$C_{-*}(\Omega_{q}Q)^{\delta}:=\left\\{\sum_{i=1}^{\infty}c_{i}x_{i}:c_{i}\in\Lambda,x_{i}\in
C_{-*}(\Omega_{q}Q,{\mathbf{k}});\lim_{i\to\infty}{\mathbf{v}}c_{i}-\delta|x_{i}|=\infty\right\\}$
(4.16)

For $x\in\pi_{1}Q$ we have defined $|x|$ as the least length of based
geodesics in $x,$ and $H^{0}(\Omega_{q}Q)^{\delta}$ is isomorphic to the
complete group ring

$\Lambda[\pi_{1}Q]^{\delta}:=\left\\{\sum_{i=1}^{\infty}c_{i}x_{i}:c_{i}\in\Lambda,x_{i}\in\pi_{1}Q;\lim_{i\to\infty}{\mathbf{v}}c_{i}-\delta|x_{i}|=\infty\right\\}.$
(4.17)

######  Proposition 4.8.

The ${\mathbin{\mathbb{Z}}}$-graded $\Lambda$-algebra isomorphism

$HW^{0}(T_{q}^{*}Q,T_{q}^{*}Q)\cong\Lambda[\pi_{1}Q]$ (4.18)

associated to the exact structure on the cotangent bundle extends continuously
to the completions. Moreover, for every single-degree object
${\mathbf{b}}\in{\mathbin{\cal F}}(T^{*}Q)$ the $\pi_{1}Q$ representation on
$HF^{*}(T_{q}^{*}Q,{\mathbf{b}})$ extends continuously to
$\Lambda[\pi_{1}Q]^{\delta}.$

###### Sketch Proof..

The first statement follows from the usual correspondence between geodesics
with fixed endpoints and Hamiltonian chords of the radial coordinate on the
cotangent bundle with endpoints on cotangent fibres. This correspondence
identifies the length with the symplectic action. The second statement is a
consequence of the convergence of operations in the ordinary Fukaya category,
and the appearance of the action in the functor from the exact to the nonexact
category (c.f. Equation (3.56)), which implies that when a Hamiltonian chord
$x$ is considered as an element of the exact Floer complex, the action of
$T^{-{\mathbin{\cal A}}(x)}x$ on $HF^{*}(T_{q}^{*}Q,{\mathbf{b}})$ has norm
bounded above by $1$. ∎

######  Remark 4.9.

The above result is a fragment of the study of _local Fukaya categories._ The
more complete account will appear in a forthcoming paper by the first author,
Groman and Varolgunes [7].

We prove now a general fact about one-dimensional representations:

######  Lemma 4.10.

Let $\rho:\pi_{1}Q\to\Lambda^{*}$ be a one-dimensional representation. This
extends continuously to $\Lambda[\pi_{1}Q]^{\delta}$ if and only if
$|{\mathbf{v}}\rho|\leqslant\nobreak\delta.$

###### Proof.

Suppose $|{\mathbf{v}}\rho|\leqslant\nobreak\delta.$ Take
$\sum_{i=1}^{\infty}c_{i}x_{i}\in\Lambda[\pi_{1}Q]^{\delta},$
$c_{i}\in\Lambda$ and $x_{i}\in\pi_{1}Q$ with
$\lim_{i\to\infty}{\mathbf{v}}c_{i}-\delta|x_{i}|=\infty.$ Then

${\mathbf{v}}[c_{i}\rho(x_{i})]={\mathbf{v}}c_{i}+{\mathbf{v}}\rho(x_{i})\geqslant{\mathbf{v}}c_{i}-\delta|x_{i}|,$
(4.19)

which tends to infinity. So $\sum_{i=1}^{\infty}c_{i}\rho(x_{i})$ exists and
$\rho$ extends to $\Lambda[\pi_{1}Q]^{\delta}.$

Suppose conversely that $\rho$ extends to $\Lambda[\pi_{1}Q]^{\delta}.$ Let
$x\in\pi_{1}Q,$ let $c>1$ and for $i=1,2,3,\cdots$ let
$y_{i}:=\sum_{j=1}^{i}T^{jc\delta|x|}x^{j}\in\Lambda[\pi_{1}Q]$ which
converges in $\Lambda[\pi_{1}Q]^{\delta}.$ We show that

$-{\mathbf{v}}\rho(x)\leqslant\nobreak\delta|x|.$ (4.20)

Write $\rho(x)=:T^{{\mathbf{v}}\rho(x)}u$ where $u$ is a unit of
$\Lambda^{0}.$ Then

$\rho(y_{i})=\sum_{j=1}^{i}T^{j[c\delta|x|+{\mathbf{v}}\rho(x)]}u^{j}.$ (4.21)

