# Hochschild cohomology of noncommutative planes and quadrics

## TL;DR

This paper describes the Hochschild cohomology of noncommutative planes and quadrics by analyzing automorphism groups of elliptic triples and quadruples, providing new insights especially for elliptic quadruples.

## Contribution

It offers a novel description of Hochschild cohomology for noncommutative geometries using automorphism groups, including new results for elliptic quadruples.

## Key findings

- Automorphism groups of elliptic triples are characterized.
- Hochschild cohomology for noncommutative planes and quadrics is explicitly described.
- New results on automorphism groups for elliptic quadruples.

## Abstract

We give a description of the Hochschild cohomology for noncommutative planes (resp. quadrics) using the automorphism groups of the elliptic triples (resp. quadruples) that classify the Artin-Schelter regular $\mathbb{Z}$-algebras used to define noncommutative planes and quadrics. For elliptic triples the description of the automorphism groups is due to Bondal-Polishchuk, for elliptic quadruples it is new.