Given a standard graded polynomial ring over a commutative Noetherian ring
A
A
, we prove that the cohomological dimension and the height of the ideals defining any of its Veronese subrings are equal. This result is due to Ogus when
A
A
is a field of characteristic zero, and follows from a result of Peskine and Szpiro when
A
A
is a field of positive characteristic; our result applies, for example, when
A
A
is the ring of integers.