### April 2009

Suppose you have a proper map $f:X \rightarrow Y$ between smooth (quasi) projective varieties.  Then suppose you have a coherent sheaf $F$ on $X$.  After viewing that sheaf as an element of the Grothendieck Group of coherent sheaves on $X$, there are two things you could do.  The first thing is that you could push forward this element to the Grothendieck Group on $Y$, and then take its chern character, to arrive at an element in the Chow Ring of $Y$.  The second option you have is to first take the chern character of $F$ in the Chow Ring of $X$, and then push it forward to the Chow Ring of $Y$.  Now we have two elements in the Chow Ring of $Y$ and we could wonder if we have the same element.  The Grothendieck Riemann Roch (GRR) Theorem tells us we don’t quite have the same element, but it tells us exactly by how much we are off!