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The Stacks project

Lemma 16.11.1. Let k be a field of characteristic p > 0. Let (\Lambda , \mathfrak m, K) be an Artinian local k-algebra. Assume that \dim H_1(L_{K/k}) < \infty . Then \Lambda is a filtered colimit of Artinian local k-algebras A with each map A \to \Lambda flat, with \mathfrak m_ A \Lambda = \mathfrak m, and with A essentially of finite type over k.

Proof. Note that the flatness of A \to \Lambda implies that A \to \Lambda is injective, so the lemma really tells us that \Lambda is a directed union of these types of subrings A \subset \Lambda . Let n be the minimal integer such that \mathfrak m^ n = 0. We will prove this lemma by induction on n. The case n = 1 is clear as a field extension is a union of finitely generated field extensions.

Pick \lambda _1, \ldots , \lambda _ d \in \mathfrak m which generate \mathfrak m. As K is formally smooth over \mathbf{F}_ p (see Algebra, Lemma 10.158.7) we can find a ring map \sigma : K \to \Lambda which is a section of the quotient map \Lambda \to K. In general \sigma is not a k-algebra map. Given \sigma we define

\Psi _\sigma : K[x_1, \ldots , x_ d] \longrightarrow \Lambda

using \sigma on elements of K and mapping x_ i to \lambda _ i. Claim: there exists a \sigma : K \to \Lambda and a subfield k \subset F \subset K finitely generated over k such that the image of k in \Lambda is contained in \Psi _\sigma (F[x_1, \ldots , x_ d]).

We will prove the claim by induction on the least integer n such that \mathfrak m^ n = 0. It is clear for n = 1. If n > 1 set I = \mathfrak m^{n - 1} and \Lambda ' = \Lambda /I. By induction we may assume given \sigma ' : K \to \Lambda ' and k \subset F' \subset K finitely generated such that the image of k \to \Lambda \to \Lambda ' is contained in A' = \Psi _{\sigma '}(F'[x_1, \ldots , x_ d]). Denote \tau ' : k \to A' the induced map. Choose a lift \sigma : K \to \Lambda of \sigma ' (this is possible by the formal smoothness of K/\mathbf{F}_ p we mentioned above). For later reference we note that we can change \sigma to \sigma + D for some derivation D : K \to I. Set A = F[x_1, \ldots , x_ d]/(x_1, \ldots , x_ d)^ n. Then \Psi _\sigma induces a ring map \Psi _\sigma : A \to \Lambda . The composition with the quotient map \Lambda \to \Lambda ' induces a surjective map A \to A' with nilpotent kernel. Choose a lift \tau : k \to A of \tau ' (possible as k/\mathbf{F}_ p is formally smooth). Thus we obtain two maps k \to \Lambda , namely \Psi _\sigma \circ \tau : k \to \Lambda and the given map i : k \to \Lambda . These maps agree modulo I, whence the difference is a derivation \theta = i - \Psi _\sigma \circ \tau : k \to I. Note that if we change \sigma into \sigma + D then we change \theta into \theta - D|_ k.

Choose a set of elements \{ y_ j\} _{j \in J} of k whose differentials \text{d}y_ j form a basis of \Omega _{k/\mathbf{F}_ p}. The Jacobi-Zariski sequence for \mathbf{F}_ p \subset k \subset K is

0 \to H_1(L_{K/k}) \to \Omega _{k/\mathbf{F}_ p} \otimes K \to \Omega _{K/\mathbf{F}_ p} \to \Omega _{K/k} \to 0

As \dim H_1(L_{K/k}) < \infty we can find a finite subset J_0 \subset J such that the image of the first map is contained in \bigoplus _{j \in J_0} K\text{d}y_ j. Hence the elements \text{d}y_ j, j \in J \setminus J_0 map to K-linearly independent elements of \Omega _{K/\mathbf{F}_ p}. Therefore we can choose a D : K \to I such that \theta - D|_ k = \xi \circ \text{d} where \xi is a composition

\Omega _{k/\mathbf{F}_ p} = \bigoplus \nolimits _{j \in J} k \text{d}y_ j \longrightarrow \bigoplus \nolimits _{j \in J_0} k \text{d}y_ j \longrightarrow I

Let f_ j = \xi (\text{d}y_ j) \in I for j \in J_0. Change \sigma into \sigma + D as above. Then we see that \theta (a) = \sum _{j \in J_0} a_ j f_ j for a \in k where \text{d}a = \sum a_ j \text{d}y_ j in \Omega _{k/\mathbf{F}_ p}. Note that I is generated by the monomials \lambda ^ E = \lambda _1^{e_1} \ldots \lambda _ d^{e_ d} of total degree |E| = \sum e_ i = n - 1 in \lambda _1, \ldots , \lambda _ d. Write f_ j = \sum _ E c_{j, E} \lambda ^ E with c_{j, E} \in K. Replace F' by F = F'(c_{j, E}). Then the claim holds.

Choose \sigma and F as in the claim. The kernel of \Psi _\sigma is generated by finitely many polynomials g_1, \ldots , g_ t \in K[x_1, \ldots , x_ d] and we may assume their coefficients are in F after enlarging F by adjoining finitely many elements. In this case it is clear that the map A = F[x_1, \ldots , x_ d]/(g_1, \ldots , g_ t) \to K[x_1, \ldots , x_ d]/(g_1, \ldots , g_ t) = \Lambda is flat. By the claim A is a k-subalgebra of \Lambda . It is clear that \Lambda is the filtered colimit of these algebras, as K is the filtered union of the subfields F. Finally, these algebras are essentially of finite type over k by Algebra, Lemma 10.54.4. \square


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