Now if $c\delta|x|+{\mathbf{v}}\rho(x)<0$ then
${\mathbf{v}}\rho(y_{i})=i[c\delta|x|+{\mathbf{v}}\rho(x)]$ which diverges in
$\Lambda.$ So $c\delta|x|+{\mathbf{v}}\rho(x)\geqslant 0$ and
$-{\mathbf{v}}\rho(x)\leqslant\nobreak c\delta|x|.$ Letting $c$ tend to $1$ we
find $-{\mathbf{v}}\rho(x)\leqslant\nobreak\delta|x|.$

Applying (4.20) to $x^{-1}$ in place of $x,$ we find also
${\mathbf{v}}\rho(x)\leqslant\nobreak\delta|x|.$ So
$|{\mathbf{v}}\rho(x)|\leqslant\nobreak\delta|x|$ for every $x\in\pi_{1}Q,$
which completes the proof. ∎

The proof of Lemma 4.3 is now complete by noting that Theorem 4.8 and Lemma
4.10 imply it. ∎

###  4.4 Proof of Theorem 1.38

We use two theorems on linear groups: one is by Tits [56, Corollary 1] and the
other is a Lie–Kolchin theorem stated for instance by Borel [13, Chapter III
10.5]. Both apply to fields of arbitrary characteristic.

######  Theorem 4.11 (Tits).

Every finitely-generated linear group has either a nonabelian free subgroup or
a finite-index solvable subgroup.

######  Theorem 4.12 (Lie–Kolchin).

Every solvable linear group over an algebraically closed field has a finite-
index triangularisable subgroup.

######  Remark 4.13.

Let $\rho:\pi_{1}Q\to\mathop{\rm GL}V$ be a representation on a finite-
dimensional $\Lambda$-vector space $V.$ Since $\pi_{1}Q$ is finitely generated
it follows that so is $\rho(\pi_{1}Q)$ to which we can apply Theorem 4.11. In
the circumstances of Theorem 1.38 we shall have the latter alternative of
Theorem 4.11. To this we can apply Theorem 4.12 because $\Lambda$ is
algebraically closed.

We come now to the proof of Theorem 1.38. Let ${\mathbf{b}}\in H{\mathbin{\cal
F}}(T^{*}Q)$ be a single-degree object, and $\rho:\pi_{1}Q\to\mathop{\rm
GL}HF^{*}(T_{q}^{*}Q,{\mathbf{b}})$ the representation given by the Yoneda
functor. By hypothesis the group $\pi_{1}Q$ has no nonabelian free subgroup,
and accordingly its image $\rho(\pi_{1}Q)\subseteq\mathop{\rm
GL}HF^{*}(T_{q}^{*}Q,{\mathbf{b}})$ does not have such a subgroup either. This
linear group has as in Remark 4.13 a finite-index triangularisable subgroup.
So there exists a finite-index subgroup of $\pi_{1}Q$ on which $\rho$ is
triangularisable. Let $f:(P,p)\to(Q,q)$ be the corresponding finite cover, and
take a one-dimensional subrepresentation $\sigma:\pi_{1}P\to\Lambda^{*}$ of
the triangularisable group. By the latter part of Lemma 4.2 this $\sigma$ is
isomorphic to the representation on $HF^{*}(E,T_{p}^{*}P)$ for some local
system $E$ on a smooth perturbation of $P.$ Hence using Proposition 1.40 we
find $HF^{0}(f^{*}{\mathbf{b}},E)\neq 0.$ By Lemma 4.3 the valuation of
$\sigma$ is bounded by means of the underlying Lagrangian of ${\mathbf{b}}.$

If $\pi_{1}Q$ is abelian, we do not have to use the representation theory
results and we can take $P=Q.$ ∎

##  Appendix A Novikov Fields in Arbitrary Characteristic

In this section we define the notion of (generalised) Novikov fields and prove
that there exist complete algebraically-closed Novikov fields in arbitrary
charactristic. We give two proofs: one is Krasner’s lemma, Lemma A.4, a
general fact about nonArchimedean valued fields, and the other is an explicit
construction in §A.2.

###  A.1 Valued Fields

######  Definition A.1.

By a a valued field we mean a field $K$ equipped with a valuation (or more
precisely a nonArchimedean valuation) which is a function
${\mathbf{v}}:K\to(-\infty,+\infty]$ such that:

  * •

for $a\in K$ we have ${\mathbf{v}}a=+\infty$ if and only if $a=0;$ and

  * •

for $a,b\in K$ we have ${\mathbf{v}}(ab)={\mathbf{v}}a+{\mathbf{v}}b$ and
${\mathbf{v}}(a+b)\geqslant\min\\{{\mathbf{v}}a,{\mathbf{v}}b\\}.$

The ring of integers of $K$ is the subring $K^{0}\subseteq K$ defined by

$K^{0}:=\\{a\in K:{\mathbf{v}}a\geqslant 0\\},$ (A.1)

which has a unique maximal ideal $K^{+}:=\\{a\in K:{\mathbf{v}}a>0\\}$ and the
residue field

$\operatorname{rf}K:=K^{0}/K^{+}.$ (A.2)

