The Decomposition of Hom _k (S, k) into Indecomposable Injectives

Abstract

Let S be an algebra essentially of finite type over a field k. Then, Hom _k (S, k) is an injective S–module, and the Matlis structure theorem (Matlis, E.: Pacific J. Math. 8, 511–528 1958) tells us that it can be written as a direct sum of indecomposable injectives. We compute the multiplicities of these injectives. Let 𝔭 be a prime ideal in S, and let k(𝔭) be the injective hull of S/𝔭. If the residue field k(𝔭) is algebraic over k, then the multiplicity of I(𝔭) is μ(𝔭) = 1. If the transcendence degree of k(𝔭) over k is ≥ 1, then $\mu (\mathfrak {p})\geq {|\#k|}^{\aleph _{0}}$, that is the multiplicity is no less than the cardinality of the field k raised to the power ℵ ₀. If S is finitely generated over k, then equality holds, that is, $\mu (\mathfrak {p}) = {|\#k|}^{\aleph _{0}}$. For k(𝔭) of transcendence degree ≤ 1, the result is not surprising, but for k(𝔭) of transcendence degree ≥ 2 it is not clear that μ(𝔭) ≠ 0. We prove the result by induction on the transcendence degree, and the key is that we produce an injective map, from a space whose dimension we know by induction and into the space whose dimension we want to estimate. The interest in the result comes from the fact that the size of μ(𝔭) measures the failure of a natural map ψ(f) : f ^× → f ^! to be an isomorphism. Here, f ^× and f ^! are the twisted inverse image functors of Grothendieck duality.