This post is a brief review of deformation theory. I personally learned how to compute deformation spaces of singularities from Jozef Steenbrink, Theo de Jong and Duco van Straten (it is very enjoyable to compute deformation spaces of singularities late at night provided one has a large supply of either coffee or beer).

Consider a field k. Set R = k[x_1, …, x_n]. Let I be an ideal of R. Choose a set of generators f_1, …, f_r of I. Denote Rel the module of relations, i.e., such that we have a short exact sequence

0 —> Rel —> R^{oplus r} —> I —> 0

Suppose that A is an Artinian local ring with residue field k. Set R_A = A[x_1, …, x_n]. An embedded deformation of R/I over A is an ideal I_A ⊂ R_A such that I_A \otimes_A k = I and such that R_A/I_A is flat over A. It turns out that this is equivalent to the following:

1. I_A can be generated by elements f’_1, …, f’_r whose reductions modulo m_A are equal to f_1, …, f_r, and
2. for any (g_1, …, g_r) ∈ Rel there exist g’_1, …, g’_r in R_A such that ∑ g’_if’_i = 0.

If you’ve ever tried to compute the deformation space of a singularity then you’ve seen this. In particular, if A = k[ε] is the ring of dual numbers, then f_i’ = f_i + ε h_i and condition 2 implies that f_i → h_i defines a map from I to R/I. Thus the tangent space to this deformation functor is

T^1 = Hom_R(I, R/I) = Hom_R(I/I^2, R/I)

Note that this is typically an infinite dimensional space (except if R/I is Artinian), which make sense because we are doing embedded deformations. To get the first order deformation space of R/I as a singularity we can divide out by the module of derivations of R over k, but for this blog post we prefer not to do so.

Next, I want to consider the obstruction space to our given deformation problem. Suppose that we have a small extension A —> B of Artinian local rings with kernel K. Suppose that f’_1, …, f’_r is a bunch of elements of R_A whose images in R_B do define an embedded deformation. To see if f’_1, …, f’_r is also a deformation we need to check if relations G = (g_1, …, g_r) ∈ Rel lift to relations among the f’_1, …, f’_r. By the assumption that we have a deformation over B we know that we can pick (g’_1, …, g’_r) such that ∑ g’_i f’_i is an element Ob(G) of KR_A = R \otimes_k K. Picking different choices of g’_i changes Ob(G) by an element of KI. Hence the obstruction is a well defined map

Rel —> R/I \otimes_k K

Note that the trivial relations f_if_j = f_jf_i of course do lift to relations among the f’_i. Hence we see that the obstruction map is an element of

Ob ∈ Hom_R(Rel/TrivRel, R/I) \otimes_k K

where TrivRel ⊂ Rel is the module of trivial relations. However, the obstruction map above depends on the initial choice of lifts f’_i to elements of R_A (with more or less given images in R_B as we’re given the deformation over B). Altering the choice of these f’_i modifies Ob by an element of Hom_R(R^{oplus r}, R/I) \otimes_k K. Combining all of the above we see that

T^2 = Coker(Hom_R(R^{oplus r}, R/I) —> Hom_R(Rel/TrivRel, R/I))

Again this is usually an infinite dimensional vector space over k.

I’m going to try and say something intelligent about this obstruction space in a future blog post. But for the moment I just make the comment that Rel/TrivRel is equal to the first Koszul homology group for the ring R and the sequence of elements f_1, …, f_r.