The value group $\operatorname{vg}K\subseteq{\mathbin{\mathbb{R}}}$ is a
subgroup of ${\mathbin{\mathbb{R}}}$ defined by

$\operatorname{vg}K:={\mathbf{v}}(K\setminus\\{0\\})\subseteq{\mathbin{\mathbb{R}}}.$
(A.3)

We say that $K$ is complete if so is the the norm

$|\bullet|:K\to[0,\infty)$ (A.4)

defined by $|a|:=e^{-{\mathbf{v}}a},$ $a\in K.$

Here are some facts about complete valued fields:

######  Proposition A.2.

Let $K$ be a complete valued field, and $L/K$ an algebraic extension; then the
valuation of $K$ extends uniquely to $L$ so that:

  * (a)

if $L/K$ has finite degree, the valuation on $L$ is complete; and

  * (b)

if $L/K$ has infinite degree, the valuation of $L$ is strictly not complete.

Also, if $L/K$ is finite then

$[L:K]\geqslant[\operatorname{rf}L:\operatorname{rf}K][\operatorname{vg}L:\operatorname{vg}K].$
(A.5)

More precisely, there exist two intermediate fields of $L/K,$ the maximal
unramified field $L^{\rm unr}$ and the maximal tamely-ramified field $L^{\rm
tame},$ such that:

  * (i)

$K\subseteq L^{\rm tame}\subseteq L^{\rm unr}\subseteq L;$

  * (ii)

$\operatorname{rf}K\subseteq\operatorname{rf}L^{\rm
unr}=\operatorname{rf}L^{\rm tame}\subseteq\operatorname{rf}L;$

  * (iii)

$\operatorname{vg}K=\operatorname{vg}L^{\rm
unr}\subseteq\operatorname{vg}L^{\rm tame}\subseteq\operatorname{vg}L,$
$[\operatorname{vg}L^{\rm tame}:\operatorname{vg}L^{\rm unr}]=[L^{\rm
tame}:L^{\rm unr}];$

  * (iv)

if $\operatorname{rf}K$ has characteristic $p=0$ then $L=L^{\rm tame},$ and if
$p>0$ then $[L:L^{\rm tame}]$ is a power of $p.$

###### Proof.

We consult Neukirch [44]; the referred statements in what follows are all from
Chapter II of this book.

The unique extension is by Theorem 4.8. By Proposition 4.9 the extended
valuation is equivalent to the $\ell^{\infty}$ norm on a direct sum of $K,$
whose elements have only finitely many nonzero components even if they are
infinite dimensional. So they are complete if and only if finite-dimensional.

The inequality (A.5) is from Proposition 6.8. The field $L^{\rm unr}$ is
defined in Definition 7.4, and $L^{\rm tame}$ in Definition 7.10 but under the
hypothesis that $\operatorname{rf}K$ has positive characteristic; if
$\operatorname{rf}K$ has characteristic $0$ we define $L^{\rm tame}:=L.$ The
parts (i), (ii) and (iv) are then obvious. The part (iii) follows from
Proposition 7.7; this is stated again in the positive-characteristic case, but
its proof applies also to the characteristic-zero case. ∎

Here the extension $L/L^{\rm tame}$ is called the wild part of $L/K,$ which is
in general difficult to study. But it has degree a power of $p;$ and for
Galois extensions of such a degree, there is a classification result:

######  Proposition A.3 (Artin–Schreier).

Let $K$ be any field of characteristic $p.$ Then for every $a\in K$ either:

  * (i)

the polynomial $f_{a}=f_{a}(x):=x^{p}-x-a\in K[x]$ has a root $b\in K,$ in
which case its roots $b,b+1,\cdots,b+p-1$ are all in $K;$ or

  * (ii)

$f_{a}\in K[x]$ is irreducible, in which case $K[x]/f_{a}$ is a degree-$p$
Galois extension of $K.$

Conversely:

  * (iii)

every degree-$p$ Galois extension of $K$ is of the form $K[x]/f_{a}$ for some
$a\in K;$ or more generally

  * (iv)

every finite Galois extension of $K$ of degree equal to a power of $p$ is a
repeated extension of this kind.

###### Proof.

For (i)–(iii) consult for instance Lang [41, Chapter VI, Theorem 6.4]; and
then for (iv) use the following fact [41, Chapter I, Corollary 6.6]: for every
finite group $G$ of order a power of $p$ there exists a normal series

$G=G_{0}\supseteq G_{1}\supseteq\cdots\supseteq G_{n}=\\{1\\}$ (A.6)

with $[G_{i}:G_{i-1}]=p$ for every $i.$ ∎

Finally we recall:

######  Lemma A.4 (Krasner).

Every valued field $K$ has a complete algebraically-closed extension.

