The Stacks project

49.3 Discriminant of a finite locally free morphism

Let $X$ be a scheme and let $\mathcal{F}$ be a finite locally free $\mathcal{O}_ X$-module. Then there is a canonical trace map

\[ \text{Trace} : \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{F}) \longrightarrow \mathcal{O}_ X \]

See Exercises, Exercise 111.22.6. This map has the property that $\text{Trace}(\text{id})$ is the locally constant function on $\mathcal{O}_ X$ corresponding to the rank of $\mathcal{F}$.

Let $\pi : X \to Y$ be a morphism of schemes which is finite locally free. Then there exists a canonical trace for $\pi $ which is an $\mathcal{O}_ Y$-linear map

\[ \text{Trace}_\pi : \pi _*\mathcal{O}_ X \longrightarrow \mathcal{O}_ Y \]

sending a local section $f$ of $\pi _*\mathcal{O}_ X$ to the trace of multiplication by $f$ on $\pi _*\mathcal{O}_ X$. Over affine opens this recovers the construction in Exercises, Exercise 111.22.7. The composition

\[ \mathcal{O}_ Y \xrightarrow {\pi ^\sharp } \pi _*\mathcal{O}_ X \xrightarrow {\text{Trace}_\pi } \mathcal{O}_ Y \]

equals multiplication by the degree of $\pi $ (which is a locally constant function on $Y$). In analogy with Fields, Section 9.20 we can define the trace pairing

\[ Q_\pi : \pi _*\mathcal{O}_ X \times \pi _*\mathcal{O}_ X \longrightarrow \mathcal{O}_ Y \]

by the rule $(f, g) \mapsto \text{Trace}_\pi (fg)$. We can think of $Q_\pi $ as a linear map $\pi _*\mathcal{O}_ X \to \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ Y}(\pi _*\mathcal{O}_ X, \mathcal{O}_ Y)$ between locally free modules of the same rank, and hence obtain a determinant

\[ \det (Q_\pi ) : \wedge ^{top}(\pi _*\mathcal{O}_ X) \longrightarrow \wedge ^{top}(\pi _*\mathcal{O}_ X)^{\otimes -1} \]

or in other words a global section

\[ \det (Q_\pi ) \in \Gamma (Y, \wedge ^{top}(\pi _*\mathcal{O}_ X)^{\otimes -2}) \]

The discriminant of $\pi $ is by definition the closed subscheme $D_\pi \subset Y$ cut out by this global section. Clearly, $D_\pi $ is a locally principal closed subscheme of $Y$.

Lemma 49.3.1. Let $\pi : X \to Y$ be a morphism of schemes which is finite locally free. Then $\pi $ is étale if and only if its discriminant is empty.

Proof. By Morphisms, Lemma 29.36.8 it suffices to check that the fibres of $\pi $ are étale. Since the construction of the trace pairing commutes with base change we reduce to the following question: Let $k$ be a field and let $A$ be a finite dimensional $k$-algebra. Show that $A$ is étale over $k$ if and only if the trace pairing $Q_{A/k} : A \times A \to k$, $(a, b) \mapsto \text{Trace}_{A/k}(ab)$ is nondegenerate.

Assume $Q_{A/k}$ is nondegenerate. If $a \in A$ is a nilpotent element, then $ab$ is nilpotent for all $b \in A$ and we conclude that $Q_{A/k}(a, -)$ is identically zero. Hence $A$ is reduced. Then we can write $A = K_1 \times \ldots \times K_ n$ as a product where each $K_ i$ is a field (see Algebra, Lemmas 10.53.2, 10.53.6, and 10.25.1). In this case the quadratic space $(A, Q_{A/k})$ is the orthogonal direct sum of the spaces $(K_ i, Q_{K_ i/k})$. It follows from Fields, Lemma 9.20.7 that each $K_ i$ is separable over $k$. This means that $A$ is étale over $k$ by Algebra, Lemma 10.143.4. The converse is proved by reading the argument backwards. $\square$

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