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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!

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    $\begingroup$ 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 $\text{CH}^d(Y)$. $\endgroup$ – Jason Starr May 25 '17 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 '17 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 '17 at 18:35
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Question:

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)

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  • $\begingroup$ 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 '17 at 21:07

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