By Burt Totaro

Crew cohomology unearths a deep courting among algebra and topology, and its fresh functions have supplied very important insights into the Hodge conjecture and algebraic geometry extra generally. This booklet provides a coherent suite of computational instruments for the research of team cohomology and algebraic cycles. Early chapters synthesize history fabric from topology, algebraic geometry, and commutative algebra so readers shouldn't have to shape connections among the literatures all alone. Later chapters exhibit Peter Symonds's influential evidence of David Benson's regularity conjecture, delivering a number of new editions and enhancements. whole with concrete examples and computations all through, and an inventory of open difficulties for additional learn, this booklet might be priceless to graduate scholars and researchers in algebraic geometry and comparable fields.

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**Extra resources for Group Cohomology and Algebraic Cycles**

**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.