Frobenius quotients, inflation categories and weighted projective lines

Frobenius Quotients, Inflation Categories, and Weighted Projective Lines

Introduction and Background

The paper "Frobenius quotients, inflation categories and weighted projective lines" (2604.02626) develops a categorical framework connecting exact and Frobenius category theory, with significant consequences for the structure theory of vector bundles over weighted projective lines (WPLs). The principal innovation is the introduction and systematic study of Frobenius quotient functors between Frobenius exact categories and the canonical extension of such quotients to the associated inflation categories. This machinery is applied to explicitly construct a new class of Frobenius quotient functors between categories of vector bundles on weighted projective lines and monomorphism categories of graded modules over truncated polynomial rings.

Weighted projective lines, as introduced by Geigle and Lenzing, provide a categorical and geometric context to study canonical algebras and bring together techniques from algebraic geometry, triangulated and exact category theory, and the representation theory of algebras. Of particular note is the relationship between vector bundle categories on a WPL and submodule (monomorphism) categories of modules over local algebras, drawing connections to classical results such as Birkhoff's work on abelian group subgroups and the Ringel–Schmidmeier "submodule problem".

Frobenius Quotients: Definitions and Recognition

A Frobenius quotient in this context is an exact functor $F: \mathcal{A} \rightarrow \mathcal{B}$ between Frobenius exact categories that factors through the quotient by a removable subcategory (essential kernel) defined by objects sent to zero, after which the resulting functor is an exact equivalence. The removable subcategory comprises summands of projective-injectives that are annihilated by $F$. The main recognition result establishes that such a functor is a Frobenius quotient if and only if it is a pretriangle-equivalence (it induces a triangulated equivalence on stable categories), is dense, full, and objective.

This axiomatization provides a powerful categorical tool—it allows the systematic passage from highly structured "source" Frobenius categories to better-understood or more "linear" target Frobenius categories, encoding much of the homological information of the source while discarding projective artifacts.

Inflation Categories and Compatibility Results

Inflation categories generalize the classical category of monomorphisms: for any exact category, the $n$-inflation category consists of composable sequences of $n$ inflations, with morphisms expressed as commutative ladders. These categories naturally inherit a Frobenius exact structure. The authors prove that Frobenius quotient functors lift canonically to functors on inflation categories ($n$-inflation categories for any $n$), and these lifted functors are again Frobenius quotients. This is formalized in Theorem~\ref{thm:inf} and is crucial for relating derived and submodule-theoretic properties of the source and target categories.

This inflation setup underlies a multitude of representation-theoretic phenomena, generalizing the classical case of monomorphism categories (submodule categories)---notably central in the Birkhoff–Ringel–Simson correspondence.

Applications to Graded Gorenstein-Projective Modules

Another essential technical development is the connection between graded Gorenstein-projective modules (a generalization of maximal Cohen–Macaulay modules) and categories of projective-module factorizations. The construction relates $(n+1)$-fold projective-module factorizations (analogues of matrix factorizations, extending Eisenbud's original formulation) over a graded ring to sequences of inflations over the associated quotient ring by a homogeneous element.

A pivotal result is that the cokernel functor from the category of projective-module factorizations to an inflation category of Gorenstein-projective modules is a Frobenius quotient. This construction underpins the main geometric application of the paper.

Explicit Frobenius Quotients for Weighted Projective Lines

The central geometric application is the construction of an explicit Frobenius quotient functor

$${\rm vect}-\mathbb{X}(p,q,r) \to \mathcal{S}^\mathbb{Z}_{p-1, q-1}(r),$$

where ${\rm vect}-\mathbb{X}(p,q,r)$ denotes vector bundles over the weighted projective line of type $(p,q,r)$, and $F$0 is a monomorphism grid category—a category whose objects are $F$1 grids of strict monomorphisms between finitely generated $F$2-graded modules over $F$3. The essential kernel of this functor is described explicitly in terms of line bundles parametrized by shifts of the projective coordinates.

The functor realizes several prior conjectures and generalizations, showing not only the existence but also providing a concrete construction via double-cokernel procedures. This directly recovers, in a more transparent form, the mysterious quotient functors constructed for special cases in previous works (notably Kussin–Lenzing–Meltzer [KLM2]). This explicitness allows one to track homological and representation-theoretic properties across these categorical equivalences.

Strong claims include a derived equivalence between the stable categories of these Frobenius categories, yielding theoretical bridges between the geometry of weighted projective lines and the module theory of certain finite-dimensional algebras.

Numerical and Structural Consequences

• For each triple of integers $F$4, the functor constructed yields a stable equivalence between $F$5 modulo the prescribed line bundles and the monomorphism grid category.
• The essential kernel is precisely the additive closure of line bundles of the form $F$6 and $F$7 for $F$8, $F$9, and $n$0.
• Functorial operations respect the inflation category structure, meaning high-level homological constructs (e.g., AR quivers, recollements) are preserved or reflected by the quotient morphisms.

Theoretical and Practical Implications

The framework of Frobenius quotients and inflation categories systematically reduces the complexity of Frobenius categories, isolating tractable "monomorphism-type" categories from geometric or homologically intricate inputs (e.g., vector bundles on WPLs). This allows for:

• Transfer of homological invariants (Ext, AR structure, triangulated structure) between categories of geometric and algebraic origin.
• Potential classification results and "modular" understanding of vector bundle categories via submodule categories, leading to new insights in both singularity theory and finite-dimensional algebra representation theory.
• The extension of classical Knörrer periodicity and Happel–Seidel symmetries to noncommutative and higher-inflation categorical settings.

Looking forward, these constructions open avenues for a more general theory of "Frobenius reduction" for categories associated to singular varieties, arbitrary graded (or DG) algebras, or higher-dimensional analogues, potentially impacting the study of singularity categories, moduli of sheaves, and categorification programs in algebraic geometry and mathematical physics.

Conclusion

This work provides a technically precise and conceptually clarifying framework for producing and analyzing Frobenius quotients between categories of geometric and representation-theoretic significance. The extension to inflation categories ensures a broad applicability, while the explicit results for weighted projective lines and their relation to monomorphism categories resolve open problems concerning the existence and nature of such functors. The new techniques and equivalences developed yield tools for further exploration in homological algebra, algebraic geometry, and representation theory.