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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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##### References:
 [1] Beilinson, A.A. : Residues and adeles , Funkt. Anal. Pril. 14(1) (1980), 44-45; English trans. in Func. Anal. Appl. 14(1) (1980), 34-35. · Zbl 0509.14018 · doi:10.1007/BF01078412 [2] Borel, A. et al.: Algebraic D-Modules , Academic Press, Boston, 1987. · Zbl 0642.32001 [3] Bourbaki, N. : Commutative Algebra , Hermann, Paris, 1972. [4] Grothendieck, A. and Dieudonné, J. : Eléments de géométrie algébrique. IV , Publ. Math. IHES 32 (1967). · Zbl 0153.22301 · numdam:PMIHES_1967__32__5_0 · eudml:103873 [5] El Zein, F. : Complexe Dualizant et Applications à la Classe Fondamentale d’un Cycle , Bull. Soc. Math. France, Mémoire 58, 1978. · Zbl 0388.14002 · numdam:MSMF_1978__58__1_0 · eudml:94780 [6] Huang, I.C. : Pseudofunctors on modules with 0-dimensional support , to appear in: Memoirs of AMS. · Zbl 0851.18007 [7] Hübl, R. and Kunz, E. : Integration of differential forms on schemes , J. Reine Angew. Math. 410 (1990), 53-83. · Zbl 0712.14006 · doi:10.1515/crll.1990.410.53 · crelle:GDZPPN002207850 · eudml:153258 [8] Hübl, R. and Kunz, E. : Regular differential forms and duality for projective morphisms , J. Reine Angew. Math. 410 (1990), 84-108. · Zbl 0709.14014 · doi:10.1515/crll.1990.410.84 · crelle:GDZPPN002207869 · eudml:153259 [9] Huber, A. : On the Parshin-Beilinson adeles for schemes , Abh. Math. Sem. Univ. Hamburg 61 (1991), 249-273. · Zbl 0763.14006 · doi:10.1007/BF02950770 [10] Hübl, R. and Sastry, P. : Regular differential forms and relative duality , Amer. J. Math. 115(4) (1993), 749-787. · Zbl 0796.14012 · doi:10.2307/2375012 [11] Hübl, R. : Traces of Differential Forms and Hochschild Homology , Lecture Notes in Math. 1368, Springer, Berlin, 1989. · Zbl 0675.13019 [12] Hübl, R. : Residues of regular and meromorphic differential forms , to appear in: Math. Annalen. · Zbl 0814.14022 · doi:10.1007/BF01450504 · eudml:165270 [13] Hübl, R. and Yekutieli, A. : Adeles and Differential Forms , preprint. · Zbl 0847.14006 [14] Köthe, G. : Topological Vector Spaces I , Vol. 1, Springer-Verlag, New York, 1969. · Zbl 0179.17001 [15] Kunz, E. : Kähler Differentials , Vieweg, Braunschweig, Wiesbaden, 1986. · Zbl 0587.13014 [16] Grothendieck, A. : Local Cohomology , Lecture Notes in Math. 41, Springer-Verlag, Berlin, 1967. · Zbl 0185.49202 · doi:10.1007/BFb0073971 [17] Lipman, J. : Dualizing sheaves, differentials and residues on algebraic varieties , Asterisque 117 (1984). · Zbl 0562.14003 [18] Lipman, J. : Residues and Traces of Differential Forms via Hochschild Homology , Contemp. Math. 61, AMS, Providence, 1987. · Zbl 0606.14015 [19] Lipman, J. and Sastry, P. : Regular differentials and equidimensional scheme-maps , J. Alg. Geom. 1 (1992), 101-130. · Zbl 0812.14011 [20] Macdonald, I.G. : Duality over complete local rings , Topology 1 (1962), 213-235. · Zbl 0108.26401 · doi:10.1016/0040-9383(62)90104-0 [21] Hartshorne, R. : Residues and Duality , Lecture Notes in Math. 20, Springer-Verlag, Berlin, 1966. · Zbl 0212.26101 · doi:10.1007/BFb0080482 · eudml:203789 [22] Sastry, P. and Yekutieli, A. : On residue complexes, dualizing sheaves and local cohomology modules , to appear in: Israel J. Math. · Zbl 0834.14003 · doi:10.1007/BF02783219 [23] Warner, F.W. : Foundations of Differrentiable Manifolds and Lie Groups , Springer-Verlag, New York, 1983. · Zbl 0516.58001 [24] Yekutieli, A. : An explicit construction of the Grothendieck residue complex (with an appendix by P. Sastry) , Astérisque 208 (1992). · Zbl 0788.14011 [25] Yekutieli, A. : Residues and differential operators on schemes , Preprint, 1994. · Zbl 0962.14010 · doi:10.1215/S0012-7094-98-09509-6 · arxiv:alg-geom/9602011
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