On the algebraic and analytic 𝑞-De Rham complexes attached to 𝑞-difference equations

Roques, Julien

doi:10.1090/tran/8540

On the algebraic and analytic $q$-De Rham complexes attached to $q$-difference equations
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by Julien Roques PDF

Trans. Amer. Math. Soc. 375 (2022), 1461-1507

Abstract:

This paper is concerned with the algebraic and analytic $q$-de Rham complexes attached to linear $q$-difference operators with Laurent polynomial coefficients over the field of complex numbers. There is a natural morphism from the former to the latter complex. Whether or not it is a quasi-isomorphism, i.e., whether or not the induced morphisms on the corresponding cohomology spaces are isomorphisms, is the basic question considered in the present paper. We study this question following three distinct approaches. The first one is based on duality, and leads to a direct connection between the problem considered in the present paper and the convergence of formal series solutions of $q$-difference equations. The second approach is sheaf theoretic, based on growth considerations. The third one relies on the local analytic theory of $q$-difference equations. The paper ends with an extension of our results to variants of the above $q$-de Rham complexes when certain $q$-spirals of poles are allowed. Our study includes the case $\vert q \vert = 1$.

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