Rational motives on pro-algebraic stacks
Can Yaylali
TL;DR
The paper develops a pro-limit motivated theory of Beilinson rational motives for pro-${\mathcal{I}}$-algebraic stacks, defining $\mathrm{DM}(x,X)$ as a colimit of finite-type motives and establishing a robust motivic six-functor formalism for suitable morphisms in the pro-setting. It provides two complementary constructions to realize the six-functor structure: a general pullback-formalism framework for diagrammatic extensions and a six-functor formalism via adjointability or cartesian correspondences, yielding base-change, projection formulas, homotopy-invariance, localization, and purity. The theory is then applied to the stack of displays $\mathrm{Disp}$, proving its motive is Tate and that $\mathrm{DM}(\mathrm{Disp}) \simeq \mathrm{DM}(\mathrm{Disp}_1)$, with explicit motivic-cohomology calculations in terms of Chow groups; a connection to Barsotti–Tate stacks $\mathrm{BT}$ is also drawn. Finally, the absolute Galois group action on motivic homology is analyzed via pro-systems of finite extensions, yielding a Galois-action interpretation of motivic invariants and a continuous-homotopy-fixed-points perspective on $A^{n}(X,m)_{\mathbb{Q}}$.
Abstract
We define an $\infty$-category of rational motives for inverse limits of algebraic stacks, so-called pro-algebraic stacks. We show that it admits a $6$-functor formalism for certain classes of morphisms. On pro-schemes, we show that this $6$-functor formalism is in the sense of Liu-Zheng. This theory yields an approach to the theory of motives for non-representable algebraic stacks and non-finite type morphisms.