Relative Hochschild Cohomology Overview

• Relative Hochschild cohomology is a Hochschild-type theory for algebra extensions, where cochains are defined via relative bar constructions and Ext functors.
• It employs relative homological algebra techniques to manage additional structures, leading to applications in derivations, Lie algebra characteristics, and geometric comparisons.
• The theory unifies various constructions—extension-relative, equivariant-relative, Lie–Rinehart, and coefficient-relative—providing a robust framework for advanced cohomological analysis.

Searching arXiv for recent and foundational papers on relative Hochschild cohomology. Relative Hochschild cohomology denotes a family of Hochschild-type theories in which the ordinary cochain complex is constrained by additional structure. In the most explicit algebraic form appearing here, it is attached to an extension $B\subseteq A$ and an $A$-bimodule $M$, and is defined by

$\HH^*(A\mid B,M):=\Ext^*_{(A^e\mid B^e)}(A,M), \qquad \HH^*(A\mid B):=\HH^*(A\mid B,A).$

Its cochains are realized by the relative bar construction as

$C^n(A\mid B,M)\cong \Hom_{B\text{-}B}(A^{\otimes_B n},M).$

At the same time, several papers use “relative” in broader senses: relative to a commutative base ring, relative to a Lie–Rinehart base algebra, relative to an equivariance algebra, or relative only in the weaker sense of coefficients or categorical comparison. This suggests that the term names a cluster of related constructions rather than a single uniform formalism (Lindell, 14 Aug 2025, Lindell et al., 2024, Stancu, 2010).

1. Core definition and terminological range

For extensions of algebras, the relative theory is built from Hochschild’s relative homological algebra. A short exact sequence of $A$-modules is $(A\mid B)$-exact when it splits as a sequence of $B$-modules; relative projectives are defined by the corresponding lifting property; and every module admits a relative projective resolution. In this setting, relative Hochschild cohomology is the derived functor $\Ext^*_{A^e\mid B^e}(A,-)$, with $A^e=A\otimes A^{\mathrm{op}}$ and $A$0 (Lindell et al., 2024, Lindell, 14 Aug 2025).

The literature represented here uses several adjacent meanings of “relative Hochschild cohomology”.

Construction	Auxiliary datum	Representative formulation
Extension-relative	$A$1	$A$2
Equivariant-relative	$A$3	$A$4
Lie–Rinehart-relative	base algebra $A$5 or $A$6 with $A$7	spectral sequence or Poisson model
Base-ring-relative	commutative ring $A$8	relative derived category $A$9
Coefficient-relative analogue	bimodule coefficients	Hochschild cochains with values in a bimodule

Two terminological cautions are explicit. First, Hochschild cohomology with coefficients in a bimodule is not automatically a genuine relative theory with respect to a subalgebra or algebra morphism. The paper on ring objects in monoidal categories constructs bimodule-valued Hochschild cochains, but states that it does not define a theory of the form $M$0 or $M$1 (Hellstrøm-Finnsen, 2016). Second, the paper on actions of central elements studies an $M$2-relative center $M$3 acting on $M$4, but this is relative to coefficients rather than to an extension $M$5 (Hermann, 2014).

2. Relative homological algebra and bar-type models

The computational backbone of the extension-relative theory is the relative bar complex. For an $M$6-bimodule $M$7,

$M$8

with differential

$M$9

This realizes $\HH^*(A\mid B,M):=\Ext^*_{(A^e\mid B^e)}(A,M), \qquad \HH^*(A\mid B):=\HH^*(A\mid B,A).$0 as cohomology of a concrete relative Hochschild complex (Lindell, 14 Aug 2025).