###### Proof.

Let $L$ be the algebraic closure of $K.$ The valuation extends uniquely to $L$
by the first assertion in Proposition A.2, because $L$ is the union of finite
subextensions of $K.$ The completion of this valued field $L$ is algebraically
closed, by Krasner’s lemma [44, Chapter II, §6, Exercise 2]. ∎

###  A.2 Novikov Fields

Let ${\mathbf{k}}$ be a field. We define two Novikov fields over
${\mathbf{k}}$ with formal variable $T$:

$\displaystyle\Lambda^{\min}:$
$\displaystyle=\left\\{\sum_{i=1}^{\infty}a_{i}T^{\gamma_{i}}:a_{i}\in{\mathbf{k}},\gamma_{i}\in{\mathbin{\mathbb{R}}};\lim_{i\to\infty}\gamma_{i}=+\infty\right\\},$
(A.7) $\displaystyle\Lambda^{\max}:$
$\displaystyle=\left\\{a=\sum_{\gamma\in{\mathbin{\mathbb{R}}}}a_{\gamma}T^{\gamma}:a_{\gamma}\in{\mathbf{k}}\text{
and }\operatorname{supp}a\subseteq{\mathbin{\mathbb{R}}}\text{ well-
ordered}\right\\}$ (A.8)

where $\operatorname{supp}a:=\\{\gamma\in{\mathbin{\mathbb{R}}}:a_{\gamma}\neq
0\\}.$ Recall that $\operatorname{supp}a$ being well ordered means that every
nonempty subset has a minimum. In particular, $\operatorname{supp}a$ is half
discrete in the sense that for every $\gamma\in\operatorname{supp}a$ there
exists $\epsilon>0$ such that

$\operatorname{supp}a\cap(\gamma,\gamma+\epsilon)=\emptyset;$ (A.9)

because we can take $\gamma+\epsilon$ to be the minimum of
$\operatorname{supp}a\cap(\gamma,\infty).$

Clearly $\Lambda^{\min}\subseteq\Lambda^{\max}.$ The sums and products in
$\Lambda^{\max}$ are defined in the natural way: for $a,b\in\Lambda^{\max}$ we
can define

$a+b:=\sum_{\gamma\in\operatorname{supp}a\cup\operatorname{supp}b}(a_{\gamma}+b_{\gamma})T^{\gamma}$
(A.10)

because $\operatorname{supp}a\cup\operatorname{supp}b$ is well ordered; and

$ab:=\sum_{\gamma}\left(\sum_{\alpha+\beta=\gamma}a_{\alpha}b_{\beta}\right)T^{\gamma}$
(A.11)

because $\sum_{\alpha+\beta=\gamma}$ is finite, which follows since
$\operatorname{supp}a,\operatorname{supp}b$ are half discrete. Clearly
$\Lambda^{\min}\subseteq\Lambda^{\max}$ is a subfield.

The valuation ${\mathbf{v}}:\Lambda^{\max}\to(-\infty,+\infty]$ is defined by
setting ${\mathbf{v}}0:=+\infty$ and by taking in (A.8) the least
$\gamma\in\operatorname{supp}a.$ With respect to this, both
$\Lambda^{\min},\Lambda^{\max}$ are complete. We have

$\operatorname{rf}\Lambda^{\min}\cong\operatorname{rf}\Lambda^{\max}\cong{\mathbf{k}}\text{
and
}\operatorname{vg}\Lambda^{\min}\cong\operatorname{vg}\Lambda^{\max}={\mathbin{\mathbb{R}}}.$
(A.12)

The respective rings of integers in $\Lambda^{\min},\Lambda^{\max},$ called
the Novikov rings, are defined by the same formulae but with $[0,\infty)$ in
place of ${\mathbin{\mathbb{R}}};$ for these to make sense, it suffices that
${\mathbf{k}}$ be a ring.

######  Definition A.5.

By a Novikov field over ${\mathbf{k}}$ we mean a valued field extension of
$\Lambda^{\min}.$

######  Remark A.6.

By Lemma A.4 there is a complete algebraically-closed Novikov field over
${\mathbf{k}}.$ We can also construct it explicitly as in the following
proposition.

######  Proposition A.7.

  * (i)

$\Lambda^{\min},\Lambda^{\max}$ are both perfect.

  * (ii)

$\Lambda^{\max}$ is maximal in Krull’s sense [40]; that is, $\Lambda^{\max}$
has no nontrivial valued field extension $K$ with
$\operatorname{rf}K\cong{\mathbf{k}}$ and
$\operatorname{vg}K={\mathbin{\mathbb{R}}}.$ In particular, if ${\mathbf{k}}$
is algebraically closed, so is $\Lambda^{\max}.$

  * (iii)

If ${\mathbf{k}}$ is algebraically closed of characteristic $0,$ so is
$\Lambda^{\min}.$
If ${\mathbf{k}}$ has positive characteristic $p$ then $\Lambda^{\min}$ is not
algebraically closed, and its algebraic closure is strictly not complete.

######  Remark A.8.

These results do not assert that $\Lambda^{\max}$ is the algebraic closure of
$\Lambda^{\min}.$ For rational-power series, the algebraic closure is
described in Kedlaya [38].

