Suppose we have more than one vertex in $\ti\cw^*$. Pick any vertex $V\in\spsystem^*$ that is maximal within $\spsystem^*$ (there may be more than one), and let $\cellgens=\setdef{W \in\spsystem^*}{W\subset V,\,\codim(W\subset V)=1}$. Since this set is a finite, and also non-empty (due to there existing a sequence of edges within $\ti\cw^*$ from $V$ to $U$), we can consider $\topcell=\topcell\combcell(V,\cellgens)$, the unique largest cell in $\ti\cw^*$ containing $V$. Since all cells of $\ti\cw^*$ that contain $V$ (and are thus rooted at $V$) are faces of $\topcell$, we can just contract it to its faces not containing $V$. More precisely, let $\topcell=\setof{V}\times [0,1]^{\cellgens}$, and for any $u\colon\cellgens\to[0,1]$, we set

\[
\]

where $u_{const}\colon\cellgens\to [0,1]$ is the constant function with value
$(1-\max u)$. Note that

* $u'(\cellgen)=1$ for at least one $\cellgen\in\cellgens$, thus $\kappa_{\combcell}(u')$ is on some smaller dimensional face of $\topcell$, but not in any open face rooted at $V$;

* $u'=u$ for all such points (where a coordinate is $1$);

* $u\mapsto u'$ is continuous.

Thus, we can obtain a homotopy $H\colon\ti\cw^*\times[0,1]\to\ti\cw^*$ where

H(κ_(u), t)     =κ_((1-t)u+tu'),          u∈[0,1]^

H(p,t)     = p,          p∈(^*∖V).

By the previous observations, this is indeed the desired strong deformation retract as it is well-defined and continuous on both $\topcell\times[0,1]$ and $\ti\cw(\spsystem^*\setminus\setof{V})\times[0,1]$ (both being closed sets whose union is $\ti\cw^*$), and for $H_t\colon p\mapsto H(p,t)$, $H_0$ is the identity, $H_1$ maps to $\ti\cw(\spsystem^*\setminus\setof{V})$, while $H_t$ is always the identity when restricted to $\ti\cw(\spsystem^*\setminus\setof{V})$.

Repeating this step, we can contract $\ti\cw^*$ to its minimal vertex. Thus in particular the original map $\psi$ is always homotopic to a constant map, hence all homotopy groups save for $\pi_0$ are trivial. Hence, given that $\ti\cw$ is a CW complex, the connected components are contractible.

The above contractibility of the connected components of $\ti\cw(\spsystem)$ basically follows from the fact that $\spsystem$ is intersection-closed. If we drop this assumption, then we still can define the set of combinatorial cells $\combcell(\cellroot,\cellgens)$, where $R$ is a root $V$, the generator $\cellgens$ is a set of subspaces $W$ of codimension one in $V$, but in this case in the definition of the cell we impose additionally that any intersection of type $\cap_{G\in {\mathcal H}} G$, for ${\mathcal H}\subset \cellgens$, should belong to $\spsystem$. Then, in a similar way as above, we glue the associated topological cells into a CW complex $\ti\cw(\spsystem)$. However, in this new situation, $\ti\cw(\spsystem)$ might have non-trivial homotopy type. E.g., assume that $V=\gen{e_1,e_2,e_3}$ is 3-dimensional, and (the non-intersection closed) $\spsystem$ consists of the subspaces $0$, $\gen{e_1}$, $\gen{e_1,e_3}$, $V$, $\gen{e_2,e_3}$, $\gen{e_2}$. Then $\ti\cw(\spsystem)\sim S^1$.

Next we introduce weights to this complex as well.

We call $(\spsystem,w)$ a weighted system of vector spaces (WSS) if $\spsystem$ is an intersection closed system of vector spaces as in Definition V-cell, and $w\colon\spsystem\to\bbz$ is a function bounded from below. Additionally, for any $V,W\in\spsystem$ with $W\subseteq V$, we require

\begin{equation}\label{eq:weightfun}
w(W)\leq w(V)+\codim(W\subseteq V).
\end{equation}

We invite the reader to recall the definition of weighted CW complex from Definition def:compat-weight.

Next, for any WSS $(\spsystem,w)$, we introduce a weight function on its associated CW complex.

Let $(\spsystem, w)$ be a weighted system of vector spaces, and consider the associated complex $\ti\cw=\ti\cw(\spsystem)$. For a $\spsystem$-cell $\combcell$, let

\[
\ti w\bigl(\topcell(\combcell)\bigr)=w\bigl(\cellroot(\combcell)\bigr)+\height\combcell.
\]

The defined $\ti w$ forms a set of compatible weight function on $\ti\cw(\spsystem)$.

This is a consequence of the definition wss-def, since for any face $\combcellface$ of a $\spsystem$-cell $\combcell$,

\begin{align*}
\ti w\bigl(\topcell(\combcellface)\bigr)
& =w\bigl(\cellroot(\combcellface)\bigr)+\height\combcellface
\leq w\bigl(\cellroot(\combcell)\bigr)+\codim\bigl(\cellroot(\combcellface)\subseteq \cellroot(\combcell)\bigr)+\codim\bigl(T(\combcellface)\subseteq R(\combcellface)\bigr)\leq \\
& \leq w\bigl(\cellroot(\combcell)\bigr)+\codim\bigl(\celltip(\combcell)\subseteq \cellroot(\combcell)\bigr)
=\ti w\bigl(\topcell(\combcell)\bigr). \qedhere
\end{align*}

The resulting pair $(\ti\cw,\ti w)$ is called the weighted CW complex associated to $(\spsystem, w)$.

§.§ The weighted system of subspaces associated with a curve singularity

Let $(C,o)$ be an isolated curve singularity and consider the associated weighted complex $(\cw, w)=(R(0,c),w)$ together with its cubes $\setof{\cellset_q}_q$, and the filtration $\setof{\filt(\ell)}_{\ell}$ introduced in section <ref>. Note that $\setof{\filt(\ell)}_{\ell}$ is a linear subspace system of $\fnring=\fnring_{C,o}$. We denote it by $\Im\filt$. We write $\grverts$ for the index set of the irreducible components of $(C,o)$, i.e. $r=\abs{\grverts}$.

