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Liftings of point-wise finite dimensional persistence modules over local commutative Artinian rings

José A. Vélez-Marulanda

Abstract

Let $\mathbf{k}$ be a field and let $V: \mathscr{C} \to \mathbf{k}\textup{-Mod}$ be a point-wise finite dimensional persistence modules, where $\mathscr{C}$ is a small category. Assume that for all local Artinian $\mathbf{k}$-algebras $R$ with residue field isomorphic to $\mathbf{k}$, there is a generalized persistence module $M: \mathscr{C} \to R\textup{-Mod}$, such that for all $x\in \mathrm{Ob}(\mathscr{C})$, $M(x)$ is free over $R$ with finite rank and $\mathbf{k}\otimes_R M(x)\cong V(x)$. If $V$ is a direct sum of indecomposable persistence modules $V_I: \mathscr{C}\to \mathbf{k}\textup{-Mod}$ with endomorphism ring isomorphic to $\mathbf{k}$, then $M$ is a direct sum of indecomposables $M_I:\mathscr{C}\to R\textup{-Mod}$ with endomorphism ring isomorphic to $R$

Liftings of point-wise finite dimensional persistence modules over local commutative Artinian rings

Abstract

Let be a field and let be a point-wise finite dimensional persistence modules, where is a small category. Assume that for all local Artinian -algebras with residue field isomorphic to , there is a generalized persistence module , such that for all , is free over with finite rank and . If is a direct sum of indecomposable persistence modules with endomorphism ring isomorphic to , then is a direct sum of indecomposables with endomorphism ring isomorphic to
Paper Structure (2 sections, 3 theorems, 2 equations)

This paper contains 2 sections, 3 theorems, 2 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm1']}

Key Result

Theorem 1.1

Let $V$ a pointwise finite dimensional persistence module such that $V$ is the direct sum of indecomposable persistence modules with endomorphism ring isomorphic to $\Bbbk$. Then for all objects $R$ in $\mathcal{A}_\Bbbk$ and all lifts $(M,\phi)$ of $V$ over $R$, the $R$-generalized persistence modu

Theorems & Definitions (9)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Definition 2.3
  • Remark 2.4
  • proof : Proof of Lemma \ref{['lemma1.2']}
  • proof : Proof of Theorem \ref{['thm1']}