Lemma 62.3.1. Let K/k be a field extension. Let Z be an integral locally algebraic scheme over k. The multiplicity m_{Z', Z_ K} of an irreducible component Z' \subset Z_ K is 1 or a power of the characteristic of k.
62.3 Cycles relative to fields
Let k be a field. Let X be a locally algebraic scheme over k. Let r \geq 0 be an integer. In this setting we have the group Z_ r(X) of r-cycles on X, see Section 62.2.
Base change. For any field extension k'/k there is a base change map Z_ r(X) \to Z_ r(X_{k'}), see Chow Homology, Section 42.67. Namely, given an integral closed subscheme Z \subset X of dimension r we send [Z] \in Z_ r(X) to the r-cycle [Z_{k'}]_ r \in Z_ r(X_{k'}) associated to the closed subscheme Z_{k'} \subset X_{k'} (of course in general Z_{k'} is neither irreducible nor reduced). The base change map Z_ r(X) \to Z_ r(X_{k'}) is always injective.
Proof. If the characteristic of k is zero, then k is perfect and the multiplicity is always 1 since X_ K is reduced by Varieties, Lemma 33.6.4. Assume the characteristic of k is p > 0. Let L be the function field of Z. Since Z is locally algebraic over k, the field extension L/k is finitely generated. The ring K \otimes _ k L is Noetherian (Algebra, Lemma 10.31.8). Translated into algebra, we have to show that the length of the artinian local ring (K \otimes _ k L)_\mathfrak q is a power of p for every minimal prime ideal \mathfrak q.
Let L'/L be a finite purely inseparable extension, say of degree p^ n. Then K \otimes _ k L \subset K \otimes _ k L' is a finite free ring map of degree p^ n which induces a homeomorphism on spectra and purely inseparable residue field extensions. Hence for every minimal prime \mathfrak q as above there is a unique minimal prime \mathfrak q' \subset K \otimes _ k L' lying over it and
p^ n \text{length}((K \otimes _ k L)_\mathfrak q) = [\kappa (\mathfrak q') : \kappa (\mathfrak q)] \text{length}((K \otimes _ k L')_{\mathfrak q'})
by Algebra, Lemma 10.52.12 applied to M = (K \otimes _ k L')_{\mathfrak q'} \cong (K \otimes _ k L)_{\mathfrak q}^{\oplus p^ n}. Since [\kappa (\mathfrak q') : \kappa (\mathfrak q)] is a power of p we conclude that it suffices to prove the statement for L' and \mathfrak q'.
By the previous paragraph and Algebra, Lemma 10.45.3 we may assume that we have a subfield L/k'/k such that L/k' is separable and k'/k is finite purely inseparable. Then K \otimes _ k k' is an Artinian local ring. The argument of the preceding paragraph (applied to L = k and L' = k') shows that \text{length}(K \otimes _ k k') is a power of p. Since L/k' is the localization of a smooth k'-algebra (Algebra, Lemma 10.158.10). Hence S = (K \otimes _ k L)_\mathfrak q is the localization of a smooth R = K \otimes _ k k'-algebra at a minimal prime. Thus R \to S is a flat local homomorphism of Artinian local rings and \mathfrak m_ R S = \mathfrak m_ S. It follows from Algebra, Lemma 10.52.13 that \text{length}(K \otimes _ k k') = \text{length}(R) = \text{length}(S) = \text{length}((K \otimes _ k L)_\mathfrak q) and the proof is finished. \square
Lemma 62.3.2. Let k be a field of characteristic p > 0 with perfect closure k^{perf}. Let X be an algebraic scheme over k. Let r \geq 0 be an integer. The cokernel of the injective map Z_ r(X) \to Z_ r(X_{k^{perf}}) is a p-power torsion module (More on Algebra, Definition 15.88.1).
Proof. Since X is quasi-compact, the abelian group Z_ r(X) is free with basis given by the integral closed subschemes of dimension r. Similarly for Z_ r(X_{k^{perf}}). Since X_{k^{perf}} \to X is a homeomorphism, it follows that Z_ r(X) \to Z_ r(X_{k^{perf}}) is injective with torsion cokernel. Every element in the cokernel is p-power torsion by Lemma 62.3.1. \square
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