Next we compare the cubes of $R(0,c)$ and of $\Im\filt$ and of the CW complex associated with $\Im\filt$. Furthermore, we enhance $\Im\filt$ with a weight system induced by the weights of $R(0,c)$ (compatibly with Definition V-cell-weight).
We start with the following notation:

\[
\ti{\cellset}_q = \setof{\text{combinatorial $\Im\filt$-cells of dimension $q$}}, \quad
\ti{\cellset}_* = \cup_q\ti{\cellset}_q.
\]

* For any $q$-cube $\cell\in\cellset_q$, the set

\[
\combcell=\setdef{\filt(\ell)}{\ell\in \cell\cap L}
\]

is a combinatorial $\Im\filt$-cell of dimension at most $q$ (where $\dim C>\dim\combcell$ can happen).

Let $\phi:\cellset_*\to\ti{\cellset}_*$ be the map that takes a cube $\cell$ to the thus associated $\combcell$.

* Any combinatorial $\Im\filt$-cell $\combcell$ can be realized as $\phi(C)$ for some $C\in \cellset_*$.

In fact, for each $\Im\filt$-cell $\combcell$ of dimension $q$, there exists (usually not unique) $q$-cube $\cell\in\cellset_q$ with $\phi(\cell)=\combcell$. We call such a cube $C$ minimal representative of $\combcell$.

In particular, all the $\Im\filt$-cells have dimension no greater than the rank of the lattice $\bbz^r$, namely $r=\text{number of components of }(C,o)$.

Fix $C=(\ell, I)$. Then consider $V=\filt(\ell)$ and $V_i=\filt(\ell+E_i)$ for all $i\in I$. Then $\cap_{i\in J}V_i=\filt(\ell+E_J)$ for any $\emptyset\neq J\subseteq I$. Hence, $\combcell$ as a combinatorial cell has root $V$ and generators $\setdef{V_i}{V_i\neq V}$. Clearly $\dim(\combcell)\leq\abs{I}$.

For the other direction, the case $q=0$ is trivial, so we will assume $q>0$. With the notation introduced after Lemma uniquemax_0, consider the lattice points $\maxpoint_V$ and $\maxpoint_\cellgen$ ($\cellgen\in\cellgens(\combcell)$). Since $\cellgen\subsetneq V$, we get $\maxpoint_\cellgen\gneqq\maxpoint_V$. Define $I_G(\ell^*_V)\subset\grverts$ as the collection of those indices $i\in \grverts$ such that $\maxpoint_V<\maxpoint_V+E_{i}\leq\maxpoint_\cellgen$. Then $I_G(\ell^*_V)\not=\emptyset$. Moreover, since ${\rm codim}(G\subset V)=1$, we have $\filt(\maxpoint_V+E_{i})=\cellgen$ for any $i\in I_G(\ell^*_V)$. (Note that $I_G(\ell^*_V)$ can be defined also as the set of indices $i\in\grverts$ such that $\filt(\maxpoint_V+E_{i})=\cellgen$, since in this case $\maxpoint_V<\maxpoint_V+E_{i}\leq\maxpoint_\cellgen$ also holds.) Now for any collection of non-empty subsets $J_G(\ell^*_V)\subset I_G(\ell^*_V)$ (indexed by $G\in\cellgens(\combcell)$) set

\[
\cell=(\maxpoint_V,\cup_{\cellgen\in\cellgens(\combcell)}J_G(\ell^*_V)).
\]

Clearly $\phi(\cell)=\combcell$. Moreover, if each $J_G(\ell^*_V)$ contains exactly one element, say $i_G$, then all the indices $i_\cellgen$ must be distinct (since the spaces $\filt(\maxpoint_V+E_{i_\cellgen})=\cellgen$ are distinct). As such, the dimension of $\cell$ is exactly $q=|\cellgens(\combcell)|$.

Next, we wish to determine all the representatives $C$ of a fixed $\combcell=\combcell(V,\cellgens)$, $\cellgens\not=\emptyset$. Clearly, if $C=(\ell, I)\in \phi^{-1}(\combcell)$, then $\filt(\ell)=V$. Moreover, for any $G\in\cellgens$ there must exist $i\in I$ such that $\filt(\ell+E_i)=G$, and for any $i\in I$ the space $\filt(\ell+E_i)$ either belongs to $\cellgens$ or it is $V$.

Let us fix $\ell$ with $\filt(\ell)=V$. Let $\phi^{-1}(\combcell)_{\ell}$ be the set of cells of type $(\ell, I)\in\phi^{-1}(\combcell)$. For any $G\in \cellgens$ set $I_G(\ell):= \setdef{i\in \grverts}{\filt(\ell+E_i)=G}$, and write $I^*(\ell)$ for the support $\abs{\ell^*_V-\ell}=\setdef{i}{E_i\leq \ell^*_V-\ell}$. Then $C=(\ell,I)\in\phi^{-1}(\combcell)_{\ell}$ if and only if $I\cap I_G(\ell)\neq\emptyset$ and $I\subset I^*(\ell)\cup(\cup_G I_G(\ell))$.

Note that the sets $\setof{I_G(\ell)}_G$ are all pairwise disjoint. Indeed, if $i\in I_G(\ell)\cap I_{G'}(\ell)$ then $G=\filt(\ell+E_i)=G'$. Furthermore, they are all determined by $I_G(\ell^*_V)$ (introduced in the proof of Lemma lattice-to-sscw) as follows.

$I_G(\ell)=I_G(\ell^*_V)\setminus\abs{\ell^*_V-\ell}$ for any $G\in\cellgens$.

If $i\in I_G(\ell)$ then $\filt(\ell+E_i)=G$, hence $i\not\in\abs{\ell^*_V-\ell}$ (otherwise, $\ell+E_i\leq \ell^*_V$, which would imply $\filt(\ell+E_i)=V$). But then $(*)$ $\filt(\ell^*_V+E_i)=\filt(\ell+E_i)\cap \filt(\ell^*_V)=G\cap V=G$.

If $i\in I_G(\ell^*_V)\setminus\abs{\ell^*_V-\ell}$, then write $W:=\filt(\ell+E_i)$. Then $W\subset\filt(\ell)=V$. But $W=V$ cannot hold since that would imply $\ell+E_i\leq \ell^*_V$, or $i\in\abs{\ell^*_V-\ell}$. Hence $W$ has codimension 1 in $V$. But then, again from $(*)$, $W\cap V=G$, hence $W=G$.

Fix $\combcell=\combcell(V,\cellgens)$ with $\cellgens\not=\emptyset$. Then the following facts hold:

* $\phi^{-1}(\combcell)_{\ell}\not=\emptyset \ \Leftrightarrow\ I_G(\ell^*_V)\setminus|\ell^*_V-\ell| \not=\emptyset$ for any $G\in\cellgens$.

