Resolutions of symmetric ideals via stratifications of derived categories

We propose a method to unify various stability results about symmetric ideals in polynomial rings by stratifying related derived categories. We execute this idea for chains of $GL_n$-equivariant modules over an infinite field $k$ of positive characteristic. We prove the Le--Nagel--Nguyen--R\"omer conjectures for such sequences and obtain stability patterns in their resolutions as corollaries of our main result, which is a semiorthogonal decomposition for the bounded derived category of $GL_{\infty}$-equivariant modules over $S = k[x_1, x_2, \ldots, x_n, \ldots]$. Our method relies on finite generation results for certain local cohomology modules. We also outline approaches (i) to investigate Koszul duality for $S$-modules taking the Frobenius homomorphism (of $GL_{\infty}$) into account, and (ii) to recover and extend Murai's results about free resolutions of symmetric monomial ideals.