Hochschild–Serre Factorization

• Hochschild–Serre Factorization Theorem is a collection of results that decompose (co)homological invariants into simpler contributions using tools like spectral sequences and functorial gluing.
• It facilitates the computation of extension classes and central invariants by expressing global structures via local substructures and explicit connection maps.
• The theorem underpins applications in algebraic geometry, representation theory, and topology by enabling the breakdown of complex invariants into additive or symmetric parts.

The Hochschild–Serre Factorization Theorem refers to a collection of results and explicit formulas showing how (co)homological invariants of algebraic, topological, and categorical structures can be factored, split, or decomposed via contributions from simpler substructures—especially under the presence of symmetry, semiorthogonal decompositions, group actions, or functorial gluing data. These factorization phenomena are frequently realized through the tools of spectral sequences, derived functors, connection cochains, and operadic or categorical splittings. The most classical form, originally in the context of group and Lie algebra cohomology, posits that the cohomology of an extension can be computed from the cohomology of its constituent subgroups or ideals together with precise connecting maps; in modern algebraic geometry, topology, and noncommutative geometry, analogous mechanisms are constructed using semiorthogonal decompositions, factorization homology, higher Hochschild chains, and Serre functors.

1. Classical Structure and Group Extensions

In the classical case, let $G$ be a group with normal subgroup $N$ and $A$ a $G$-module with trivial $N$-action. The Lyndon–Hochschild–Serre (LHS) spectral sequence yields: $$E_2^{p,q} = H^p(G/N; H^q(N; A)) \implies H^{p+q}(G; A).$$ The theorem allows explicit computation of extension classes and central extensions through connection cochains. Given a $G$-invariant homomorphism $f: N \to A$, a connection cochain $\tau: G \to A$ with $\tau(n g) = f(n) + \tau(g)$ satisfies that $-\delta\tau$ represents the extension class in $H^2(G/N; A)$: $e(G_A) = -d_2f = [ -\delta\tau ].$ This construction factors extension classes as pushforwards of lower-level invariants, yielding a conceptual and computational model for extension and transgression phenomena (Fujitani, 2018).

2. Spectral Sequences and Additivity in Derived Structures

In categorical and geometric contexts, the theorem generalizes to the behavior of Hochschild (co)homology under semiorthogonal decompositions. For a smooth projective variety $X$ with derived category $\mathrm{D}(X) = \langle A_1, \ldots, A_n \rangle$, one obtains: $$HH_\bullet(X) \cong \bigoplus_{i=1}^n HH_\bullet(A_i)$$ with each summand a canonical subspace. For cohomology, additivity fails; instead, long exact sequences relate $HH^\bullet(A)$ and $HH^\bullet(A_i)$ with gluing functor corrections: $$\cdots \to HH^t(A) \to HH^t(A_1) \oplus HH^t(A_2) \to \operatorname{Hom}^{t+1}(\varphi, \varphi) \to HH^{t+1}(A) \to \cdots$$ This mirrors the classical spectral sequence's filtration, and the correction terms encode the obstruction to split additivity (0904.4330).

3. Serre Functor, Hochschild–Serre Cohomology, and Symmetric Products

A refined formulation involves the Hochschild–Serre cohomology, which tracks all Serre functor powers. For a smooth projective variety $X$ of dimension $d_X$: $\hochserre_k(X) := \mathrm{RHom}_{X \times X}\bigl( \Delta_* \mathcal{O}_X,\, \Delta_*(\omega_X^{\otimes k}[k d_X]) \bigr ),$ with $k = 0$ corresponding to Hochschild cohomology and $k = 1$ to homology. For symmetric quotient stacks $[X^n / \mathfrak{S}_n]$ (derived equivalent to Hilbert schemes), the factorization theorem gives: $\bigoplus_{n\geq 0}\hochserre_k([X^n/\mathfrak{S}_n])\, t^n \cong \mathrm{Sym}^\bullet\Big( \bigoplus_{i \geq 1} \hochserre_{1+(k-1)i}(X) t^i \Big ),$ demonstrating that the invariants of Hilbert schemes are symmetric powers of Serre-twisted invariants of $X$ (Belmans et al., 2023).

