=q$, it is compatible with
$q_{d_{\alpha},0}$. Since $\gamma$ was arbitrary, the claim is established. ∎

The $\delta_{i}$ are currently used as placeholders which must be replaced by
other ordinals. The following claim is the main mechanism by which we will
replace the $\delta_{i}$.

######  Claim 3.

Suppose that $c_{0},c_{1}\in[H]^{n-1}$, $k_{0},k_{1}\leq n-1$, $\alpha\in H$
and $\ell_{0},\ell_{1}<\omega$ are such that

  * •

$\alpha$ is $k_{0}$-addable for $c_{0}$

  * •

$\alpha$ is $k_{1}$-addable for $c_{1}$

  * •

$q_{c_{0}\cup\\{\alpha\\},\ell_{0}}$ and $q_{c_{1}\cup\\{\alpha\\},\ell_{1}}$
are compatible.

Then for every $\alpha^{\prime}$ such that $\alpha^{\prime}$ is
$k_{0}$-addable for $c_{0}$ and $k_{1}$-addable for $c_{1}$,
$q_{c_{0}\cup\\{\alpha^{\prime}\\},\ell_{0}}$ and
$q_{c_{1}\cup\\{\alpha^{\prime}\\},\ell_{1}}$ are compatible.

###### Proof.

We may assume that $\alpha\neq\alpha^{\prime}$. Let $c=c_{0}\cup c_{1}$; then
$|c|\leq 2n-2$. Let $b\in[H]^{2n-1}$ be a (possibly trivial) end-extension of
$c\cup\\{\alpha\\}$ such that every element of $b\setminus(c\cup\\{\alpha\\})$
is greater than $\alpha$. Let $i^{*}$ be such that $b(i^{*})=\alpha$ and let
$b^{\prime}=b\setminus\\{\alpha\\}\cup\\{\alpha^{\prime}\\}$. Then $b$ and
$b^{\prime}$ are aligned with
$\mathbf{r}(b,b^{\prime})=2n-1\setminus\\{i^{*}\\}$. Let
$\mathbf{m}_{0},\mathbf{m}_{1}$ be such that
$c_{i}\cup\\{\alpha\\}=b[\mathbf{m}_{i}]$; then
$c_{i}\cup\\{\alpha^{\prime}\\}=b^{*}[\mathbf{m}_{i}]$.

Suppose that
$(\gamma^{\prime},\delta)\in\operatorname{dom}(q_{c_{0}\cup\\{\alpha^{\prime}\\},\ell_{0}})\cap\operatorname{dom}(q_{c_{1}\cup\\{\alpha^{\prime}},\ell_{1})$.
Let $\gamma^{\prime}=u_{b^{\prime}}(\eta)$. By homogeneity of $H$, the
relative position of $u(c_{i}\cup\\{\alpha\\}\ell_{0})$ within $v_{b}$ is the
same as the relative position of $u(c_{i}\cup\\{\alpha^{\prime}\\}\ell_{0})$
within $v_{b^{\prime}}$. Moreover,
$\overline{q}_{c_{i}\cup\\{\alpha\\},\ell_{i}}=\overline{q}_{c_{i}\cup\\{\alpha^{\prime}\\},\ell_{i}}$.
In particular, if $\gamma=u_{b}(\eta)$ then
$(\gamma,\delta)\in\operatorname{dom}(q_{c_{0}\cup\\{\alpha\\},\ell_{0}})\cap\operatorname{dom}(q_{c_{1}\cup\\{\alpha},\ell_{1})$
and
$q_{c_{i}\cup\\{\alpha\\},\ell_{i}}(\gamma,\delta)=q_{c_{i}\cup\\{\alpha^{\prime}\\},\ell_{i}}(\gamma^{\prime},\delta)$.
Since $q_{c_{0}\cup\\{\alpha\\},\ell_{0}}$ and
$q_{c_{1}\cup\\{\alpha\\},\ell_{1}}$ are compatible,

