The Decomposition of Hom k ( S , k ) into Indecomposable Injectives
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The Decomposition of Homk (S, k) into Indecomposable Injectives
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Amnon Neeman
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Received: 26 April 2014 / Accepted: 9 May 2014 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2015
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Abstract Let S be an algebra essentially of finite type over a field k. Then, Homk (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 I (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 μ(p) ≥ |#k|ℵ0 , that is the multiplicity is no less than the cardinality of the field k raised to the power ℵ0 . If S is finitely generated over k, then equality holds, that is, μ(p) = |#k|ℵ0 . 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.
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Keywords Injective modules · Commutative rings
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Mathematics Subject Classification (2010) 13C11
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1 Introduction
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Let S be an algebra essentially of finite type over a field k. If M is a flat S–module, then Homk (M, k) is an injective S–module, in particular Homk (S, k) is injective. Since S is noetherian, the injective module Homk (S, k) can be decomposed as a direct sum of
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A. Neeman () Centre for Mathematics and its Applications, Mathematical Sciences Institute, John Dedman Building, The Australian National University, Canberra, ACT 2601, Australia e-mail: [email protected]
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indecomposable injectives. The indecomposable injectives are all of the form I (p), where p ⊂ S is a prime ideal and I (p) is the injective hull of S/p. In other words, classical theory, which may be found (for example) in Lam [6, Section 3F], tells us that there exists a decomposition Homk (S, k) ∼ I (p)μ(p) = p
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and it becomes interesting to compute the μ(p). In this article, we do that.
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Reminder 1.1 Let k(p) be the residue field at p, that is, the quotient field of the integral domain S/p. Adopt the convention that, for any S–module M, we denote by Mp the localization of M at p. By [3, Theorem 3.2.8], the multiplicity μ(p) is equal to the dimension of the k(p)–vector space HomSp k(p), Homk (S, k)p . The isomorphisms HomSp k(p), Homk (S, k)p ∼ =
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