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

TL;DR

This paper develops an explicit method to construct differential $ abla$-graded varieties from algebraic data, reducing computational complexity and providing concrete descriptions, with applications to Lie-Rinehart algebras.

Contribution

It introduces an algorithm for constructing $ abla$-graded extensions of differential varieties using homotopy retracts, simplifying homological computations and enabling explicit descriptions.

Findings

Reduced homological computations via an explicit algorithm.

Provided concrete examples illustrating the construction.

Linked Lie-Rinehart algebras to explicit differential graded varieties.

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.