The Riemann-Roch without denominators can be expressed as follows: Let $f: X\rightarrow Y$ be a closed embedding of quasi-projective smooth $k$-varieties of codimension $d$ for some field $k$. Let $E$ be a vector bundle of rank $r$ on $X$. Then we have in $\mathrm{CH}^{\bullet}(Y)$ that \begin{equation*} c_{j}(f_{*}([E])) = \begin{cases} 0, & \text{if $0<j<d$}; \\ r(-1)^{d-1}(d-1)!\cdot[X], & \text{if $j=d$}. \end{cases} \end{equation*} This can be found in Fulton's book 'Intersection Theory', Theorem 15.3 and the examples following it. My question is:

Does this hold when $X$ is just a closed (not smooth) subvariety of $Y$ ($Y$ is still smooth)? I mainly want to know is it true that $$c_{j}(f_{*}([E])) = 0 \ \text{if $0<j<d$}?$$

I want to use this to show that the $p$-th Chern class maps the $p+1$-th filtration of $K$-group to $0$, as claimed in Example 15.3.6 in that book. All the sources I found don't mention this point or rather sloppy about this. Could you explain this, or give a proper reference? Thanks!

  • 3
    $\begingroup$ Because $Y$ is smooth, the natural map $\text{CH}^d(Y)\to \text{CH}_{n-d}(Y)$ is an isomorphism, $n=\text{dim}(Y)$. The singular locus $X_{\text{sing}}$ of your subvariety (not subscheme) has dimension $< n-d$. Thus, for the open subscheme $U\subset Y$ that is the complement of $X_{\text{sing}}$, the flat pullback map $\text{CH}_{n-d}(Y)\to \text{CH}_{n-d}(U)$ is an isomorphism. Since $X\cap U$ is smooth, the result is true in $\text{CH}_{n-d}(U)$. Thus, the result is also true in $\text{CH}^d(Y)$. $\endgroup$ May 25, 2017 at 10:54
  • $\begingroup$ Thanks a lot, this is perfect! It works for all $\mathrm{CH}_{j}$ with $0<j<=d$, together with the commutativity of push-forward and pull-back (in some pull-back diagram) for K-theory. $\endgroup$
    – Lao-tzu
    May 25, 2017 at 12:28
  • $\begingroup$ Sorry, in the above it should be $n−d<=j<n$ (in case others may doubt). $\endgroup$
    – Lao-tzu
    May 25, 2017 at 18:35

1 Answer 1



Does this hold when $X$ is just a closed (not smooth) subvariety of $Y$ ($Y$ is still smooth)?

Answer: No, there is not such a formula for a general closed immersion, regardless $Y$ being smooth or not. Note that for a general closed immersion $f_*$ is not defined.

Remark: For the Gysin morphism $f_*$ to be defined it is required that the closed immersion $f$ is regular. If $f$ is a regular closed immersion then the Riemann-Roch without denominators holds and it gives the same formula as in the smooth case. Reference are Grothendieck's Appendix at Expose 0 of SGA6 for $k$ of characteristic zero and a paper from Jouanolou for arbitrary characteristic (cf. Theorem 2.1)

  • $\begingroup$ I guess you are saying the formula $c(f_*E)=1+f_*(P(N, E))$ holds for regular closed immersions (of course, this formula makes sense only for regular immersions). I'm asking a slightly different thing, a corollary of it. If $f$ is only a closed immersion, $f_*$ can still be defined using finite locally free resolution, as $Y$ is smooth. The total Chern class can be defined for sheaves, hence the Chern classes in each degree. $\endgroup$
    – Lao-tzu
    Jun 10, 2017 at 21:07

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.