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The first published version of the Grothendieck-Riemann-Roch theorem, GRR for short, was written in the form $$ \operatorname{ch}(f_!\alpha).\operatorname{Td}(Y) = f_*\left(\operatorname{ch}(\alpha).\operatorname{Td}(X)\right) $$ for $f\colon X\to Y$ and $\alpha\in K(X),$ where of course $\operatorname{Td}(X)=\operatorname{td}(\mathscr T_X).$ However it seems to me that the vast majority of later publications, most notably Fulton's Intersection theory, the formulation $$ \operatorname{ch}(f_!\alpha) = f_*\left(\operatorname{ch}(\alpha).\operatorname{td}(\mathscr T_f)\right) $$ is used, where $\mathscr T_f$ is the relative tangent sheaf or something similar to it (the vitual tangent sheaf etc.).

Q1: Why is the second formulation preferable?

I would think the first one is better as it does not only measure to what extent the chern character fails to be a natural transformation between the covariant K-homology and Chow homology functors, but actually shows a canonical way how to modify it to be one.

Q2: Are the two formulations equivalent?

Surely they are when we are discussing proper maps between sonsingular quasprojective schemes over a field, but how about in more general contexts? I would think they are always equivalent, by the virue of the multiplicativity of the Todd character and the projection formula, but I am very likely missing something.

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    $\begingroup$ One reason is just because people may want to compute $ch(f_!a)$ and the second formulation tells you how to do that. $\endgroup$
    – Will Sawin
    Commented Apr 20, 2015 at 20:36
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    $\begingroup$ The second version is also more general. If $X$ or $Y$ are singular, then the terms in first statement are undefined, but $\mathcal{T}_f$ may still be defined, e.g. when $f$ is smooth and proper. $\endgroup$
    – Donu Arapura
    Commented Apr 21, 2015 at 6:26
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    $\begingroup$ Well, there are ways of defining Todd classes even for singular varieties. One can take the Baum-Fulton-MacPhearson natural transformation $\tau\colon K_0\to A_{\mathbb Q}$ and define the homology Todd class $\tau_X(\mathscr O_X).$ For locally complete intersections in nonsingular varieties that reduces to $\operatorname{td}(\mathscr T_X)\cap [X],$ the Todd class of the virtual tangent bundle capped with the fundamental class. Provided, GRR in the form above fails to hold for these new Todd classes, but still. $\endgroup$
    – A Rock and a Hard Place
    Commented Apr 26, 2015 at 20:33
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    $\begingroup$ In fact the Baum-Fulton-MacPhearson Riemann-Roch theory seems to me to be an argument for writing the theorem in the first version, aka. as exhibiting a natural transformation: that is the version that generalizes to nonsingular varieties and even to separated schemes of finite type over a field. $\endgroup$
    – A Rock and a Hard Place
    Commented Apr 26, 2015 at 20:37
  • $\begingroup$ Only a comment instead of an answer, but in the introduction of SGA 6 they write that they will use the second formulation because it is "more useful for their needs". I'm not familiar with this part of SGA 6 so I can't say exactly why how this is true, but maybe an answer is to be found there :-) $\endgroup$
    – Tim
    Commented Feb 19 at 22:24

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