An equivalent description appears through relative Ext. If $\HH^*(A\mid B,M):=\Ext^*_{(A^e\mid B^e)}(A,M), \qquad \HH^*(A\mid B):=\HH^*(A\mid B,A).$1 is a relative $\HH^*(A\mid B,M):=\Ext^*_{(A^e\mid B^e)}(A,M), \qquad \HH^*(A\mid B):=\HH^*(A\mid B,A).$2-projective resolution, then

$\HH^*(A\mid B,M):=\Ext^*_{(A^e\mid B^e)}(A,M), \qquad \HH^*(A\mid B):=\HH^*(A\mid B,A).$3

The equivariant paper recalls that if $\HH^*(A\mid B,M):=\Ext^*_{(A^e\mid B^e)}(A,M), \qquad \HH^*(A\mid B):=\HH^*(A\mid B,A).$4, there is a natural map

$\HH^*(A\mid B,M):=\Ext^*_{(A^e\mid B^e)}(A,M), \qquad \HH^*(A\mid B):=\HH^*(A\mid B,A).$5

and that if $\HH^*(A\mid B,M):=\Ext^*_{(A^e\mid B^e)}(A,M), \qquad \HH^*(A\mid B):=\HH^*(A\mid B,A).$6 is $\HH^*(A\mid B,M):=\Ext^*_{(A^e\mid B^e)}(A,M), \qquad \HH^*(A\mid B):=\HH^*(A\mid B,A).$7-separable, then relative Ext agrees with ordinary Ext: $\HH^*(A\mid B,M):=\Ext^*_{(A^e\mid B^e)}(A,M), \qquad \HH^*(A\mid B):=\HH^*(A\mid B,A).$8 These facts explain when a relative theory reduces to the absolute one (Pojar et al., 11 May 2026).

A further generalization replaces a single algebra by a presheaf of algebras. In that setting, Stancu constructs a relative derived category $\HH^*(A\mid B,M):=\Ext^*_{(A^e\mid B^e)}(A,M), \qquad \HH^*(A\mid B):=\HH^*(A\mid B,A).$9 by inverting maps whose cones are contractible as complexes of $C^n(A\mid B,M)\cong \Hom_{B\text{-}B}(A^{\otimes_B n},M).$0-bimodules. The resulting morphism groups recover relative Yoneda cohomology and, in particular,

$C^n(A\mid B,M)\cong \Hom_{B\text{-}B}(A^{\otimes_B n},M).$1

The functor $C^n(A\mid B,M)\cong \Hom_{B\text{-}B}(A^{\otimes_B n},M).$2 to the associated algebra $C^n(A\mid B,M)\cong \Hom_{B\text{-}B}(A^{\otimes_B n},M).$3 then induces a full and faithful functor

$C^n(A\mid B,M)\cong \Hom_{B\text{-}B}(A^{\otimes_B n},M).$4

so the Gerstenhaber–Schack comparison becomes a consequence of a stronger derived-category statement (Stancu, 2010).

3. The first relative cohomology and its Lie algebra

The first relative Hochschild cohomology is particularly rigid. For a pair $C^n(A\mid B,M)\cong \Hom_{B\text{-}B}(A^{\otimes_B n},M).$5,

$C^n(A\mid B,M)\cong \Hom_{B\text{-}B}(A^{\otimes_B n},M).$6

where $C^n(A\mid B,M)\cong \Hom_{B\text{-}B}(A^{\otimes_B n},M).$7 are the $C^n(A\mid B,M)\cong \Hom_{B\text{-}B}(A^{\otimes_B n},M).$8-derivations that are $C^n(A\mid B,M)\cong \Hom_{B\text{-}B}(A^{\otimes_B n},M).$9-module maps, equivalently derivations vanishing on $A$0. Thus $A$1 consists of derivations of $A$2 that kill $A$3, modulo the corresponding relative inner derivations (Lindell et al., 2024).

This quotient inherits a Lie bracket

$A$4

The same paper proves that $A$5 is a Lie subalgebra of $A$6, that $A$7 is a Lie ideal, and that there is a natural injective Lie map

$A$8

A direct consequence stated there is that Lie-theoretic properties of $A$9, such as being abelian, nilpotent, or solvable, pass to $(A\mid B)$0 (Lindell et al., 2024).