* $I_G(\ell)\subset I_G(\bar{\ell})$ for any $\bar{\ell}$ with $\ell\leq \bar{\ell}\leq \ell^*_V$.

* $\phi^{-1}(\combcell)_{\ell}\not=\emptyset \ \Rightarrow\ \phi^{-1}(\combcell)_{\bar{\ell}}\not=\emptyset$ for any $\bar{\ell}$ with $\ell\leq \bar{\ell}\leq \ell^*_V$.

* If $(\ell, I)\in \phi^{-1}(\combcell)$ and $\ell\leq \bar{\ell}\leq \ell^*_V$ then $(\bar{\ell},J)\in \phi^{-1}(\combcell) $ for any $J$ with $I\subset J\subset I\cup|\ell^*_V-\bar{\ell}|$.

* $(\ell, I)\in \phi^{-1}(\combcell)$ if and only if $\filt(\ell)=V$ and $I\subset |\ell^*_V-\ell|\cup (\cup_GI_G(\ell^*_V)) $, with $I\cap(I_G(\ell^*_V)\setminus |\ell^*_V-\ell|)\not=\emptyset$ for any $G$.

The above statements imply the following facts as well.

* Any cell $\cell\in \phi^{-1}(\combcell)$ (a not necessarily minimal representative of $\combcell$) contains a minimal representative as a face (not necessarily of codimension $1$).

* Given an $\Im\filt$-cell $\combcell$ and any representative $\cell$ of $\combcell$, we have

\[
\setdef{\phi(\cellface)}{\text{$\cellface$ is a face of $\cell$}}=\setdef*[\big]{\combcellface\subseteq\combcell}{\combcellface\text{ is an $\Im\filt$-cell}}.
\]

In particular, the left hand side is independent of the choice of $\cell$.

Next we compare the weights appearing at different levels. Firstly, the CW complex $\cw=R(0,c)$ ($c$ finite or infinite) has a weight function defined as $w_0(\ell)=2\hh(\ell)-\abs{\ell}$ for any lattice point $\ell$, and $w_q(\topcell)=\max\setdef{w_0(\ell)}{\text{$\ell$ vertex of $\topcell$}}$.

On the other hand, on $\spsystem=\Im\filt$ associated with $(C,o)$, we set the following weights. First note that $V$ determines uniquely $\maxpoint_V$ (it is maximal with $V=\filt(\maxpoint_V)$). Then, we set $w_{\spsystem}(V):= w_0(\maxpoint_V)$.

Let us verify that $(\spsystem, w_{\spsystem})$ defines a weighted system of vector spaces, i.e. $ w_{\spsystem}$ satisfies eq:weightfun. Indeed, for any $V,W\in\Im\filt$ with $W\subseteq V$, we have $\filt\bigl(\max\setof{\maxpoint_V,\maxpoint_W}\bigr)=W$, so $\maxpoint_W\geq\maxpoint_V$. Following an increasing path in the lattice from $\maxpoint_V$ to $\maxpoint_W$, the value of $\filt$ can only decrease in codimension-$1$ steps, and $w_0(\ell)$ increases by $1$ at each such step (and otherwise decreases by $1$).

Next, by Definition V-cell-weight, for any combinatorial or topological cell of $\spsystem$ or $\ti\cw(\spsystem)$ respectively, we have $\ti w_{\spsystem}(\topcell(\combcell))=w_{\spsystem}(R(\combcell))+\height\combcell$. This defines a set of compatible weight functions on $\ti\cw(\spsystem)$, or on the set of combinatorial cells of $\spsystem$.

It is worth to emphasize that $\ti w_{\spsystem}(\combcell)=w_{\spsystem}(R(\combcell))+\height\combcell$ defined in this way is not the maximum of the $\ti w_{\spsystem}$-weights of the vertices of $\combcell$. Consider e.g. the situation from Example ex:delta2, the segment $[0,3]$: $\filt(0)=\fnring$, $\hh(0)=0$, $\maxpoint_{\fnring}=0$, $\filt(3)=\mathfrak{m}$ (the maximal ideal), $\hh(3)=1$, $\maxpoint_{\mathfrak{m}}=3$. The segment $[0,1]$ is sent to a 1-cell $\combcell$ in $\ti\cw$ with end-vertices $\maxpoint_{\fnring}$ and $\maxpoint_{\mathfrak{m}}$ (and the segment $[1,3]$ to $\maxpoint_{\mathfrak{m}}$). Moreover, $\ti w(\fnring)=0$, $\ti w(\mathfrak{m})=-1$, but $\ti w(\combcell)=\ti w(\fnring)+\codim (\mathfrak{m}\subset \fnring)=1$.

The next statement connects the set of weight functions of $(\cw,w)$ and $(\ti\cw(\spsystem), \ti w_{\spsystem})=(\ti\cw, \ti w)$ in the spirit of the comparisons of statements from lattice-to-sscw–V-cell-faces.

Let $V$ and $W$ be the root and tip of $\combcell\in\ti\cellset_q$.
Then the following facts hold:

* For any representative $\cell=(\ell,I)$ of $\combcell$, we have

\[
w(\cell)=w(\ell)+\height\combcell= w(\ell)+{\rm codim}(W\subset V).
\]

\[
\min\setdef*[\big]{w(\cell)}{\cell\in\cellset_*, \phi(\cell)=\combcell}=
w(\maxpoint_V)+\height\combcell= \ti w_{\spsystem}(\combcell).
\]

(a) Take a sequence of lattice points $\setof{x_i}_{i=0}^t$ such that $x_0=\ell$, $x_t=\ell+E_I$, and for any $i$ one has $x_{i+1}=x_i+E_{v(i)}$ for a certain $v(i)$. Hence $\filt(x_0)=V$ and $\filt(x_t)=W$. Then $w(x_{i+1})-w(x_i)\in\setof{1,-1}$, and it is 1 exactly when ${\rm codim}(\filt(x_{i+1})\subset \filt(x_i))=1$. Hence $w(x_i)\leq w(\ell)+{\rm codim}(W\subset V)$ for any $x_i$. Since any vertex of $C$ appears as such an $x_i$ we get $w(C)\leq w(\ell)+{\rm codim}(W\subset V)$.

