Quasi locally free sheaves on primitive double schemes
Abstract
Let X be a primitive double scheme, X = X red , L = I X the ideal sheaf of X, which is a line bundle on X. Let E be a coherent sheaf on X. Then there exists a nonempty open subset U of X and integers m 1 , m 2 , uniquely determined by E, such that E |U is locally isomorphic to
). If we can take U = X, we say the E is quasi locally free. If m 1 = 1 we say that E is rigid. In this case, we show that deformations of E are of the same type. So locally free sheaves and rigid quasi locally free sheaves on X play the same role on X as locally free sheaves do on smooth schemes.
Let E be a quasi locally free sheaf on X. Then E 1 = I X E and E |X are vector bundles on X, and we have a canonical surjective morphism
Similarly there is a canonical exact sequence Σ(E) of length 4 of vector bundles on X associated to E. Given a surjective morphism Φ : E → E of vector bundles on X, we define an element ∇ E,E (Φ) ∈ Ext 2 X (E, E ⊗ L) such that there exists a quasi locally free sheaf E on X such that Ψ E ≃ Φ if and only ∇ E,E (Φ) = 0, and prove that if E or E can be extended to a vector bundle on X, then E exists. Two ways are given to define ∇ E,E , a direct one and a more complicated one, using the cohomology theory of sheaves of non abelian groups, that can also be used to define an obstruction to the existence of a quasi locally free sheaf E corresponding to a given exact sequence of length 4.
We study also the existence and properties of flat families of quasi locally free sheaves, giving the possibility to construct universal families.
Related
- Refined proper descent for left completed sheaves on locally compact Hausdorff spaces
- Moduli spaces of rank 2 sheaves on ℙ3 via Quot schemes
- The moduli space of torsion-free sheaves with quasi-maximal third Chern class
- Quasi-F-split primitive symplectic varieties in positive characteristic
- Quasi-modularity of q-traces and integrals over Hilbert schemes
BibTeX
@misc{drezet2026quasi,
title = {{Quasi locally free sheaves on primitive double schemes}},
author = {Jean-Marc Drézet},
howpublished = {HAL},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35033},
eprint = {2610.35033},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05838872v1},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Quasi locally free sheaves on primitive double schemes
AU - Jean-Marc Drézet
PY - 2026
DA - 2026/10/26
PB - HAL
DO - 10.48550/arxiv.2610.35033
UR - https://hal.science/hal-05838872v1
AB - Let X be a primitive double scheme, X = X red , L = I X the ideal sheaf of X, which is a line bundle on X. Let E be a coherent sheaf on X. Then there exists a nonempty open subset U of X and integers m 1 , m 2 , uniquely determined by E, such that E |U is locally isomorphic to ). If we can take U = X, we say the E is quasi locally free. If m 1 = 1 we say that E is rigid. In this case, we show that deformations of E are of the same type. So locally free sheaves and rigid quasi locally free sheaves on X play the same role on X as locally free sheaves do on smooth schemes. Let E be a quasi locally free sheaf on X. Then E 1 = I X E and E |X are vector bundles on X, and we have a canonical surjective morphism Similarly there is a canonical exact sequence Σ(E) of length 4 of vector bundles on X associated to E. Given a surjective morphism Φ : E → E of vector bundles on X, we define an element ∇ E,E (Φ) ∈ Ext 2 X (E, E ⊗ L) such that there exists a quasi locally free sheaf E on X such that Ψ E ≃ Φ if and only ∇ E,E (Φ) = 0, and prove that if E or E can be extended to a vector bundle on X, then E exists. Two ways are given to define ∇ E,E , a direct one and a more complicated one, using the cohomology theory of sheaves of non abelian groups, that can also be used to define an obstruction to the existence of a quasi locally free sheaf E corresponding to a given exact sequence of length 4. We study also the existence and properties of flat families of quasi locally free sheaves, giving the possibility to construct universal families.
ER -