On modules over valuations

arXiv:1102.1241v2 [math.MG] 5 Apr 2011 On modules over valuations. Semyon Alesker ∗ Department of Mathematics, Tel Aviv University, Ramat Aviv 69978 Tel Aviv, Israel e-mail: semyon@post.tau.ac.il Abstract To any smooth manifold X an algebra of smooth valuations V ∞(X) was associated in [1]-[3],[5]. In this note we initiate a study of V ∞(X)-modules. More specifically we study finitely generated projective modules in analogy to the study of vector bundles on a manifold. In particular it is shown that for a compact manifold X there exists a canonical isomorphism between the K-ring constructed out of finitely generated projective V ∞(X)-modules and the classical topological K0-ring constructed out of vector bundles over X. 1 Introduction. Let X be a smooth manifold of dimension n.1 In [1]-[3],[5] the notion of a smooth valuation on X was introduced. Roughly put, a smooth valuation is a C-valued finitely additive measure on compact submanifolds of X with corners, which satisfies in addition some extra conditions. We omit here the precise description of the conditions due to their technical nature. Let us notice that basic examples of smooth valuations include any smooth measure on X and the Euler characteristic. There are many other natural examples of valuations coming from convexity, integral, and differential geometry. We refer to recent lecture notes [4], [8], [7] for an overview of the subject, examples, and applications. The space V ∞(X) of all smooth valuations is a Fr´echet space. It has a canonical product making V ∞(X) a commutative associative algebra over C with a unit element (which is the Euler characteristic). In this note we initiate a study of modules over V ∞(X). Our starting point is the analogy to the following well known fact due to Serre and Swan [11], [12]: if X is compact, then the category of smooth vector bundles of finite rank over X is equivalent to the category of finitely generated projective modules over the algebra C∞(X) of smooth functions (the functor in one direction is given by taking global smooth sections of a vector bundle). ∗Partially supported by ISF grant 701/08. 1All manifolds are assumed to be countable at infinity, i.e. presentable as a union of countably many compact subsets. In particular they are paracompact. 1 In order to state our main results we need to remind a few general facts about valuations on manifolds. We have a canonical homomorphism of algebras V ∞(X) →C∞(X) (1) given by the evaluation on points, i.e. φ 7→[x 7→φ({x})]. This is an epimorphism. The kernel, denoted by W1, is a nilpotent ideal of V ∞(X): (W1)n+1 = 0. Next, smooth valuations form a sheaf of algebras which is denoted by V∞ X : for an open subset U ⊂X, V∞ X (U) = V ∞(U), where the restriction maps are obvious. We denote by OX the sheaf of C∞-smooth functions on X. Then the map (1) gives rise to the epimorphism of sheaves V∞ X ։ OX. (2) Recall now the notion of a projective module. Let A be a commutative associative algebra with a unit. An A-module M is called projective if M is a direct summand of a free A-module, i.e. there exists an A-module N such that M ⊕N is a free A-module (not necessarily of finite rank). It is easy to see that if M is in addition finitely generated then M is a direct summand of a free A-module of finite rank. Let A be a sheaf of algebras on a topological space X. A sheaf M of A-modules is called a locally projective A-module if any point x ∈X has an open neighborhood U such that M(U) is a projective A(U)-module. Let us denote by ProjfV∞ X −mod the full subcategory of V∞ X -modules consisting of locally projective V∞ X -modules of finite rank. Let us denote by ProjfV ∞(X) −mod the full subcategory of the category of V ∞(X)-modules consisting of projective V ∞(X)-modules of finite rank. In Section 2 we prove the following result. 1.1 Theorem. Let X be a smooth manifold. (1) Any locally projective V∞ X -module of finite rank is locally free. (2) Assume in addition that X is compact. Let E be a locally free V∞ X -module of finite rank. Then there exists another locally free V∞ X -module H of finite rank such that E ⊕H is isomorphic to (V∞ X )N for some natural number N. (3) Assume again that X is compact. Then the functor of global sections Γ: ProjfV∞ X −mod →ProjfV ∞(X) −mod is an equivalence of categories. Notice that all the statements of the theorem are completely analogous to the classical situation of vector bundles (whose spaces of sections are projective finitely generated C∞(X)- modules). For example a version of (2) for vector bundles says that any vector bundle is a direct summand of a free bundle. A classical version of (3) is the above mentioned theorem 2 of Serre-Swan. The method of proof of Theorem 1.1 is a minor modification of the proof for the analogous statement for vector bundles. To formulate our next main result observe that to any V∞ X -module we can associate an OX-module via M 7→M ⊗V∞ X OX, (3) where OX is considered as V∞ X -module via the epimorphism (2). Clear

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