Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On construction of differential $\mathbb Z$-graded varieties

Aliaksandr Hancharuk, Ruben Louis

TL;DR

The paper develops a computable framework for constructing $ abla$-graded $Q$-varieties that encode both the Koszul–Tate resolution of a quotient $ abla ext{O}/ abla ext{I}$ and a positively graded $Q$-variety associated with Lie–Rinehart algebras. The central innovation is an arborescent Koszul–Tate resolution that feeds into an explicit arborescent extension, with a refined homological perturbation algorithm that terminates in finite steps under finite projective resolutions. The main results establish existence and explicit descriptions of these extensions over $ abla ext{O}/ abla ext{I}$ and over $ abla ext{O}$ preserving $ abla ext{I}$, including a robust set of geometric applications to universal Lie $ abla$-∞ algebroids and singular foliations. Concrete examples demonstrate the methodology on Lie algebroids, vector fields vanishing on subspaces, and quadratic-ideal preserving derivations, highlighting new multipart algebraic structures and multiplications arising from hook maps. This framework provides a computable bridge between singular spaces, Lie–Rinehart theories, and higher graded geometric structures with potential impact on BV/BFV formalisms and deformation theory.

Abstract

Given a commutative unital algebra $\mathcal O$, a proper ideal $\mathcal I$ in $\mathcal O$, and a positively graded differential variety over $\mathcal O/\mathcal I$, we provide a $\mathbb Z$-graded extension, whose negative part is an arborescent Koszul-Tate resolution of $\mathcal O/ \mathcal I$. This extension is obtained through an algorithm exploiting the explicit homotopy retract data of the arborescent Koszul-Tate resolution, so that the number of homological computations in the construction is significantly reduced. For a positively graded differential variety over $\mathcal O$ that preserves the ideal $\mathcal I$, the extension admits a manifest description in terms of decorated trees and computed data. As a by-product, to every Lie--Rinehart algebra over the coordinate ring of an affine variety $ W \subseteq M = \mathbb{C}^d $, one associates an explicit differential $\mathbb{Z}$-graded variety over $M$ whose negative component is the arborescent Koszul--Tate resolution of the coordinate ring $\mathbb C[x_1, \ldots, x_d]/\mathcal I_W$ of $W$, and whose positive component is the universal dg-variety of the given Lie--Rinehart algebra. Concrete examples are given.

On construction of differential $\mathbb Z$-graded varieties

TL;DR

The paper develops a computable framework for constructing -graded -varieties that encode both the Koszul–Tate resolution of a quotient and a positively graded -variety associated with Lie–Rinehart algebras. The central innovation is an arborescent Koszul–Tate resolution that feeds into an explicit arborescent extension, with a refined homological perturbation algorithm that terminates in finite steps under finite projective resolutions. The main results establish existence and explicit descriptions of these extensions over and over preserving , including a robust set of geometric applications to universal Lie -∞ algebroids and singular foliations. Concrete examples demonstrate the methodology on Lie algebroids, vector fields vanishing on subspaces, and quadratic-ideal preserving derivations, highlighting new multipart algebraic structures and multiplications arising from hook maps. This framework provides a computable bridge between singular spaces, Lie–Rinehart theories, and higher graded geometric structures with potential impact on BV/BFV formalisms and deformation theory.

Abstract

Given a commutative unital algebra , a proper ideal in , and a positively graded differential variety over , we provide a -graded extension, whose negative part is an arborescent Koszul-Tate resolution of . This extension is obtained through an algorithm exploiting the explicit homotopy retract data of the arborescent Koszul-Tate resolution, so that the number of homological computations in the construction is significantly reduced. For a positively graded differential variety over that preserves the ideal , the extension admits a manifest description in terms of decorated trees and computed data. As a by-product, to every Lie--Rinehart algebra over the coordinate ring of an affine variety , one associates an explicit differential -graded variety over whose negative component is the arborescent Koszul--Tate resolution of the coordinate ring of , and whose positive component is the universal dg-variety of the given Lie--Rinehart algebra. Concrete examples are given.
Paper Structure (20 sections, 18 theorems, 131 equations, 1 figure)

This paper contains 20 sections, 18 theorems, 131 equations, 1 figure.

Table of Contents

  1. Definitions
  2. $\mathbb Z$-graded $Q$-varieties
  3. The negative and positive components of a $\mathbb Z$-graded $Q$-variety
  4. Lie-Rinehart algebras and positively graded $Q$-varieties
  5. Arborescent Koszul-Tate resolutions
  6. The construction
  7. Main results
  8. Arborescent extension of $\mathbb N$-graded $Q$-varieties over $\mathcal{O}/\mathcal{I}$
  9. Proof of Theorem \ref{['thm:gen.case.computations']}
  10. Arborescent extension of $\mathbb N$-graded $Q$-varieties over $\mathcal{O}$
  11. Applications and results in the geometric setting
  12. Proof of Theorem \ref{['thm:main2']}
  13. Examples
  14. Arborescent extention of an $\mathcal{I}$-Lie algebroid
  15. Vector fields vanishing on a subspace of $\mathbb K^2$
  16. ...and 5 more sections

Key Result

Proposition 1.5

Let $\mathcal{G} = \oplus_{j\in \mathbb Z} \mathcal{G}_j$ be a $\mathbb Z$-graded commutative unital algebra, and $(F^i\mathcal{G})_{i\in \mathbb N}$ be the negative filtration. Every derivation $Z \colon \mathcal{G}\to \mathcal{G}$ of degree $k$ extends to a well-defined derivation $\bar{Z}\colon \

Figures (1)

  • Figure 2: Admissible planar trees

Theorems & Definitions (56)

  • Definition 1.1
  • Remark 1.2
  • Remark 1.3
  • Proposition 1.5
  • proof
  • Definition 1.6: KOTOV2023104908
  • Definition 1.8
  • Definition 1.10
  • Remark 1.11
  • Example 1.12
  • ...and 46 more