Contravariant Hochschild Cohomology

• Contravariant Hochschild cohomology is a generalized cohomological theory that encodes Ext-groups and duality for associative algebras and schemes.
• It features a rich algebraic structure with cup products, Gerstenhaber brackets, and BV operators that are key to analyzing deformation and Poisson structures.
• The theory underpins practical applications in deformation theory, noncommutative geometry, and string topology through explicit constructions from DG algebras and polyvector fields.

Contravariant Hochschild cohomology is a generalized cohomological theory arising in both algebra and algebraic geometry, with a defining feature that the functoriality is contravariant with respect to morphisms. It plays a foundational role in deformation theory, noncommutative geometry, and derived algebraic geometry. Contravariant Hochschild cohomology may be constructed for associative algebras or for schemes, typically encoding Ext-groups of derived invariants that naturally admit rich algebraic structures—most notably, Gerstenhaber and Batalin–Vilkovisky (BV) algebra structures. Modern developments place contravariant Hochschild (co)homology in a bivariant framework with deep connections to duality, deformation quantization, and string topology.

1. Definitions and Foundational Constructions

For a unital differential graded (DG) $k$-algebra $A$, the contravariant Hochschild cochain complex is given by

$$C^*(A, A^\vee) := \mathrm{Hom}_{A^e}(B(A,A,A), A^\vee) \cong \mathrm{Hom}_k((s\,\overline{A})^{\otimes *}, A^\vee)$$

where $A^\vee = \mathrm{Hom}_k(A, k)$ is the $k$-dual bimodule. The total differential splits into internal and external components, and the cohomology

$HH^*(A, A^\vee) = \ker \delta / \mathrm{im} \delta$

is the contravariant Hochschild cohomology. The construction generalizes to the setting of schemes: given a separated, essentially finite type morphism $x: X \to S$ of noetherian schemes, the derived Hochschild complex is

$$H_x := L\Delta_x^*\,R\Delta_{x*}\,\mathcal{O}_X \in D_{qc}(X)$$

and the contravariant Hochschild groups are

$$HH^i(X/S) = \mathrm{Ext}^i_{\mathcal{O}_X}(H_x, H_x)$$

which are modules over the ring $\bigoplus_{j} H^j(S, \mathcal{O}_S)$ (Tarrío et al., 2010, Abbaspour, 2013).

2. Algebraic Structures: Cup Product, Gerstenhaber Bracket, and BV Operator

The cup product in $C^*(A, A^\vee)$ is defined by

$$(\phi \star \psi)(a_1, ..., a_{p+q}) = \phi(a_1,...,a_p)\big(\psi(a_{p+1},...,a_{p+q})\big)$$

which is associative and graded-commutative on cohomology:

$$[\phi]\star[\psi] = (-1)^{|\phi||\psi|} [\psi] \star [\phi]$$

A Gerstenhaber bracket is given by explicit insertion operations, making $HH^*(A, A^\vee)$ a Gerstenhaber algebra. When $A$ is Calabi–Yau or satisfies Poincaré duality (e.g., $A \simeq A^\vee[d]$ in the derived category), a BV-operator $\Delta$ of degree $+1$ may be constructed via duality from the Connes operator, giving

$$[f,g] = (-1)^{|f|}\,\Delta(f\!\smile\!g) - (-1)^{|f|}\,\Delta(f)\!\smile\!g - f\!\smile\!\Delta(g), \quad \Delta^2=0$$

Thus $(HH^*(A, A^\vee), \star, \Delta)$ is a BV algebra (Abbaspour, 2013). In the geometric (scheme-theoretic) picture, cup product and graded-commutativity are similarly realized, and the unit is the identity map $HH^0(X/S) \ni \mathrm{id}_{H_x}$ (Tarrío et al., 2010).

