Lemma 37.75.1 (0F39)—The Stacks project

Lemma 37.75.1. Given a cartesian square

\[ \xymatrix{ U \ar[d]_\pi & U' \ar[l]^ h \ar[d]^{\pi '} \\ V & V' \ar[l]_ g } \]

with $\pi $ locally quasi-finite with finite fibres and a function $w : U \to \mathbf{Z}$ we have $(\int _\pi w) \circ g = \int _{\pi '} (w \circ h)$.

Proof. This follows immediately from the second description of $\int _\pi w$ given above. To prove it from the definition, you use that if $E/F$ is a finite extension of fields and $F'/F$ is another field extension, then writing $(E \otimes _ F F')_{red} = \prod E'_ i$ as a product of fields finite over $F'$, we have

\[ [E : F]_ s = \sum [E'_ i : F']_ s \]

To prove this equality pick an algebraically closed field extension $\Omega /F'$ and observe that

\begin{align*} [E : F]_ s & = |\mathop{\mathrm{Mor}}\nolimits _ F(E, \Omega )| \\ & = |\mathop{\mathrm{Mor}}\nolimits _{F'}(E \otimes _ F F', \Omega )| \\ & = |\mathop{\mathrm{Mor}}\nolimits _{F'}((E \otimes _ F F')_{red}, \Omega )| \\ & = \sum |\mathop{\mathrm{Mor}}\nolimits _{F'}(E'_ i, \Omega )| \\ & = \sum [E'_ i : F']_ s \end{align*}

where we have used Fields, Lemma 9.14.8. $\square$

The Stacks project

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