Hochschild Cohomological Dimension

• Hochschild cohomological dimension is a homological invariant that measures the maximal degree in which Hochschild cohomology is nonzero.
• It is defined as the projective dimension of an algebra over its enveloping algebra and finds applications in deformation theory, Hopf algebras, and quantum groups.
• The invariant exhibits diverse behavior, ranging from finite values in certain algebras to infinity in cases with periodicity or complex noncommutative properties.

Searching arXiv for recent and foundational papers on Hochschild cohomological dimension. Hochschild cohomological dimension is a homological invariant of an associative algebra that measures the highest degree in which Hochschild cohomology can be nonzero, or equivalently the projective dimension of the algebra as a module over its enveloping algebra. In the standard formulation for a $k$-algebra $A$ and an $A$-bimodule $M$, Hochschild cohomology is defined by

$$HH^n(A,M)=\operatorname{Ext}^n_{A^e}(A,M),\qquad A^e=A\otimes A^{\mathrm{op}},$$

and the Hochschild cohomological dimension is

$$\operatorname{cd}(A)=\operatorname{pd}_{A^e}(A)=\sup\{\,n\mid HH^n(A,M)\neq 0\text{ for some }A\text{-bimodule }M\,\}.$$

This invariant appears across Hopf algebra theory, braided tensor categories, deformation theory, representation theory, monoid homology, Banach algebra cohomology, and the study of quiver and surface algebras (Bichon, 2014, Bichon et al., 2024).

1. Definition and equivalent formulations

For an associative algebra $A$, Hochschild cohomological dimension is defined as the projective dimension of $A$ as an $A^e$-module, where $A^e=A\otimes A^{\mathrm{op}}$ (Bichon, 2014). The same invariant can be described as the supremum of integers $A$0 for which there exists an $A$1-bimodule $A$2 with $A$3 (Bichon et al., 2024). In this sense it is a noncommutative dimension invariant: it records the maximal cohomological degree detectable by Hochschild theory, rather than geometric dimension in the commutative sense.

For Hopf algebras, the invariant admits a one-sided reformulation. If $A$4 is a Hopf algebra and $A$5 an $A$6-bimodule, one forms a right $A$7-module $A$8 on the same vector space by

$A$9

Then

$A$0

and therefore

$A$1

This reformulation is central in Hopf-algebraic computations because it replaces bimodule homological algebra by the homological algebra of the trivial module (Bichon, 2014).

A relative variant also occurs for commutative $A$2-algebras over a general commutative base ring. In that setting the paper "A Lower-Bound on the Hochschild Cohomological Dimension" defines

$A$3

and identifies Hochschild cohomology with a relative Ext functor for the class of $A$4-split epimorphisms (Kratsios, 2016). This produces a relative projective-dimension interpretation that parallels the absolute one.

In Banach algebra theory, the notion is adapted to continuous Hochschild cohomology. For a Banach algebra $A$5, one defines

$A$6

and all Banach $A$7-bimodules $A$8; if no such $A$9 exists, one sets $M$0 (Behnamian et al., 4 Oct 2025). This is Helemskii’s continuous version and differs in formulation from the algebraic finite-dimensional setting, though the controlling idea is the same.

2. General behavior and basic examples

Several standard examples illustrate how varied Hochschild cohomological dimension can be. If $M$1 is a linear algebraic group with coordinate Hopf algebra $M$2, then

$M$3

and if $M$4 is a discrete group, then $M$5 equals the group cohomological dimension with coefficients $M$6 (Bichon, 2014). These examples connect Hochschild cohomology with classical geometric and group-theoretic dimensions.

For finite-dimensional Hopf algebras, the behavior is sharply dichotomic: either $M$7 in the semisimple case or $M$8, since finite-dimensional Hopf algebras are Frobenius and self-injective (Bichon, 2014). This contrasts with braided Hopf algebras in suitable semisimple braided comodule categories, where finite positive dimensions are common and computable (Bichon et al., 2024).