###### Proof of Proposition A.7.

For (i) let $\Lambda$ be either $\Lambda^{\min}$ or $\Lambda^{\max}.$ By
definition, a field $\Lambda$ is perfect if either it has characteristic $p=0$
or the ring homomorphism $\Lambda\to\Lambda$ defined by $x\mapsto x^{p}$ is
surjective. Suppose $p>0$ and consider the polynomial $x^{p}-a\in\Lambda[x],$
$a\in\Lambda.$ We show that this has a root in $\Lambda.$ If $a=0$ it has root
$0\in\Lambda.$ If $a\neq 0$ we can write $a=T^{{\mathbf{v}}a}b(1+c),$
$b\in{\mathbf{k}}^{*}$ and $c\in\Lambda^{+}.$ Let $b^{1/p}\in{\mathbf{k}}$ be
a root of $x^{p}-b\in{\mathbf{k}}[x],$ which exists because ${\mathbf{k}}$ is
algebraically closed. Define

$(1+c)^{1/p}:=\sum_{k=0}^{\infty}\binom{1/p}{k}c^{k}\in\Lambda.$ (A.13)

Then $T^{{\mathbf{v}}a/p}b^{1/p}(1+c)^{1/p}\in\Lambda$ is a desired root.

The former part of (ii) is an old result [36, 40]. For the latter, if
${\mathbf{k}}$ is algebraically closed, then
$\operatorname{rf}\Lambda\cong{\mathbf{k}}$ is algebraically maximal and
$\operatorname{vg}\Lambda={\mathbin{\mathbb{R}}}$ is already maximal in
${\mathbin{\mathbb{R}}};$ so $\Lambda$ has no nontrivial finite extension, or
equivalently $\Lambda$ is algebraically closed.

The former part of (iii) is proved by Fukaya, Oh, Ohta and Ono [23, Lemma A.1]
but we give another proof, using Proposition A.2 with $K=\Lambda^{\min}.$
Suppose therefore that $L/K$ is a finite extension, and note that:

  * •

since $\operatorname{rf}K$ is algebraically closed it follows that the
inclusions in (ii) are all equalities, so $K=L^{\rm unr};$

  * •

since $\operatorname{vg}K={\mathbin{\mathbb{R}}}$ it follows that the
inclusions in (iii) are all equalities, so $L^{\rm unr}=L^{\rm tame};$ and

  * •

$L=L^{\rm tame}$ because $\operatorname{rf}K$ has characteristic zero.

Consequently, $K=L;$ that is, $K=\Lambda$ has no nontrivial finite extension,
or equivalently, $K$ is algebraically closed.

For the latter part of (iii) let $\gamma<0$ and consider the Artin–Schreier
polynomial

$x^{p}-x-T^{\gamma}\in\Lambda^{\min}[x].$ (A.14)

If this had a root of the form $x=\sum_{n=1}^{\infty}x_{n}T^{\gamma_{n}}$ then
by induction on $n$ it must be of the form

$\sum_{n=1}^{\infty}T^{-\gamma/p^{n}}\in\Lambda^{\max}\setminus\Lambda^{\min};$
(A.15)

so $\Lambda^{\min}$ is not algebraically closed.

We show next that the algebraic closure of $\Lambda^{\min}$ has infinite
degree over $\Lambda^{\min}.$ By an Artin–Schreier element over
$K:=\Lambda^{\min}$ we shall mean a root of an Artin–Schreier polynomial over
$K.$ Let $L/K$ be the field extension which contains all the Artin–Schreier
elements over $K.$ We prove the following three facts in this order:

  * (I)

if $a_{0},a_{1},a_{2}\in K$ with $a_{0}=a_{1}+a_{2}$ and if
$\alpha_{1},\alpha_{2},\alpha_{3}\in L$ are respectively roots of the
polynomials $x^{p}-x-a_{i}\in K[x],$ $i=0,1,2,$ then $K(\alpha_{0})\subseteq
K(\alpha_{1},\alpha_{2});$

  * (II)

if $\alpha$ is an Artin–Schreier element over $K$ then $K(\alpha)\subseteq
K(\beta;\delta_{1},\cdots,\delta_{n})$ where $\beta$ is a root of the
polynomial $x^{p}-x-b\in K[x]$ for some $b\in K$ with ${\mathbf{v}}b\geqslant
0,$ and every $\delta_{i}$ a root of the polynomial
$x^{p}-x-c_{i}T^{\gamma_{i}}\in K[x]$ for some $c_{i}\in{\mathbf{k}}$ and
$\gamma_{i}<0;$ and

  * (III)

if the algebraic closure of $\Lambda^{\min}$ had finite degree over
$\Lambda^{\min},$ we should have
$L=K(\beta_{1},\cdots,\beta_{m};\delta_{1},\cdots,\delta_{n})$ where
$\beta_{1},\cdots,\beta_{m}$ are such as $\beta$ in (II) and
$\delta_{1},\cdots,\delta_{n}$ are under the same condition as in (II) but
that $n$ may of course be different.

