3 - Divisor Classes, Ramification and Hyperplanes

Summary

In this chapter, we calculate the divisor class group Cl(K[V]^G). The approach uses Samuel's theory of Galois descent [86]. The final theorem, proved in Section 3.9, states that Cl(K[V]^G) is isomorphic to the subgroup of Hom(G,K^×) consisting of those homomorphisms which take the value one on every pseudoreflection. As a consequence, we prove a theorem of Nakajima, which states that K[V]^G is a unique factorization domain if and only if this subgroup of Hom(G,K^×) is trivial. In contrast, the Picard group Pic(K[V]^G) is always trivial. We prove this theorem of Kang in Section 3.6.

We also give a ramification formula for the invariant ψ introduced in Section 2.4, and use this to prove the Carlisle–Kropholler conjecture in Section 3.13.

We begin with some generalities on divisors. Our treatment follows Samuel [86], with the exception that there are some minor simplifications arising from the fact that we are not interested in non-Noetherian rings here.

Divisors

We say that a ring A is a normal domain if it is a commutative Noetherian integrally closed domain (these hypotheses are satisfied by K[V] and K[V]^G by Proposition 1.1.1 and Theorem 1.3.1). Let A be a normal domain, and let L be the field of fractions of A. A fractional ideal a of A is a non-zero A-submodule of L with the property that there exists a non-zero element x ∈ A with xa ⊆ A.