nLab Tate sphere

Idea

The Tate sphere, in the strict sense, is the object in a sheaf topos (or (∞,1)-sheaf (∞,1)-topos, or its stabilization) over a gros site of algebraic varieties, which is one of the following: $\mathbb{P}^{1}$ (the projective line); the quotient

$$S^1_{Tate} \;\coloneqq\; \mathbb{A}^1 / (\mathbb{A}^1 \setminus \{0\}) \,,$$

which denotes the (homotopy-)cofiber of the inclusion morphism into the sheaf represented by the affine line $\mathbb{A}^1$ from the subobject $\mathbb{A}^1 \setminus \{0\}$ represented by the affine line with origin removed (equivalently the object underlying the multiplicative group $\mathbb{G}_m$); or $S^{1} \wedge \mathbb{G}_{m}$.

These constructions are understood by default in the context of motivic homotopy theory working over the Nisnevich site (VRO 07, Remark 2.22), where they are all $\mathbb{A}^{1}$-homotopy equivalent. But the construction principle, especially that of the homotopy cofibre definition, is clearly more general.

In formal constructions of categories of motives, one typically ‘inverts the Tate sphere’ in some sense in order to represent Tate twists correctly in Weil cohomology theories such as étale cohomology, or in motivic cohomology theories. This is the case, for example, both for the stable motivic homotopy categories à la Voevodsky, where one more precisely inverts the operation of smashing with the Tate sphere, and for the category of pure motives, where one inverts the Lefschetz motive.

Related concepts

• motivic homotopy theory

• motivic sphere

• n-sphere

• Tate twist

References

• Vladimir Voevodsky, Oliver Röndigs, Paul Arne Østvær, Voevodsky’s Nordfjordeid Lectures: Motivic Homotopy Theory, In: Bjørn Ian Dundas, Marc Levine, Paul Arne Østvær, Oliver Röndigs, Vladimir Voevodsky (eds.), Motivic Homotopy Theory, Springer 2007 (doi:10.1007/978-3-540-45897-5_7, pdf)

On (stable) motivic Cohomotopy of schemes (as motivic homotopy classes of maps into motivic Tate spheres):

• Aravind Asok, Jean Fasel, Mrinal Kanti Das, Euler class groups and motivic stable cohomotopy, Journal of the EMS 24 8 (2022) 2775–2822 [arXiv:1601.05723, doi:10.4171/jems/1156]

• Samuel Lerbet, Motivic stable cohomotopy and unimodular rows, Advances in Mathematics 436 109415 (2024) [arXiv:2206.11688, doi:10.1016/j.aim.2023.109415]