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*}

15.33 Cartier's equality and geometric regularity

A reference for this section and the next is [Section 39, MatCA]. In order to comfortably read this section the reader should be familiar with the naive cotangent complex and its properties, see Algebra, Section 10.132.

Lemma 15.33.1 (Cartier equality). Let $K/k$ be a finitely generated field extension. Then $\Omega _{K/k}$ and $H_1(L_{K/k})$ are finite dimensional and $\text{trdeg}_ k(K) = \dim _ K \Omega _{K/k} - \dim _ K H_1(L_{K/k})$.

Proof. We can find a global complete intersection $A = k[x_1, \ldots , x_ n]/(f_1, \ldots , f_ c)$ over $k$ such that $K$ is isomorphic to the fraction field of $A$, see Algebra, Lemma 10.152.11 and its proof. In this case we see that $\mathop{N\! L}\nolimits _{K/k}$ is homotopy equivalent to the complex

\[ \bigoplus \nolimits _{j = 1, \ldots , c} K \longrightarrow \bigoplus \nolimits _{i = 1, \ldots , n} K\text{d}x_ i \]

by Algebra, Lemmas 10.132.2 and 10.132.13. The transcendence degree of $K$ over $k$ is the dimension of $A$ (by Algebra, Lemma 10.115.1) which is $n - c$ and we win. $\square$

Lemma 15.33.2. Let $K \subset L \subset M$ be field extensions. Then the Jacobi-Zariski sequence

\[ 0 \to H_1(L_{L/K}) \otimes _ L M \to H_1(L_{M/K}) \to H_1(L_{M/L}) \to \Omega _{L/K} \otimes _ L M \to \Omega _{M/K} \to \Omega _{M/L} \to 0 \]

is exact.

Lemma 15.33.3. Given a commutative diagram of fields

\[ \xymatrix{ K \ar[r] & K' \\ k \ar[u] \ar[r] & k' \ar[u] } \]

with $k \subset k'$ and $K \subset K'$ finitely generated field extensions the kernel and cokernel of the maps

\[ \alpha : \Omega _{K/k} \otimes _ K K' \to \Omega _{K'/k'} \quad \text{and}\quad \beta : H_1(L_{K/k}) \otimes _ K K' \to H_1(L_{K'/k'}) \]

are finite dimensional and

\[ \dim \mathop{\mathrm{Ker}}(\alpha ) - \dim \mathop{\mathrm{Coker}}(\alpha ) -\dim \mathop{\mathrm{Ker}}(\beta ) + \dim \mathop{\mathrm{Coker}}(\beta ) = \text{trdeg}_ k(k') - \text{trdeg}_ K(K') \]

Proof. The Jacobi-Zariski sequences for $k \subset k' \subset K'$ and $k \subset K \subset K'$ are

\[ 0 \to H_1(L_{k'/k}) \otimes K' \to H_1(L_{K'/k}) \to H_1(L_{K'/k'}) \to \Omega _{k'/k} \otimes K' \to \Omega _{K'/k} \to \Omega _{K'/k} \to 0 \]

and

\[ 0 \to H_1(L_{K/k}) \otimes K' \to H_1(L_{K'/k}) \to H_1(L_{K'/K}) \to \Omega _{K/k} \otimes K' \to \Omega _{K'/k} \to \Omega _{K'/K} \to 0 \]

By Lemma 15.33.1 the vector spaces $\Omega _{k'/k}$, $\Omega _{K'/K}$, $H_1(L_{K'/K})$, and $H_1(L_{k'/k})$ are finite dimensional and the alternating sum of their dimensions is $\text{trdeg}_ k(k') - \text{trdeg}_ K(K')$. The lemma follows. $\square$


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