$q_{c_{0}\cup\\{\alpha^{\prime}\\},\ell_{0}}(\gamma^{\prime},\delta)=q_{c_{0}\cup\\{\alpha\\},\ell_{0}}(\gamma,\delta)=q_{c_{1}\cup\\{\alpha\\},\ell_{1}}(\gamma,\delta)=q_{c_{1}\cup\\{\alpha^{\prime}\\},\ell_{1}}(\gamma^{\prime},\delta),$

as desired. ∎

We now turn more directly to constructing the objects in the statement of the
lemma. If $r\Vdash\dot{a}\in[\dot{A}]^{\leq n}$, note that
$r\Vdash\forall\alpha\in a(p_{\alpha}\in\dot{G})$, so let $\dot{\ell}_{a}$ be
a name for the unique $\dot{\ell}$ such that $q_{d_{a},\ell}\in G$. Moreover,
let $\dot{S}_{\dot{a}}$ be a name for

$\\{\gamma<\kappa\,\mid\,(\gamma,\delta)\in\operatorname{dom}(q_{d_{a},\ell_{a}})\text{
for some }\delta<\chi\\},$

observing that $\dot{S}_{a}$ is forced to be finite. Moreover, by homogeneity,
$\dot{S}_{a}$ is simply

$\\{\gamma<\kappa\,\mid\,(\gamma,\delta)\in\operatorname{dom}(\overline{q}_{\ell_{a}})\text{
for some }\delta<\kappa\\}.$

We will show that $q$ forces the existence of ordinals

$\langle\varepsilon_{a}^{\alpha}\,\mid\,a\in[A]^{\leq
n}\setminus\varnothing,\alpha\in\kappa\setminus S_{\varnothing}\rangle$

with the intention of setting $F_{\alpha}(a)=f_{\varepsilon_{a}^{\alpha}}$.
These will satisfy a few additional properties; to state these, we need some
further notation. If $\vec{a}=\langle a_{i}\mid 1\leq i\leq m\rangle$ is a
string on $[A]^{\leq j}$ for some $j\leq n$, we denote

$d_{\vec{a}}^{\alpha}=a_{1}\cup\\{\varepsilon_{a_{i}}^{\alpha}\,\mid\,2\leq
i\leq m\\}\cup\\{\delta_{k}\mid j\leq k<n\\}$

######  Claim 4.

$q$ forces that there exist ordinals

$\langle\dot{\varepsilon}_{a}^{\alpha}\,\mid\,a\in[\dot{A}]^{\leq
n}\setminus\varnothing,\alpha\in\kappa\setminus\dot{S}_{\varnothing}\rangle$

satisfying

  1. (1)

$\dot{\varepsilon}_{a}^{\alpha}\in H_{|a|-1}$ for all nonempty
$\dot{a}\in[\dot{A}]^{\leq n}$ and
$\alpha\in\kappa\setminus\dot{S}_{\varnothing}$;

  2. (2)

$\dot{\varepsilon}_{\\{\beta\\}}^{\alpha}=\beta$ for all $\beta\in\dot{A}$ and
all $\alpha\in\kappa\setminus\dot{S}_{\varnothing}$;

  3. (3)

For $\varnothing\subsetneq a\subsetneq b$,
$\dot{f}_{\dot{\varepsilon}_{a}^{\alpha}}(\alpha)<\dot{f}_{\dot{\varepsilon}_{b}^{\alpha}}(\alpha)$;

  4. (4)

For every $1\leq m\leq n$, every string $\dot{\vec{a}}$ on $[\dot{A}]^{\leq
m}$, and every $\alpha\not\in\dot{S}_{a_{1}}$,

$q_{d_{\vec{a}}^{\alpha},\dot{\ell}_{a_{1}}}\in\dot{G}.$

###### Proof.

The construction is by induction on $|a|$. If $|a|=1$, so $a=\\{\beta\\}$ for
some $\beta$, then setting $\varepsilon_{a}^{\alpha}=\beta$ will do; property
$(1)$ follows from definition of $A$, property $(2)$ is immediate, property
$(3)$ is vacuously satisfied, and property $(4)$ also follows from definition
of $A$, noting that $\ell_{\\{\beta\\}}=0$.

