The Stacks project

Proof. This follows from the surjectivity of the map $g^*\mathcal{C}_{Z/X} \to \mathcal{C}_{Z'/X'}$ proved in Morphisms, Lemma 29.31.4. $\square$

Proof. Since both $i_* \circ j^* \circ C$ and $p^*$ are bivariant classes (see Lemmas 42.33.2 and 42.33.4) we can use the equation

(suitably interpreted) to define $c(Z \to X, \mathcal{N})$ as a bivariant class. This works because $p^*$ is always bijective on chow groups by Lemma 42.36.3.

Let $X' \to X$, $Z' \to X'$, and $\mathcal{N}'$ be as in Lemma 42.54.1. Write $c = c(Z \to X, \mathcal{N})$ and $c' = c(Z' \to X', \mathcal{N}')$. The second statement of the lemma means that $c'$ is the restriction of $c$ as in Remark 42.33.5. Since we claim this is true for all $X'/X$ locally of finite type, a formal argument shows that it suffices to check that $c' \cap \alpha ' = c \cap \alpha '$ for $\alpha ' \in \mathop{\mathrm{CH}}\nolimits _ k(X')$. To see this, note that we have a commutative diagram

which induces closed immersions:

To get $c \cap \alpha '$ we use the class $C \cap \alpha '$ defined using the morphism $W \times _{\mathbf{P}^1_ X} \mathbf{P}^1_{X'} \to \mathbf{P}^1_{X'}$ in Lemma 42.48.1. To get $c' \cap \alpha '$ on the other hand, we use the class $C' \cap \alpha '$ defined using the morphism $W' \to \mathbf{P}^1_{X'}$. By Lemma 42.48.3 the pushforward of $C' \cap \alpha '$ by the closed immersion $W'_\infty \to (W \times _{\mathbf{P}^1_ X} \mathbf{P}^1_{X'})_\infty $, is equal to $C \cap \alpha '$. Hence the same is true for the pullbacks to the opens

by Lemma 42.15.1. Since we have a commutative diagram

these classes pushforward to the same class on $N'$ which proves that we obtain the same element $c \cap \alpha ' = c' \cap \alpha '$ in $\mathop{\mathrm{CH}}\nolimits _*(Z')$. $\square$

Proof. Denote $N' \to N$ the closed immersion of vector bundles corresponding to the surjection $\mathcal{N}^\vee \to (\mathcal{N}')^\vee $. Then we have closed immersions

Thus the desired relationship between the bivariant classes follows immediately from Lemma 42.44.2. $\square$

Proof. By Lemma 42.54.2 we have $res(c(Z \to X, \mathcal{N})) = c(Z' \to X', g^*\mathcal{N})$ and the equality follows from Lemma 42.54.3. $\square$

Proof. The bivariant class $c_{top}(\mathcal{E}) \in A^*(Z \times _ X Y)$ was defined in Remark 42.38.11. By Lemma 42.54.2 we may replace $X$ by $Y$. Thus we may assume $Z \to X$ is a regular closed immersion of codimension $r$, we have $\dim _\delta (X) = n$, and we have to show that $c(Z \to X, \mathcal{N}) \cap [X]_ n = c_{top}(\mathcal{E}) \cap [Z]_{n - r}$ in $\mathop{\mathrm{CH}}\nolimits _*(Z)$. By Lemma 42.54.3 we may even assume $\mathcal{N}^\vee \to \mathcal{C}_{Z/X}$ is an isomorphism. In other words, we have to show $c(Z \to X, \mathcal{C}_{Z/X}^\vee ) \cap [X]_ n = [Z]_{n - r}$ in $\mathop{\mathrm{CH}}\nolimits _*(Z)$.

