Indecomposability of graded modules over a graded ring

Abstract

Let $R=\bigoplus_{i\geq 0}R_i$ be a Noetherian commutative non-negatively graded ring such that $(R_0,\mathfrak{m}_0)$ is a Henselian local ring. Let $\mathfrak{m}$ be its unique graded maximal ideal $\mathfrak{m}_0+\bigoplus_{i>0}R_i$. Let $T$ be a module-finite (non-commutative) graded $R$-algebra. Let $T\mathop{\mathrm{grmod}}$ denote the category of finite graded left $T$-modules, and $M\in T\mathop{\mathrm{grmod}}$. Then the following are equivalent: (1) $\hat M$ is an indecomposable $\hat T$-module, where $\widehat{(-)}$ denotes the $\mathfrak{m}$-adic completion; (2) $M_{\mathfrak{m}}$ is an indecomposable $T_{\mathfrak{m}}$-module; (3) $M$ is an indecomposable $T$-module; (4) $M$ is indecomposable as a graded $T$-module. As a corollary we prove that for two finite graded left $T$-modules $M$ and $N$, the following are equivalent: (1) If $M=M_1\oplus\cdots\oplus M_s$ and $N=N_1\oplus\cdots\oplus N_t$ are decompositions into indecomposable objects in $T\mathop{\mathrm{grmod}}$, then $s=t$, and there exist some permutation $\sigma\in \frak S_s$ and integers $d_1,\ldots,d_s$ such that $N_i\cong M_{\sigma i}(d_i)$, where $-(d_i)$ denotes the shift of degree; (2) $M\cong N$ as $T$-modules; (3) $M_{\mathfrak{m}}\cong N_{\mathfrak{m}}$ as $T_{\mathfrak{m}}$-modules; (4) $\hat M\cong \hat N$ as $\hat T$-modules. As an application, we compare the FFRT property of rings of characteristic $p$ in the graded sense and in the local sense.