4. Cohomology under Group Actions and Categorical Factorization

For a finite group $G$ acting on a $k$-linear triangulated category $C$, the Hochschild cohomology of the invariant category splits canonically: $$HH^\bullet(C^G) \cong \left( \bigoplus_{g \in G} HH^\bullet(C, \varphi_g) \right)^G,$$ where $\varphi_g$ are the autoequivalences corresponding to $g \in G$; the $g = 1$ summand yields $HH^\bullet(C)^G$. In fractional Calabi–Yau contexts (Serre functor $Se \cong o \circ [n]$), the invariant part and twisted summand are sharply distinguished (Perry, 2018). The factorization theorem thus extends from group cohomology to derived categories and orbifolds.

5. Factorization Homology, Higher Hochschild Chains, and Exponential Laws

In topological and derived settings, the Hochschild–Serre factorization theorem is realized through functorial gluing and local-to-global principles. The higher Hochschild functor $CH_X(A)$ for a space $X$ and CDGA $A$ satisfies:

• $CH_{pt}(A) \simeq A$,
• $CH_{X \sqcup Y}(A) \simeq CH_X(A) \otimes CH_Y(A)$,
• $CH_{W}(A) \simeq CH_X(A) \otimes_{CH_Z(A)} CH_Y(A)$ for spaces $W \simeq X \cup_Z Y$.

There is an exponential law (Fubini theorem): $$CH_{X \times Y}(A) \simeq CH_X\big( CH_Y(A) \big ),$$ interpreted as a factorization of global invariants via iterated local data (Ginot et al., 2010). Topological chiral homology satisfies analogous factorization, reinforcing the point that such invariants are fundamentally assembled from local pieces and their gluing, precisely echoing the philosophy of Hochschild–Serre.

6. Spectral Sequence Differentials and Arithmetic Factorization

The explicit formula for the first nonzero differential in the Hochschild–Serre spectral sequence for semiabelian varieties is given by the connecting homomorphism: $d_2^{0,2}: H^0(k, \wedge^2 M) \to H^2(k, M),$ from the short exact sequence: $0 \to M \to Q_2(M) \to \wedge^2 M \to 0,$ where $Q_2(M)$ is the module of quadratic functions on the dual $M^\vee$. For Jacobians, $d_2^{0,2}(c_1) = \operatorname{Bockstein}(c_1')$ relates the canonical principal polarization to the theta-characteristic torsor and determines degeneration of the spectral sequence at $E_2$ (Petrov et al., 22 Nov 2024). For tori, analogous differentials compute the Brauer group via explicit Bockstein and “halving” maps.

7. Extensions, Infinite-Dimensional Cases, and Physical Realizations

The standard theorem, for finite-dimensional Lie algebras $\mathfrak{g} = \mathfrak{g}_0 \ltimes \mathcal{I}$, asserts: $$H^2(\mathfrak{g}; \mathfrak{g}) \cong \bigoplus_{p+q=2} H^p(\mathfrak{g}/\mathcal{I}; \mathcal{F}) \otimes H^q(\mathcal{I}; \mathfrak{g}),$$ with all nontrivial deformations localized in the ideal. For infinite-dimensional symmetry algebras, such as BMS$_3$, BMS$_4$, and Maxwell–BMS, the theorem is conjecturally generalized: nontrivial deformations still appear only through coefficients multiplying terms in the ideal, with the "rigid" subalgebra remaining undeformed. This framework organizes the classification of deformations and central extensions for gravitational symmetry algebras (Safari, 2020).

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In summary, the Hochschild–Serre Factorization Theorem and its generalizations encompass a broad spectrum of results linking the cohomology of algebraic, categorical, and topological constructs to decompositions or extensions. Through spectral sequences, semiorthogonal decompositions, categorical and operadic splittings, and explicit functorial formulas, these theorems provide foundational tools for understanding the local structure and global assembly of invariants, facilitating computations in algebraic geometry, representation theory, topology, and mathematical physics.