Given a nonsingular projective variety $X$ with a close subvariety $Y \subset X$, let the inclusion map be $i : Y \rightarrow X$. Let $A(X)$ and $A(Y)$ be the Chow ring of $X$ and $Y$ respectively, is the push forward map $i\_\* : A(Y) \rightarrow A(X)$ a ring homomorphism? (ref: Hartshorne AG, appendix A, top of pg 429). Thanks.



Is the Chow ring's push forward of inclusion map a ring homomorphism?
No. Try $\{0\} \hookrightarrow {\mathbb P}^1$.`


