Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2007.06350

Modules determined by their composition factors in higher homological algebra

Joseph Reid
View original source ↗Read source document ↗Export metadata ↓

Abstract

ABSTRACT. Let $Φ$ be a finite dimensional $K$-algebra and let $\mathscr{C} = \textrm{mod}\: Φ$ be the abelian category of finitely generated right $Φ$-modules. In their 1985 paper ``Modules determined by their composition factors'', Auslander and Reiten showed that under certain conditions modules in $\textrm{mod}\: Φ$ are determined by their composition factors, and show an important formula related to the Auslander-Reiten translation. Let $\mathscr{T}$ be a $d$-cluster tilting subcategory of $\mathscr{C}$, which by definition is also $d$-abelian. In this paper we will define the Grothendieck group for a $d$-abelian category, and show that the Grothendieck groups of $\mathscr{C}$ and $\mathscr{T}$ are isomorphic. We show also that under certain conditions, the indecomposable objects of $\mathscr{T}$ are determined up to isomorphism by their composition factors in $\mathscr{C}$. Finally, we generalise the formula from Auslander and Reiten involving the higher dimensional Auslander-Reiten translation.

Explore related subjects

math.RT

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Joseph Reid. 2020-07-13. Modules determined by their composition factors in higher homological algebra. https://arxiv.org/abs/2007.06350

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A local relative trace formula for F*\SL(2,F)

In this note, we derive explicitly the local relative trace formula for the symmetric space F*\SL(2,F) at the level of Lie algebras, where F is a p-adic field of residue characteristic greater than two and F* is the set of invertible elements in F. This is perhaps one of the simplest non-trivial analogs of the trace formula, and also a motivating example for the author's work (in preparation) on the relative trace formula.

math.RT↗
arXiv2026

Semi-infinite parabolic IC-sheaf

Let G be a connected reductive group, P its parabolic subgroup. We consider the parabolic semi-infinite category of sheaves on the affine Grassmanian of G and construct the parabolic version of the semi-infinite IC-sheaf of each orbit. We establish some of its properties and relate it to sheaves on the Drinfeld compactification of the moduli stack Bun_P of P-torsors on a curve. We also relate the parabolic semi-infinite IC-sheaf with the dual baby Verma object on the spectral side.

math.RT↗
arXiv2026

The Grothendieck group of an extriangulated category

In this paper, we investigate the split Grothendieck group $K^{\rm sp}_{0}(\mathcal{M})$ of a $d$-rigid subcategory $\mathcal{M}$ in an extriangulated category $\mathscr{C}$. As applications, we prove the following results: (1) If $\mathcal{M}$ is a silting subcategory, then the Grothendieck group $K_{0}(\mathscr{C})$ is isomorphic to $K_{0}^{\rm sp}(\mathcal{M})$; (2) If $\mathcal{M}$ is a $d$-cluster tilting subcategory, then $K_{0}(\mathscr{C})$ is isomorphic to the index Grothendieck group $K_{0}^{\rm in}(\mathcal{M})$; (3) Let $\mathcal{C}_{A_{n}}^{d}$ be the $d$-cluster category of type $A_n$. If $d$ is even, then $K_0(\mathcal{C}_{A_{n}}^{d})\cong \mathbb{Z}/(n+1)\mathbb{Z}$. If $d$ is odd, then $K_0(\mathcal{C}_{A_{n}}^{d})\cong \mathbb{Z}$ if $n$ is odd; $K_0(\mathcal{C}_{A_{n}}^{d})\cong 0$ if $n$ is even.

math.RT↗