For (I) note that $\alpha_{1}+\alpha_{2}$ is a root of the polynomial
$x^{p}-x-a_{0}\in K[x],$ so $\alpha_{0}=\alpha_{1}+\alpha_{2}+c$ for some
$c\in\\{0,1,\cdots,p-1\\},$ which implies $K(\alpha_{0})\subseteq
K(\alpha_{1},\alpha_{2})$ as we want. For (II) write $\alpha$ as a root of the
polynomial $x^{p}-x-a\in K[x],$ $a\in K=\Lambda^{\min},$ and recall that $a$
has by definition only finitely many terms of negative powers with respect to
$T;$ that is, $a=b+\sum c_{i}T^{\gamma_{i}},$ ${\mathbf{v}}b\geqslant 0$ and
$\gamma_{i}<0.$ Applying to this the part (I) repeatedly, we see that
$K(\alpha)\subseteq K(\beta;\delta_{1},\cdots,\delta_{n})$ as we want. For
(III) it is clear from definition that $L\supseteq
K(\beta_{1},\cdots,\beta_{m};\delta_{1},\cdots,\delta_{n}).$ On the other
hand, by our hypothesis $L/K$ is a finite extension and so is obtained from
$K$ by adding finite Artin–Schreier elements. Hence using (II) repeatedly we
get the other inclusion we want.

Now in (III) write $\delta_{i}$ as a root of the polynomial
$x^{p}-x-c_{i}T^{\gamma_{i}}\in K[x]$ for some $c_{i}\in{\mathbf{k}}$ and
$\gamma_{i}<0.$ Then
$\delta_{i}=\sum_{n=1}^{\infty}c_{i}^{1/p^{n}}T^{\gamma_{i}/p^{n}}$ for some
$c_{i}^{1/p^{n}}\in\overline{\mathbf{k}},$ the algebraic closure of
${\mathbf{k}},$ so

$L\subseteq\left\\{\sum
a_{\gamma}T^{\gamma}:a_{\gamma}\in\overline{\mathbf{k}}\text{ and if
$\gamma<0$ then
$\gamma\in{\mathbin{\mathbb{Q}}}(p;\gamma_{1},\cdots,\gamma_{n})$}\right\\}.$
(A.16)

But as ${\mathbin{\mathbb{R}}}/{\mathbin{\mathbb{Q}}}$ is an infinite
extension there is some $\epsilon<0$ that is not in
${\mathbin{\mathbb{Q}}}(p;\gamma_{1},\cdots,\gamma_{n}).$ Then by (A.16) the
Artin–Schreier element $\sum_{n=1}^{\infty}T^{\epsilon/p^{n}},$ a root of the
polynomial $x^{p}-x-T^{\epsilon}\in K[x],$ is not in $L,$ which contradicts
the definition of $L.$

So the algebraic closure of $\Lambda^{\min}$ has infinite degree over
$\Lambda^{\min}$ and by Proposition A.2 (b) it is strictly not complete. ∎

######  Remark A.9.

The rôle of Artin–Schreier polynonimals is essential in the proof above. In
fact, if ${\mathbf{k}}$ is algebraically closed then every finite normal
extension $L/\Lambda^{\min}$ is a repeated Artin–Schreier extension, as we
show now. In the notation of Proposition A.2 with $K=\Lambda^{\min},$ we have
again $K=L^{\rm tame}.$ On the other hand, by (i) above, $L/L^{\rm tame}$ is
automatically Galois. To this we can apply Proposition A.3 so
$L/\Lambda^{\min}$ is a repeated Artin–Schreier extension as we want.

##  Appendix B Proof of Theorem 1.31

In this section we sketch the proof of Theorem 1.31, a version of the
generation theorem for the derived category $D{\mathbin{\cal F}}(T^{*}Q).$ We
follow Fukaya, Seidel and Smith’s method [25] using Lefschetz fibrations,
Seidel long exact sequences and Viterbo restriction functors. When working
with nonexact Lagrangians, only the last ingredient is missing from the
literature. After providing the argument for this generalisation below, the
remainder of the proof extends fairly immediately.

We could instead follow the first author’s method [2] too, in which we should
introduce the symplectic cohomology (an invariant associated to the time-one
Hamiltonian orbits) and prove that it has the same relation as in the exact
case to the algebraic invariants obtained from the Fukaya category. We can do
this in closed symplectic manifolds (as will appear in [6]) and for monotone
Lagrangians (as is done already in [48]) but not yet in our setting.