Next, we analyse three consecutive points $x_{i-1}, \ x_i,\ x_{i+1}$. Assume that $w(x_{i})-w(x_{i-1})=-1$ and $w(x_{i+1})-w(x_i)=1$. Then if we replace $x_i$ by $\bar{x}_i=x_{i-1}+E_{v(i)}$, by the matroid inequality we will have $w(\bar{x}_i)+w(x_i)\geq w(x_{i-1})+w(x_{i+1})$. Hence if we replace the path $x_{i-1}, \ x_i,\ x_{i+1}$ by $x_{i-1}, \ \bar{x}_i,\ x_{i+1}$, the new one will satisfy $w(\bar{x}_{i})-w(x_{i-1})=1$ and $w(x_{i+1})-w(\bar{x}_i)=-1$. That is, in the sequence of $w$-jumps, the sequence $-1$, 1 is replaced by 1, $-1$. In particular there exists a sequence $\setof*{x_i}_i$ such that we have all the 1's at the first $k={\rm codim}(W\subset V)$ steps. In particular, $w(x_k)=w(\ell)+k$, hence $w(C)\geq w(\ell)+{ \rm codim}(W\subset V)$.

(b) Assume that $\phi(C)=\combcell$, $C=(\ell, I)$. Then $\filt(\ell)=V$, hence $\ell\leq \maxpoint_V$. Hence $w(\ell)\geq w(\maxpoint_V)$ and $w(C)\geq \ti w(\combcell)$. On the other hand, by the proof of part (b) of lattice-to-sscw, there exists some $C=(\maxpoint_V,I)$ with $\phi(C)=\combcell$, which realizes the equality $w(C)=\ti w(\combcell)$.

§.§ The weighted homotopy equivalence of (X,w) and (X,w)

For an isolated curve singularity $(C,o)$ we considered two weighted CW complexes. The first one is $(\cw,w)=(R(0,c),w)$ ($c\geq c_{\semigp}$), the other one in $(\ti\cw,\ti w)$ associated with $\Im\filt$. In this section we prove that the induced lattice cohomologies and graded roots are canonically isomorphic. To this end, we will find a mapping that induces a homotopy equivalence on all level sets $\setof*{S_n}_n$ and $\setof*{\ti S_n}_n$ in the two complexes.

Note that any point of $\cw$ belongs to a unique (relative) open cell $(\ell, I)^\circ$.

We define $\alpha\colon \cw\to\ti\cw$ as follows. Fix a cell $\topcell=C=(\ell, I)\in \cellset_*$, and set $\combcell=\combcell(V, \cellgens):=\phi(C)$. Regard $\topcell$ embedded in $(\bbr_{\geq 0})^r$ with its natural coordinates. Then the restriction of $\alpha$ to $C^\circ=\topcell^\circ$ is the composition $\kappa_{\combcell}\circ \beta_{C^\circ}$, where $\beta_{C^\circ}:\topcell^\circ\to \topcell^\circ (\combcell(V,\cellgens))$ is given by

\begin{equation}\label{eq:alpha}
\beta_{C^\circ}(p)=\Bigl(V,\bigl(\cellgens\ni\cellgen\mapsto\max\setdef{p_i-\ell_i}{i\in I, \filt(\ell+E_i)=\cellgen}\bigr)\Bigr)\in
\topcell^\circ(\combcell(V,\cellgens)).
\end{equation}

For all level sets $\levelset_n\subseteq\cw$ and $\ti\levelset_n\subseteq\ti\cw$ $(n\in \bbz$), we have

\begin{equation}\label{eq:Imalpha}
\Im\bigl(\alpha|_{\levelset_n}\bigr)\subseteq\ti\levelset_n,
\end{equation}

and $\alpha|_{\levelset_n}\colon\levelset_n\to\ti\levelset_n$ is a homotopy equivalence.

The proof consists of several steps. We start with the following.

The map $\alpha$ is continuous.

It is sufficient to prove that $\alpha|_\cell$ is continuous for any closed cell $\cell=(\ell,I)$ in $\cw$. We prove this by induction on $\dim\cell=\abs{I}$. For $\dim\cell=0$, there is nothing to prove. Otherwise, assume that we know this continuity whenever $\dim\cell<q$ for some integer $q>0$. Then fix some $C=(\ell, I)$ of dimension $q$. At a point $p\in(\ell, I)^\circ$, a small neighborhood $U$ is mapped into the open cell $\kappa_{\combcell}(\topcell^\circ (\combcell(V,\cellgens)))$, so $\alpha(x)=\kappa_{\combcell}(V,u)$ for all $x\in U$, with $u\in(0,1)^\cellgens$ defined as $u(\cellgen)$ being the maximum in the [text=definition]CW-hom-equiv-def. This is clearly continuous in $x$. If $p\in\D\cell$, then by the induction hypothesis, $\alpha|_\cellface$ is continuous at $p$ for all proper faces $\cellface\subseteq\cell$ containing $p$, hence $\alpha|_{\D\cell}$ is continuous at $p$.

We then only need to verify the following fact. Let $\bar{\alpha}:C\to \ti\cw$ be the natural continuous extension of $\alpha|_{C^\circ}:C^\circ \to \ti\cw$ to $C$, that is, $\bar{\alpha}=\kappa_{\combcell}\circ \beta_{C}$, where $\beta_{C}:C\to \topcell (\combcell(V,\cellgens))$ is given by the very same formula as in (<ref>) as a continuous extension of $\beta_{C^\circ}$. Since both $\beta_{C}$ and $\kappa_{\combcell}$ are continuous, $\bar{\alpha}$ is continuous. Then we have to show that $\bar{\alpha}|_{\partial C}=\alpha|_{\partial C}$.

Fix $p\in \D C$. Using the notations

\[
I_0=\setdef{i\in I}{p_i-\ell_i=0}, \quad
I_1=\setdef{i\in I}{p_i-\ell_i=1}, \quad
I_*=\setdef{i\in I}{0<p_i-\ell_i<1},
\]

we have $p\in D^\circ $ for $D=(\ell+E_{I_1},I_*)$. Let us also introduce

\[
\cellgens_1=\setdef{\filt(\ell+E_i)}{i\in I_1}, \quad
\ti\cellgens_*=\setdef{\filt(\ell+E_{I_1}+E_i)}{i\in I_*}\setminus\setof{\filt(\ell+E_{I_1})}.
\]

