TITLE: About Riemann-Roch without denominators
QUESTION [9 upvotes]: 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!

REPLY [2 votes]: 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)