###  B.1 Generation and Restriction of Fukaya categories

Let $X$ be a Liouville domain and $U\Subset X$ a Liouville subdomain (i.e. a
codimension $0$ submanifold of $X$, with smooth contact boundary). We shall
assume that all objects of ${\mathbin{\cal W}}_{{\rm ex}}(X)$ are supported on
Lagrangians which are equipped with a primitive function for the Liouville
$1$-form, which is constant along the intersection with $\partial U$. In this
setting, Abouzaid and Seidel [8] constructed a Viterbo restriction functor

${\mathbin{\cal W}}_{{\rm ex}}(X)\to{\mathbin{\cal W}}_{{\rm ex}}(U).$ (B.1)

On the other hand, the maximum principle for pseudoholomorphic discs in $X$
bounded by Lagrangians in $U$ yields an $A_{\infty}$ functor

${\mathbin{\cal F}}(U)\to{\mathbin{\cal F}}(X)$ (B.2)

on compact Lagrangians. The compatibility between these two functors is given
by the following result:

######  Proposition B.1.

The left and right modules over ${\mathbin{\cal W}}_{{\rm ex}}(X)$ associated
to the image of each object of ${\mathbin{\cal F}}(U)$ in ${\mathbin{\cal
F}}(X)$ are equivalent to the pullbacks of the associated modules over
${\mathbin{\cal F}}(U)$ under the Viterbo restriction map.

###### Sketch Proof.

We make a little modification of Abouzaid–Seidel’s techniques [8]. We begin by
constructing a Hamiltonian $H$ such that for each exact Lagrangian $L\subset
X$ whose primitive vanishes away from $U$, the generators of the Floer complex
$CF^{*}(L,L;H)$ which lie away from $U$ have strictly positive action. In [8]
one constructs a family depending on a parameter $\rho\in(0,1]$ by rescaling
$H$ along the Liouville flow inside $U$, yielding a function that does not
depend on $\rho$ away from $U$; this has the property that the action of all
generators in $U$ are scaled by $\rho$, while those outside have constant
action. We consider instead a family parametrised by $\rho\in[1,\infty)$ which
is constant in $U$, and obtained by rescaling outside; now the action in $U$
is constant, but the action outside is scaled.

Consider now a closed (unobstructed) Lagrangian $K$ in $U$; we shall omit all
notation for the choices of local systems and bounding cochains because they
are not relevant to our argument. The goal is to show that the $A_{\infty}$
module structure of $CF^{*}(K,L)$ over the wrapped Floer cochains of $L$ in
$X$ is equivalent to the module structure obtained by pullback from the
corresponding module structure after restriction to $U$. This amounts to
constructing a homotopy between the given module structure and one for which
we can prove the vanishing of any $A_{\infty}$ module operation with input
including one chord with boundary on $L$ lying away from $U$. To see this, we
use the fact that such operations (with fixed inputs and outputs) are given by
sums of contributions indexed by homotopy classes of discs with the chosen
boundary conditions and asymptotics. The key point is then to show that, for a
given homotopy class, we may choose $\rho$ large enough so that the relevant
contributions vanish. The argument is exactly the same as the proof of [8,
Lemma 7.4]: the topological energy of any homotopy class that contributes must
positive, and since $X$ is exact it can be expressed by Stokes’s theorem as a
sum of actions of the intersection point and of the homotopy class of the
boundary component along $K$. All of these terms are independent of $\rho$,
except those corresponding to inputs lying outside $U$; since these are
rescaled by $\rho$ and are negative, we find that the topological energy is
negative for sufficiently large $\rho$, hence that the moduli space is empty.
∎

We shall use the following consequence:

######  Corollary B.2.

If ${\mathbf{b}}$ is an object of ${\mathbin{\cal F}}(U)$ whose image in
${\mathbin{\cal W}}(X)$ lies in the category split-generated by a collection
of exact Lagrangians, then ${\mathbf{b}}$ lies in the subcategory of
${\mathbin{\cal W}}(U)$ split-generated by their images under the restriction
functor.

###### Proof.

Let ${\mathbin{\cal L}}_{{\mathbf{b}}}(X)$ and ${\mathbin{\cal
R}}_{{\mathbf{b}}}(X)$ denote the left and right modules over the exact
category ${\mathbin{\cal W}}_{ex}(X)$ associated to ${\mathbf{b}}$. The
criterion for split-generation is that the map from the tensor product of
these modules to the self-Floer cohomology of ${\mathbf{b}}$

${\mathbin{\cal L}}_{{\mathbf{b}}}(X)\otimes_{{\mathbin{\cal
W}}_{ex}(X)}{\mathbin{\cal R}}_{{\mathbf{b}}}(X)\to
CF^{*}({\mathbf{b}},{\mathbf{b}})$ (B.3)

hits the unit on cohomology. By the above result, these modules are isomorphic
to the pullbacks of the modules ${\mathbin{\cal L}}_{{\mathbf{b}}}(U)$ and
${\mathbin{\cal R}}_{{\mathbf{b}}}(U)$ over the image of ${\mathbin{\cal
W}}_{ex}(X)$ (which we denote ${\mathbin{\cal W}}_{ex}(U)$ associated to
${\mathbf{b}}$. The proof amounts to the homotopy commutativity of the
following triangle