Suppose that $r\leq q$ forces that $\dot{a}\in[A]^{m}$. By extending $r$ if
necessary, we may assume that $r$ decides

  * •

$\dot{a}=\check{a}$ for some $a\subseteq H_{0}$

  * •

the values of $\varepsilon_{a^{\prime}}^{\alpha}$ and $\ell_{a^{\prime}}$ for
each $a^{\prime}\subsetneq a$

  * •

the value of $\ell_{a}$

  * •

the value of $\dot{f}_{\varepsilon_{a^{\prime}}^{\alpha}}(\alpha)$ for each
$a^{\prime}\subsetneq a$

If $\vec{a}$ is a string on $[a]^{\leq|a|-1}$, then since $r\Vdash
q_{d_{\vec{a}}^{\alpha},\ell_{a_{1}}}\in\dot{G}$, we must have $r\leq
q_{d_{\vec{a}}^{\alpha},\ell_{a_{1}}}\in\dot{G}$. Now, note that if $\vec{a}$
and $\vec{b}$ are strings on $[a]^{\leq|a|-1}$, there is an $\alpha$ (namely
$\delta_{|a|-1}$) such that
$q_{d_{\vec{a}}\setminus\delta_{i}\cup\\{\alpha\\},\ell_{a_{1}}}$ and
$q_{d_{\vec{a}}\setminus\delta_{i}\cup\\{\alpha\\},\ell_{b_{1}}}$ are
compatible. Therefore, by Claim 3, for every $\varepsilon\in H_{|a|-1}$,
$q_{d_{\vec{a}}\setminus\delta_{i}\cup\\{\varepsilon\\},\ell_{a_{1}}}$ and
$q_{d_{\vec{a}}\setminus\delta_{i}\cup\\{\varepsilon\\},\ell_{b_{1}}}$ are
compatible. Now, by choice of the $v_{a^{\prime},k}$, note that for every
string $\vec{a}$,
$\\{v_{d_{\vec{a}}\setminus\delta_{i}\cup\\{\varepsilon\\},\ell_{a_{1}}}\,\mid\,\varepsilon\in
H_{|a|-1}\\}$ is an infinite $\Delta$-system with root
$v_{d_{\vec{a}}\setminus\\{\delta_{|a|-1}\\},|a|-1}$, so there is an
$\varepsilon$ such that
$v_{d_{\vec{a}}\setminus\delta_{i}\cup\\{\varepsilon\\},\ell_{a_{1}}}\setminus
v_{d_{\vec{a}}\setminus\\{\delta_{|a|-1}\\},|a|-1}$ is disjoint from
$\\{\gamma<\chi\mid\exists\delta(\delta,\gamma)\in\operatorname{dom}(r)\\}$
for every such $\vec{a}$. By Claim 1, note that
$v_{d_{\vec{a}}\setminus\delta_{i}\cup\\{\varepsilon\\},\ell_{a_{1}}}\upharpoonright(v_{d_{\vec{a}}\setminus\\{\delta_{|a|-1}\\},|a|-1}\times\kappa)=v_{d_{\vec{a}},\ell_{a_{1}}}\upharpoonright(v_{d_{\vec{a}}\setminus\\{\delta_{|a|-1}\\},|a|-1}\times\kappa)$.
Therefore,

$s_{0}=r\cup\bigcup_{\begin{subarray}{c}\vec{a}\\\ \alpha\not\in
S_{a_{1}}\end{subarray}}v_{d_{\vec{a}}\setminus\delta_{i}\cup\\{\varepsilon\\},\ell_{a_{1}}}$

is a condition extending $r$. Note that, by homogeneity of $H$ and definition
of $S_{a_{1}}$, $(\varepsilon,\alpha)\not\in\operatorname{dom}(s_{0})$.
Therefore, we extend $s_{0}$ to define the value at $(\varepsilon,\alpha)$ to
be larger than $\dot{f}_{\varepsilon_{a^{\prime}}^{\alpha}}(\alpha)$ for any
$a^{\prime}\subsetneq a$. This $s$ is readily verified to force that
$\varepsilon_{a}^{j,k}=\varepsilon$ satisfies all of the desired conditions. ∎