Let us trace through the steps in the definition of $c(Z \to X, \mathcal{C}_{Z/X}^\vee ) \cap [X]_ n$. Let $b : W \to \mathbf{P}^1_ X$ be the blowing up of $\infty (Z)$. We first have to compute $C \cap [X]_ n$ where $C \in A^0(W_\infty \to X)$ is the class of Lemma 42.48.1. To do this, note that $[W]_{n + 1}$ is a cycle on $W$ whose restriction to $\mathbf{A}^1_ X$ is equal to the flat pullback of $[X]_ n$. Hence $C \cap [X]_ n$ is equal to $i_\infty ^*[W]_{n + 1}$. Since $W_\infty $ is an effective Cartier divisor on $W$ we have $i_\infty ^*[W]_{n + 1} = [W_\infty ]_ n$, see Lemma 42.29.5. The restriction of this class to the open $C_ ZX \subset W_\infty $ is of course just $[C_ ZX]_ n$. Because $Z \subset X$ is regularly embedded we have

as graded $\mathcal{O}_ Z$-algebras, see Divisors, Lemma 31.21.5. Hence $p : N = C_ ZX \to Z$ is the structure morphism of the vector bundle associated to the finite locally free module $\mathcal{C}_{Z/X}$ of rank $r$. Then it is clear that $p^*[Z]_{n - r} = [C_ ZX]_ n$ and the proof is complete. $\square$

Proof. The statement makes sense as $Z \times _ X Y$ is a closed subscheme of $Y$. Because $\mathcal{N}$ has rank $r$ we know that $c(Z \to X, \mathcal{N}) \cap [Y]_ n$ is in $\mathop{\mathrm{CH}}\nolimits _{n - r}(Z \times _ X Y)$. Since $\dim _\delta (Z \cap Y) \leq n - r$ the chow group $\mathop{\mathrm{CH}}\nolimits _{n - r}(Z \times _ X Y)$ is freely generated by the cycle classes of the irreducible components $W \subset Z \times _ X Y$ of $\delta $-dimension $n - r$. Let $\xi \in W$ be the generic point. By assumption (2) we see that $\dim (\mathcal{O}_{Y, \xi }) = r$. On the other hand, since $\mathcal{N}$ has rank $r$ and since $\mathcal{N}^\vee \to \mathcal{C}_{Z/X}$ is surjective, we see that the ideal sheaf of $Z$ is locally cut out by $r$ equations. Hence the quasi-coherent ideal sheaf $\mathcal{I} \subset \mathcal{O}_ Y$ of $Z \times _ X Y$ in $Y$ is locally generated by $r$ elements. Since $\mathcal{O}_{Y, \xi }$ is Cohen-Macaulay of dimension $r$ and since $\mathcal{I}_\xi $ is an ideal of definition (as $\xi $ is a generic point of $Z \times _ X Y$) it follows that $\mathcal{I}_\xi $ is generated by a regular sequence (Algebra, Lemma 10.104.2). By Divisors, Lemma 31.20.8 we see that $\mathcal{I}$ is generated by a regular sequence over an open neighbourhood $V \subset Y$ of $\xi $. By our description of $\mathop{\mathrm{CH}}\nolimits _{n - r}(Z \times _ X Y)$ it suffices to show that $c(Z \to X, \mathcal{N}) \cap [V]_ n = [Z \times _ X V]_{n - r}$ in $\mathop{\mathrm{CH}}\nolimits _{n - r}(Z \times _ X V)$. This follows from Lemma 42.54.5 because the excess normal sheaf is $0$ over $V$. $\square$

Proof. We will use the criterion of Lemma 42.35.3. Thus we may assume that $X$ is an integral scheme and we have to show that $i^*[X]$ is equal to $c \cap [X]$. Let $n = \dim _\delta (X)$. As usual, there are two cases.

If $X = D$, then we see that both classes are represented by $c_1(\mathcal{N}) \cap [X]_ n$. See Lemma 42.54.5 and Definition 42.29.1.

