The Stacks project

43.7 Flat pullback

Suppose that $f : X \to Y$ is a flat morphism of varieties. By Morphisms, Lemma 29.28.2 every fibre of $f$ has dimension $r = \dim (X) - \dim (Y)$1. Let $Z \subset X$ be a $k$-dimensional closed subvariety. We define $f^*[Z]$ to be the $(k + r)$-cycle associated to the scheme theoretic inverse image: $f^*[Z] = [f^{-1}(Z)]_{k + r}$. Let $\alpha = \sum n_ i [Z_ i]$ be a $k$-cycle on $Y$. The pullback of $\alpha $ is the sum $f_* \alpha = \sum n_ i f^*[Z_ i]$ where each $f^*[Z_ i]$ is defined as above. This defines a homomorphism

\[ f^* : Z_ k(Y) \longrightarrow Z_{k + r}(X) \]

See Chow Homology, Section 42.14.

Lemma 43.7.1. Let $f : X \to Y$ be a flat morphism of varieties. Set $r = \dim (X) - \dim (Y)$. Then $f^*[\mathcal{F}]_ k = [f^*\mathcal{F}]_{k + r}$ if $\mathcal{F}$ is a coherent sheaf on $Y$ and the dimension of the support of $\mathcal{F}$ is at most $k$.

Proof. See Chow Homology, Lemma 42.14.4. $\square$

Lemma 43.7.2. Let $f : X \to Y$ and $g : Y \to Z$ be flat morphisms of varieties. Then $g \circ f$ is flat and $f^* \circ g^* = (g \circ f)^*$ as maps $Z_ k(Z) \to Z_{k + \dim (X) - \dim (Z)}(X)$.

Proof. Special case of Chow Homology, Lemma 42.14.3. $\square$

[1] Conversely, if $f : X \to Y$ is a dominant morphism of varieties, $X$ is Cohen-Macaulay, $Y$ is nonsingular, and all fibres have the same dimension $r$, then $f$ is flat. This follows from Algebra, Lemma 10.128.1 and Varieties, Lemma 33.20.4 showing $\dim (X) = \dim (Y) + r$.

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