Many finite-dimensional algebras have infinite Hochschild cohomological dimension because Hochschild cohomology is nonzero in infinitely many degrees. For periodic self-injective algebras of polynomial growth, the quotient $M$9 is isomorphic to $$HH^n(A,M)=\operatorname{Ext}^n_{A^e}(A,M),\qquad A^e=A\otimes A^{\mathrm{op}},$$0, where $$HH^n(A,M)=\operatorname{Ext}^n_{A^e}(A,M),\qquad A^e=A\otimes A^{\mathrm{op}},$$1 is the ideal generated by homogeneous nilpotents and $$HH^n(A,M)=\operatorname{Ext}^n_{A^e}(A,M),\qquad A^e=A\otimes A^{\mathrm{op}},$$2 is the period. Hence $$HH^n(A,M)=\operatorname{Ext}^n_{A^e}(A,M),\qquad A^e=A\otimes A^{\mathrm{op}},$$3 for all $$HH^n(A,M)=\operatorname{Ext}^n_{A^e}(A,M),\qquad A^e=A\otimes A^{\mathrm{op}},$$4, so $$HH^n(A,M)=\operatorname{Ext}^n_{A^e}(A,M),\qquad A^e=A\otimes A^{\mathrm{op}},$$5 (Bialkowski et al., 2017). Similar infinite-dimensional behavior appears for Jacobian algebras from triangulated unpunctured surfaces whenever the triangulation has at least one internal triangle: then $$HH^n(A,M)=\operatorname{Ext}^n_{A^e}(A,M),\qquad A^e=A\otimes A^{\mathrm{op}},$$6 in infinitely many arithmetic progressions of degrees, so $$HH^n(A,M)=\operatorname{Ext}^n_{A^e}(A,M),\qquad A^e=A\otimes A^{\mathrm{op}},$$7 (Valdivieso-Diaz, 2015).

The invariant can also be finite in nontrivial settings. For a family of associative $$HH^n(A,M)=\operatorname{Ext}^n_{A^e}(A,M),\qquad A^e=A\otimes A^{\mathrm{op}},$$8-algebras

$$HH^n(A,M)=\operatorname{Ext}^n_{A^e}(A,M),\qquad A^e=A\otimes A^{\mathrm{op}},$$9

one has $$\operatorname{cd}(A)=\operatorname{pd}_{A^e}(A)=\sup\{\,n\mid HH^n(A,M)\neq 0\text{ for some }A\text{-bimodule }M\,\}.$$0 for $$\operatorname{cd}(A)=\operatorname{pd}_{A^e}(A)=\sup\{\,n\mid HH^n(A,M)\neq 0\text{ for some }A\text{-bimodule }M\,\}.$$1, while $$\operatorname{cd}(A)=\operatorname{pd}_{A^e}(A)=\sup\{\,n\mid HH^n(A,M)\neq 0\text{ for some }A\text{-bimodule }M\,\}.$$2 and $$\operatorname{cd}(A)=\operatorname{pd}_{A^e}(A)=\sup\{\,n\mid HH^n(A,M)\neq 0\text{ for some }A\text{-bimodule }M\,\}.$$3 (Kratsios, 2014). This example shows that the invariant can take small finite values even in noncommutative families.

3. Hopf algebras, Gerstenhaber–Schack theory, and monoidal methods

A major development is the systematic comparison between Hochschild cohomology and Gerstenhaber–Schack cohomology for Hopf algebras. For a Hopf algebra $$\operatorname{cd}(A)=\operatorname{pd}_{A^e}(A)=\sup\{\,n\mid HH^n(A,M)\neq 0\text{ for some }A\text{-bimodule }M\,\}.$$4, Taillefer’s result identifies Gerstenhaber–Schack cohomology as

$$\operatorname{cd}(A)=\operatorname{pd}_{A^e}(A)=\sup\{\,n\mid HH^n(A,M)\neq 0\text{ for some }A\text{-bimodule }M\,\}.$$5

where $$\operatorname{cd}(A)=\operatorname{pd}_{A^e}(A)=\sup\{\,n\mid HH^n(A,M)\neq 0\text{ for some }A\text{-bimodule }M\,\}.$$6 denotes the category of Yetter–Drinfeld modules (Bichon, 2014). The corresponding Gerstenhaber–Schack cohomological dimension is

$$\operatorname{cd}(A)=\operatorname{pd}_{A^e}(A)=\sup\{\,n\mid HH^n(A,M)\neq 0\text{ for some }A\text{-bimodule }M\,\}.$$7

