Relative Semi-Infinite Cohomology

• Relative Semi-Infinite Cohomology is a generalized cohomology theory computed relative to a substructure, bridging absolute and semi-infinite invariants.
• It employs homological tools such as bar resolutions, spectral sequences, and operadic formalism to manage complex filtrations and graded splittings.
• Its applications span geometric representation theory, quantum groups, vertex operator algebras, and prismatic cohomology, offering unified insights across advanced mathematical domains.

Relative semi-infinite cohomology is an advanced homological invariant arising across infinite-dimensional algebra, geometric representation theory, and algebraic geometry. It generalizes classical cohomology theories and semi-infinite cohomology by introducing a "relative" parameter: a substructure (typically a subalgebra, subgroup, or geometric stratum) with respect to which the cohomology is computed. The relative theory interpolates between absolute and semi-infinite regimes, encompasses complex inductive and filtrational techniques, and is foundational in geometric representation theory, quantum groups, and prismatic/cohomological frameworks.

1. Foundational Constructions of Relative Semi-Infinite Cohomology

Relative semi-infinite cohomology typically involves:

• Two algebraic objects $A$ and $B$ (often $B\subset A$),
• Complexes built from resolutions that incorporate both left and right derived functor aspects,
• Graded splittings or filtrations that control the passage from "negative to positive" subalgebras or degrees.

Operadic formalism yields the most general construction: for a morphism of $P$-algebras $f:B\to A$, with $P$ an operad, relative cohomology is computed by applying the cotriple (comonadic) formalism to the associated module categories via enveloping algebras. The standard semi-injective resolution for a $P$-algebra $A$ with a semi-infinite structure is

$$\text{Bar}_P(A, M) := \text{Hom}_{\operatorname{Mod}_{U_P(A)}}(\text{Bar}_\bullet(A,B,A), M) \otimes_{U_P(A)} \text{Bar}_\bullet(A,N,A),$$

where $U_P(A)\simeq U_P(B)\otimes_k U_P(N)$ with $N$ the "complement" to $B$ in $A$ under a semi-infinite splitting (Yanagida, 2018).

In geometric settings, such as infinite-dimensional flag varieties and Grassmannians, relative semi-infinite cohomology arises as the hypercohomology of categories of sheaves or D-modules equipped with an equivariance or support condition relative to a group action or moduli parameter, often constructed via colimits over families of subgroups or coweights (Gaitsgory, 2017, Gaitsgory, 2017).

2. Relative Semi-Infinite Cohomology in Representation Theory and Flag Varieties

Relative semi-infinite cohomology is central in geometric representation theory, especially for objects like semi-infinite flag varieties, affine Grassmannians, and Zastava spaces. For instance, it provides an explicit combinatorial and geometric approach to local intersection cohomology via "semi-infinite moment graphs," whose stalks encode the relative semi-infinite cohomology modules attached to Schubert cells (Lanini, 2015, Hayash, 2023). The Braden–MacPherson algorithm computes stalks (graded $S$-modules) by recursively propagating data along the moment graph structure, capturing key combinatorial invariants:

$$\operatorname{rk} \mathscr{B}(A)_B = v^{\delta(A,B)}q_{B,A}(v),$$

where $q_{B,A}(v)$ are Lusztig's generic polynomials, and $\delta(A,B)$ is a semi-infinite length function.

In the theory of moduli of bundles and parabolic structures, the construction of semi-infinite IC sheaves on finite-dimensional approximations like Zastava spaces and their factorization over the Ran space realizes relative semi-infinite t-structures and organizes cohomology in families (Hayash, 2023, Dhillon et al., 3 Aug 2025). These IC sheaves satisfy factorization and generically serve as intermediate extensions in semi-infinite t-structures.

3. Homological Methods: Resolutions, Spectral Sequences, and Operadic Formalisms

Relative semi-infinite cohomology exploits bar resolutions, spectral sequences, and cotriple methods:

• For algebraic groups, the relative bar resolution and the concept of $(G,H)$-injectivity (injectivity with respect to a subgroup $H$) define relative Ext-groups $\operatorname{Ext}^i_{G,H}(M,N)$ via relative injective resolutions (Loos, 1 May 2025).
• A relative Grothendieck spectral sequence is available when compositions of left-exact functors respect acyclicity and splitting properties:

$$E_2^{i,j} = R^i_{G',H'}F'(R^j_{G,H}F(M)) \Longrightarrow R^{i+j}_{G,H}(F'\circ F)(M).$$

• In operadic contexts, by leveraging semi-infinite splitting of enveloping algebras, one constructs a semi-injective resolution as above, which, when specialized to associative and Lie algebras, recovers classical semi-infinite cohomology theories (Yanagida, 2018).

These homological constructions support explicit computations and establish deep analogies between relative and semi-infinite regimes.

