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Transfer of A-infinity structures to projective resolutions

Janina C. Letz

TL;DR

The article provides a complete, constructive proof that A_infty-structures can be transferred along quasi-isomorphisms over a commutative ring, generalizing earlier field-specific results. It develops obstruction theory within the bar/tensor coalgebra framework to inductively build higher multiplications and morphisms, yielding existence and uniqueness (up to A_infty-homotopy) of transferred structures, including strictly unital variants. The work extends the transfer to A_infty-modules and demonstrates how units can be preserved under mild hypotheses, with explicit, implementable steps. This has practical implications for constructing projective resolutions with A_infty-structure in algebraic settings and for maintaining homotopy-invariant algebraic information.

Abstract

In this article we prove various results about transferring or lifting $\mathrm{A}_\infty$-algebra structures along quasi-isomorphisms over a commutative ring.

Transfer of A-infinity structures to projective resolutions

TL;DR

The article provides a complete, constructive proof that A_infty-structures can be transferred along quasi-isomorphisms over a commutative ring, generalizing earlier field-specific results. It develops obstruction theory within the bar/tensor coalgebra framework to inductively build higher multiplications and morphisms, yielding existence and uniqueness (up to A_infty-homotopy) of transferred structures, including strictly unital variants. The work extends the transfer to A_infty-modules and demonstrates how units can be preserved under mild hypotheses, with explicit, implementable steps. This has practical implications for constructing projective resolutions with A_infty-structure in algebraic settings and for maintaining homotopy-invariant algebraic information.

Abstract

In this article we prove various results about transferring or lifting -algebra structures along quasi-isomorphisms over a commutative ring.
Paper Structure (7 sections, 22 theorems, 86 equations)

This paper contains 7 sections, 22 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. The tensor coalgebra
  3. A-infinity-algebras
  4. Obstructions
  5. Transfer of A-infinity-structures
  6. Transfer of strictly unital A-infinity-structures
  7. A-infinity-modules

Key Result

Theorem A

Let $Q$ be a commutative ring and $B$ an $\mathrm{A}_{\infty}$-algebra over $Q$. Given a quasi-isomorphism $\varepsilon_1 \colon A \to B$ of complexes with $A$ a bounded below complex of projective modules, there exists an $\mathrm{A}_{\infty}$-algebra structure on $A$ and a quasi-isomorphism $\vare

Theorems & Definitions (36)

  • Theorem A
  • Lemma 4.2
  • proof
  • Lemma 4.4
  • proof
  • Lemma 4.5
  • proof
  • Lemma 4.7
  • proof
  • Lemma 5.2
  • ...and 26 more