The central bridge theorem states that for any Hopf algebra $$\operatorname{cd}(A)=\operatorname{pd}_{A^e}(A)=\sup\{\,n\mid HH^n(A,M)\neq 0\text{ for some }A\text{-bimodule }M\,\}.$$8 and any $$\operatorname{cd}(A)=\operatorname{pd}_{A^e}(A)=\sup\{\,n\mid HH^n(A,M)\neq 0\text{ for some }A\text{-bimodule }M\,\}.$$9-bimodule $A$0,

$A$1

where $A$2 is a cofree Yetter–Drinfeld module built from the twisted right module $A$3 (Bichon, 2014). An immediate consequence is the inequality

$A$4

This is one of the principal structural tools in the area, because Gerstenhaber–Schack cohomology is often more tractable under tensor-categorical operations.

A second major result is monoidal invariance. If Hopf algebras $A$5 and $A$6 have tensor-equivalent comodule categories, then Gerstenhaber–Schack cohomology is monoidally invariant and

$A$7

Together with the bridge theorem, this yields

$A$8

for monoidally equivalent Hopf algebras (Bichon, 2014). This gives a way to transfer upper bounds and often exact values between quantum groups with equivalent tensor categories.

Equality between Hochschild and Gerstenhaber–Schack dimensions is known in important classes. For cosemisimple Hopf algebras of Kac type, one has

$A$9

proved via an averaging argument using the Haar integral and the assumption $A$0 (Bichon, 2014). For universal cosovereign Hopf algebras, the equality extends under the milder condition $A$1 (Bichon, 2016). The general cosemisimple case remains open in the source material.

4. Braided and monoidal generalizations

The braided analogue of the classical Hopf-algebra identity is developed for Hopf algebras in braided comodule categories. If $A$2 is a cosemisimple coquasitriangular Hopf algebra and $A$3 is a Hopf algebra in the braided category $A$4 of right $A$5-comodules, then

$A$6

This is the central equality theorem of "Cohomological dimension of braided Hopf algebras" (Bichon et al., 2024). It generalizes the well-known ordinary Hopf algebra result to a braided setting.

The proof uses a braided version of the Gerstenhaber–Schack–Kassel–Ginzburg–Kapranov Ext isomorphism, based on an adjunction between a functor $A$7 from left modules to bimodules and a right adjoint $A$8 that twists the left action using the antipode (Bichon et al., 2024). Cosemisimplicity of $A$9 is used to show that the forgetful functors from $A^e$0-comodules to ordinary modules are separable, so projective dimensions agree before and after forgetting the coaction.

The same paper gives criteria for smoothness and twisted Calabi–Yau duality in terms of the trivial module. If $A^e$1 is of type FP in the braided module category, then $A^e$2 is smooth as an ordinary algebra (Bichon et al., 2024). Under an additional one-dimensional Ext hypothesis, $A^e$3 is twisted Calabi–Yau of dimension $A^e$4 with an explicit braided Nakayama automorphism

$A^e$5

This situates Hochschild cohomological dimension within the broader duality theory of braided quantum groups (Bichon et al., 2024).

5. Computed dimensions in quantum groups and related Hopf-algebraic families

A striking feature of the Hopf-algebraic literature is the repeated appearance of cohomological dimension $A^e$6 in free and quantum symmetry contexts. The paper "Gerstenhaber-Schack and Hochschild cohomologies of Hopf algebras" proves that for the coordinate algebra of the quantum permutation group,

$A^e$7

under the stated cosemisimplicity and Kac-type hypotheses (Bichon, 2014). More generally, for $A^e$8 with $A^e$9, the paper shows $A^e=A\otimes A^{\mathrm{op}}$0 and $A^e=A\otimes A^{\mathrm{op}}$1 when $A^e=A\otimes A^{\mathrm{op}}$2 is a trace (Bichon, 2014).

