The RiemannRoch Theorem, the GrothendieckRiemannRoch Theorem , the GrothendieckHirzebruchRiemannRoch Theorem , all of them are well explained at Wikipedia .I would like to understand the meaning of the "RiemannRoch without denominators". Who first established such a theorem ? What was the motive ?

$\begingroup$ This doesn't look like a research question... $\endgroup$ – Fernando Muro Mar 26 '14 at 21:00
Some History
I believe the theorem was formulated by Grothendeick, Theorie des intersections et theoreme de RiemannRoch. Seminaire de geometrie algebrique du BoisMarie 1966/67 (SGA 6), Lect. Notes Math. 225, SpringerVerlag, BerlinHeidelbergNew York (1971), Expose XIV, (3.1), p. 670. It was proved by Jouanolou in Jouanolou, J.P., RiemannRoch sans denominateurs. (French) Invent. Math. 11, (1970) 1526. In Gillet, H., RiemannRoch theorems for higher algebraic Ktheory. Adv. Math. 40, (1981) 203289., Gillet extended it , but kept the name, to the Chern class maps for the higher algebraic Kgroups with values in cohomology theories satisfying certain axioms.
Meaning of the ''RiemannRoch without denominators''
Roughly, the following is what the RiemannRoch theorem without denominators means (from O. B. Podkopaev, E. K. Shinder, On the Riemann–Roch Theorem Without Denominators, St. Petersburg Math. J., 6 Vol. 18 (2007), No. 6, Pages 1021–1027):
The Riemann–Roch formula without denominators for a closed embedding $i : Y \hookrightarrow X$ of codimension $d$ expresses the Chern class $c_d(i_∗\mathcal{O}_Y)$ in terms of the class $[Y ] ∈ CH^d(X)$.
The following is the RiemannRoch theorem without denominators:
Theorem: Let $\frak X$ be a nonsingular variety over a field $\mathbb{F}$, and let $i:Y\hookrightarrow\frak X$ be a closed embedding of an irreducible subvariety $Y$, which has codimension $d$. Then, in $CH^d(\frak X)$, $$c_d(i_*\mathcal{O}_Y)=(1)^{d1}(d1)![Y]$$
The proof is very long and hence I refer you to the paper cited above, http://www.maths.ed.ac.uk/~aar/papers/gillet.pdf, and W. Fulton, Intersection Theory, Second Edition, SpringerVerlag, 1998.


$\begingroup$ Dear @Sanath Devalapurkar: To write bold text, you can use markdown as follows:
**bold text**
produces bold text. To produce headers, you can use# header
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. Mathjax$...$
code is not meant to format text. $\endgroup$ – Ricardo Andrade Mar 27 '14 at 20:51 
$\begingroup$ @RicardoAndrade Thanks  I will edit my answer. $\endgroup$ – user62675 Mar 27 '14 at 22:22

$\begingroup$ The mentioned "RiemannRoch theorem without denominators" by O. B. Podkopaev, E. K. Shinder (St. Petersburg Math. J., 6 Vol. 18 (2007), No. 6, Pages 1021–1027),could be renamed "RiemannRoch without Jouanolou" ! $\endgroup$ – AlAmrani Apr 7 '14 at 7:15

$\begingroup$ I am sorry for my uncalled (onthespot) reaction above. $\endgroup$ – AlAmrani Nov 28 '14 at 19:51
The theorem was conjectured by Grothendieck and proved by J.P. Jouanolou, RiemannRoch sans dénominateurs (1970). By clearing the denominators you can compute Chern classes, not just the Chern character.