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Z n F. I p is a free F V I -module and the sequence splits as F V I -modules. Finally I p = n p − 1 + p · f dim F V = f dim F z1 , . . , z np I + f dim FV I . PROOF : As before let : F z1 , . . , z n be the map deﬁned by ing on F z1 , . . , z n F z1 , . . , z n f = f p for any f F z1 , . . , z n . The gradhas been arranged to make this a map of alge- §6] FROBENIUS POWERS 51 bras. 4 Since there is an inclusion I I p , there is also an induced p map of algebras F V I F V I given by f +I fp + I p , where f F V .

A generator for the dual principal system z1n , . . 1 tells us that e1 , . . , e n = n∩ n−1 u1 · · · n−1 un = sgn n−1 u1 , . . , u n u1 · · · 0 un n is a Macaulay dual for the ideal e1 , . . , e n generated by the elementary symmetric polynomials. The support of e1 , . . , en on the set of monomials is 0 n−1 {z1 · · · z n n } and hence the monomials in this set are precisely the monomials representing a fundamental class for F z1 , . . , z n n . As is well known when the ground ﬁeld F is of characteristic zero, n F z1 , .

These provide yet another perspective in our study, derived from one which was pioneered by Wu Wen-Ts¨ un [111]and [112] for smooth manifolds. Those Poincar´e duality quotients of F q V supporting Steenrod operations which have trivial Wu classes are closely related to the Hit Problem of ﬁnding a minimal generating set for F q V as a module over P∗ . 5) to study families of Poincar´ e duality quotients with trivial Wu classes at the end of this Part as well as in Parts IV, V and VI. g. [68] Chapter 8, [87] Chapters 10 and 11, or [90]).