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Theory of the Bernstein-Sato polynomial for Tate and Dwork-Monsky-Washnitzer algebras. (La théorie du polynôme de Bernstein-Sato pour les algèbres de Tate et de Dwork-Monsky-Washnitzer.) (French) Zbl 0765.14009
Let \(k\) be a field of characteristic zero, \(A=k[x_ 1,\dots,x_ n]\) the \(k\)-algebra of polynomials in \(n\) indeterminates and \(D=D_{A/k}=A[\partial_ 1,\dots,\partial_ n]\). Let \(s\) be an indeterminate over \(k\); for every non-zero \(f\in A\) denote by \({\mathbf F}[s]\) the free \(A_ f[s]\)-module generated by a symbol \({\mathbf f}^ s\); \({\mathbf F}[s]\) carries a canonical structure as \(D_ f[s]\)-module defined by \(\partial g{\mathbf f}^ s=(\partial g+sgf^{-1}){\mathbf f}^ s\) for every \(\partial\in\text{Der}_ k(A)\) and every \(g\in A_ f[s]\). I. N. Bernstein has shown [cf. Funct. Anal. Appl. 6(1972), 273-285 (1973); translation from Funkts. Anal. Prilozh. 6, No. 4, 26-40 (1972; Zbl 0282.46038)] that there exists a polynomial \(b(s)\in k[s]\backslash\{0\}\) and a differential operator \(P(s)\in D(s)\) such that \(b(s){\mathbf f}^ s=P(s)f{\mathbf f}^ s\). This result plays an important role in the theory of \({\mathcal D}_ X\)-modules, \(X\) a variety over \(k\) or \(k=\mathbb{C}\) and \(X\) an analytic variety. In particular, it can be used to prove the finiteness of De Rham cohomology for non-singular varieties. The authors of the present paper generalize the functional equation given above for the case of Tate algebras and Dwork-Monsky-Washnitzer algebras. They consider commutative \(k\)-algebras \(A\) which are noetherian and regular, equicodimensional of dimension \(n\), such that \(A/{\mathfrak m}\) is algebraic over \(k\) for every maximal ideal \({\mathfrak m}\) of \(A\) and such that there exist \(x_ 1,\dots,x_ n\in A\) and \(\partial_ 1,\dots,\partial_ n\in\text{Der}_ k(A)\) satisfying \(\partial_ i(x_ j)=\delta_{ij}\). They prove a functional equation as above [theorem 3.1.1]; in particular, for every \(D\)-module of minimal dimension, \(M_ f\) is a \(D\)-module of finite type. In section 4 the authors apply this result to the algebraic and formal case, and to the case of Tate-algebras and algebras of Dwork-Monsky-Washnitzer.

MSC:
14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
14F40 de Rham cohomology and algebraic geometry
32C38 Sheaves of differential operators and their modules, \(D\)-modules
14G20 Local ground fields in algebraic geometry
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