To verify the hypotheses of the theorem, we work in $V[G]$. For $a\in[A]^{\leq
n}\setminus\emptyset$, set $F_{\alpha}(\\{f_{\gamma}\,\mid\,\gamma\in
a\\})=f_{\varepsilon_{a}^{\alpha}}$. The first conclusion is immediate from
requirement (2) on $\varepsilon_{\\{\beta\\}}^{\alpha}$. The second is
immediate from requirement $(3)$. For the final requirement, if $\vec{a}$ is a
string on $[A]^{\leq n}$, then since $q_{d_{\vec{a}},\ell_{a}}\in\dot{G}$, we
see that

$c\left(a_{1}\cup\bigcup_{2\leq i\leq
m}F_{\alpha}(a_{i})\right)=\beta_{\ell_{a_{1}}}$

by our choice of $q_{d_{\vec{a}},\ell_{a}}$.

∎

##  6\. From Partition to Trivialization

###  6.1. Trivialization on a Large Set

Suppose that $\mathcal{G}$ is an $\Omega_{\kappa}$ system and let $\Phi$ be a
corresponding $n$-coherent family indexed over $X^{n}$.

######  Definition 13.

$c_{\Phi}\colon[X]^{n+1}\to[\kappa]^{<\omega}$ is defined as
$c_{\Phi}(\mathbf{x})$ is the support of $\Phi(\mathbf{x})$.

We now demonstrate how to produce a trivialization of $\Phi\upharpoonright A$
from the conclusions of Lemma 8 applied to $c_{\Phi}$.

######  Lemma 9.

Suppose that $\lambda$ is $<\kappa^{+}$-inaccessible and
$\mu=\sigma(\lambda^{+},2n+1)$. Then for any $\chi\geq\mu$, the following is
forced by $\mathbb{P}=\operatorname{Add}(\omega,\mu)$: Suppose that
$\mathcal{G}$ is an $\Omega_{\kappa}$ system and $\Phi$ is an $n$-coherent
family corresponding to $\mathcal{G}$ indexed by $X^{n}$ with $X$ containing
at least $\mu$ many of the Cohen sequences added by $\mathbb{P}$. Then there
is an $A\subseteq X$ containing at least $\lambda$ many Cohen sequences such
that $\Phi\upharpoonright A$ is type $II$ trivial.

###### Proof.

Let $A,F_{j},\\{T_{a}\,\mid\,a\in[A]^{\leq n+1}\\},\\{S_{a}\mid a\in[A]^{\leq
n+1}\\}$ are as in the conclusions of Corollary 8 applied to $c_{\Phi}$; note
that since $c_{\Phi}$ colors using finite sets, we use $T_{a}$ rather than
$\beta_{a}$. Let $S=S_{\varnothing}\cup T_{a}$. In the $\lim{}^{1}$ case this
is fairly easy: for any $x,y\in A$, after applying appropriate maps $p$, for
any $\alpha\not\in S$,
$\Phi_{x}(\alpha)-\Phi_{y}(\alpha)=\Phi_{x}(\alpha)-\Phi_{y}(\alpha)+\Phi_{F_{\alpha}(x,y)}(\alpha)-\Phi_{F_{\alpha}(x,y)}(\alpha)=0$
as $\Phi_{x}-\Phi_{F_{\alpha}(x,y)}$ and $\Phi_{y}-\Phi_{F_{\alpha}(x,y)}$ are
supported on $S$, so the constant $0$ system will do as a trivialization
outside of $S$. In general, $\Phi\upharpoonright\bigoplus_{i\in
S}\mathcal{G}_{i}$ will therefore trivialize $\Phi$. The general case will
take a bit more work. We will make use of formal expressions; denote
$\mathrm{Free}(Y)$ as the free abelian group over $Y$ and denote the basis
element corresponding to $y$ as $e(y)$. We will define formal expressions for
$s\leq n$ of the form, for $\vec{\rho}\in A^{[s]}$ and $\alpha\not\in
S_{\varnothing}$

$A_{s}^{\alpha}(\rho)\in\mathrm{Free}(\\{f\mid\bigwedge\rho(\alpha)\leq f\leq
F(\rho)(\alpha)\\}^{[s+1]})$

as well as, for $\tau\in A^{[s+1]}$ and $\alpha\not\in S_{\varnothing}$,
expressions

