Theorem 15.40.1. Let $k$ be a field. Let $(A, \mathfrak m, K)$ be a Noetherian local $k$-algebra. If the characteristic of $k$ is zero then the following are equivalent

1. $A$ is a regular local ring, and

2. $k \to A$ is formally smooth in the $\mathfrak m$-adic topology.

If the characteristic of $k$ is $p > 0$ then the following are equivalent

1. $A$ is geometrically regular over $k$,

2. $k \to A$ is formally smooth in the $\mathfrak m$-adic topology.

3. for all $k \subset k' \subset k^{1/p}$ finite over $k$ the ring $A \otimes _ k k'$ is regular,

4. $A$ is regular and the canonical map $H_1(L_{K/k}) \to \mathfrak m/\mathfrak m^2$ is injective, and

5. $A$ is regular and the map $\Omega _{k/\mathbf{F}_ p} \otimes _ k K \to \Omega _{A/\mathbf{F}_ p} \otimes _ A K$ is injective.

Proof. If the characteristic of $k$ is zero, then the equivalence of (1) and (2) follows from Lemmas 15.38.2 and 15.38.5.

If the characteristic of $k$ is $p > 0$, then it follows from Proposition 15.35.1 that (1), (3), (4), and (5) are equivalent. Assume (2) holds. By Lemma 15.37.8 we see that $k' \to A' = A \otimes _ k k'$ is formally smooth for the $\mathfrak m' = \mathfrak mA'$-adic topology. Hence if $k \subset k'$ is finite purely inseparable, then $A'$ is a regular local ring by Lemma 15.38.2. Thus we see that (1) holds.

Finally, we will prove that (5) implies (2). Choose a solid diagram

$\xymatrix{ A \ar[r]_{\bar\psi } \ar@{-->}[rd] & B/J \\ k \ar[u]^ i \ar[r]^\varphi & B \ar[u]_\pi }$

as in Definition 15.37.1. As $J^2 = 0$ we see that $J$ has a canonical $B/J$ module structure and via $\bar\psi$ an $A$-module structure. As $\bar\psi$ is continuous for the $\mathfrak m$-adic topology we see that $\mathfrak m^ nJ = 0$ for some $n$. Hence we can filter $J$ by $B/J$-submodules $0 \subset J_1 \subset J_2 \subset \ldots \subset J_ n = J$ such that each quotient $J_{t + 1}/J_ t$ is annihilated by $\mathfrak m$. Considering the sequence of ring maps $B \to B/J_1 \to B/J_2 \to \ldots \to B/J$ we see that it suffices to prove the existence of the dotted arrow when $J$ is annihilated by $\mathfrak m$, i.e., when $J$ is a $K$-vector space.

Assume given a diagram as above such that $J$ is annihilated by $\mathfrak m$. By Lemma 15.38.5 we see that $\mathbf{F}_ p \to A$ is formally smooth in the $\mathfrak m$-adic topology. Hence we can find a ring map $\psi : A \to B$ such that $\pi \circ \psi = \bar\psi$. Then $\psi \circ i, \varphi : k \to B$ are two maps whose compositions with $\pi$ are equal. Hence $D = \psi \circ i - \varphi : k \to J$ is a derivation. By Algebra, Lemma 10.131.3 we can write $D = \xi \circ \text{d}$ for some $k$-linear map $\xi : \Omega _{k/\mathbf{F}_ p} \to J$. Using the $K$-vector space structure on $J$ we extend $\xi$ to a $K$-linear map $\xi ' : \Omega _{k/\mathbf{F}_ p} \otimes _ k K \to J$. Using (5) we can find a $K$-linear map $\xi '' : \Omega _{A/\mathbf{F}_ p} \otimes _ A K$ whose restriction to $\Omega _{k/\mathbf{F}_ p} \otimes _ k K$ is $\xi '$. Write

$D' : A \xrightarrow {\text{d}} \Omega _{A/\mathbf{F}_ p} \to \Omega _{A/\mathbf{F}_ p} \otimes _ A K \xrightarrow {\xi ''} J.$

Finally, set $\psi ' = \psi - D' : A \to B$. The reader verifies that $\psi '$ is a ring map such that $\pi \circ \psi ' = \bar\psi$ and such that $\psi ' \circ i = \varphi$ as desired. $\square$

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