Question

Suppose that $Y$ is a subvariety of $X$, and both are nonsingular. I'm interested in knowing a nice condition to make the inclusion $i\colon Y\to X$ induce an isomorphism of Chow rings $A^{.}(X)\cong A^{.}(Y)$. For example, if $X$ and $Y$ were spaces and I wanted an isomorphism in cohomology, I might hope to show that $Y$ is a deformation retract of $X$.

So my novice question (not very precise, sorry) is if intersection theory offers an analogous or at least comparably useful technique. I'd appreciate a pointer to an example, say, in Fulton's book. (Which, if I'd read and understood fully, of course, surely I could answer my own question!)

Answer 1

Hi Graham,

I can't think of too many interesting situations where this happens, but here is one. If $i:Y\to X$ is the inclusion of the zero section of an algebraic vector bundle, then restriction gives an isomorphism of Chow groups $CH^*(X)\cong CH^*(Y)$ (indexed by codimension). A proof can be found in [Fulton, chap 1, prop 1.9]. This fits with your intuition because Chow groups are $\mathbb{A}^1$ homotopy invariants ($CH^*(X\times \mathbb{A}^1)\cong CH^*(X)$), and $Y$ would be a deformation retract of $X$ in this sense.