$C_{s}^{\alpha}(\tau)\in\mathrm{Free}([\\{f\mid\bigwedge\tau(\alpha)\leq
f(\alpha)<F_{\alpha}(\alpha)\\}]^{s+1})$
$S_{s}^{\alpha}(\tau)\in\mathrm{Free}([\\{f\mid\bigwedge\tau_{0}(\alpha)\leq
f(\alpha)\leq F(\alpha)\\}]^{s+1}).$

We also will make use of the operations

$d\colon\mathrm{Free}([\\{f\mid\bigwedge\rho(\alpha)\leq
f(\alpha)\\}^{[s+1]})\to\mathrm{Free}(\\{f\mid\bigwedge\rho(\alpha)\leq
f(\alpha)\\}]^{s})$

given by
$de(f_{0},...,f_{s})=\sum_{i}(-1)^{i}e(f_{0},...,\widehat{f_{i}},...,f_{s})$.
Also, given $g\in\omega^{\kappa}$ and an expression $\rho$ in
$\mathrm{Free}(\\{f\mid\bigwedge\rho(\alpha)\leq
f(\alpha)<g(\alpha)\\}^{[s]})$, we define $\rho*g$ on the standard basis by
setting $e(f_{0},...,f_{s-1})*g=e(f_{0},...,f_{s-1},g)$. These two operations
satisfy the relation

$d(x*y)=d(x)*y+(-1)^{s}x.$

The idea is that $A_{s}$ is the $s$-stage approximation to a trivialization,
$S_{s}$ is the boundary of $A_{s+1}$ with a sign, and $C_{s}$ is the error
between the boundary of the $s$ stage approximation and the boundary of
$\Phi$. Note that $d\Phi$ provides a map
$E_{\Phi}\colon\mathrm{Free}(\\{f\mid\bigwedge\rho\leq^{k}f\\}^{[n+1]})\to\bigoplus_{i}G_{i,\bigwedge\rho(i)}$
by interpreting $e(\sigma)$ as $d\Phi(\sigma)$ and projecting by using $0$
maps in the first $k$ coordinates and afterwards using the structure maps. As
the base case $A_{1}$, we set $A_{1}=0$. In general, we define

$C_{s}^{\alpha}(\tau)=e(\tau)-\sum_{i<n+1}(-1)^{i}A_{s}(\tau^{i})$
$S_{s}^{\alpha}(\tau)=d\left(C_{s}(\tau)*F_{\alpha}(\tau)\right)$
$A_{s+1}(\tau)=(-1)^{s+1}C_{s}(\tau)*F_{\alpha}(\tau).$

The following lemma is the key point in ensuring our approximations converge
to a trivialization. Eventually, we will set
$\Psi(\tau)=E_{\Phi}(A_{n}(\tau))$ The point will be that when $s=n$,
$C_{n}(\tau)$ will be supported within $S$; a consequence will be that
$d\Psi=d\Phi$ outside of $S^{n}$.

######  Claim 5.

$C_{s}(\tau)$ is of the form $(-1)^{s+1}S_{s}(\tau)$ plus terms of the form
$e(F^{*}(\vec{\sigma}))$ for $\vec{\sigma}\in\tau^{{}^{[\\![s+1]\\!]}}$.

###### Proof.

Observe that $S_{s}^{\alpha}$ can be rewritten as follows:

$\displaystyle S_{s}(\tau)$
$\displaystyle=d(C_{s}(\tau))*F_{\alpha}(\tau)+(-1)^{s+1}C_{s}(\tau)$
$\displaystyle=d(e(\tau))*F_{\alpha}(\tau)-d\left(\sum_{i<n+1}(-1)^{i}A_{s}(\tau^{i})\right)*F_{\alpha}(\tau)+(-1)^{s+1}C_{s}(\tau)$
$\displaystyle(*)$
$\displaystyle=\sum_{i<s+1}(-1)^{i}e(\tau^{i},F(\tau))-\sum_{i<s+1}(-1)^{i}\sum_{j<s+1}[A_{s}^{\alpha}(\tau^{i})*F_{\alpha}(\tau)]^{j}$
$\displaystyle\quad+(-1)^{s+1}C_{s}(\tau).$

The proof proceeds by induction on $s$ to show that $(*)$ is a sum of terms of
the form $e(\bigcup_{2\leq i\leq m}F_{\alpha}(a_{i}))$ for
$\vec{a}\in\tau^{{}^{[\\![s+1]\\!]}}$. The $s=1$ case is immediate: $A_{1}=0$
and the two terms on the left hand sum are of the appropriate form.

