  In this paper, we conjecture an extension of the Hilbert basis theorem and
the finite generation of invariants to commutative algebras in symmetric finite
tensor categories over fields of positive characteristic. We prove the
conjecture in the case of semisimple categories and more generally in the case
of categories with fiber functors to the characteristic $p > 0$ Verlinde
category of $SL_{2}$. We also construct a symmetric finite tensor category
$\sVec_{2}$ over fields of characteristic $2$ and show that it is a candidate
for the category of supervector spaces in this characteristic. We further show
that $\sVec_{2}$ does not fiber over the characteristic $2$ Verlinde category
of $SL_{2}$ and then prove the conjecture for any category fibered over
$\sVec_{2}$.
