Clarification on statement of a decomposition for graded ring

Question

Above is a decomposition of an element $f \in R$ for $R$ a graded ring. I am confused by the statement $f_j = 0$ for $j.$ Did Professor Eisenbud mean for all but finitely many $j$ as would be the case for infinite direct sums?

More context.

Answer 1

This is just a typo, and it should read

... and $f_j=0$ for all but finitely many $j$;