Hochschild cohomology of qgr

The category \(\operatorname{qgr} A\) is the noncommutative projective scheme attached to \(A\) — the noncommutative \(\mathbb{P}^3\) itself, as opposed to its homogeneous coordinate ring. Its Hochschild cohomology \(\mathrm{HH}^\bullet(\operatorname{qgr} A)\) is the deformation theory of the variety rather than the ring.

It can be computed from a truncated Koszul-bimodule resolution of the Beilinson algebra \(\operatorname{End}(\mathcal{O} \oplus \mathcal{O}(1) \oplus \mathcal{O}(2) \oplus \mathcal{O}(3))\). For every four-dimensional family it satisfies \(\mathrm{HH}^0 = k\) and has Euler characteristic \(-4\); the remaining dimensions \(\mathrm{HH}^1, \mathrm{HH}^2, \mathrm{HH}^3\) are recorded on this site where they have been computed directly.

This is in general different from the Hochschild cohomology of the graded algebra: they agree for \(\mathbb{P}^3\) and the Sklyanin algebra but differ for the skew, twisted and Clifford families, so the qgr cohomology is computed in its own right.

Overview

\(\dim \mathrm{HH}^i(\operatorname{qgr} A)\) for \(i = 1, 2, 3\), where computed (\(\mathrm{HH}^0 = k\) always):