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Differential graded algebras with divided powers and homotopy Lie algebras

Antoine Caradot, Zongzhu Lin

TL;DR

The paper develops a framework for differential graded algebras with divided powers (pd dg algebras) over pd dg rings, enabling pd dg resolutions of R/𝔪 without Noetherian hypotheses via symmetric-tensor constructions.It builds Koszul-Tate resolutions in this context, establishing PD algebra and Hopf-algebra structures, and proves a left adjoint/freeness property for the Tate-like functor, including lifting and extension results essential for homological constructions.The authors connect these pd dg resolutions to homotopy Lie algebras and the Yoneda algebra, providing explicit presentations in the complete-intersection case and showing how to reconstruct CI rings from restricted Lie data, with criteria for finite generation and structural consequences.Overall, the work links divided-power dg-algebra techniques to complete intersection theory and the structure of Ext algebras, offering tools to study non-Noetherian settings and to recover CI data from homotopy Lie information.

Abstract

Given a commutative ring $R$ and a quotient module $R/\mathfrak{m}$, we study the structure of differential graded algebra with divided powers that the resolution of $R/\mathfrak{m}$ can possess. We provide a construction of such a resolution by making use of symmetric tensors, which does not require a Noetherian assumption on $R$. We then apply those results to study the homotopy Lie algebra associated to a the pair $(R,R/\mathfrak{m})$, and then investigate in more details the complete intersection case.

Differential graded algebras with divided powers and homotopy Lie algebras

TL;DR

The paper develops a framework for differential graded algebras with divided powers (pd dg algebras) over pd dg rings, enabling pd dg resolutions of R/𝔪 without Noetherian hypotheses via symmetric-tensor constructions.It builds Koszul-Tate resolutions in this context, establishing PD algebra and Hopf-algebra structures, and proves a left adjoint/freeness property for the Tate-like functor, including lifting and extension results essential for homological constructions.The authors connect these pd dg resolutions to homotopy Lie algebras and the Yoneda algebra, providing explicit presentations in the complete-intersection case and showing how to reconstruct CI rings from restricted Lie data, with criteria for finite generation and structural consequences.Overall, the work links divided-power dg-algebra techniques to complete intersection theory and the structure of Ext algebras, offering tools to study non-Noetherian settings and to recover CI data from homotopy Lie information.

Abstract

Given a commutative ring and a quotient module , we study the structure of differential graded algebra with divided powers that the resolution of can possess. We provide a construction of such a resolution by making use of symmetric tensors, which does not require a Noetherian assumption on . We then apply those results to study the homotopy Lie algebra associated to a the pair , and then investigate in more details the complete intersection case.

Paper Structure

This paper contains 22 sections, 41 theorems, 155 equations.

Table of Contents

  1. Introduction
  2. The category of pd dg algebras over a pd dg ring
  3. pd dg rings
  4. Koszul-Tate Resolutions for arbitrary commutative rings
  5. Symmetric tensors of differential complexes
  6. PD algebra structure
  7. Hopf algebra structures
  8. Freeness of the strictly commutative dg algebra Lg
  9. Lifting properties
  10. pd dg algebra extensions
  11. Derivations of pd dg algebras
  12. Homotopy Lie algebras and complete intersection rings
  13. The Yoneda algebra of a complete intersection ring
  14. Reconstruction of complete intersections
  15. Finite generation of the Yoneda algebra
  16. ...and 7 more sections

Key Result

Lemma 2.1

If $\mathsf{k}\to \mathsf{k}'$ is a ring homomorphism, then for any pd dg algebra $(A^*,I,\gamma)$, the pd dg algebra structure $(A^*_{\mathsf{k}'},I_{\mathsf{k}'}, \gamma_{\mathsf{k}'})$ is isomorphic to the tensor product of pd dg algebras $(A^*,I,\gamma)$ and $(\mathsf{k}',\{0\},0)$.

Theorems & Definitions (80)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Theorem 3.3
  • proof
  • Lemma 3.4
  • Theorem 3.5
  • ...and 70 more