If $D \not= X$, then $D \to X$ is an effective Cartier divisor and in particular a regular closed immersion of codimension $1$. Again by Lemma 42.54.5 we conclude $c(D \to X, \mathcal{N}) \cap [X]_ n = [D]_{n - 1}$. The same is true by definition for the gysin homomorphism and we conclude once again. $\square$

Proof. To check this we may use Lemma 42.35.3. Thus we may assume $X$ is an integral scheme and we have to show $c \cap c(Z \to X, \mathcal{N}) \cap [X] = c(Z \to X, \mathcal{N}) \cap c \cap [X]$ in $\mathop{\mathrm{CH}}\nolimits _*(Z \times _ X Y)$.

If $Z = X$, then $c(Z \to X, \mathcal{N}) = c_{top}(\mathcal{N})$ by Lemma 42.54.5 which commutes with the bivariant class $c$, see Lemma 42.38.9.

Assume that $Z$ is not equal to $X$. By Lemma 42.35.3 it even suffices to prove the result after blowing up $X$ (in a nonzero ideal). Let us blowup $X$ in the ideal sheaf of $Z$. This reduces us to the case where $Z$ is an effective Cartier divisor, see Divisors, Lemma 31.32.4,

If $Z$ is an effective Cartier divisor, then we have

where $i^* \in A^1(Z \to X)$ is the gysin homomorphism associated to $i : Z \to X$ (Lemma 42.33.3) and $\mathcal{E}$ is the dual of the kernel of $\mathcal{N}^\vee \to \mathcal{C}_{Z/X}$, see Lemmas 42.54.3 and 42.54.7. Then we conclude because Chern classes are in the center of the bivariant ring (in the strong sense formulated in Lemma 42.38.9) and $c$ commutes with the gysin homomorphism $i^*$ by definition of bivariant classes. $\square$

Proof. Denote $W \to \mathbf{P}^1_ X$ the blowing up of $\infty (Z)$ as in Section 42.53. Similarly, denote $W' \to \mathbf{P}^1_ X$ the blowing up of $\infty (Y)$. Since $\infty (Z) \subset \infty (Y)$ we get an opposite inclusion of ideal sheaves and hence a map of the graded algebras defining these blowups. This produces a rational morphism from $W$ to $W'$ which in fact has a canonical representative

See Constructions, Lemma 27.18.1. A local calculation (omitted) shows that $U$ contains at least all points of $W$ not lying over $\infty $ and the open subscheme $C_ Z X$ of the special fibre. After shrinking $U$ we may assume $U_\infty = C_ Z X$ and $\mathbf{A}^1_ X \subset U$. Another local calculation (omitted) shows that the morphism $U_\infty \to W'_\infty $ induces the canonical morphism $C_ Z X \to C_ Y X \subset W'_\infty $ of normal cones induced by the inclusion of ideals sheaves coming from $Z \subset Y$. Denote $W'' \subset W$ the strict transform of $\mathbf{P}^1_ Y \subset \mathbf{P}^1_ X$ in $W$. Then $W''$ is the blowing up of $\mathbf{P}^1_ Y$ in $\infty (Z)$ by Divisors, Lemma 31.33.2 and hence $(W'' \cap U)_\infty = C_ ZY$.

Consider the effective Cartier divisor $i : \mathbf{P}^1_ Y \to W'$ from (8) and its associated bivariant class $i^* \in A^1(\mathbf{P}^1_ Y \to W')$ from Lemma 42.33.3. We similarly denote $(i'_\infty )^* \in A^1(W'_\infty \to W')$ the gysin map at infinity. Observe that the restriction of $i'_\infty $ (Remark 42.33.5) to $U$ is the restriction of $i_\infty ^* \in A^1(W_\infty \to W)$ to $U$. On the one hand we have

because $i_\infty ^*$ kills all classes supported over $\infty $, because $i^*[U]$ and $[W'']$ agree as cycles over $\mathbf{A}^1$, and because $C_ ZY$ is the fibre of $W'' \cap U$ over $\infty $. On the other hand, we have

because $(i'_\infty )^*$ and $i^*$ commute (Lemma 42.30.5) and because the fibre of $i : \mathbf{P}^1_ Y \to W'$ over $\infty $ factors as $o : Y \to C_ YX$ and the open immersion $C_ YX \to W'_\infty $. The lemma follows. $\square$

