Lubin Tate Cohomology and Deformations of n-buds

We reprove Lazard’s result that every commutative n-bud is extendible to an n+ 1 bud, from an obstruction theoretic point of view. We locate the obstruction to extending an arbitrary n-bud in a certain cohomology group, and classify isomorphism classes of n-bud extensions for low degrees. 1 Lubin-Tate cohomology Definition 1. Let R be a commutative, unital ring and F a formal group law on R (associated to a formal group G). We define the Lubin-Tate cosimplicial ring of R with coefficients in G by the following: i. A = R[[x1, . . . , xn]], where we set A 0 = R. ii. The coface operators δ n : A n → A are defined by δ n(f)(x1, . . . , xn+1) = f(x2, . . . , xn+1), δ n+1 n (f)(x1, . . . , xn+1) = f(x1, . . . , xn) and δ i n(f)(x1, . . . , xn+1) = f(x1, . . . , xi +F xi+1, . . . , xn+1) for i otherwise. iii. The codegeneracy operators σ n : A n → A are given by σ n(f)(x1, . . . , xn−1) = f(x1, . . . , 0, . . . , xn−1) for 1 ≤ i ≤ n where the ith entry is replaced by a zero. It is not hard to check that these maps define a cosimplicial object. The Lubin-Tate complex of R will be the associated alternating sign complex. Let LT (R,G) or LT (R,F ) both denote the graded cohomology ring of this complex, called the Lubin-Tate cohomology of R with coefficients in G or F respectively (where the use G or F will depend on context). Remark 1. Note that the above is just an extension of the second cohomology ring of a formal group law constructed in [11]. We are primarily in this note interested in LT ∗(R,Ga), and primarily in degrees 2 and 3. As such, the only advancement we have past [11] is some investigation of the third degree, and perhaps firmer theoretical footing. Jonathan Lubin suggested that we avoid naming things after mathematicians, so we might want to call it something something like the “formal cohomology” of R, though this is still not a satisfying name. Remark 2. Also notice that LT (R,G) can be realized as the Hochschild cohomology of G with coefficients in the formal affine R-line (in the sense of II, §3, no. 4.4 of [2]), where the “action” G× A1R → A 1 R is the trivial one.