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Motivic tubular neighborhoods. (English) Zbl 1130.14009
Let $$i: A\to B$$ 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.

##### MSC:
 14C25 Algebraic cycles 55P42 Stable homotopy theory, spectra 18F20 Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) 14F42 Motivic cohomology; motivic homotopy theory
##### Keywords:
limit motives; moduli of curves; Tate motives
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