The deformation as a functor

Question

I'm studying some deformation theory of associative algebras and my professor told me that the association $A \mapsto \text{Def}(A)$(I associate to an associative algebra the deformation space of $A$) can be made functorial.

Let's consider $\mathbf{C}$ the category of some commutative algebras, and let's consider $\text{Def}(A)$ the deformation space of an element $A$ in our category.

How $\text{Def}:\mathbf{C}\to\mathbf{Set}$ is a functor?

If the question is too vage could you at least give me some more details?

Hope someone can help me.

Thanks!

Answer 1

A convenient way to study the infinitesimal/formal deformation theory of associative algebras is by studying the "classical" deformation functor $$ \mathrm{Def}_A \colon \mathbf{Art}_\Bbbk \to \mathbf{Set} $$ where

• $\Bbbk$ is a field of characteristic $0$
• $A = (V, \mu)$ is the associative $\Bbbk$-algebra you want to deform (here $V$ is a $\Bbbk$-vector space and $\mu \colon V \otimes_\Bbbk V \to V$ defines an associative multiplication)
• $\mathbf{Art}_\Bbbk$ is the category of local commutative Artinian $\Bbbk$-algebras which function as the "base" of the deformation
• $\mathrm{Def}_A (B)$ is the set of "deformations of $A$ over $B$ up to equivalence", i.e. the set of $B$-algebras with underlying $\Bbbk$-vector space $A \otimes_\Bbbk B$ and a multiplication $\widetilde \mu$ which reduces to $\mu$ modulo the maximal ideal $\mathfrak m$ of $B$, where two deformations over $B$, given by $\widetilde \mu$ and $\widetilde \mu'$ are called equivalent if there is a $B$-algebra automorphism $T \colon A \otimes_\Bbbk B \to A \otimes_\Bbbk B$ such that $\widetilde \mu' (T (f), T (g)) = T (\widetilde \mu (f, g))$ and $T$ is the identity modulo $\mathfrak m$.

Useful exercises

1. Write out the above definition for $B = \Bbbk [t] / (t^{n+1})$ with maximal ideal $(t)$ and convince yourself that this is a natural notion of "one-parameter deformation of order $n$".

2. Check that any deformation over $B = \Bbbk [t] / (t^2)$, called first-order deformation, is given by a Hochschild 2-cocycle and two first-order deformations are equivalent precisely when they differ by a Hochschild 2-coboundary. Thus, one gets a 1-to-1 correspondence between first-order deformations up to equivalence and the cohomology group $\mathrm{HH}^2 (A, A)$. Informally, $\mathrm{HH}^2 (A, A)$ measures the number of nontrivial directions in which $A$ can be deformed (with the caveat that if $A$ is infinite-dimensional, then $\mathrm{HH}^2 (A, A) = 0$ does not necessarily imply that there are no nontrivial deformations).

Note that higher-order deformations may be obstructed, so it is not true that any first-order deformation can necessarily be extended to higher orders. The deformation functor $\mathrm{Def}_A$ asks for associative deformations over $B$, so the obstruction theory is in some sense encoded abstractly into $\mathrm{Def}_A$.

The deformation functor $\mathrm{Def}_A$ is in fact isomorphic to the deformation functor that is naturally associated to the DG Lie algebra $$ (\mathrm{Hom}_\Bbbk (A^{\otimes_\Bbbk \bullet + 1}, A), d, [-{,}-]) $$ where

• $\bullet$ denotes the degree
• $d$ is the usual Hochschild differential
• $[-{,}-]$ is the Gerstenhaber bracket.

In the setting of DG Lie algebras, the obstruction theory (here associativity) is encoded into the DG Lie bracket (the Gerstenhaber bracket). In more general deformation problems, infinitely many brackets are sometimes needed, giving rise to L$_\infty$ algebras.

If $A = (V, \mu)$ is a finite-dimensional algebra, i.e. $V$ is a finite-dimensional vector space, then $A$ can be viewed as a point in the variety $\mathrm{Alg}_V \subset \Bbbk^{(\dim V)^3}$ of all associative multiplications on $V$. There is a natural action of $\mathrm{GL} (V)$ on $\mathrm{Alg}_V$ given by change of basis, where two algebras $A, A' \in \mathrm{Alg}_V$ are isomorphic if and only if they lie in the same $\mathrm{GL} (V)$-orbit. In this case, studying $\mathrm{Def}_A$ is in some sense completely determined by understanding the local structure of $\mathrm{Alg}_V$ (with its $\mathrm{GL} (V)$-action) around $A$.

One more interesting observation: the tangent space at $A$ to $\mathrm{Alg}_V$ is the space of Hochschild 2-cocycles and the tangent space to the $\mathrm{GL} (V)$-orbit at $A$ (a subvariety of $\mathrm{Alg}_V$) is the space of Hochschild 2-coboundaries, so that $\mathrm{HH}^2 (A, A)$ appears as a quotient of (geometric) tangent spaces.

Recommended literature

Gerstenhaber, Murray, On the deformation of rings and algebras, Ann. Math. (2) 79, No. 1, 59-103 (1964). ZBL0123.03101.

Doubek, M.; Markl, M.; Zima, P., Deformation theory (lecture notes)., Arch. Math., Brno 43, No. 5, 333-371 (2007). ZBL1199.13015.

Markl, Martin, Deformation theory of algebras and their diagrams., CBMS Regional Conference Series in Mathematics 116. Providence, RI: American Mathematical Society (AMS); Washington, DC: Conference Board of the Mathematical Sciences (CBMS) (ISBN 978-0-8218-8979-4/pbk). ix, 129 p. (2012). ZBL1267.16029.