Around the Gysin triangle I

模动机范畴中的Gysin三角

Gysin triangles in the category of motifs with modulus

本文研究模动机范畴中的Gysin三角，将其视为对数晶体上同调Gysin三角的动机提升，并比较模动机与Voevodsky动机，得出对象分解为p-挠部分与Voevodsky动机部分的推论。

Quaternionic Projective Bundle Theorem and Gysin Triangle in MW-Motivic Cohomology

In this paper, we show that the motive of the quaternionic Grassmannian $HP^n$ (as defined by I. Panin and C. Walter) splits in the category of effective MW-motives (as defined by B. Calmès, F. Déglise and J. Fasel). Moreover, we extend this result to an arbitrary symplectic bundle, obtaining the so-called quaternionic projective bundle theorem. Finally, we give the Gysin triangle in MW-motivic cohomology.

Around the Gysin triangle II

We study the construction and properties of the Gysin triangle in an axiomatic framework which covers triangulated mixed motives and MGl-modules over an arbitrary base S. This allows to define the Gysin morphism associated to a projective morphism between smooth S-schemes and prove duality for projective smooth S-schemes. As part of the construction, cobordism classes are considered and we give a proof of the Myschenko theorem generalized in our context - this in fact gives another proof of the latter theorem in classical stable homotopy through complex realization. Finally, these construction…

The Gysin triangle via localization and A1-homotopy invariance

Let X be a smooth scheme, Z a smooth closed subscheme, and U the open complement. Given any localizing and A1-homotopy invariant of dg categories E, we construct an associated Gysin triangle relating the value of E at the dg categories of perfect complexes of X, Z, and U. In the particular case where E is homotopy K-theory, this Gysin triangle yields a new proof of Quillen's localization theorem, which avoids the use of devissage. As a first application, we prove that the value of E at a smooth scheme belongs to the smallest (thick) triangulated subcategory generated by the values of E at the …

Künneth formulas for motives and additivity of traces

We prove several Künneth formulas in motivic homotopy categories and deduce a Verdier pairing in these categories following SGA5, which leads to the characteristic class of a constructible motive, an invariant closely related to the Euler-Poincaré characteristic. We prove an additivity property of the Verdier pairing using the language of derivators, following the approach of May and Groth-Ponto-Shulman; using such a result we show that in the presence of a Chow weight structure, the characteristic class for all constructible motives is uniquely characterized by proper covariance, additivity a…

On the Poncelet triangle condition over finite fields

Let ${\mathbf P}^2$ denote the projective plane over a finite field ${\mathbb F}_q$. A pair of nonsingular conics $({\mathcal A}, {\mathcal B})$ in the plane is said to satisfy the Poncelet triangle condition if, considered as conics in ${\mathbf P}^2({\overline{\mathbb F}}_q)$, they intersect transverally and there exists a triangle inscribed in ${\mathcal A}$ and circumscribed around ${\mathcal B}$. It is shown in this article that a randomly chosen pair of conics satisfies the triangle condition with asymptotic probability $1/q$. We also make a conjecture based upon computer experimentation…