s and reduced curves in $\PP^2$ introducing ideals of quasi-adjunction \[rem:qaideals\]. Here we will introduce the analogous objects for the cyclic covers of $\PP^2_w$ ramified along non-reduced curves. These objects are not ideals in general, but modules over the local rings $\cO_{\PP^2_w,P}$ which will be called quasi-adjunction modules.

Consider the local situation at $0\in X=\frac{1}{w_2}(w_0,w_1)\cong \CC^2/G$ a cyclic singularity, as defined in §\[subsec:CyclicQuotient\], fixing $\zeta\in \CC$ a $w_2$-th primitive root of unity. In §\[subsec:CyclicQuotient\] we defined the class $[g]:k\in\ZZ_{w_2}$ of a quasi-germ $g\in\cO_{X,\zeta^{k}}$ (which depends on the choice of $\zeta$). Also note that any $g\in \cO_{X,\zeta^{k}}$ defines a Weil divisor in $X$ denoted by $\operatorname{div}_X(g)$.

Let $C=\sum_{i=1}^r n_iC_i$ be a local Weil divisor at $0$ given by the set of zeros of quasi-germs $C_i=\{f_i=0\}$. Denote by $\pi: Y \to X$ a good $\Q$-resolution of $C\subset (X,0)$ and $\Gamma$ the dual graph associated with $\pi$. Then the total transform of $C$ and the relative canonical divisor can be written as $$\label{eq:notationEi-local}
\begin{aligned}
\pi^{*} C &= \hat{C} + \sum_{\v \in \Gamma} m_\v E_\v,
& \quad \pi^{*} C_i &= \hat{C}_i + \sum_{\v \in \Gamma} m_{\v i} E_\v, 
& \quad K_\pi &= \sum_{\v \in \Gamma} (\nu_\v-1) E_\v,
\end{aligned}$$ where $m_\v=\sum_{i=1}^r n_im_{\v i}$.

\[def:M\] For any integer $k\in \ZZ$ and rational $\lambda\in \QQ$ we introduce the following *quasi-adjunction $\cO_X$-modules*: $$\label{eq:Mlocal}
\cM_\pi(C^\lambda,k)=
\left\{ g \in \cO_{X,\zeta^{\s_{\lambda,k}}}
\vphantom{\sum_{i=1}^r}\right. 
\left|\ 
\operatorname{mult}_{E_\v} (\pi^{*}\operatorname{div}_X(g)) > \sum_{i=1}^r {\left \{ \lambda n_i \right \}} m_{\v i} - \nu_\v, \
\forall \v \in \Gamma
\right\}$$ where $\s_{\lambda,k}:=k - |w| - \sum_{i=1}^r {\left \lfloor \lambda n_i \right \rfloor} [f_i]$ and $|w|:=w_0+w_1+w_2$.

In section \[sec:examples\] (Proposition \[prop:indres\]) it will be shown that $\cM_\