  We compute the graded polynomial identities of the infinite dimensional upper
triangular matrix algebra over an arbitrary field. If the grading group is
finite, we prove that the set of graded polynomial identities admits a finite
basis. We find conditions under which a grading on such an algebra satisfies a
nontrivial graded polynomial identity. Finally, we provide examples showing
that two nonisomorphic gradings can have the same set of graded polynomial
identities.
