By Dagmar M. Meyer

ISBN-10: 0521850649

ISBN-13: 9780521850643

Poincaré duality algebras originated within the paintings of topologists at the cohomology of closed manifolds, and Macaulay's twin platforms within the research of irreducible beliefs in polynomial algebras. those principles are tied jointly utilizing simple commutative algebra related to Gorenstein algebras. Steenrod operations additionally originated in algebraic topology, yet could top be considered as a method of encoding the knowledge frequently hidden in the back of the Frobenius map in attribute p<>0. they supply a noncommutative software to check commutative algebras over a Galois box. during this Tract the authors skilfully collect those rules and practice them to difficulties in invariant thought. a few striking and unforeseen interdisciplinary connections are published that might curiosity researchers within the components of commutative algebra, invariant concept or algebraic topology.

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3 the ideal K : L is P∗ -invariant as both K and L are. The principal ideal generated by u, the image of a in A/K, is K : L K and is therefore a P∗ -invariant ideal. Hence u A/K is a Thom class. g. [89], and [38] and [113] for the group theoretic background). 2 The §1] ´ DUALITY QUOTIENTS OF F q V P∗ -UNSTABLE POINCARE 57 the image of a in A/K is P∗ -invariant. Since the quotient map A A/K is a map of P∗ -algebras, the preimage a + K of the ideal u is P∗ -invariant and hence so is L = K : a . 1 it is also A-primary and irreducible.

1 e1 , . . , e n , so z1n , . . , z nn e1 . . en . e1 , . . , e n . §5] THE K L PARADIGM: A COMPUTATIONAL TOOL 43 Set n −z1 . . −z n = det n−1 n−1 · · · −z1 . . . · · · −z n 1 . . 1 = −1 n 2 sgn z1 n−1 · · · zn 0 , n where the symmetric group n acts on the set {0, 1 , . . , n − 1} by permutation of the elements. is up to sign the discriminant and Cramer’s rule implies as in Example 1 that n z1n , . . , z nn : e1 , . . , e n . Note n has degree n2 which is also the degree of a transition element for e1 , .

1 it follows that I p is irreducible. Since F V I p is totally ﬁnite I p is also F V -primary. 5 has been proven by N. 1). 5 allows us to deduce much more about the nature of the relation between F V I and F V I p . 6 mensional F -vector space. Let e N0 and set q = p e . If I F V is an F V -primary irreducible ideal, then so is I q . If z1 , . . , z n V ∗ is a basis and z D a monomial representing a fundamental class of F V I then z1 · · · z n q−1 z q D represents a fundamental class of F V I q , so f dim F V I q = n q − 1 + q · f dim F V I .

### Poincare Duality Algebras, Macaulay's Dual Systems, and Steenrod Operations by Dagmar M. Meyer

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