add refinement of theorem about decompositions

ncr-of-monomial-curves.tex

@@ -92,6 +92,34 @@
 
 \section{Introduction}
 
+\begin{lemma}
+For $A \in \Min(S)$, the $\QQ_{\geq 0}$-graded direct summand $I_{q,r}$ of $R^{1/q}$ is, up to a degree shift, isomorphic to $I(A)$ if and only if 
+\begin{equation}\label{eq:char-indec-summand}
+ \forall \, \alpha \in A : \; m_{q,r} + q\alpha \in \SGR \qquad \text{and} \qquad \forall \, \beta \in \NN_0 \setminus \SGR_A : \; m_{q,r} + q\beta \notin \SGR,
+\end{equation}
+where $\frac{m_{q,r}}{q}$ denotes the minimal degree of a monomial in $I_{q,r}$.
+\end{lemma}
+
+\begin{proof}
+In view of Corollary~\ref{cor:indec-summ}, we may write $I_{q,r} = x^{\frac{m_{q,r}}{q}} \cdot I(A^\prime)$ for an $A^\prime \in \Min(\SGR)$. In particular, $I_{q,r} \shcong I(A^\prime)$. Consequently, by Lemma~\ref{lemma:rank-1}, the $\QQ_{\geq 0}$-graded direct summand $I_{q,r}$ of $R^{1/q}$ is, up to a degree shift, isomorphic to $I(A)$ if and only if $A^\prime = A$, that is,
+\[ I_{q,r} = x^{\frac{m_{q,r}}{q}} \cdot I(A). \]
+At the level of $\kk$-vector spaces, this is equivalent to $I_{q,r}$ being generated by monomials of degrees in $\frac{m_{q,r}}{q} + \SGR_A$. Using the description~\eqref{eq:ind-summand-as-vector-space} of $I_{q,r}$, we see that this is precisely the case when
+\[ \{ a \in \SGR \mid a \equiv r \pmod{q} \} = m_{q,r} + q \cdot \SGR_A. \]
+One readily checks that this is equivalent to the condition~\eqref{eq:char-indec-summand}.
+\end{proof}
+
+Let $\overline{A}$ denote the set of minimal elements of $\gapset$ with respect to $\preceq_\SGR$, so that the corresponding gap module $I(\overline{A})$ is equal to the normalization $\overline{R} = \kk[x]$ of $R$.\todo{add note that $R$ and $\overline{R}$ are gap modules}
+
+\begin{corollary}
+The number of $\QQ_{\geq 0}$-graded direct summands of $R^{1/q}$ that are, up to degree shift, isomorphic to the normalization $\overline{R} = \kk[x]$ of $R$ is precisely the number of $r \in \intval{0,q-1}$ satisfying
+$m_{q,r} + q \cdot \overline{A} \subseteq \SGR$.
+\end{corollary}
+
+\begin{proof}
+Observing that $\SGR_{\overline{A}} = \NN_0$, the result is an immediate consequence of the previous lemma.
+\end{proof}
+
+
 % some results/terminology in section 2 are made independent of the characteristic
 
 \section{Preliminaries}