For the adjoint Hopf subalgebra $A^e=A\otimes A^{\mathrm{op}}$3 of the universal Hopf algebra $A^e=A\otimes A^{\mathrm{op}}$4 of a bilinear form, one has

$A^e=A\otimes A^{\mathrm{op}}$5

and in the cosemisimple case

$A^e=A\otimes A^{\mathrm{op}}$6

(Bichon, 2014). These computations rely on restriction of free Yetter–Drinfeld resolutions across adjoint Hopf subalgebras and on strict exact sequences of Hopf algebras.

Universal cosovereign Hopf algebras provide another family with exact dimension $A^e=A\otimes A^{\mathrm{op}}$7. If $A^e=A\otimes A^{\mathrm{op}}$8 is an asymmetry, then

$A^e=A\otimes A^{\mathrm{op}}$9

and if $A$00 is generic, then

$A$01

Hence for generic asymmetry $A$02,

$A$03

(Bichon, 2016). The proof combines two structural tools: invariance under graded twisting by a finite abelian group and decomposition theorems for Hochschild and Gerstenhaber–Schack cohomology of free products (Bichon, 2016).

The same paper records examples such as $A$04, for which

$A$05

It also notes that the "free" compact quantum groups $A$06, $A$07, and $A$08 all have cohomological dimension $A$09 in this sense (Bichon, 2016). This establishes a stable homological dimension pattern across several quantum-group families.

A braided example appears in the two-parameter braided quantum group $A$10. The coordinate algebra $A$11 admits an explicit free resolution of the trivial module of length $A$12, giving

$A$13

and therefore

$A$14

(Bichon et al., 2024). The same example is also shown to be twisted Calabi–Yau of dimension $A$15 with an explicit Nakayama automorphism (Bichon et al., 2024).

6. Finite-dimensional, quiver, and surface algebras

Outside Hopf theory, Hochschild cohomological dimension is often studied through explicit bimodule resolutions. For Jacobian algebras $A$16 arising from triangulated unpunctured surfaces, one has

$A$17

whenever the triangulation has at least one internal triangle, because $A$18 in infinitely many degrees (Valdivieso-Diaz, 2015). If there are no internal triangles, then $A$19 for all $A$20, so $A$21 (Valdivieso-Diaz, 2015). The multiplicative structure is nontrivial exactly when internal triangles are present.

For $A$22-cluster tilted algebras of type $A$23, the criterion is again combinatorial. If the bound quiver has at least one $A$24-saturated cycle, then $A$25 is nonzero in infinitely many arithmetic progressions of degrees and the Hochschild cohomological dimension is infinite. If there is no $A$26-saturated cycle, then $A$27 for all $A$28, and the paper concludes that the cohomological dimension is $A$29 (Gubitosi, 2017). In this family the existence of $A$30-saturated cycles also controls whether the Gerstenhaber algebra structure is nontrivial.

Some self-injective special biserial algebras likewise have infinite Hochschild cohomological dimension. For the algebras $A$31 defined from a circular quiver with double arrows and relations $A$32, the paper gives closed formulas for $A$33 in every degree and deduces

$A$34

(Furuya, 2014). When $A$35, the only systematic vanishing occurs in degrees $A$36, but there remain nonzero groups in arbitrarily large degrees (Furuya, 2014).

For the self-injective special biserial algebra $A$37, the paper computes $A$38 explicitly and shows

$A$39

(Itaba, 2014). The quotient ring $A$40 is finitely generated, and its positive-degree generators furnish another proof that cohomology is nonzero in infinitely many degrees (Itaba, 2014).

