By Wolfgang Lück

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We will be looking now at rings over an Artin ring as base, and we put conditions on them by specifying that they should be flat over the Artin ring and have desired properties along the closed fiber. We will see that the same structure theorem persists in this situation, so that we can always extend deformations of codimension 2 Cohen–Macaulay subschemes in this local setting, by lifting the elements of the matrix ϕ. Here is the situation. We suppose given C a local Artin ring with residue field k, an ideal J ⊆ C , and its quotient C = C /J.

Suppose given a structure S over a field k. S could be a scheme, or a closed subscheme of a given scheme, or a vector bundle on a given scheme, etc. We look for deformations of S over the ring An = k[t]/tn+1 . These would be called n-th order deformations of S. Since it is hard to classify these all at once, we consider an easier problem. Suppose Sn is a given deformation of S over An . Then we seek to classify all deformations Sn+1 over An+1 whose restriction to An is the given deformation Sn .

Then the isomorphism classes of extensions of B by M as A-algebras CHAPTER 1: GETTING STARTED 27 are in natural one-to-one correspondence with elements of the group T 1 (B/A, M ). The trivial extension corresponds to the zero element. Proof. Let A[x] → B be a surjective map of a polynomial ring over A to B, let {ei } be a set of generators of the B-module M , and let y = {yi } be a set of indeterminates with the same index set as {ei }. Then we consider the polynomial ring A[x, y], and note that if B is any extension of B by M , then one can find a surjective ring homomorphism f : A[x, y] → B , not unique, which makes a commutative diagram 0 → (y) → A[x, y] → A[x] → 0 ↓ ↓f ↓ 0 → M → B → B → 0 ↓ ↓ ↓ 0 0 0 where the two outer vertical arrows are determined by the construction.

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