In a small neighborhood $U$ of $p$, we again have $\alpha(x)=\kappa_{\combcell}(V,u_x)$ for all $x\in U\cap C^\circ $ where $u_x$ is defined as the maximum from above. Thus,

\[
u_p=\lim_{\substack{x\to p \\ x\in\ C^\circ}}u_x=\bigl(\cellgens\ni\cellgen\mapsto\max\setdef{p_i-\ell_i}{i\in I,\filt(\ell+E_i)=\cellgen}\bigr),
\]

corresponding to the point $\beta_{C}(p)=(V,u_p)\in\topcell(\combcell(V,\cellgens))$. Hence $\bar{\alpha}(p)=\kappa_{\combcell}(V,u_p)$. Meanwhile, by its very definition, $\alpha(p)$ is obtained by evaluating $\kappa_{\combcell_{\ti D}}$ at $\beta_{D^\circ}(p)$, namely at

\[
\Bigl(V\cap(\cap_{W\in \cellgens_1}W),\bigl(\ti\cellgens_*\ni\cellgen\to\max\setdef{p_i-\ell_i}{\filt(\ell+E_{I_1}+E_i)=\cellgen}\bigr)\Bigr).
\]

But $\beta_{D^\circ}(p)=\delta_{\ti D}(\beta_{C}|_{D_0}(p))$ (for the definition of $\delta_{\ti D}$, $\combcell_{\ti D}$ and for the identity $\kappa_{\ti C_{\ti D}}\circ \delta_{\ti D}=\kappa_{\ti C}|_{\ti D}$ see the proof of Proposition gluing-cont), hence $\bar{\alpha}(p)=\alpha(p)$ follows from the next commutative diagram

(100,100)(0,0)$\ti D\ (\mbox{as face of $\topcell(\combcell)$})$
(290,100)(0,0)$\topcell^\circ(\combcell_{\ti D})$
(150,50)(0,0)$\kappa_{\combcell}|_{\ti D}$
(250,50)(0,0)$\kappa_{\combcell_{\ti D}}$
(335,160)(0,0)$\alpha|_{D^\circ}=\kappa_{\combcell_{\ti D}}\circ \beta_{D^\circ}$
(50,160)(0,0)$\bar{\alpha}|_{D^\circ}=\kappa_{\combcell}|_{\ti D}\circ \beta_{C}|_{D^\circ}$
(200,110)(0,0)$\delta_{\ti D}$

Next, we verify the inclusion (<ref>).

Fix $C=(\ell, I)$. If $\phi(C)=\combcell$, then by the continuity of $\alpha$ we also have $\alpha((\ell, I))\subset \topcell(\combcell)$. But, $\ti w(\combcell)\leq w(C)$ by Proposition Im-F-cell-weight. Hence, if $C\subset S_n$ then $\topcell(\combcell)\subset\ti S_n$ and $\alpha(C)\subset\topcell(\combcell)\subset \ti S_n$.

Next, we show that $\alpha|_{\levelset_n}:S_n\to \ti S_n$ is a homotopy equivalence. We do this by checking that $\alpha|_{\levelset_n}$ is onto $\ti\levelset_n$, and the fiber $\alpha^{-1}(\ti p)\cap\levelset_n$ is contractible for all $\ti p\in\ti S_n$ and $n\in\bbz$. Since any $\ti p\in \ti S_n$ has a small neighbourhood $U$ in $\ti S_n$ such that the inclusion $(\alpha|_{S_n})^{-1}(\ti p)\hookrightarrow(\alpha|_{S_n})^{-1}(U)$ is a homotopy equivalence, the map $\alpha|_{S_n}$ is a quasifibration in the sense of Dold and Thom <cit.>, hence the contractibility of the fibers imply the homotopy equivalence.

First note that if $\phi(C)=\combcell$ then $\alpha$ maps the cube $C$ surjectively onto $\topcell(\combcell)$. Therefore, if $\ti p\in \ti S_n$ belongs to $\topcell^\circ (\combcell)$, then from Corollary cor-felosrolas and Proposition Im-F-cell-weight follows that there exists $C$ with $\phi(C)=\combcell$ and $w(C)=\ti w(\combcell)$. Hence $\topcell(\combcell)$ (and $\ti p$ too) belongs to the image of $\alpha|_{S_n}$.

Next we fix $\ti p\in \ti S_n\cap \topcell^\circ (\combcell)$ and we wish to describe the fiber $\alpha^{-1}(\ti p)\cap S_n$. Set $V:=R(\combcell)$.

If $p\in\alpha^{-1}(\ti p)\cap S_n $ and $p\in (\ell, I)^\circ$ then $\phi((\ell,I))=\combcell$, $\alpha((\ell,I))= \topcell(\combcell)$, and $w((\ell, I))\leq n$. Then $\ell\in S_n(V)$ (cf. <ref>) and, in fact, (using the notation $T^*(\ell)=|\maxpoint_V-\ell|$ from the discussions in front of Lemma lemma-ig) $(\ell, I^*(\ell))\subset S_n(V)$. Let $S_n(V,\combcell)$ be the subspace of $S_n(V)$ consisting of cubes of type $(\ell, J)$ such that $J\subset I^*(\ell)$ and there exists a certain cube $(\ell, I)\in\phi^{-1}(\combcell)$ with $J\subset I$ and $w((\ell, I))\leq n$. From Proposition Im-F-cell-weight follows that if $\ell\in S_n(V,\combcell)$ then the cube $(\ell, I^*(\ell))\subset S_n(V,\combcell)$, hence the whole real rectangle $R(\ell, \maxpoint_V)$ belongs to $S_n(V, \combcell)$. In particular, the space $S_n(V,\combcell)$ is contractible.

Then consider the projection $pr:\phi^{-1}(\combcell)\cap S_n\to S_n(V, \combcell)$ induced by the natural coordinate-wise projections $(\ell, I)\to (\ell, J)$, where $J=I\cap I^*(\ell)$. One verifies that these `local cube projections' glue together into a continuous map. In particular, the composition $\alpha^{-1}(\ti p)\cap S_n\hookrightarrow \phi^{-1}(\combcell)\cap S_n\stackrel{pr}{\longrightarrow} S_n(V, \combcell)$, denoted by $pr(\ti p)$, is also continuous. But, by a local study at the level of cubes shows that each fiber of $pr(\ti p)$ is contractible. Since $pr(\ti p)$ is a quasifibration with contractible fibers and contractible base space $S_n(V, \combcell)$, its total space $(\alpha|_{S_n})^{-1}(\ti p)=\alpha^{-1}(\ti p)\cap S_n$ is contractible too. But this fact is what we wish to proof.

This finishes the proof of Theorem CW-hom-equiv.

