By Burichenko V.P.

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**Example text**

1 it is also A-primary and irreducible. 5 stable Gorenstein algebra over the Galois ﬁeld F q and I a P∗ -invariant A -primary ideal of A . Then I contains P∗ -invariant parameter ideals. It is irreducible if and only if for every P∗ -invariant parameter ideal Q I there is an element a A with Q : I = a + Q such that a becomes a Thom class in A/Q . 4. We can apply these results to the study of P∗ -Poincar´e duality quotient algebras of F q V as follows: let F q V H be a P∗ -Poincar´e duality quotient of F q V with I = ker F q V .

A n ≤ q − 1. Let z A h q be such a term of degree n q − 1 + q d. 2) we obtain zA · hq ∩ A = z ∩ = = q−1−a1 u1 · · · q−1 q−1 u1 · · · u1 · · · q−1−a1 q−1−a1 q−1 un q−1 q−1−a n u1 · · · u1 · · · un un q−1−a n q−1−a n un un q I · hq ∩ q · h ∩ · · q q h q h ∩ I I I I if deg h = d if deg h = d. Since h I it annihilates I under the ∩-product and I evaluates to zero on h. Hence the element q−1 u1 · · · q−1 u n q I belongs to I q ⊥n q−1 +q d as was to be shown. DEFINITION: If V is an n -dimensional vector space over the ﬁeld F of characteristic p , q is a power of p , and u1 , .

3: GL 2, F2 From this we derive the following formula for the number of isomorphism classes of Poincar´e duality quotients of F2 x, y of formal dimension d: 1 d+1 ✣ 2 + 3 · 2dimF2 F2 x, y d + 2 · 2dimF2 F2 x, y d − 1. 6 So we need to compute the dimensions of F2 x, y d and F2 x, y d . Let us handle the involution ﬁrst. Since permutes x and y the ring of invariants is F2 x + y, x y and this has Poincar´ e series P F2 x + y, x y , t = 1 = 1 − t 1 − t2 i + 1 t 2i + i=0 i + 1 t 2i+1 . i=0 Therefore we obtain dimF2 F2 x, y d = d + 1.