The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

Proposition 15.34.1. Let $k$ be a field of characteristic $p > 0$. Let $(A, \mathfrak m, K)$ be a Noetherian local $k$-algebra. The following are equivalent

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

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

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

  4. $A$ 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. Proof of (3) $\Rightarrow $ (1). Assume (3). Let $k \subset k'$ be a finite purely inseparable extension. Set $A' = A \otimes _ k k'$. This is a local ring with maximal ideal $\mathfrak m'$. Set $K' = A'/\mathfrak m'$. We get a commutative diagram

\[ \xymatrix{ 0 \ar[r] & H_1(L_{K/k}) \otimes K' \ar[r] \ar[d]_\beta & \mathfrak m/\mathfrak m^2 \otimes K' \ar[r] \ar[d] & \Omega _{A/k} \otimes _ A K' \ar[r] \ar[d]_{\cong } & \Omega _{K/k} \otimes K' \ar[r] \ar[d]_\alpha & 0 \\ & H_1(L_{K'/k'}) \ar[r] & \mathfrak m'/(\mathfrak m')^2 \ar[r] & \Omega _{A'/k'} \otimes _{A'} K' \ar[r] & \Omega _{K'/k'} \ar[r] & 0 } \]

with exact rows. The third vertical arrow is an isomorphism by base change for modules of differentials (Algebra, Lemma 10.130.12). Thus $\alpha $ is surjective. By Lemma 15.33.3 we have

\[ \dim \mathop{\mathrm{Ker}}(\alpha ) - \dim \mathop{\mathrm{Ker}}(\beta ) + \dim \mathop{\mathrm{Coker}}(\beta ) = 0 \]

(and these dimensions are all finite). A diagram chase shows that $\dim \mathfrak m'/(\mathfrak m')^2 \leq \dim \mathfrak m/\mathfrak m^2$. However, since $A \to A'$ is finite flat we see that $\dim (A) = \dim (A')$, see Algebra, Lemma 10.111.6. Hence $A'$ is regular by definition.

Equivalence of (3) and (4). Consider the Jacobi-Zariski sequences for rows of the commutative diagram

\[ \xymatrix{ \mathbf{F}_ p \ar[r] & A \ar[r] & K \\ \mathbf{F}_ p \ar[r] \ar[u] & k \ar[r] \ar[u] & K \ar[u] } \]

to get a commutative diagram

\[ \xymatrix{ 0 \ar[r] & \mathfrak m/\mathfrak m^2 \ar[r] & \Omega _{A/\mathbf{F}_ p} \otimes _ A K \ar[r] & \Omega _{K/\mathbf{F}_ p} \ar[r] & 0 & \\ 0 \ar[r] & H_1(L_{K/k}) \ar[r] \ar[u] & \Omega _{k/\mathbf{F}_ p} \otimes _ k K \ar[r] \ar[u] & \Omega _{K/\mathbf{F}_ p} \ar[r] \ar[u] & \Omega _{K/k} \ar[r] \ar[u] & 0 } \]

with exact rows. We have used that $H_1(L_{K/A}) = \mathfrak m/\mathfrak m^2$ and that $H_1(L_{K/\mathbf{F}_ p}) = 0$ as $K/\mathbf{F}_ p$ is separable, see Algebra, Proposition 10.152.9. Thus it is clear that the kernels of $H_1(L_{K/k}) \to \mathfrak m/\mathfrak m^2$ and $\Omega _{k/\mathbf{F}_ p} \otimes _ k K \to \Omega _{A/\mathbf{F}_ p} \otimes _ A K$ have the same dimension.

Proof of (2) $\Rightarrow $ (4) following Faltings, see [Faltings-einfacher]. Let $a_1, \ldots , a_ n \in k$ be elements such that $\text{d}a_1, \ldots , \text{d}a_ n$ are linearly independent in $\Omega _{k/\mathbf{F}_ p}$. Consider the field extension $k' = k(a_1^{1/p}, \ldots , a_ n^{1/p})$. By Algebra, Lemma 10.152.3 we see that $k' = k[x_1, \ldots , x_ n]/(x_1^ p - a_1, \ldots , x_ n^ p - a_ n)$. In particular we see that the naive cotangent complex of $k'/k$ is homotopic to the complex $\bigoplus _{j = 1, \ldots , n} k' \rightarrow \bigoplus _{i = 1, \ldots , n} k'$ with the zero differential as $\text{d}(x_ j^ p - a_ j) = 0$ in $\Omega _{k[x_1, \ldots , x_ n]/k}$. Set $A' = A \otimes _ k k'$ and $K' = A'/\mathfrak m'$ as above. By Algebra, Lemma 10.132.8 we see that $\mathop{N\! L}\nolimits _{A'/A}$ is homotopy equivalent to the complex $\bigoplus _{j = 1, \ldots , n} A' \rightarrow \bigoplus _{i = 1, \ldots , n} A'$ with the zero differential, i.e., $H_1(L_{A'/A})$ and $\Omega _{A'/A}$ are free of rank $n$. The Jacobi-Zariski sequence for $\mathbf{F}_ p \to A \to A'$ is

\[ H_1(L_{A'/A}) \to \Omega _{A/\mathbf{F}_ p} \otimes _ A A' \to \Omega _{A'/\mathbf{F}_ p} \to \Omega _{A'/A} \to 0 \]

Using the presentation $A[x_1, \ldots , x_ n] \to A'$ with kernel $(x_ j^ p - a_ j)$ we see, unwinding the maps in Algebra, Lemma 10.132.4, that the $j$th basis vector of $H_1(L_{A'/A})$ maps to $\text{d}a_ j \otimes 1$ in $\Omega _{A/\mathbf{F}_ p} \otimes A'$. As $\Omega _{A'/A}$ is free (hence flat) we get on tensoring with $K'$ an exact sequence

\[ K'^{\oplus n} \to \Omega _{A/\mathbf{F}_ p} \otimes _ A K' \xrightarrow {\beta } \Omega _{A'/\mathbf{F}_ p} \otimes _{A'} K' \to K'^{\oplus n} \to 0 \]

We conclude that the elements $\text{d}a_ j \otimes 1$ generate $\mathop{\mathrm{Ker}}(\beta )$ and we have to show that are linearly independent, i.e., we have to show $\dim (\mathop{\mathrm{Ker}}(\beta )) = n$. Consider the following big diagram

\[ \xymatrix{ 0 \ar[r] & \mathfrak m'/(\mathfrak m')^2 \ar[r] & \Omega _{A'/\mathbf{F}_ p} \otimes K' \ar[r] & \Omega _{K'/\mathbf{F}_ p} \ar[r] & 0 \\ 0 \ar[r] & \mathfrak m/\mathfrak m^2 \otimes K' \ar[r] \ar[u]^\alpha & \Omega _{A/\mathbf{F}_ p} \otimes K' \ar[r] \ar[u]^\beta & \Omega _{K/\mathbf{F}_ p} \otimes K' \ar[r] \ar[u]^\gamma & 0 } \]

By Lemma 15.33.1 and the Jacobi-Zariski sequence for $\mathbf{F}_ p \to K \to K'$ we see that the kernel and cokernel of $\gamma $ have the same finite dimension. By assumption $A'$ is regular (and of the same dimension as $A$, see above) hence the kernel and cokernel of $\alpha $ have the same dimension. It follows that the kernel and cokernel of $\beta $ have the same dimension which is what we wanted to show.

The implication (1) $\Rightarrow $ (2) is trivial. This finishes the proof of the proposition. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 07E5. Beware of the difference between the letter 'O' and the digit '0'.