Now assume that $s>1$ and the induction hypothesis holds for $s-1$. Note that,
by definition of $S_{s}^{\alpha}$, that for $s>1$ $(*)$ is equal to

$\sum_{i<s+1}(-1)^{i}e(\tau^{i},F_{\alpha}(\tau))-(-1)^{s}\sum_{i<s+1}(-1)^{i}S_{s-1}(\tau^{i})*F(\tau);$

this is perhaps more apparent from the line above in our earlier calculation
plus the observation that

$S_{s}(\tau)=(-1)^{s+1}d(A_{s+1}(\tau)).$

Note that the induction hypothesis immediately implies that
$(-1)^{s}S_{s-1}(\tau^{i})*F(\tau)$ is of the form
$C_{s-1}(\tau^{i})*F_{\alpha}(\tau)$ plus terms of the form
$e(F^{*}(\vec{a}))$ for $\vec{a}\in[\tau]^{{}^{[\\![s+1]\\!]}}$ as
$e(\bigcup_{2\leq i\leq m}F_{\alpha}^{*}(a_{i}))*F(\tau)$ is just appending
$\tau$ to the string and applying $F$. Then $(*)$ reduces to terms of the form
$e(\bigcup_{2\leq i\leq m}F_{\alpha}(a_{i}))$ plus

$\displaystyle\sum_{i<s+1}(-1)^{i}e\left(\tau^{i},F_{\alpha}({\tau})\right)-\sum_{i<s+1}(-1)^{i}\left[\mathcal{C}_{s-1}^{\alpha}\left(\tau^{i}\right)*F_{\alpha}({\tau})\right]$
$\displaystyle=$
$\displaystyle\sum_{i<s+1}(-1)^{i}e\left(\tau^{i},F_{\alpha}({\tau})\right)-\sum_{i<s+1}(-1)^{i}\left[e\left(\tau^{i},F_{\alpha}({\tau})\right)-\sum_{j<s}(-1)^{j}\mathcal{A}_{s-1}^{\alpha}\left(\left(\tau^{i}\right)^{j}\right)*F_{\alpha}({\tau})\right]$
$\displaystyle=$
$\displaystyle\sum_{i<s+1}\sum_{j<s}(-1)^{i+j}\mathcal{A}_{s-1}^{\alpha}\left(\left(\tau^{i}\right)^{j}\right)*F_{\alpha}({\tau}).$

The key observation is that for any term with $i\leq j$, the term with
$i^{\prime}=j+1$ and $j^{\prime}=i$ is the same but with opposite sign, thus
showing that

$\sum_{i<s+1}\sum_{j<s}(-1)^{i+j}\mathcal{A}_{s-1}\left(\left(\tau^{i}\right)^{j}\right)*F({\tau})=0.$

∎

By hypothesis on $F$ plus noting that $d\Phi^{*}(S_{n})=0$ as $S_{n}$ is a
boundary, we will find that the system
$\Psi(\rho)(\alpha)=d\Phi^{*}(A_{n}^{\alpha}(\rho))(\alpha)$ satisfies
$d\Psi=d\Phi$ outside of $S$. If we can show that each $\Psi_{\vec{f}}$ is
finitely supported, we will complete the proof as
$\Psi+(\Phi-\Psi)\upharpoonright\bigoplus_{\alpha\in S}\mathcal{G}_{\alpha}$
will trivialize $\Phi$. This, however, follows from an easy induction that
$A_{s}^{\alpha}(\tau)$ is always a sum of the form $\bigcup_{2\leq i\leq
m}F_{\alpha}(a_{i})$ for $\vec{a}$ a string on $[\tau]^{\leq n}$. Thus, for

$\gamma\not\in\bigcup_{\sigma\subseteq\tau}\left(S_{\sigma}\cup
T_{\sigma}\right),$

we have $\Psi_{\tau}(\gamma)=0$. ∎

###  6.2. Trivialization Everywhere

We will now complete the proof of our main results. Let
$\lambda_{0}=\beth_{1}(\kappa)^{+}$ and
$\lambda_{n+1}=\sigma(\lambda_{n}^{+},2n+1)$; observe that
$\sup(\lambda_{n})=\beth_{\omega}(\kappa)$ and each $\lambda_{n}$ is $<\kappa$
inaccessible.

