Traces and Differential Operators over Beilinson Completion Algebras
Traces and Differential Operators over Beilinson Completion Algebras
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Beilinson 完备代数上的迹和微分算子
DOI：
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发表时间：
1995
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通讯作者：
Amnon Yekutieli
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作者：
Amnon Yekutieli
贝林森完成代数 (BCA) A 是完美域 k 上的完全半局部代数，其剩余域是高维局部域。另外A是半拓扑代数。沿饱和点链完成代数 k 簇的结构束是 BCA 的典型示例。我们挑选出 BCA 之间的两种同态：态射和强化。第一种包括剩余有限局部同态，而第二种是一种局部同态。我们证明每个 BCA A 都有一个对偶模块 K(A)，并且这些对偶模块是逆变的。态射和协变 w.r.t.强化。对于任何半拓扑 A 模块 M，我们定义其对偶 Dual_{A} M := Hom_{A}^{cont}(M, K(A))。这种对偶运算具有尊重微分算子的显着特性：给定连续 DO D : M --> N，则存在对偶 DO Dual_{A}(D) : Dual_{A} N --> Dual_{A} M。上述结果（在后续论文中）用于在任何有限类型 k 方案 X 上构造 Grothendieck 留数复形 K_{X}^{.}，并导出其许多属性。
A Beilinson completion algebra (BCA) A is a complete semilocal algebra over a perfect field k, whose residue fields are high dimensional local fields. In addition A is a semi-topological algebra. The completion of the structure sheaf of an algebraic k-variety along a saturated chain of points is the prototypical example of a BCA. We single out two kinds of homomorphisms between BCAs: morphisms and intensifications. The first kind includes residually finite local homomorphisms, whereas the second kind is a sort of localization. We prove that every BCA A has a dual module K(A), and these dual modules are contravariant w.r.t. morphisms and covariant w.r.t. intensifications. For any semi-topological A-module M we define its dual Dual_{A} M := Hom_{A}^{cont}(M, K(A)). This duality operation has the remarkable property of respecting differential operators: given a continuous DO D : M --> N, there is a dual DO Dual_{A}(D) : Dual_{A} N --> Dual_{A} M. The results above are used (in a subsequent paper) to construct the Grothendieck residue complex K_{X}^{.} on any finite type k-scheme X, and to derive many of its properties.