Section 43.7 (0AZE): Flat pullback

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

$\dim (X) = \dim (Y) + r$ .

Comments (2)

Comment #8538 by Minki Lee on June 28, 2023 at 04:08

I think I see a typo in the definition; the pullback of $\alpha$ should be $f^{\ast}\alpha$ , not $f_{\ast}\alpha$ , right?

Comment #9124 by Stacks project on May 13, 2024 at 08:39

Thanks and fixed here.

The Stacks project

43.7 Flat pullback

Comments (2)

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