######  Lemma 10.

Suppose that $\chi\geq\beth_{\omega}(\kappa)$ and let
$\mathbb{P}=\operatorname{Add}(\omega,\kappa)$. The following holds in
$V^{\mathbb{P}}$: Suppose that $\mathcal{I}$ is an injective $\Omega_{\kappa}$
system and $\Phi$ is an $n$-coherent family corresponding to $\mathcal{I}$
indexed by a set $X\subseteq\omega^{\kappa}$ containing at least $\lambda_{n}$
many of the Cohen sequences added by $\mathbb{P}$. Then $\Phi$ is $n$-trivial.

###### Proof.

The proof proceeds by induction on $n$. By Lemma 9, there is an $A\subseteq X$
containing at least $\lambda_{n-1}$ many Cohen sequences such that
$\Phi\upharpoonright A$ is $n$-trivial. By Lemma 7, the inductive hypothesis,
and Proposition 3, $\Phi$ is $n$-trivial. ∎

We obtain Theorem 3 as an immediate corollary.

##  7\. Conclusion

We conclude by noting several questions related to the work here. One
conspicuous example is the following.

######  Question 1.

Is it consistent that derived limits of $\Omega_{\kappa}$ systems are additive
for every $\kappa$?

The answer is yes in the case of $\lim^{1}\mathbf{A}_{\kappa}$, as shown in
[3, Theorem 5.1]; in fact, it is implied by $\lim^{1}\mathbf{A}=0$. We may ask
whether this holds for higher values of $n$.

######  Question 2.

Does $\lim^{n}\mathbf{A}=0$ imply $\lim^{n}\mathbf{A}_{\kappa}=0$?

Worth noting is that the forcings involved in this paper necessarily increase
the size of the continuum; in particular, the case of $\kappa=2^{\aleph_{0}}$
is unclear.

######  Question 3.

Is it consistent that $\lim^{n}\mathbf{A}_{2^{\aleph_{0}}}=0$ for all $n>0$?

There are two questions about $\Omega_{\kappa}$ systems which serve as a
reasonable analogue. The first is on additivity.

######  Question 4.

Does additivity of $\lim^{n}$ for $\Omega_{\omega}$ systems imply additivity
of $\Omega_{\kappa}$ systems for all uncountable $\kappa$?

The second is about coherent families.

######  Question 5.

Suppose that every $n$-coherent family is $n$-trivial for all
$\Omega_{\omega}$ systems. Must the same hold for all uncountable $\kappa$?

At the moment, we believe the proof of [3, Theorem 5.1] will generalize to
show the answer to Question 5 is yes if $n=1$.

We may also ask about how large the continuum must be for additivity of
derived limits, both for $\mathbf{A}$ and for arbitrary $\Omega_{\omega}$
systems.

######  Question 6.

What is the minimum value of the continuum compatible with
$\lim^{n}\mathbf{A}=0$ for every $n>0$?

######  Question 7.

What is the minimum value of the continuum compatible with additivity of
derived limits for $\Omega_{\omega}$ systems?

Note that if $GCH$ holds in the ground model, then after forcing with
$\operatorname{Add}(\omega,\beth_{\omega})$ the continuum will be of size
$\aleph_{\omega+1}$, giving an upper bound. In light of the results of [11],
of particular interest is when $\mathfrak{b}<\mathfrak{d}$, for instance the
Mitchell and Miller models. A related question, which is true when $n=1$, is
the following.

######  Question 8.

Suppose $X$ is a $<^{*}$-increasing $\omega_{n}$ sequence. Is
$\lim^{n}\mathbf{A}\upharpoonright X\neq 0$?

This is true when $n=1$, as shown in [3]. A positive result would also
eliminate the need for the weak diamond principles used in [11] to obtain the
nonvanishing of higher derived limits.

## References

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