For finite-dimensional basic algebras with $(A\mid B)$1, the first relative group admits explicit combinatorial descriptions. In the monomial case it is a subquotient of Strametz’s model for $(A\mid B)$2, obtained by imposing vanishing on the arrows of $(A\mid B)$3. In radical square zero cases one gets decompositions involving semidirect products such as

$(A\mid B)$4

for an $(A\mid B)$5-Kronecker quiver with an $(A\mid B)$6-Kronecker subquiver. More generally, the paper identifies semisimple quotients by products of $(A\mid B)$7 over parallelism classes in the complement quiver (Lindell et al., 2024).

The same work relates $(A\mid B)$8 to a contracted fundamental group. It defines a normal subgroup $(A\mid B)$9 of $B$0 and constructs an injective map

$B$1

For monomial algebras, the quotient $B$2 is identified with the fundamental group of a contracted quiver $B$3, yielding

$B$4

This furnishes a lower-bound mechanism measuring the “new cycles” present in $B$5 beyond those already in $B$6 (Lindell et al., 2024).

4. Equivariant and Lie–Rinehart forms of relativity

One important relative mechanism comes from group actions. For a $B$7-algebra $B$8 and a $B$9-equivariant $\Ext^*_{A^e\mid B^e}(A,-)$0-bimodule $\Ext^*_{A^e\mid B^e}(A,-)$1, Jensen’s equivariant Hochschild complex is formed from the invariant cochains

$\Ext^*_{A^e\mid B^e}(A,-)$2

The central theorem identifies its cohomology with a relative Ext group: $\Ext^*_{A^e\mid B^e}(A,-)$3 where $\Ext^*_{A^e\mid B^e}(A,-)$4 is the $\Ext^*_{A^e\mid B^e}(A,-)$5-enveloping algebra, isomorphic to the skew group algebra $\Ext^*_{A^e\mid B^e}(A,-)$6. In this formulation, equivariant Hochschild cohomology is literally a relative derived functor with respect to the subalgebra $\Ext^*_{A^e\mid B^e}(A,-)$7 (Pojar et al., 11 May 2026).

For group algebras $\Ext^*_{A^e\mid B^e}(A,-)$8, this relative viewpoint is used to inject equivariant group cohomology of centralizers into equivariant Hochschild cohomology. The same paper derives a nonvanishing criterion for $\Ext^*_{A^e\mid B^e}(A,-)$9: if $A^e=A\otimes A^{\mathrm{op}}$0, $A^e=A\otimes A^{\mathrm{op}}$1, and $A^e=A\otimes A^{\mathrm{op}}$2 is $A^e=A\otimes A^{\mathrm{op}}$3-good, then

$A^e=A\otimes A^{\mathrm{op}}$4

The proof proceeds through the quotient $A^e=A\otimes A^{\mathrm{op}}$5 and a resulting nonzero class in $A^e=A\otimes A^{\mathrm{op}}$6 (Pojar et al., 11 May 2026).

A second major relative framework is furnished by Lie–Rinehart algebras. If $A^e=A\otimes A^{\mathrm{op}}$7 is a Lie–Rinehart algebra with $A^e=A\otimes A^{\mathrm{op}}$8 projective as an $A^e=A\otimes A^{\mathrm{op}}$9-module and $A$00, then there is a spectral sequence

$A$01

Here $A$02 is Hochschild cohomology of the commutative base algebra $A$03, and the Lie–Rinehart structure induces the $A$04-action on these groups. This decomposes the computation of $A$05 into base Hochschild cohomology and Lie–Rinehart cohomology (Kordon et al., 2020).

The 2023 paper on Lie–Rinehart algebras pushes this relative pattern further. For $A$06 smooth and $A$07 projective over $A$08, it proves

$A$09

The passage from ordinary Hochschild cochains on $A$10 to Poisson cohomology is mediated by “nonlinear Chevalley–Eilenberg” complexes and the notion of an $A$11-quasi-module, where the failure of strict $A$12-linearity is recorded by homotopies $A$13 satisfying

$A$14

This is a relative Hochschild mechanism in which the base algebra $A$15 is not external data but part of the cochain-level correction (Kosmeijer et al., 2023).

