INTEGRAL INVARIANTS AND COHOMOLOGY OF BSpin(n): A CORRECTION

D. J. BENSON AND JAY A. WOOD

Jaume Aguadé has pointed out that the proof of Theorem 7.1 does not make sense. The problem is that the proposed short exact sequence is not exact. Here is a proof of the theorem:

(i) Let R be the ring ℤ[x_{1}^{2}, …, x_{r}^{2}, q_{1}, …, q_{r−2}, η_{r−1}] and S be the subring ℤ[p_{1}, …, p_{r}, q_{1}, …, q_{r−2}, η_{r−1}]. We claim that R is free as an S-module, with free basis the collection of elements of the form x_{1}^{2a_{1}} ⋯ x_{r}^{2a_{r}} with 0 ≤ a_{i} < i.

The fact that x_{i}^{2i} is equal to p_{1}x_{i}^{2i−2} −p_{2}x_{i}^{2i−4} +⋯±p_{i} ∓ the ith elementary symmetric function of x_{1}^{2}, …, x_{i−1}^{2}, x_{i+1}^{2}, …, x_{r}^{2} may be used inductively to show that the purported basis elements generate R as an S-module. To show that replacing x_{i}^{2i} by this expression is a process that must terminate in a finite number of steps, use a weighting on monomials where x_{1}^{2a_{1}} … x_{n}^{2a_{n}} has weight ∑ λ_{i}a_{i}, and where the λ_{i} are positive numbers chosen so that iλ_{i} is greater than the sum of any i of the λ_{j} with j ≠ i. For example, we could take λ_{i} = ∑_{j=i}^{n} j!.

If these elements were to satisfy a linear relation over S then the same linear relation would still hold after tensoring with ℚ and passing to fields of fractions, because R is torsion free. But ℚ⊗R = ℚ[x_{1}^{2}, …, x_{r}^{2}] and ℚ⊗S = ℚ[p_{1}, …, p_{r}], so no such linear relation holds by dimension counting.

It now follows that S is ℤ-pure in R (i.e., S = R ∩ ℚ ⊗ S). Since (ℚ ⊗ R)^{W} = ℚ ⊗ S, it follows that R^{W} = S.

The proof of (ii) is similar.

Department of Mathematics, University of Georgia, Athens, GA 30602, USA

  E-mail address: djb@byrd.math.uga.edu

Department of Mathematics, Computer Science and Statistics, Purdue University Calumet, Hammond, IN 46323, USA

  E-mail address: wood@calumet.purdue.edu