Indecomposability of graded modules over a graded ring

Abstract

Let R=⊕i≥0Ri be a Noetherian commutative nonnegatively graded ring such that (R0,m0) is a Henselian local ring. Let m be its unique graded maximal ideal m0+⊕i>0Ri. Let T be a module-finite (noncommutative) graded R-algebra. Let Tgrmod denote the category of finite graded left T-modules and M∈Tgrmod. Then the following are equivalent: (1) Mˆ is an indecomposable Tˆ-module, where (−)ˆ denotes the m-adic completion; (2) Mm is an indecomposable Tm-module; (3) M is an indecomposable T-module; and (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=M1⊕⋯⊕Ms and N=N1⊕⋯⊕Nt are decompositions into indecomposable objects in Tgrmod, then s=t, and there exist some permutation σ∈Ss and integers d1,…,ds such that Ni≅Mσi(di), where −(di) denotes the shift of degree; (2) M≅N as T-modules; (3) Mm≅Nm as Tm-modules; and (4) Mˆ≅Nˆ as Tˆ-modules. As an application, we compare the FFRT property of rings of characteristic p in the graded sense and in the local sense.