${{\mathbin{\cal L}}_{{\mathbf{b}}}(X)\otimes_{{\mathbin{\cal
W}}_{ex}(X)}{\mathbin{\cal
R}}_{{\mathbf{b}}}(X)}$${CF^{*}({\mathbf{b}},{\mathbf{b}})}$${{\mathbin{\cal
L}}_{{\mathbf{b}}}(U)\otimes_{{\mathbin{\cal W}}_{ex}(U)}{\mathbin{\cal
R}}_{{\mathbf{b}}}(U)}$ (B.4)

which follows by the same argument as the above proposition. ∎

###  B.2 Fukaya, Seidel and Smith’s Method

Combining Fukaya, Seidel and Smith’s construction [25] and Seidel’s doubling
construction [51, §18] we get, for every closed connected manifold $Q$:

  * •

a Liouville manifold $X$, containing $Q$ as a exact Lagrangian so that the map
$H^{*}(X)\to H^{*}(Q)$ is equipped with a split surjection; and

  * •

a collection $S_{1},\cdots,S_{k}\subset X$ of Lagrangian spheres which
intersect $Q$ transversely in one point, and for which there is a composition
$\phi$ of Dehn twists about $S_{i}$ such that $\phi Q$ is disjoint from $Q$
(up to Hamiltonian isotopy).

######  Remark B.3.

The only property which is not explicitly stated in [51, §18] is the assertion
that the restriction map to $Q$ splits on cohomology, but this follows from
the geometry of the construction because $X$ is a branched cover of a disc
cotangent bundle, yielding the desired splitting of $H^{*}(X)\to H^{*}(Q)$.

From these properties we get:

######  Lemma B.4.

If $U\subset X$ is a sufficiently small Weinstein neighbourhood of $Q$, then

  * •

the intersection of each Lagrangian $S_{1}$ with $U$ is Hamiltonian isotopic
to a cotangent fibre, and

  * •

the image of every closed Lagrangian in $U$ under $\phi$ is disjoint from $U$
(up to Hamiltonian isotopy).∎

$\pi(X)$$0$$\pi(S_{1})$$\pi(S_{k})$$\pi(\Sigma)$ Figure B.1: the line segments
are $\pi(S_{1}),\cdots,\pi(S_{k})$ and the dashed curve is $\pi(\Sigma).$

Fix a background class on $U$ and the class induced in $X$ by the splitting
above. Introduce the Fukaya category ${\mathbin{\cal F}}(X)$ which contains
the Lagrangians $S_{i}$ with the trivial relative spin structures (which
exists because every $S_{i}$ is the double of a disc in $U$), the trivial
rank-one local systems and the trivial bounding cochains.

######  Remark B.5.

Here we allow the ground ring ${\mathbf{k}}$ to have characteristic $2,$
because we do not take the ${\mathbin{\mathbb{Z}}}_{2}$-invariant Fukaya
category as Seidel does [51].

The key result about this Fukaya category is:

######  Lemma B.6.

Every object of ${\mathbin{\cal F}}(X)$ supported in $U$ lies in the category
split-generated by the spheres $S_{i}$.

###### Sketch Proof.

The proof is the same as that for exact Lagrangians by Fukaya, Seidel and
Smith [25], using Oh’s result [45, Section 9] for Seidel sequences in the
nonexact case. Oh proves indeed that given two objects
${\mathbf{b}},{\mathbf{b}}^{\prime}$ and a Lagrangian sphere $S$, with
corresponding Dehn twist $\tau_{S}$, we have a homotopy for the composition

$CF^{*}(\tau_{S}{\mathbf{b}},S)\otimes CF^{*}(S,{\mathbf{b}}^{\prime})\to
CF^{*}(\tau_{S}{\mathbf{b}},{\mathbf{b}}^{\prime})\to
CF^{*}({\mathbf{b}},{\mathbf{b}}^{\prime})$ (B.5)

where the first map is given by composition and the second by multiplication
with a distinguished closed morphism in
$CF^{*}({\mathbf{b}},\tau_{S}{\mathbf{b}})$. So the cone of this morphism lies
in the category generated by $S$.

Using this sequence repeatedly we see that if ${\mathbf{b}}$ is an object of
${\mathbin{\cal F}}(X)$ supported in $U$ then there is some closed morphism
$\alpha$ from ${\mathbf{b}}$ to $\phi{\mathbf{b}}$ whose cone lies in the
category generated by $S_{1},\cdots,S_{k}.$ But the underlying Lagrangians of
${\mathbf{b}},\phi_{*}{\mathbf{b}}$ may be made disjoint after a Hamiltonian
isotopy, so $\alpha$ is exact and the cone of $\alpha$ is isomorphic to a
direct sum of ${\mathbf{b}}$ and $\phi_{*}{\mathbf{b}}$ (with shifts). This
implies that ${\mathbf{b}}$ is a summand of the object lying in the category
generated by $S_{1},\cdots,S_{k}$. ∎

Combining this result with Corollary B.2 completes the proof of Theorem 1.31,
since our starting data is that the collection of Lagrangians spheres $S_{i}$
intersect $U$ in cotangent fibres.

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