Hochschild cohomology of the algebra

The Hochschild cohomology \(\mathrm{HH}^\bullet(A)\) of an algebra \(A\) is the cohomology of the complex \(\operatorname{Hom}_{A^e}(P_\bullet, A)\), where \(P_\bullet\) is the bar (or, for a Koszul algebra, the Koszul) resolution of \(A\) as an \(A\)-bimodule. It governs the deformation theory of \(A\): \(\mathrm{HH}^0(A)\) is the centre, \(\mathrm{HH}^1(A)\) the outer derivations, and \(\mathrm{HH}^2(A)\) the infinitesimal deformations.

The graded refinement \(\mathrm{HH}^\bullet_0\)

For a connected graded algebra the Hochschild cohomology is itself graded, and the part of internal degree \(0\), written \(\mathrm{HH}^\bullet_0(A)\), is the piece relevant to graded deformations — those that keep the grading and the number of generators and relations. The dimensions \(\dim \mathrm{HH}^i_0(A)\) for \(i = 0, 1, 2, 3\) are the invariants tabulated on this site (with \(\mathrm{HH}^0_0 = 1\) always).

In particular \(\mathrm{HH}^2_0(A)\) is the space of first-order graded deformations, and is identified with the tangent space to the moduli stack at \([A]\) — this is what the Kodaira–Spencer map measures.

This is the deformation theory of the ring \(A\). The deformation theory of the associated variety is captured instead by the Hochschild cohomology of qgr.

Overview

\(\dim \mathrm{HH}^i_0(A)\) for \(i = 1, 2, 3\), for every family (\(\mathrm{HH}^0_0 = 1\) always), as computed in arXiv:2511.08390: