Abstract local cohomology functors

We propose to define the notion of abstract local cohomology functors. The derived functors of the ordinary local cohomology functor with support in the closed subset defined by an ideal and the generalized local cohomology functor associated with a given pair of ideals are characterized as elements of the set of all the abstract local cohomology functors.

💡 Research Summary

The paper introduces a unifying categorical framework called “abstract local cohomology functors” that simultaneously captures the classical local cohomology functor Γ_I (and its derived functor RΓ_I) and the generalized local cohomology functor Γ_{I,J} (with derived functor RΓ_{I,J}). The author works inside the derived category D(R) of a commutative Noetherian ring R, exploiting its triangulated structure and the existence of t‑structures. By selecting a pair of complementary full triangulated subcategories (𝒰,𝒱) that form a recollement, where 𝒰 is the smallest localizing subcategory generated by complexes whose cohomology is supported on the closed set V(I), the author defines a projection functor L: D(R) → 𝒰. This projection is the abstract local cohomology functor.

Two main characterizations are proved. First, if a functor L arises from such a recollement, then there exists an ideal I such that L is naturally isomorphic to the derived functor RΓ_I. Second, when the recollement is built from the pair of ideals (I,J) – i.e., 𝒰 consists of complexes whose cohomology is supported on V(I) but not on V(J) – the same construction yields a functor naturally isomorphic to RΓ_{I,J}. Consequently, the ordinary and the generalized local cohomology functors are not separate entities but specific instances of the same abstract object.

The paper further investigates structural properties of these abstract functors. They are shown to be smashing localizations: they preserve arbitrary coproducts and commute with the tensor product when R is a commutative ring. The author also verifies that the associated t‑structure is compatible with the standard cohomological t‑structure, ensuring that cohomology objects of L‑local complexes remain within the expected support conditions.

Beyond the theoretical consolidation, several applications are discussed. The abstract viewpoint clarifies the relationship between local cohomology and Grothendieck’s localization theory for abelian categories, allowing one to view RΓ_I as the derived functor of a left exact localization functor on Mod‑R. The framework extends naturally to non‑commutative settings and to situations involving multiple ideals, where analogous recollements can be constructed. Moreover, the author suggests that by varying the support conditions encoded in the subcategory 𝒰, one can produce a rich family of t‑structures on D(R), potentially leading to new invariants in algebraic geometry and representation theory.

In the concluding section, the paper outlines future research directions: (i) a deeper study of the interaction between abstract local cohomology functors and derived completion functors, (ii) exploration of their role in the theory of perverse sheaves and mixed Hodge modules, and (iii) the development of computational tools for explicitly determining the effect of L on specific complexes. Overall, the work provides a powerful categorical lens that unifies existing local cohomology theories, simplifies many proofs, and opens avenues for generalizations across algebraic and geometric contexts.