Proof. Observe that the assumptions remain satisfied after any base change by a morphism $X' \to X$ which is locally of finite type (the short exact sequence of virtual normal sheaves is locally split hence remains exact after any base change). Thus to check the equality of bivariant classes we may use Lemma 42.35.3. Thus we may assume $X$ is an integral scheme and we have to show $c(Z \to X, \mathcal{N}) \cap [X] = c(Z \to Y, \mathcal{N}') \cap c(Y \to X, \mathcal{N}'') \cap [X]$.

If $Y = X$, then we have

The first equality by Lemma 42.54.3. The second because Chern classes commute with bivariant classes (Lemma 42.38.9). The third equality by Lemma 42.54.3.

Assume $Y \not= X$. By Lemma 42.35.3 it even suffices to prove the result after blowing up $X$ in a nonzero ideal. Let us blowup $X$ in the product of the ideal sheaf of $Y$ and the ideal sheaf of $Z$. This reduces us to the case where both $Y$ and $Z$ are effective Cartier divisors on $X$, see Divisors, Lemmas 31.32.4 and 31.32.12.

Denote $\mathcal{N}'' \to \mathcal{E}$ the surjection of finite locally free $\mathcal{O}_ Z$-modules such that $0 \to \mathcal{E}^\vee \to (\mathcal{N}'')^\vee \to \mathcal{C}_{Y/X} \to 0$ is a short exact sequence. Then $\mathcal{N} \to \mathcal{E}|_ Z$ is a surjection as well. Denote $\mathcal{N}_1$ the finite locally free kernel of this map and observe that $\mathcal{N}^\vee \to \mathcal{C}_{Z/X}$ factors through $\mathcal{N}_1$. By Lemma 42.54.3 we have

and

Since Chern classes of bundles commute with bivariant classes (Lemma 42.38.9) it suffices to prove

in $A^*(Z \to X)$. This we may assume that $\mathcal{N}'' = \mathcal{C}_{Y/X}$. This reduces us to the case discussed in the next paragraph.

In this paragraph $Z$ and $Y$ are effective Cartier divisors on $X$ integral of dimension $n$, we have $\mathcal{N}'' = \mathcal{C}_{Y/X}$. In this case $c(Y \to X, \mathcal{C}_{Y/X}^\vee ) \cap [X] = [Y]_{n - 1}$ by Lemma 42.54.5. Thus we have to prove that $c(Z \to X, \mathcal{N}) \cap [X] = c(Z \to Y, \mathcal{N}') \cap [Y]_{n - 1}$. Denote $N$ and $N'$ the vector bundles over $Z$ associated to $\mathcal{N}$ and $\mathcal{N}'$. Consider the commutative diagram

of cones and vector bundles over $Z$. Observe that $N'$ is a relative effective Cartier divisor in $N$ over $Z$ and that

is cartesian where $o$ is the zero section of the line bundle $C_ Y X$ over $Y$. By Lemma 42.54.9 we have $o^*[C_ ZX]_ n = [C_ Z Y]_{n - 1}$ in

By the cartesian property of the square above this implies that

in $\mathop{\mathrm{CH}}\nolimits _{n - 1}(N')$. Now observe that $\gamma = c(Z \to X, \mathcal{N}) \cap [X]$ and $\gamma ' = c(Z \to Y, \mathcal{N}') \cap [Y]_{n - 1}$ are characterized by $p^*\gamma = [C_ Z X]_ n$ in $\mathop{\mathrm{CH}}\nolimits _ n(N)$ and by $(p')^*\gamma ' = [C_ Z Y]_{n - 1}$ in $\mathop{\mathrm{CH}}\nolimits _{n - 1}(N')$. Hence the proof is finished as $i^* \circ p^* = (p')^*$ by Lemma 42.31.1. $\square$