Indecomposability of graded modules over a graded ring

TL;DR

This paper establishes equivalences between various notions of indecomposability for graded modules over a graded ring, and explores their implications for module decompositions and properties like FFRT in characteristic p.

Contribution

It proves that indecomposability of modules over a graded ring is equivalent to indecomposability over its completion and localizations, providing new insights into module structure and decomposition.

Findings

Indecomposability is equivalent across different module categories.

Decomposition uniqueness is characterized by module isomorphisms and shifts.

Comparison of FFRT property in graded and local contexts.

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.