4. Extensions to Infinite-Dimensional and Vertex Algebra Settings

Extensions of semi-infinite cohomology to vertex operator algebras (VOAs), quantum groups, and SCFT-inspired graded-unitary settings are prominent:

• In the construction of quantum groups (e.g., $GL_q(2)$), $0$-th semi-infinite cohomology of tensor products of braided VOAs with complementary central charges yields the algebraic structure, with the cancelation condition $c+\bar{c}=28$ critical for anomaly cancellation in the BRST complex (Frenkel et al., 2011).
• For graded-unitary VOAs arising from four-dimensional SCFTs, the relative semi-infinite cohomology chain complex $C(g_{-2h^\vee},g,V_{matter})$ inherits a Kähler package of commuting BRST differentials $Q^+$, $Q^-$, Lefschetz $sl(2)$ automorphisms, and a formality property:

$$C = \operatorname{ker}\Delta \oplus \operatorname{im}Q^+ \oplus \operatorname{im}\overline{Q}_+.$$

The cohomology is quasi-isomorphic to the complex itself, establishing graded unitarity and supporting an outer $\operatorname{USp}(2)$ symmetry (Beem et al., 12 Sep 2025).

• In the paper of W-algebras and Weyl modules with multiple singular points, relative semi-infinite cohomology reveals discrepancies in naive constructions and motivates refined module definitions (e.g., $\tilde{\mathbb{V}}^{\lambda,\mu}$ so that $$H^0(\tilde{\mathbb{V}}^{\lambda,\mu}) \simeq Z_2^{\lambda,\mu}$$) (Fortuna et al., 2023, Dhillon, 2019).

5. Relative Cohomology in Algebraic Geometry and Prismatic Theory

Recent advances in prismatic and $A_{\text{inf}}$-cohomology provide robust frameworks for relative theories in $p$-adic and derived geometry:

• The relative $A_{\text{inf}}$-cohomology of a morphism $f:X\to Y$ of $p$-adic formal schemes is constructed as

$$R\Gamma_{A_{\text{inf}}}(X/Y) = Rp_* A\Omega_{X/Y}, \quad A\Omega_{X/Y} = L\eta_\mu(R\nu_{f*}A_{\text{inf},X}),$$

where the fiber product of topoi is fundamental to the relative picture (Gaisin et al., 2022).

• There is a canonical comparison with relative prismatic ($q$-crystalline) cohomology after inverting $\mu$:

$$(\mu_{\text{proét}}^{-1}R\Gamma_{q-\text{CRYS}}(X/Y))^{\wedge} \otimes^L_{A_{\text{inf}}}A_{\text{inf}}[1/\mu] \simeq Rf_{\eta,\text{proét}*}\mathbb{Z}_p \otimes^L_{\mathbb{Z}_p}A_{\text{inf},Y}[1/\mu].$$

This implies that, post-inversion, period comparison across de Rham, crystalline, prismatic, and étale frameworks unifies the cohomological picture.

Such constructions suggest that spectral sequences and "infinite-slope" truncations in this context could motivate and structure relative semi-infinite cohomology in geometric applications.

6. Comparative Contexts and Analogies

Relative semi-infinite cohomology shares formal features and analogies with classical and semi-infinite cohomology:

• Both theories employ resolutions controlled by substructures (subgroups, subalgebras, topoi),
• Spectral sequences and derived functors admit direct analogies,
• The passage from absolute to relative "cuts off" infinite or "half-infinite" contributions, sharpening invariants,
• Many computations (Ext, intersection cohomology stalks, BRST cohomology groups) streamline by imposing (relative) splitting.

In particular, the intersection with representation theory (as in the relation between stalks of Braden–MacPherson sheaves and multiplicities in category $\mathcal{O}$) and with geometric approaches (e.g., Zastava models for semi-infinite flag varieties) reveals the power and flexibility of the relative semi-infinite concept (Hayash, 2023, Lanini, 2015).

7. Applications, Impact, and Future Directions

Relative semi-infinite cohomology is pivotal in:

• Quantum group and VOA constructions,
• Geometric and categorical Langlands program (global vs local compatibility, factorization, t-structures),
• Derived and prismatic algebraic geometry, period comparison, and $p$-adic Hodge theory,
• Representation theory of algebraic groups, highest-weight theory, and linkage principles.

Ongoing research pursues:

• Extensions to Tate categories and non-finite-type objects (Yanagida, 2018),
• Enhanced factorization and compatibility of IC sheaves in infinite-dimensional settings (Hayash, 2023, Dhillon et al., 3 Aug 2025),
• Interpretations and computations in SCFT/VOA correspondence and derived Poisson reductions (Beem et al., 12 Sep 2025).

Relative semi-infinite cohomology thus functions as an organizing principle across advanced mathematical domains, linking homological algebra, infinite-dimensional geometry, quantum and vertex algebra, and arithmetic geometry in a unified, flexible formalism.