3. The Contravariant Viewpoint and Polyvector Fields

For the exterior algebra $A = \Lambda(V)$ on a finite-dimensional vector space $V$, the Hochschild cohomology is controlled by contravariant data:

$$HH^*(\Lambda(V)) \cong T_{\mathrm{poly}}^{\text{even}}(V^*)[wt]\,\, \oplus\,\, (k\;\text{if}\;\dim V\;\text{odd})$$

where $T_{\mathrm{poly}}^{\text{even}}(V^*)$ denotes even-weight polyvector fields on $V^*$. This is underpinned by Koszul duality and the Hochschild–Kostant–Rosenberg theorem, with the key property that each cohomology class is represented by a polyvector field acting contravariantly by differentiation on $V^*$ (Wong, 2016).

The Gerstenhaber algebra structure on $HH^*(\Lambda(V))$ corresponds to the Schouten–Nijenhuis bracket on these polyvector fields, and the cup product matches the wedge product. If $\dim V$ is even, the divergence operator becomes a BV-operator, resulting in a BV structure (Wong, 2016).

4. Functoriality, Duality, and the Bivariant Framework

Contravariant Hochschild cohomology fits in a three-functor formalism based on Grothendieck duality:

• $f^*$: (derived) inverse image
• $Rf_*$: derived direct image
• $f^!$: extraordinary inverse image

In schemes, the cup product and all structure maps are compatible with functorial pullbacks, base change, projection formulas, and canonical orientations/fundamental classes

$$c_x \in HH^0(X/S) = \mathrm{Hom}_{D(X)}(H_x, x^!\mathcal{O}_S)$$

These orientations are central in realizing Gysin maps and in defining a general Fulton–MacPherson–type bivariant theory (Tarrío et al., 2010).

5. Formality, Deformation Theory, and Poisson Structures

For the exterior algebra in even dimension, the differential graded Lie algebra (DGLA) of Hochschild cochains is $L_\infty$-formal, i.e., it is quasi-isomorphic as an $L_\infty$-algebra to its cohomology, and all higher Massey brackets vanish. Thus the formal deformation space is governed by Maurer–Cartan elements of the polyvector field algebra, i.e., by even-weight formal Poisson bivector fields:

$$\gamma = \sum \gamma_i r_i, \qquad [\gamma, \gamma]_{SN}=0$$

giving a geometric classification of deformations in terms of Poisson structures on the dual space (Wong, 2016). In contrast, when $\dim V$ is odd, formality may fail, as evidenced by nontrivial Massey products (Wong, 2016).

6. Geometric and Topological Applications

When $A = C_*(\Omega M)$ for a closed oriented manifold $M$, $A$ is a homologically smooth DG Hopf algebra (up to homotopy) with Poincaré duality, and $HH^*(A, A) \cong HH^*(A, A^\vee)$ acquires a BV structure. This recovers, via Burghelea–Fiedorowicz–Goodwillie, the Chas–Sullivan BV structure on the homology of the free loop space. Further, Sullivan chord diagrams act naturally on the Hochschild chain and cochain complexes, encoding cup product, Connes operator, BV operator, and relevant homotopies in a combinatorial, geometric model for string topology operations (Abbaspour, 2013).

7. Summary Table: Contravariant Hochschild Cohomology in Key Settings

Setting	Complex/Group	Structure
DG algebra $A$	$C^*(A, A^\vee)$	Gerstenhaber/BV algebra
Scheme $X \to S$	$HH^i(X/S) = \mathrm{Ext}^i(H_x, H_x)$	Graded-commutative $R$-algebra, orientation classes
$\Lambda(V)$, $\dim V$ even	$T_{\mathrm{poly}}^{\text{even}}(V^*)$	$L_\infty$-formal, BV structure
$\Lambda(V)$, $\dim V$ odd	$T_{\mathrm{poly}}^{\text{even}}(V^*) \oplus k$	BV structure, may not be formal

Contravariant Hochschild cohomology unifies and generalizes cohomological, geometric, and topological invariants, encoding deformation, duality, and string-topological phenomena through a robust algebraic framework (Wong, 2016, Abbaspour, 2013, Tarrío et al., 2010).