For the above weighted CW complexes, we have an isomorphism

\[
\lcoh^*(\cw,w)\simeq\lcoh^*(\ol\cw,\ol w), \quad
{\Got R}(\cw,w)\simeq {\Got R}(\ol\cw,\ol w)
\]

§.§ The deformation map

Let $\setof{(C_t,o)}_{t\in(\bbc,0)}$ be a flat deformation of isolated curve singularities. We assume that $(C_t,o)\subset (\bbc^N,o)$ for any $t$, and we consider the ideal filtration $\widetilde{ \filt}_t $ of $\fnring_N=\fnring_{(\bbc^N,0)} $ as in <ref>. We can assume that for $t\not=0$ the lattice $L(C_t,o)=\bbz^{r_t}$ of $(C_t,o)$ is constant, and for fixed $\ell\in L(C_t,o) $ and $t\not=0$ the Hilbert function $\hh_{t}(\ell)$ is constant. Hence, the corresponding weighted CW complex $\ti \cw_{t\not=0}$ is constant as well. Let $\ti \cw_{t=0}$ be the weighted complex of $(C_0,o)$ too.

First, we wish to define a map $\sk_0\ti\cw_{t\not=0}\to \sk_0\ti\cw_{t=0}$. Recall that if $\ti\cw$ is the complex associated with a curve $(C,o)$ then its 0-skeleton $\sk_0\ti\cw$ is in bijection with the set of lattice points of type $\maxpoint _V\in L(C,o)$, and these correspond precisely to the semigroup elements of $(C,o)$, cf. Lemma ss-set-props.

Let us fix such a lattice point $\maxpoint=\maxpoint_V\in L(C_{t\not=0})$ (or, differently said, $\maxpoint \in \sk_0\ti\cw_{t\not=0}$).

For each $t\not=0$ consider the $\bbc$-subvector space $\widetilde{\filt}_{t\not=0}(\maxpoint)\subset\fnring_N $ of finite and constant codimension. This family considered in a convenient Grassmannian has a limit $\widetilde{\filt}_{lim}(\maxpoint)$, where $\widetilde{\filt}_{lim}(\maxpoint)\subset \fnring_N$ is a subvector space of the same codimension $\hh_{t\not=0}(\maxpoint)$. In fact, the above infinite Grassmannian can be replaced by a finite Grassmannian as follows. Let $J$ be an ideal of $\fnring_N$ (e.g. a power $\mathfrak{m}_N^d$ of the maximal ideal with sufficiently large $d$) such that $J\subset \widetilde{\filt}_{t\not=0}(\maxpoint)$ for any $t\not=0$. Then $\widetilde{\filt}_{t\not=0}(\maxpoint)/J\subset \fnring_N/J$ has a limit in the compact Grassmannian, which can be lifted as $\widetilde{\filt}_{lim}(\maxpoint)$ into $ \fnring_N$ via $\fnring_N\to \fnring_N/J$.

Even more, by the Curve Selection Lemma (see e.g. <cit.>), applied in this finite model, for a generic element $g\in \widetilde{\filt}_{lim}(\maxpoint)$ (regarded as a germ of $\fnring_N$), there exists a family of germs $\setof*{g_t}_{t\in (\bbc,0)}$ such that $g_0=g$ and $g_t\in\widetilde{\filt}_{t}(\maxpoint)$ for $t\not=0$, and $g_t$ is generic in $\widetilde{\filt}_{t}(\maxpoint)$.

For $t\neq 0$, since $\maxpoint\in\semigp_{(C_t,o)}$ and $g_t$ is generic in $\widetilde{\filt}_{t}(\maxpoint)$, for its class $[g_t]\in\filt_t(\maxpoint)\subset\fnring_{(C_t,o)}$ we get

\begin{equation}\label{eq:vege1}
\Got v_{(C_t,o)}[g_t]=\maxpoint \in\sk_0\ti\cw_{t\not=0}=\semigp_{(C_t,o)}.
\end{equation}

Let us define $\maxpoint_{lim}$ as $\Got v_{(C_0,o)}[g]\in \sk_0\ti\cw_{t=0}=\semigp_{(C_0,o)}$.

The main result of this subsection is the following.

Let $\setof{(C_t,o)}_{t\in(\bbc,0)}$ be a flat deformation of isolated curve singularities. Then the map $\sk_0\ti\cw_{t\not=0}\to \sk_0\ti\cw_{t=0}$, $\ell^*\mapsto \ell^*_{lim}$, induces a degree zero graded $\bbz[U]$-module morphism $\lcoh^0(C_{t=0},o)\to \lcoh^0(C_{t\not=0},o)$, and similarly a graded (graph) map of degree zero at the level of graded roots ${\Got R}(C_{t\not=0},o)\to{\Got R}(C_{t=0},o)$.

Note that $|\maxpoint|$ (respectively $|\maxpoint_{lim}|$) is the intersection multiplicity in $(\bbc^N,o)$ of $(C_t,o)$ with $\setof{g_t=0}$, $t\neq 0$, (respectively of $(C_0,o)$ with $\setof{g=0}$). Therefore, by semicontinuity,

\begin{equation}\label{eq:vege2}
\abs{\maxpoint}=(C_t, g_t)_{o\in\bbc^N}\leq (C_0, g_0)_{o\in\bbc^N}=\abs{\maxpoint_{lim}}.
\end{equation}

* $\widetilde{\filt}_{lim}(\maxpoint)\subset \widetilde{\filt}_{t=0}(\maxpoint_{lim})$.

* $\widetilde{\filt}_{t=0}(\maxpoint_{lim})$ is the smallest subspace of type $\widetilde{\filt}_{t=0}(\ti \ell)\subset \fnring $ ($ \ti \ell\in L(C_{t=0},o)$) with $\widetilde{\filt}_{lim}(\maxpoint)\subset \widetilde{\filt}_{t=0}(\ti \ell)$.

(a) For the generic element $g_0\in \widetilde{\filt}_{lim}(\maxpoint)$ we have $\Got v_{(C_0,o)}[g_0]=\maxpoint_{lim}$, hence by semicontinuity, for any $f\in \widetilde{\filt}_{lim}(\maxpoint)$ we have $\Got v_{(C_0,o)}[f]\geq \maxpoint_{lim}$.

(b) $\widetilde{\filt}_{lim}(\maxpoint)\subset \widetilde{\filt}_{t=0}(\ti \ell)$ implies $\Got v_{(C_0,o)}[g_0]=\maxpoint_{lim}\geq \ti \ell$. Hence $\widetilde{\filt}_{t=0}(\maxpoint_{lim})\subset \widetilde{\filt}_{t=0}(\ti \ell)$.

