Division theorem over the Dwork-Monsky-Washnitzer completion of polynomial rings and Weyl algebras. (English) Zbl 0944.13018

Caenepeel, Stefaan (ed.) et al., Rings, Hopf algebras, and Brauer groups. Proceedings of the fourth week on algebra and algebraic geometry, SAGA-4, Antwerp and Brussels, Belgium, September 12-17, 1996. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 197, 175-191 (1998).
The aim of this paper is to give proofs of the Weierstraß-Hironaka division theorem, in the sense of the book by I. M. Aroca, H. Hironaka and J. L. Vicente [“The theory of the maximal contact”, Mem. Mat. Inst. J. Juan 29 (Madrid 1975; Zbl 0366.32008)] over the Dwork-Monsky-Washnitzer completion [DMW-completion] of polynomial and Weyl algebras in several indeterminates over a complete discrete valuation ring \(W\) and its field of quotients. Let \(W\langle\underline {x}\rangle\) be the ring of strictly convergent power series over \(W\) in \(n\) indeterminates, and let \(W\langle \underline {x}\rangle^{†}\) be the \(W\)-subalgebra of \(W\langle \underline {x}\rangle\) of those power series \(f=\sum_{\alpha\in{\mathbb N}^n}f_\alpha\underline {x}^\alpha\) with \(v(f_\alpha)\geq\lambda|\alpha|\) for all large enough \(\alpha\) [here \(\lambda>0\) and \(v\) is the valuation defined by \(W\)]; \(W\langle \underline {x}\rangle^{†}\) is called the DMW-completion of the polynomial ring \(W[\underline {x}]\). Let \(\prec\) be a monomial order in \(\{1,\ldots,m\}\times {\mathbb N^n}\).
The main results of the first part are theorem 3.6 [which generalizes the Weierstraß division theorem for strictly convergent power series with coefficients in a complete ultrametric valued field: see S. Bosch, U. Güntzer and R. Remmert, “Non-archimedean analysis” (1984; Zbl 0539.14017); §5.2.1] and theorem 3.7:
Let \(\underline {f}^1,\ldots,\underline {f}^p\) be non-zero elements in \(K\langle \underline {x}\rangle^m\); for every \(\underline g\in K\langle\underline {x}\rangle^m\) there is a unique family of strict convergent power series \(q_1,\ldots,q_p\in K\langle \underline {x}\rangle\) and a unique vector \(\underline r\in K\langle \underline {x}\rangle^m\) such that \(\underline g=\sum_{i=1}^pq_i\underline f^i+\underline r\) where \(\underline r\) is “smaller” with respect to \(\prec\) than \((\underline f^1,\ldots,\underline f^p)\) [we do not give a precise formulation]. A similar result [cf. theorem 3.7] holds when \(K\langle \underline {x}\rangle\) is replaced by \(K\langle \underline {x}\rangle^{†}\) and \(\prec\) by the diagonal order \(<_d\). In particular, these results imply that \(W\langle \underline {x}\rangle\) and \(W\langle \underline {x}\rangle^{†}\) are noetherian. In the second part the author considers the Weyl algebra \(\text{W}_n(W)\), its completion \(\widehat{\text{W}_n(W)}\) and its DMW-completion \(\text{W}_n(W)^{†}\) [cf. W. Fulton, Bull. Am. Math. Soc. 75, 591-593 (1969; Zbl 0205.34303) and P. Monsky and G. Washnitzer, Ann. Math., II. Ser. 88, 181-217 (1968; Zbl 0162.52504)] and he proves in theorem 5.11 a similar theorem for the algebra \(\widehat{\text{W}_n(W)}\) and \(\text{W}_n(W)^{†}\).
For the entire collection see [Zbl 0884.00036].


13J07 Analytical algebras and rings
32B05 Analytic algebras and generalizations, preparation theorems
13J05 Power series rings
12J05 Normed fields
12J10 Valued fields
16W60 Valuations, completions, formal power series and related constructions (associative rings and algebras)