Local Beilinson–Tate operators

Abstract

Abstract. In 1968 Tate introduced a new approach to residues on algebraic curves, based on a certain ring of operators that acts on the completion at a point of the function field of the curve. This approach was generalized to higher dimensional algebraic varieties by Beilinson in 1980. However Beilinson's paper had very few details, and his operator-theoretic construction remained cryptic for many years. Currently there is a renewed interest in the Beilinson-Tate approach to residues in higher dimensions.Our paper presents a variant of Beilinson's operator-theoretic construction. We consider an n-dimensional topological local field K, and define a ring of operators E(K) that acts on K, which we call the ring of local Beilinson-Tate operators. Our definition is of an analytic nature (as opposed to the original geometric definition of Beilinson). We study various properties of the ring E(K). In particular we show that E(K) has an n-dimensional cubical decomposition, and this gives rise to a residue functional in the style of Beilinson-Tate. Presumably this residue functional coincides with the residue functional that we had constructed in 1992; but we leave this as a conjecture.

Geometric and analytic structures on the higher adèles

“…This was already shown by Yekutieli, albeit in a slightly different language [53,Lemma 4.3,(2) and (4)]. For the second claim, we only need to know that the respective limits and colimits exist in ST modules; this is [53,Lemma 4.3,(3) and (6)]. If one is happy with plain field isomorphisms without extra structure, this is of course part of the original results of Parshin and Beilinson.…”

“…Lifting this morphism along the algebraic extension k((t 1 ))/k({b i } i∈I ) is the subtle point and hinges on char(k) = 0 [53]). We may assume b 0 = t 1 and c 0 = 0 for some index 0 ∈ I, so that σ maps t 1 to itself.…”

“…In this case one has to choose a presentation as an n-Tate object. This makes the comparison a little more involved, but thanks to the results of Yekutieli's paper [53], one still arrives at an isomorphism.…”

“…, b r in Lemma 3.3. Part (3) is deep in principle, but easy for us since we can rely on the theory set up in [53]. In Definition 3.10, part (2), we can read the finite O 1 -module has not yet been proven that this is independent of σ , but of course we may already use this here).…”

“…If one is happy with plain field isomorphisms without extra structure, this is of course part of the original results of Parshin and Beilinson. The construction and very definition of the canonical TLF structure/ST module structure is due to Yekutieli [51,53]. However, we know from Example 1.29, going back to Yekutieli's work, that a general field isomorphism will not preserve this structure, and from its variation Example 1.30 that it would also not preserve the n-Tate structure.…”