We give necessary and sufficient conditions for stratification and costratification to descend along a coproduct preserving, tensor-exact R-linear functor between R-linear tensor-triangulated categories which are rigidly-compactly generated by their tensor units. We then apply these results to non-positive commutative DG-rings and connective ring spectra. In particular, this gives a support-theoretic classification of (co)localizing subcategories, and thick subcategories of compact objects of the derived category of a non-positive commutative DG-ring with finite amplitude, and provides a formal justification for the principle that the space associated to an eventually coconnective derived scheme is its underlying classical scheme. For a non-positive commutative DG-ring A, we also investigate whether certain finiteness conditions in D(A) (for example, proxy-smallness) can be reduced to questions in the better understood category D(H⁰A).

我们给出了分层和共分层沿着由张量单元严格紧生成的 R 线性张量三角类别之间的余积保留、张量精确 R 线性函子下降的必要和充分条件。然后我们将这些结果应用于非正交换 DG 环和连接环谱。特别是，这给出了（共）局域化子类别和具有有限幅度的非正交换 DG 环的派生类别的紧凑对象的厚子类别的支持理论分类，并为以下原则提供了形式证明：与最终共连接派生方案相关的空间是其基础经典方案。对于非正交换 DG 环A ，我们还研究 D( A )中的某些有限性条件（例如，代理小性）是否可以简化为更好理解的类别 D( H ⁰ A ) 中的问题。