The Kodaira–Spencer map

Hochschild cohomology controls deformations of an algebra: for an Artin–Schelter regular algebra \(A\) the degree-zero part \(\mathrm{HH}^2_0(A)\) is the space of first-order graded deformations, and is identified with the Zariski tangent space of the moduli stack \(\mathcal{A}_4\) of four-dimensional Koszul AS-regular algebras at the point \([A]\).

Given a flat family of such algebras over a base scheme \(S\), with fibre \(A_p\) at a point \(p\), the Kodaira–Spencer map sends a tangent vector \(v \in \mathrm{T}_pS\) to the class in \(\mathrm{HH}^2_0(A_p)\) of the first-order deformation that \(v\) induces. It measures how moving in the base deforms the fibre.

Following arXiv:2511.08390, this gives a criterion for recognising components of the moduli stack:

• if the Kodaira–Spencer map at \(p\) is surjective, the closure of the image of \(S \to \mathcal{A}_4\) is an irreducible component of the moduli stack;
• if it is a bijection, the map to the moduli stack is generically finite.

Computing \(\mathrm{HH}^2_0\) and this map for each known family is how the paper identifies which families sweep out components of \(\mathcal{A}_4\). The columns \(\mathrm{HH}^i_0\) in the table are exactly these Hochschild dimensions.

Overview

The rank of the Kodaira–Spencer map and whether it is injective / surjective, for every family (surjective ⇒ the family is a component; bijective ⇒ generically finite):