On construction of differential $\mathbb Z$-graded varieties
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866917220093788160 |
|---|---|
| author | Hancharuk, Aliaksandr Louis, Ruben |
| author_facet | Hancharuk, Aliaksandr Louis, Ruben |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_23148 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On construction of differential $\mathbb Z$-graded varieties Hancharuk, Aliaksandr Louis, Ruben Mathematical Physics Commutative Algebra Differential Geometry 13D02, 14F08, 18G10, 53C12 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. |
| title | On construction of differential $\mathbb Z$-graded varieties |
| topic | Mathematical Physics Commutative Algebra Differential Geometry 13D02, 14F08, 18G10, 53C12 |
| url | https://arxiv.org/abs/2512.23148 |