Geometric and homological interpretations of tensor flattenings

@misc{dolezalek2026geometric,
  title = {{Geometric and homological interpretations of tensor flattenings}},
  author = {Matěj Doležálek and Jakub Jagiełła and Piotr Oszer and Tim Seynnaeve and Colin Stastny},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.17875},
  eprint = {2610.17875},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.17875},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Geometric and homological interpretations of tensor flattenings
AU  - Matěj Doležálek
AU  - Jakub Jagiełła
AU  - Piotr Oszer
AU  - Tim Seynnaeve
AU  - Colin Stastny
PY  - 2026
DA  - 2026/10/13
PB  - arXiv
DO  - 10.48550/arxiv.2610.17875
UR  - https://arxiv.org/abs/2610.17875
AB  - We relate several nonlinear flattenings of structure tensors of finite algebras and modules to their algebraic and homological invariants. Since smoothability of algebras is equivalent to their structure tensors having minimal border rank, this shows equivalence of several tensor-theoretic nonsmoothability certificates to known classical ones. Particularly, we prove two conjectures of Doležálek and Michałek relating the tangency flattening and the 3×3 minor flattening to dimensions of tangent spaces to the Hilbert scheme of points. In the process, we generalize the tangency flattening to a mixed regime with two tensor inputs, relating this to dimensions of Ext modules, and introduce the dual notion of the cotangency flattening, related to dimensions of Tor modules. In the case of algebras, we also realize the kernel of the dualized tangency flattening within the setting of Harrison cohomology, making the connection to the tangent space of the Hilbert scheme explicit.
ER  -