LHS-Spectral Sequences

• LHS-spectral sequences are a class of spectral sequence constructions that compute cohomological invariants for extension groups and categories.
• They utilize the Grothendieck construction to combine base category invariants with local module and bimodule data.
• This framework unifies classical group cohomology with categorical extensions to enable precise computations of Ext groups and cohomology rings.

LHS-Spectral Sequences

The term "LHS-spectral sequence" (Lyndon–Hochschild–Serre spectral sequence) refers to a broad suite of spectral sequence constructions, originally developed for group extensions, that have become fundamental invariants in cohomology, category theory, homological algebra, representation theory, and adjacent fields. These spectral sequences allow the computation of cohomological invariants (such as Ext groups and cohomology rings) for an "extension" category, group, or algebra, expressed in terms of invariants computed over a base structure and additional local data encoded via modules, precosheaves, or bimodules. The construction in the context of extension category algebras, as developed in (Wu, 28 Jul 2025), unifies and generalizes classical variants and provides new computational frameworks for homological invariants.

1. Construction Principles of LHS-Spectral Sequences

At the heart of the LHS-spectral sequence lies the extension of an algebraic or categorical object by auxiliary data. The original example is the extension of a group $G$ by a normal subgroup $N$, with quotient $Q = G/N$, yielding a spectral sequence: $$E_2^{p,q} = H^p(Q; H^q(N; M)) \implies H^{p+q}(G; M)$$ for a $G$-module $M$.

In the categorical generalization (Wu, 28 Jul 2025), consider:

• A small category $\mathcal{C}$,
• A precosheaf of unital $k$-algebras $\mathfrak{A}$ on $\mathcal{C}$,
• An $\mathfrak{A}$-bimodule $\mathfrak{M}$ or a right module $\mathfrak{N}$.

The Grothendieck construction $Gr_{\mathcal{C}}(\mathfrak{A}, \mathfrak{M})$ forms an "extension" category, with morphisms and compositions defined to generalize the algebraic extension formalism.

Key ingredients in the construction:

• The "extension category algebra" $\mathfrak{A} \ltimes \mathfrak{M}$, defined as $k$-linear combinations of triples $(r, m, f)$, encoding the action of $\mathcal{C}$, $\mathfrak{A}$, and $\mathfrak{M}$ with a precise multiplication law.
• The module category over $Gr_{\mathcal{C}}(\mathfrak{A}, \mathfrak{M})$ is shown (for object-finite $\mathcal{C}$) to be equivalent to the module category over $\mathfrak{A} \ltimes \mathfrak{M}$.

The LHS-type spectral sequences constructed in this setting resolve the cohomology or Ext groups over the extension category in terms of the base Grothendieck construction with coefficients in functors summarizing the $\mathfrak{N}$-data.

2. Grothendieck Construction and Extension Category Algebras

The Grothendieck construction $Gr_{\mathcal{C}}(\mathfrak{A}, \mathfrak{M})$ is defined as follows:

• Objects: coincide with those of $\mathcal{C}$.
• Morphisms: from $x$ to $y$ are triples $(r, m, f)$, with $f: x \to y$ in $\mathcal{C}$, $r \in \mathfrak{A}(y)$, $m \in \mathfrak{M}(y)$.
• Composition: $$(r, m, f) \circ (s, n, g) = (\mathfrak{A}(g)(r) s, \mathfrak{A}(g)(r) \cdot n + \mathfrak{M}(g)(m) \cdot s, f g)$$, capturing both the algebra and module actions.

The extension category algebra $\mathfrak{A} \ltimes \mathfrak{M}$ is the $k$-algebra generated by these morphisms, with multiplication matching the above composition whenever the source and target align and 0 otherwise. If $\mathcal{C}$ is object-finite, the module categories over $Gr_{\mathcal{C}}(\mathfrak{A}, \mathfrak{M})$ and over $\mathfrak{A} \ltimes \mathfrak{M}$ are equivalent.

This construction subsumes classical cases:

• Trivial extension algebras (where $\mathfrak{M}$ is a module over a single algebra) and
• Skew category algebras as particular examples.

3. Spectral Sequence Formulation and Key Results

Two principal LHS-spectral sequences arise from the extension category framework for a right $\mathfrak{A}$-module $\mathfrak{N}$:

Ext-Spectral Sequence:

$$E_2^{p,q} = \mathrm{Ext}^p_{Gr_{\mathcal{C}}(\mathfrak{A})}\left(\mathfrak{G}, \mathcal{H}^q(\mathfrak{N}_{\bigsqcup}; \mathfrak{F})\right) \implies \mathrm{Ext}^{p+q}_{Gr_{\mathcal{C}}(\mathfrak{A}, \mathfrak{N})}(\!{}_{\pi}\!\mathfrak{G}, \mathfrak{F})$$

Here, $\mathfrak{F}$ and $\mathfrak{G}$ are modules over appropriate categories, $\mathfrak{N}_{\bigsqcup}$ denotes the disjoint union of the modules $\mathfrak{N}(x)$, and the functor $\mathcal{H}^q$ is designed to keep track of homological data in the fiber direction.

