Abelian categories of Artin \'etale Motives with integral coefficients

We study the motivic t-structure on the category $\mathcal{DM}^A_{\acute{e}t,c}(S,\mathbb{Z})$ of constructible Artin \'etale motives over a base scheme $S$. We first show how smooth Artin motives are related to Artin representations of the \'etale fundamental group of $S$ when $S$ is regular. We then construct an ordinary motivic t-structure, and show that if $S$ allows resolutions of singularities by alteration, the $\ell$-adic realization functor is t-exact. Finally, we show that the perverse homotopy t-structure is the best possible approximation of a perverse motivic t-structure on the category of Artin motives, that it is well defined when the ring of coefficients is $\mathbb{Q}$ or when $S$ is of dimension $3$ or less and show that it cannot exist when $S=\mathbb{A}^4_k$ with $k$ a field. If $S$ is of dimension $2$ or less, we prove an analog of the affine Lefschetz theorem and we prove that the $\ell$-adic realization functor is t-exact and it is not t-exact when $S=\mathbb{A}^3_k$. When $S$ is of finite type over $\mathbb{F}_p$, and with rational coefficients, the $\ell$-adic realization functor is t-exact when we restrict its codomain to the category of Artin $\ell$-adic sheaves endowed with its perverse homotopy t-structure. We describe the category of perverse Artin \'etale motives $M^A_{perv}(S,\mathbb{Z})$, which is the heart of the perverse homotopy t-structure. This description can be formulated purely in terms of representations of \'etale fundamental groups of regular locally closed subschemes of $S$ in the case when $S$ is $1$-dimensional; we describe the $\ell$-adic realization in this case. In addition, the category $M^A_{perv}(S,\mathbb{Q})$ has nice properties: its objects are of finite length and we can describe its simple objects. For instance, the motive $\mathbb{E}_S$ of Ayoub-Zucker is a simple Artin perverse sheaf.Comment: 83 pages, comments welcom