5. Corings, Cartier cohomology, and higher operations

Relative Hochschild cohomology also appears as one side of duality statements for corings. If $A$16 is a $A$17-coring and is finitely generated projective as a left $A$18-module, the paper on corings defines the right algebra $A$19 and proves an Ext-level comparison

$A$20

for left finitely generated projective bicomodules $A$21. Specializing to $A$22 gives

$A$23

Thus Cartier cohomology of a coring becomes a relative Hochschild theory of its right algebra (Lindell, 14 Aug 2025).

The same paper strengthens this from an isomorphism of cohomology groups to an isomorphism of cochain-level higher structures. It constructs a strict $A$24-morphism and, under finite generation and projectivity, a strict $A$25-isomorphism

$A$26

Consequently,

$A$27

as Gerstenhaber algebras. In the application to an entwining structure $A$28, Brzeziński’s equivariant cohomology $A$29 is identified with the relative Hochschild cohomology of the twisted convolution algebra $A$30, again up to opposite structure (Lindell, 14 Aug 2025).

A different higher-structure result concerns coefficients rather than subalgebra extensions. For an $A$31-bimodule $A$32, the $A$33-relative center is

$A$34

The paper on exact sequences and the $A$35-relative center defines an action

$A$36

interprets it via loops in extension categories through Retakh’s theorem, and proves that it makes $A$37 into a right Gerstenhaber module over $A$38. The construction is compatible with exact monoidal functors and therefore Morita invariant (Hermann, 2014).

6. Categorical and dg extensions, and limits of the term

In categorical settings, Hochschild cohomology is often generalized farther than the extension-relative model, and the boundary of the word “relative” becomes important. For ring objects $A$39 in an $A$40-enriched monoidal category, the Hochschild complex is defined by

$A$41

with the Hochschild-style differential modified by associators to account for non-strict monoidal structure. The paper also defines the analogous complex $A$42 for an $A$43-bimodule object $A$44. It proves $A$45, identifies $A$46 with the center and $A$47 with derivations modulo inner derivations, interprets $A$48 via extensions in the additive case, and establishes graded commutativity and a Gerstenhaber bracket. But it explicitly does not define a genuine relative theory with respect to a subring, ring morphism, or module category (Hellstrøm-Finnsen, 2016).

For dg categories, the emphasis shifts to coefficient objects and spectral sequences. If $A$49 is a small dg category and $A$50 an $A$51-bimodule, the dg Hochschild complex is the product total complex of a bicomplex $A$52. This yields the characteristic spectral sequence

$A$53

and the forgetful spectral sequence

$A$54

The characteristic homomorphism to the graded center is interpreted as the edge map of the first spectral sequence. The paper is explicit that it does not introduce a formal relative Hochschild cohomology for a morphism $A$55, although it does treat bimodule coefficients and “derived natural transformations” as relative data in a broader sense (Neumann et al., 2016).

A geometric variant appears for dg categories of $A$56-modules on stacks. For a QCA stack $A$57,

$A$58

while the loop-space expression

$A$59

is presented as a “naive expectation” that fails in general. The paper studies the comparison morphism

$A$60

through a relative compactification of the diagonal, proves base-change criteria under vanishing conditions on the boundary, and shows that for torus quotients $A$61,

$A$62

Here “relative” refers not to a subalgebra extension but to a geometric comparison between diagonal and loop-space models of Hochschild cohomology, motivated by support theory in the sense of Benson–Iyengar–Krause (Koppensteiner, 2015).

The resulting picture is sharply stratified. In one stratum, relative Hochschild cohomology is literally $A$63 for an extension $A$64. In another, it is a relative derived or equivariant theory with respect to a base ring, a group algebra $A$65, or a Lie–Rinehart base algebra. In a third, “relative” designates coefficient, functorial, or geometric comparison data without producing a subalgebra-relative complex. Much of the modern literature is devoted to making these distinctions precise, because the algebraic consequences—Lie brackets, Gerstenhaber structures, spectral sequences, and deformation interpretations—depend on which sense of relativity is actually in force.