Cohomological Spectral Sequence:

$$E_2^{p,q} = H^p\big( Gr_{\mathcal{C}}(\mathfrak{A}); \mathcal{H}^q(\mathfrak{N}_{\bigsqcup}; \mathfrak{F}) \big) \implies H^{p+q}(Gr_{\mathcal{C}}(\mathfrak{A}, \mathfrak{N}); \mathfrak{F})$$

These spectral sequences reflect a filtration of $\mathrm{Ext}$ or cohomology groups over the extension category, with the $E_2$-terms computed as Ext or cohomology over the base Grothendieck construction with coefficients in a derived functor $$\mathcal{H}^q(\mathfrak{N}_{\bigsqcup}; \mathfrak{F})$$ capturing the contribution of the fiber (i.e., the local module structure associated with $\mathfrak{N}$).

The foundational composition law underlying these constructions is: $$(r, m, f) \circ (s, n, g) = (\mathfrak{A}(g)(r) s, \mathfrak{A}(g)(r) \cdot n + \mathfrak{M}(g)(m) \cdot s, fg)$$ This governs both the definition of morphisms in the Grothendieck construction and multiplication in the extension algebra.

4. Functorial Properties and Module Category Equivalence

The module category equivalence

$$Gr_{\mathcal{C}}(\mathfrak{A}, \mathfrak{M})\text{-Mod} \cong (\mathfrak{A} \ltimes \mathfrak{M})\text{-Mod}$$

is achieved by k-linearizing $Gr_{\mathcal{C}}(\mathfrak{A}, \mathfrak{M})$, establishing a direct correspondence between the representations of the extended category and modules over the extension algebra.

This equivalence ensures that all computations of homological invariants, such as Ext-groups and cohomology groups, for the extension algebra can be translated into the Grothendieck construction context, where the spectral sequence machinery is applicable. This is significant because it permits a two-step computation:

• First compute invariants for the base category,
• Then "lift" the result via the spectral sequence to the extension.

Such a mechanism is directly analogous to the classical LHS procedure for group cohomology, where invariants for a subgroup and a quotient are assembled to compute invariants for the group.

5. Applications, Special Cases, and Generality

The general framework encapsulates key prior constructions:

• Trivial extension algebras are recovered when $\mathcal{C}$ is a singleton category.
• Skew category algebras correspond to situations with nontrivial groupoid or category structure and appropriate module data.

Because $\mathfrak{A}$ and $\mathfrak{M}$ or $\mathfrak{N}$ are allowed to vary over objects of $\mathcal{C}$ and to be arbitrary as long as compatible actions are present, the theory can handle "local-to-global" phenomena in representation theory, homological algebra, and topology.

By providing LHS-spectral sequences in this context, the methodology enables:

• Decomposition of complex homological invariants via base and fiber data,
• The possibility to approach new questions about module categories over structured algebras affording fibered or extension-like presentations,
• Unified treatment of invariants across a wider class of algebras and categories than previously handled.

This advancement merges multiple traditions—bar resolutions, Grothendieck constructions, spectral sequences of group extensions—into a categorical, functorial, and computable apparatus.

6. Future Directions and Open Problems

Several avenues for future research are indicated:

• Further explicit calculations of the LHS-spectral sequence in concrete examples, particularly those arising in representation theory or algebraic topology.
• Extension of the formalism to categories that are not object-finite, potentially allowing for spectral sequence constructions in settings with infinitely many objects or more sophisticated site-theoretic behaviors.
• Generalization to differential graded (dg) or derived categorical frameworks, where extension algebras have a chain complex or derived enrichment, potentially impacting the computation of invariants in derived categories or stable $\infty$-categories.
• Connecting the theory more deeply with Thomason cohomology and related topological constructions, leveraging the categorical approach to resolve further questions in the intersection of cohomology, geometry, and category theory.

A plausible implication is that these spectral sequences, through their functorial and compositional flexibility, will allow for systematic explorations of "fibered" algebraic situations well beyond previously tractable cases, particularly those for which classical bar resolution or LHS spectral sequence arguments are insufficient or cumbersome.

7. Summary Table: Main Constructs and Correspondences

Concept	Definition/Role	Context in (Wu, 28 Jul 2025)
$Gr_{\mathcal{C}}(\mathfrak{A},\mathfrak{M})$	Grothendieck construction of a category with algebra and bimodule data	Core construction for extension
$\mathfrak{A} \ltimes \mathfrak{M}$	Extension category algebra	Module category equivalent to Groth.
LHS-spectral sequence	Computation of Ext/cohomology in terms of base and fiber invariants	Main theoretical result
$\mathcal{H}^q(\mathfrak{N}_\bigsqcup; \mathfrak{F})$	Derived functor capturing fiber data in spectral sequence coefficients	$E_2$-page coefficient system
Module category equivalence	Equivalence for object-finite $\mathcal{C}$ between extension algebra and Grothendieck construction	Foundational to theory

The development and deployment of LHS-spectral sequences in the context of extension category algebras represent a substantive generalization of classical spectral sequence frameworks, with formal, computational, and conceptual advances opening further lines of inquiry into the structure of module categories over intricate categorical and algebraic extensions.