A contrasting phenomenon occurs in algebras built from categories with zero compositions. The paper "Hochschild cohomology of algebras arising from categories and from bounded quivers" develops long exact sequences relating the Hochschild cohomology of the whole algebra to vertex algebras and pathwise Ext groups (Claude et al., 2017). In certain null-square projective situations with no efficient cycles, there is an integer $A$41 such that

$A$42

and consequently

$A$43

(Claude et al., 2017). In other cases, such as square algebras with free rank-one corners, odd-degree Hochschild cohomology remains nonzero for all large degrees, forcing $A$44 (Claude et al., 2017). This indicates that finite or infinite cohomological dimension can depend delicately on corner bimodules rather than only on the diagonal algebras.

7. Bounds, pathologies, and variations

One line of work studies lower bounds for commutative algebras over arbitrary commutative bases. If $A$45 is a commutative $A$46-algebra, the paper (Kratsios, 2016) proves a chain of inequalities involving flat dimension, projective dimension, relative $A$47-projective dimension, and Hochschild cohomological dimension. In particular, under suitable finiteness assumptions and a Cohen–Macaulay hypothesis at a maximal ideal $A$48,

$A$49

This yields explicit obstructions to quasi-freeness, since the same paper proves

$A$50

(Kratsios, 2016). The result generalizes a theorem of Cuntz–Quillen from the case of base field $A$51 to arbitrary commutative base rings.

A different direction concerns non-semicontinuity. The family

$A$52

satisfies $A$53 for all $A$54 and $A$55, so Hochschild cohomological dimension is not upper semi-continuous in this noncommutative family (Kratsios, 2014). The source material emphasizes that the degeneration is not flat, but the example nevertheless shows that naive semicontinuity expectations fail in noncommutative algebra.

In modular representation theory, the emphasis is often on degreewise bounds rather than on finiteness of the supremum. For a block algebra $A$56 of a finite group over an algebraically closed field of characteristic $A$57, the paper (Kessar et al., 2010) proves that for each $A$58 there is a bound

$A$59

where $A$60 is the defect. The paper does not prove finiteness of the Hochschild cohomological dimension of blocks, but it shows strong uniform control on the size of $A$61 in terms only of degree and defect (Kessar et al., 2010).

For blocks of symmetric groups, there is a sharper nonvanishing statement. If $A$62 is a positive defect block of $A$63, then $A$64 (Benson et al., 2023). More strongly, the generating-function formulas in that paper imply that for every positive defect block and every $A$65, $A$66, so the Hochschild cohomological dimension is infinite for every positive defect block of a symmetric group (Benson et al., 2023). This is a degreewise nonvanishing theorem, not merely a lower bound.

The exterior algebra furnishes another instance of infinite cohomological dimension. Over a field of characteristic $A$67, the paper (Wong, 2016) identifies $A$68 with an algebra of even-weight polyvector fields, up to a one-dimensional correction in odd dimension. Because one can produce nonzero classes in every degree, the paper’s computation implies

$A$69

(Wong, 2016).

Monoid-theoretic analogues have also appeared. For a monoid $A$70, the paper "Two-sided homological properties of special and one-relator monoids" defines Hochschild cohomological dimension as the projective dimension of $A$71 as a module over $A$72 and proves, for finitely presented special monoids,

$A$73

For one-relator special monoids $A$74, it further shows that $A$75 if $A$76 is not a proper power, while $A$77 if $A$78 is a proper power (Gray et al., 7 Jul 2025). These are monoid analogues of classical one-relator group phenomena.

In Banach algebra theory, twisted triangular Banach algebras satisfy an exact formula

$A$79

with amenability equivalent to $A$80 and amenability of both diagonal algebras (Behnamian et al., 4 Oct 2025). This shows that extension-theoretic control over cohomological dimension persists in continuous settings, even when individual Hochschild groups depend on a twist parameter.

Hochschild cohomological dimension thus serves as a unifying invariant linking projective resolutions, tensor-categorical structures, deformation theory, and geometric or combinatorial features of algebras. The source material shows that it can be exactly computable in quantum groups, braided Hopf algebras, and monoids; combinatorially determined for many quiver and surface algebras; bounded below by commutative invariants such as Krull dimension; unstable in noncommutative families; and frequently infinite because of periodicity, polyvector-field models, or repeated nonvanishing in arithmetic progressions of degrees (Bichon, 2014, Bichon et al., 2024).