# A Brief History Of Algebra And Computing [jnl article] by Bowen J. By Bowen J.

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Extra info for A Brief History Of Algebra And Computing [jnl article]

Example text

That is, U is open in X, f is a smooth proper morphism, X − U is a divisor with simple normal crossings ∪i Di , and all intersections DI = ∩i∈I Di are smooth over B. Let n be the dimension of the fibers. Define a modified pushforward g : CH j U → CH j B, Becker-Gottlieb transfer on Chow groups, by g (x) = f∗ (xcn (TX/B (− log D))) for any lift x of x to CH j X. Here the relative logarithmic tangent bundle TX/B (− log D) is a vector bundle of rank n on X. 1), to show that g is well-defined (independent of the lift x), it suffices to show that the formula gives zero for the pushforward to X of a cycle on Di for some i.

10 Questions about the Chow ring of a finite group 31 where |eir | = 2ir − 1 and |cir | = 2ir. If l = 2 and q ≡ 1 (mod 4), then ∗ ∼ HGL(n,F = F2 [e1 , e2 , . . en , c1 , c2 , . . , cn ]/(ei2 = 0), q) where |ei | = 2i − 1 and |ci | = 2i. If l = 2 and q ≡ 3 (mod 4), then ∗ ∼ HGL(n,F = F2 [e1 , e2 , . . en , c1 , c2 , . . , cn ]/ ei2 = q) i−1 ca c2i−1−a , a=0 where c0 = 1 and ci = 0 for i > n. The standard representation of GL(n, Fq ) on (Fq )n has a natural lift to a virtual complex representation of GL(n, Fq ), called the Brauer lift ρ; see Serre [124, theorem 43] or Benson [12, vol.

In particular, the ∗ description shows that if CHH is generated by transferred Euler classes, then ∗ so is CHZ/p H [138, section 11]. This gives the results we want on generation of the Chow ring by transferred Euler classes, and injectivity of the mod p ∗ cycle map. The description also shows that CHZ/p H maps onto the invariants ∗ Z/p (CHH p ) . ∗ Under the same assumptions on H , [138, section 9] also shows that CHZ/p H is detected on the subgroups H p and Z/p × H . 12 on the Chow K¨unneth formula, it follows by induction that CHG∗ is detected on elementary abelian subgroups for the groups in the theorem.