The structural conjectures for the category $\mathcal A$

Let $F/k$ be the field extension in the source, let $\mathfrak m$ be the object and $\mathcal C$ the category appearing in the source, let $\mathcal A$ be the category of admissible objects, and let $\Omega^1_{F/k}$ be the space of Kähler differentials. Structural conjectures. The following assertions are expected: for every $q\geq 0$, the functor

$$\operatorname{Hom}_{\mathcal C}(\otimes_F^q\mathfrak m,-)$$

is exact on $\mathcal A$; every irreducible object of $\mathcal A$ is a direct summand of the tensor algebra

$$\bigotimes_F^{\bullet}\Omega^1_{F/k};$$

and $\mathcal A$ is equivalent to the category of “coherent” sheaves on $\mathfrak{Sm}_k$. These assertions describe the expected exactness, generators, and geometric realization of the admissible category; the source gives no resolution.