Strong uniqueness of enhancements for the dual numbers: a case study

TL;DR

This work proves that the bounded and bounded-below derived categories ${\mathbf D}^{b}(\mathrm{Mod}(\Bbbk[\varepsilon]))$ and ${\mathbf D}^{+}(\mathrm{Mod}(\Bbbk[\varepsilon]))$ possess strongly unique dg enhancements. It achieves this by relating these triangulated categories to the category of sequences ${\bf S}$, classifying indecomposables via this equivalence, and then transferring the problem to the dg-framework through ${\widehat{{\bf S}}}$; it further shows that the derived categories of any hereditary abelian category have strongly unique enhancements. The main technical contribution is a detailed analysis of dg enhancements using the sequence-category model ${\widehat{{\bf S}}}$ and the subcategories ${\widehat{{\bf S}}}_{\mathbf p}$ and ${\widehat{{\bf S}}}_{\mathbf p,fg}^{-}$, together with coproduct decompositions and control of phantom maps. The paper also provides partial results and open questions for the unbounded and bounded-above cases, outlining a program to extend strong uniqueness beyond the currently resolved bounded/bounded-below setting.

Abstract

We prove that the bounded and bounded below derived categories of (all) modules over the dual numbers have strongly unique (dg) enhancements. To this end we relate those categories to the category of sequences of vector spaces, which allows a complete classification of indecomposable objects. Along the way we also prove that all the derived categories of any hereditary category have strongly unique enhancements.

Theorems & Definitions (78)

• Theorem A
• Remark 1
• Remark 2
• Remark 3
• Lemma 1
• proof
• Proposition 1
• proof
• Remark 4
• Remark 5