A mixed associativity formula for tensor products over two Lie-Rinehart algebras

A duality approach to the symmetry of Bernstein–Sato polynomials of free divisors

“…The following theorem gives the associativity law needed in the proof of the duality theorem (A.32). The proof we give here simplifies the original proof in [8].…”

“…This appendix contains some basic notions and results about Lie-Rinehart algebras and their modules, a simple proof of the associativity law in [8] (see Theorem (A.11)) and a detailed proof of the duality theorem in [8] (see Theorem (A.32)). To be brief, we have included only the statements and results strictly needed for the proof of the cited results and of Propositions (A.…”

“…The main ingredient of the proof is the explicit description of the D[s]-dual of D[s]h s by means of the logarithmic duality formula in [7,8].…”

“…In section §2 we recall the logarithmic Bernstein construction and we study the hypotheses we will need later to prove our main results. In section §3 we apply the duality formula in [8] to describe the D[s]-dual of D[s]h ϕ(s) , where ϕ is a C-algebra automorphism of C[s], under the hypotheses studied in section §2. In section §4 we prove the symmetry property b(−s − 2) = ±b(s) under the above hypotheses.…”

“…Finally, and for the ease of the reader, we have included an Appendix A with a detailed proof of the duality formula (A.32), and the needed notions and results about Lie-Rinehart algebras. This material includes a simple proof of the associativity law (see Theorem (A.11) and Corollary (A.14) ), the original proof in [8] being unpleasant.…”