Motivic tubular neighborhoods.

*(English)*Zbl 1130.14009Let \(i: A\to B\) be a closed embedding of finite CW complexes. One useful fact (tubular neighborhood) is that \(A\) admits a cofinal system of neighborhoods \(T\) in \(B\) with \(A\to T\) a deformation retract. This is often used in the case where \(B\) is a differentiable manifold, showing that \(A\) has the homotopy type of \(T\).

The paper under review is devoted to an algebraic version of the tubular neighborhood which has the basic properties of the topological construction, at least for a reasonably large class of cohomology theories. It turns out that a “homotopy invariant” version of the Hensel neighborhood provides such construction for theories which are homotopy invariant. Moreover, if the given cohomology theory has a Mayer-Vietoris property for the Nisnevich toplogy, then one also has an algebraic version of the punctured tubular neighborhood. The author extends these constructions providing the (punctured) tubular neighborhood of a normal crossing subscheme of a smooth scheme. He also gives applications to the construction of tangential base-points for mixed Tate motives, algebraic gluing of curves with boundary components, and limit motives.

The paper under review is devoted to an algebraic version of the tubular neighborhood which has the basic properties of the topological construction, at least for a reasonably large class of cohomology theories. It turns out that a “homotopy invariant” version of the Hensel neighborhood provides such construction for theories which are homotopy invariant. Moreover, if the given cohomology theory has a Mayer-Vietoris property for the Nisnevich toplogy, then one also has an algebraic version of the punctured tubular neighborhood. The author extends these constructions providing the (punctured) tubular neighborhood of a normal crossing subscheme of a smooth scheme. He also gives applications to the construction of tangential base-points for mixed Tate motives, algebraic gluing of curves with boundary components, and limit motives.

Reviewer: Kirill Zainoulline (München)