In particular, we also have for the codimensions in $\fnring_N$

\begin{equation}\label{eq:vege3}
\hh_{t\not=0}(\maxpoint)={\rm codim}\widetilde{\filt}_{t\not=0}(\maxpoint)=
{\rm codim}\widetilde{\filt}_{lim}(\maxpoint)\geq
{\rm codim}\widetilde{\filt}_{t=0}(\maxpoint_{lim})= \hh_{t=0}(\maxpoint_{lim}).
\end{equation}

Let us introduce the notations $d:=|\maxpoint_{lim}|-|\maxpoint|\in \bbz_{\geq 0}$ (cf. (<ref>)) and

\[
e:= \hh_{t\not=0}(\maxpoint)- \hh_{t=0}(\maxpoint_{lim})={\rm codim } (\widetilde{\filt}_{lim}(\maxpoint)\subset \widetilde{\filt}_{t=0}(\maxpoint_{lim}))\in \bbz_{\geq 0}.
\]

Then the identities (<ref>) and (<ref>) combined give

\begin{equation}\label{eq:vege4}
w_{t\not=0}(\maxpoint)= w_{t=0}(\maxpoint_{lim})+2e+d\geq w_{t=0}(\maxpoint_{lim}).
\end{equation}

Therefore, for any $n\in \bbz$, $\maxpoint\mapsto \maxpoint_{lim}$ maps $\sk_0\ti\cw_{t\not=0}\cap \ti S_n$ to $\sk_0\ti\cw_{t=0}\cap \ti S_n$.

Let us consider next an 1-cell $(\maxpoint, \setof{i})\in\cw_{t\neq 0}$, where $E_i$ is a base vector of $L(C_{t\neq 0})$. It induces the 1-cell $\combcell$ with endpoints $\maxpoint$ and $(\maxpoint+E_i)^*$, where $(\maxpoint+E_i)^*$ is associated with $\maxpoint+E_i$ as in Lemma uniquemax_0 (it is the smallest semigroup element larger than $\maxpoint+E_i$). By the above map $\maxpoint $ is sent to $\maxpoint_{lim}$ and $(\maxpoint+E_i)^* $ to $(\maxpoint+E_i)_{lim}$. Moreover, by the definition from subsection <ref> we have

\begin{equation}\label{eq:vege5}
w_{t\not=0}(\combcell)= w_{t\not =0}(\maxpoint)+1.
\end{equation}

Note that all the 1-cells of $\ti\cw_{t\not=0}$ are of this type

For any $n\in\bbz$ we will prove the following fact. If $\maxpoint$ and $(\maxpoint+E_i)^*$ and $\kappa (\topcell(\combcell))$ belong to $\ti\cw_{t\not=0}\cap \ti S_n$, then $\maxpoint_{lim}$ and $(\maxpoint+E_i)_{lim}$ sit in the same connected component of $\ti\cw_{t=0}\cap \ti S_n$.

Indeed, from the assumption we have

\begin{equation}\label{eq:vege6}
w_{t\not=0}(\maxpoint )+1\leq n.
\end{equation}

From Lemma lem_F(b) we obtain that $\widetilde{\filt}_{t=0}((\maxpoint+E_i)_{lim})\subset \widetilde{\filt}_{t=0}(\maxpoint_{lim})$, hence $\maxpoint_{lim}\leq (\maxpoint+E_i)_{lim}$ too. In particular, we can consider the inclusions

\[
\widetilde{\filt}_{lim}((\maxpoint+E_i)^*)\subset
\widetilde{\filt}_{t=0}((\maxpoint+E_i)_{lim})\subset
\widetilde{\filt}_{t=0}(\maxpoint_{lim}).
\]

\[
\codim(\widetilde{\filt}_{t=0}((\maxpoint+E_i)_{lim})\subset \widetilde{\filt}_{t=0}(\maxpoint_{lim}))\leq
\codim(\widetilde{\filt}_{lim }(((\maxpoint+E_i)^*)\subset \widetilde{\filt}_{t=0}(\maxpoint_{lim}))
\]

where the right hand side equals

\[
\codim(\widetilde{\filt}_{lim }(((\maxpoint+E_i)^*)\subset\widetilde{\filt}_{lim }(\maxpoint))+
\codim(\widetilde{\filt}_{lim }(\maxpoint))\subset \widetilde{\filt}_{t=0}(\maxpoint_{lim}))
\]

Thus, we have two points $\maxpoint_{lim}$ and $(\maxpoint+E_i)_{lim}$ in $\ti S_n$ such that $w_{t=0}(\maxpoint_{lim})+2e+d\leq n-1$ and ${\rm codim}(\widetilde{\filt}_{t=0}((\maxpoint+E_i)_{lim})\subset \widetilde{\filt}_{t=0}(\maxpoint_{lim}))\leq 1+e$. Then for any increasing path in $\cw_{t=0}$ which connects $\maxpoint_{lim}$ and $(\maxpoint+E_i)_{lim}$ we have the following facts: the length of the path is $\leq 1+e$, and at any consecutive step along the path $w_{t=0}$ can increase maximum by 1. Hence along any such path $w_{t=0}$ is $\leq w_{t=0}(\maxpoint_{lim})+e+1$. Since

\[
w_{t=0}(\maxpoint_{lim})+e+1\leq n-1-2e-d+e+1=n-e-d\leq n
\]

we obtain that $\maxpoint_{lim}$ and $(\maxpoint+E_i)_{lim}$ belong to the same connected component of $\ti\cw_{t=0}\cap \ti S_n$.

Summarized, $\maxpoint\mapsto \maxpoint_{lim}$ induces a well-defined map $\pi_0(\ti\cw_{t\not=0}\cap \ti S_n)\to \pi_0(\ti\cw_{t=0}\cap \ti S_n)$ for any $n$, and they are compatible with respect to the inclusions $\ti S_n\hookrightarrow\ti S_{n+1}$.

Usually it can happen that $\widetilde{\filt}_{lim}(\maxpoint)\not= \widetilde{\filt}_{t=0}(\maxpoint_{lim})$. It is also possible that $\maxpoint$ and $(\maxpoint+E_i)^*$ are not the endpoints of an 1-cell of $\ti\cw_{t=0}$.

