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Traces and differential operators over Beilinson completion algebras. (English) Zbl 0856.13019
Let \(k\) be a perfect field. The author calls a field \(K\) together with a topology and valuation rings \({\mathcal O}_1, \dots, {\mathcal O}_n\) a topological local field (TLF) over \(k\) if the following is true:
The residue field \(\kappa_i\) of \({\mathcal O}_i\) is the fraction field of \({\mathcal O}_{i+1}\) and \(K\) is the fraction field of \({\mathcal O}_1\);
there is a discrete field \(F\) such that \(\Omega^1_{F/k}\) has finite rank and an isomorphism \(K \cong F((t_1, \dots, t_n))\) of semi-topological \(k\)-algebras inducing \({\mathcal O}_i \cong F((t_{i+1}, \dots, t_n)) [[t_i]]\).
A commutative semi-topological local ring \(A\) together with a structure of TLF over \(k\) on the residue field \(A/{\mathfrak m}\) is called Beilinson completion algebra (BCA) if there is a strict surjective homomorphism of \(k\)-algebras \(F((s_1, \dots, s_m)) [[t_1, \dots, t_n]] \to A\) inducing an isomorphism \(F((s_1, \dots, s_m)) \cong A/{\mathfrak m}\) of TLF’s. A (general) BCA is a finite product of local BCA’s. A finite type ST module over a BCA \(A\) is a quotient of \(A^n\) for some \(n\) with the quotient topology.
In this paper the general properties of these notions are exploited. Among other things for a finite type ST module over a local BCA the concept of a dual is introduced and its relationship with the Matlis dual is examined. For morphisms of BCA’s a trace map is defined and a duality theory for continuous differential operators between ST modules over a BCA is given.
One of motivations of these concepts is the explicit construction of residue complexes on \(k\)-schemes which the author carries out elsewhere.
Reviewer: H.Wiebe (Bochum)

MSC:
13J10 Complete rings, completion
13N10 Commutative rings of differential operators and their modules
12J99 Topological fields
13B35 Completion of commutative rings
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