Arc-meromorphous functions

Wojciech Kucharz, Krzysztof Kurdyka
2020 Annales Polonici Mathematici  
We introduce arc-meromorphous functions, which are continuous functions representable as quotients of semialgebraic arc-analytic functions, and develop the theory of arc-meromorphous sheaves on Nash manifolds. Our main results are Cartan's theorems A and B for quasi-coherent arc-meromorphous sheaves. 0. Introduction. In this note, building on the theory of arc-analytic functions initiated by the second named author [16], we introduce arcmeromorphous functions and arc-meromorphous sheaves on
more » ... hous sheaves on Nash manifolds. Arc-meromorphous functions are analogs for regulous and Nash regulous functions studied in [8] and [13], respectively. The term "regulous" is derived from "regular" and "continuous", whereas "meromorphous" comes from "meromorphic" and "continuous". Our theory of arc-meromorphous sheaves is developed in parallel to the theories of regulous sheaves [8] (see also the recent survey [14] ) and Nash regulous sheaves [13] . It is established in [8] and [13] that Cartan's theorems A and B hold for quasi-coherent regulous sheaves and quasi-coherent Nash regulous sheaves. Our main results are Theorem 2.4 (Cartan's theorem A) and Theorem 2.5 (Cartan's theorem B) for quasi-coherent arc-meromorphous sheaves. Recall that Cartan's theorems A and B fail for coherent real algebraic sheaves [6, Example 12.1.5], [7, Theorem 1] and coherent Nash sheaves [11] . We refer to [6] for the general theory of semialgebraic sets, semialgebraic functions, and related concepts. Recall that a Nash manifold is an analytic submanifold X ⊂ R n , for some n, which is also a semialgebraic set. A realvalued function on X is called a Nash function if it is both analytic and semialgebraic. By [22, Theorem VI.2.1, Remark VI.2.11], each Nash manifold is Nash isomorphic to a nonsingular algebraic set in R m , for some m.
doi:10.4064/ap200517-7-8 fatcat:2qk5rsvj6bddbpz2hrrkk3g3zq