Indeed, consider the deformation of isolated irreducible plane curves $\setof{x^2+ty^3+y^5=0}_{t\in(\bbc,0)}$. For $t\neq 0$ and $\ell=0,1,\ldots,6$ we have the following semigroup elements and ($t$-independent) ideals $\widetilde{\filt}_{t\not=0}(\ell)$, which equal the limits $\widetilde{\filt}_{lim}(\ell)$ as well:

[fill] (0,0) circle [radius=0.04];
(1,0) circle [radius=0.04];
[fill] (2,0) circle [radius=0.04];
[fill] (3,0) circle [radius=0.04];
[fill] (4,0) circle [radius=0.04];
[fill] (5,0) circle [radius=0.04];
[fill] (6,0) circle [radius=0.04];

(0,0) – (6,0);

at (0,-0.4) $\fnring_N$;
at (1,-0.4) $\mathfrak{m}_N$;
at (2,-0.4) $\mathfrak{m}_N$;
at (3,-0.4) $(x,y^2)$;
at (4,-0.4) $\mathfrak{m}_N^2$;
at (5,-0.4) $(x^2,xy,y^3)$;
at (6,-0.4) $(x^2,xy^2,y^3)$;

For $t=0$ the semigroup elements and ideals $\widetilde{\filt}_{t=0}(\ell)$ are:

[fill] (0,0) circle [radius=0.04];
(1,0) circle [radius=0.04];
[fill] (2,0) circle [radius=0.04];
(3,0) circle [radius=0.04];
[fill] (4,0) circle [radius=0.04];
[fill] (5,0) circle [radius=0.04];
[fill] (6,0) circle [radius=0.04];

(0,0) – (6,0);

at (0,-0.4) $\fnring_N$;
at (1,-0.4) $\mathfrak{m}_N$;
at (2,-0.4) $\mathfrak{m}_N$;
at (3,-0.4) $(x,y^2)$;
at (4,-0.4) $(x,y^2)$;
at (5,-0.4) $(x,y^3)$;
at (6,-0.4) $(x^2,xy,y^3)$;

Hence, $4_{lim}=4$ but $\widetilde{\filt}_{lim}(4)=\mathfrak{m}^2\varsubsetneq (x,y^2)= \widetilde{\filt}_{t=0}(4_{lim})$. Moreover, $5_{lim}=6$, hence $4_{lim}$ and $5_{lim}$ are not endpoints of an 1-cell of $\ti\cw_{t=0}$.

If $(C_{t=0},o)$ is irreducible then we have constructed two maps $\pi_0(\ti\cw_{t\not=0}\cap \ti S_n)\to\pi_0(\ti\cw_{t=0}\cap \ti S_n)$. The first one, considered for plane curves, is induced by $\maxpoint\mapsto |\maxpoint|$, and in Theorem th:DEF we probed that it induces a graded graph map of degree zero at the level of graded roots ${\Got R}(C_{t\not=0},o)\to{\Got R}(C_{t=0},o)$. In fact, in Remark rem:GN we noticed that the statement can be extended to non-plane curves as well. (This fact will follow from the computations from below.) The second map, constructed in Theorem th:DEF2, is induced by $\maxpoint\mapsto \maxpoint_{lim}$. Recall that by (<ref>) $|\maxpoint|\leq \maxpoint_{lim}$.

Let us consider any lattice point $ \ell\in L(C_{t=0})=\bbz$ with $|\maxpoint|\leq \ell\leq\maxpoint_{lim}$. It satisfies

\[
\widetilde{\filt}_{t=0}(|\maxpoint|)\supset
\widetilde{\filt}_{t=0}( \ell)\supset
\widetilde{\filt}_{t=0}(\maxpoint_{lim})\supset
\widetilde{\filt}_{lim}(\maxpoint).
\]

\begin{equation}\label{eq:hdefdef}
\hh_{t=0}(|\maxpoint|)\leq
\hh_{t=0}(\ell)\leq
\hh_{t=0}(\maxpoint_{lim})\leq
\hh_{t\not=0}(\maxpoint).
\end{equation}

Moreover, since $-\ell \leq -|\maxpoint|$ we get $ w_{t=0}(\ell)\leq w_{t\not=0}(\maxpoint)$ for any such $\ell$. Hence, indeed (even if $(C_t,o)$ are not plane curves), both $\ell^*\mapsto |\ell^*|$ and $\ell^*\mapsto \ell^*_{lim}$ induce maps $\sk_0\ti\cw_{t\not=0}\cap \ti S_n\to\sk_0\ti\cw_{t=0}\cap \ti S_n$, and by the original arguments from above, maps $\pi_0(\ti\cw_{t\not=0}\cap \ti S_n)\to\pi_0(\ti\cw_{t=0}\cap \ti S_n)$.

Furthermore, any 0 and 1-dimensional cell contained in the segment $[|\maxpoint|, \maxpoint_{lim}]$ of $\ti\cw_{t=0}$ has weight $\leq w_{t\not=0}(\maxpoint )$. Therefore, if $\maxpoint\in \ti\cw_{t\not=0}\cap\ti S_n$ then the segment $[|\maxpoint|, \maxpoint_{lim}]$ belongs to $\ti\cw_{t=0}\cap\ti S_n$ hence $|\maxpoint|$ and $\maxpoint_{lim}$ belong to the same connected component of $\ti\cw_{t=0}\cap\ti S_n$. In particular, the two maps induces the same graded graph map at the level of graded roots ${\Got R}(C_{t\not=0},o)\to{\Got R}(C_{t=0},o)$ (and similarly at the level of $\lcoh^0$).

In the case of a Gorenstein curve $(C,o)$ both $\lcoh^*(C,o)$ and ${\Got R}(C,o)$ have an additional $\bbz_2$-symmetry, cf. <ref>. One can ask if in the case of a flat deformation of Gorenstein curves the graded graph map at the level of graded roots ${\Got R}(C_{t\not=0},o)\to{\Got R}(C_{t=0},o)$ is $\bbz_2$-equivariant? The answer is no, a fact which can be checked already at the deformation considered in Example ex:an.

However, we formulate the following:

In the case of a $\delta$-constant flat deformation of Gorenstein curves the graded graph map at the level of graded roots ${\Got R}(C_{t\not=0},o)\to{\Got R}(C_{t=0},o)$ (constructed above) is $\bbz_2$-equivariant.

As an enhancement of Theorem th:DEF2 we also formulate the following general conjecture.

Let $\setof*{(C_t,o)}_{t\in(\bbc,0)}$ be a flat deformation of isolated curve singularities. Then the map $\ell^*\mapsto \ell^*_{lim}$ from the level $\sk_0\ti\cw_{t\not=0}\cap \ti S_n\to \sk_0\ti\cw_{t=0}\cap \ti S_n$ can be lifted to $\ti\cw_{t\not=0}\cap \ti S_n\to \ti\cw_{t=0}\cap \ti S_n$ inducing a degree zero graded $\bbz[U]$-module morphism $\lcoh^q(C_{t=0},o)\to \lcoh^q(C_{t\not=0},o)$ for any $q\geq 1$.

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