Question in Fulton's Intersection Theory, Theorem 6.7 (b)

Question

I have a question about Theorem 6.7 (b) in Fulton's "Intersection Theory".

Notation: $X$ is a regularly embedded sub-scheme in $Y$ of co-dimension $d$. Let $\tilde Y\xrightarrow{f} Y$ be the blow-up of $Y$ along $X$ with exceptional divisor $\tilde X\xrightarrow{j}\tilde Y$. Then $f$ is a locally complete intersection (assume all schemes are quasi-projective) of codimension $0$, so we have the refined Gysin homomorphism $f^*: A_\bullet (Y)\rightarrow A_\bullet ({\tilde Y})$.

Now say $V$ is a $k$-dimensional sub-variety which is not contained in $X$. Let $\tilde V$ be the blowup of $V$ along $V\cap X$. The book claims that $f^*[V]=[\tilde V] +j_*(\tilde x) $ where $\tilde x$ is supported on $\tilde V\cap \tilde X$. I am unable to see this.

The book claims it follows from a previous theorem about refined Gysin commuting with flat pull-back so I suspect, one has to use the excision short exact sequence but I am unable to work out the details. Any help will be greatly appreciated. Thank you.

Answer 1

Here's a possible way out: Would be happy if someone checks out if it works.

We look at the commutative diagram $$\begin{array} A \tilde X & \stackrel{g}{\longrightarrow} & X \\ \downarrow{j} & & \downarrow{i} \\ \tilde Y & \stackrel{f}{\longrightarrow} & Y \end{array} $$ We show that $f^*[V]=[\tilde V]+j_* (\tilde x)$ where $\tilde x$ is supported on $g^{-1}(W)$ where $W=V\cap X$.

We see that $$ {\left (f^*[V]-[\tilde V] \right)}_{\tilde Y\backslash g^{-1}(W)}=\left (f^*[V] \right)_{f^{-1}(Y\backslash W)}-[\tilde V\cap f^{-1}(Y\backslash W)]= [f^{-1}(V\backslash W)]-[\tilde V\cap f^{-1}(Y\backslash W)]$$ Now we see that $\tilde V\cap f^{-1}(Y\backslash W)\subset f^{-1}(V\backslash W)$. For the other direction, we look at the commutative diagram keeping in mind that $f^{-1}(V\backslash W)\rightarrow V\backslash W$ is an isomorphism.

So $f^{-1}(V\backslash W)\subset \tilde V \cap f^{-1}(Y\backslash W)$ whence we get $ {\left (f^*[V]-[\tilde V] \right)}_{\tilde Y\backslash g^{-1}(W)}=0$. The result now follows from the excision exact sequence.