The Hochschild-Serre spectral sequence for a Lie-algebra ideal

Intuition Beginner

Suppose you want to understand the cohomology of a Lie algebra, but the algebra is big and complicated. The Hochschild-Serre spectral sequence is a bookkeeping machine that computes the cohomology of a Lie algebra from the cohomology of an ideal inside it together with the cohomology of the quotient. You break the hard object into two easier ones and then stitch the answers back together.

Think of an ideal as a normal subgroup-style piece you can divide out. If $\mathfrak{h}$ sits inside $\mathfrak{g}$ as an ideal, the quotient $\mathfrak{g}/\mathfrak{h}$ is itself a Lie algebra. The machine takes the cohomology of $\mathfrak{h}$, lets the quotient act on it, and then takes the cohomology of the quotient with those values as coefficients.

The payoff is that solvable and nilpotent Lie algebras can be built up one abelian layer at a time. Each layer is simple to handle, and the machine assembles the layers into the cohomology of the whole.

Visual Beginner

Picture a grid of boxes. The horizontal axis counts cohomology degrees coming from the quotient $\mathfrak{g}/\mathfrak{h}$, and the vertical axis counts degrees coming from the ideal $\mathfrak{h}$. The box in column $p$ and row $q$ holds $H^p(\mathfrak{g}/\mathfrak{h},H^q(\mathfrak{h},M))$. Arrows step two columns right and one row down, knocking out pieces in successive rounds until the grid settles. Reading the surviving boxes along each diagonal recovers the cohomology of $\mathfrak{g}$.

The bottom row alone, where $q=0$, already carries the cohomology of the quotient. The left column, where $p=0$, carries the part of the ideal's cohomology that the quotient leaves fixed.

Worked example Beginner

Take the smallest interesting case: the three-dimensional Heisenberg algebra. It has a basis $X,Y,Z$ with the single bracket relation that $X$ and $Y$ produce $Z$, while $Z$ commutes with everything. The center, spanned by $Z$, is an ideal $\mathfrak{h}$. The quotient $\mathfrak{g}/\mathfrak{h}$ is the two-dimensional abelian algebra spanned by the images of $X$ and $Y$.

For coefficients in the field $k$ the cohomology of a one-dimensional algebra is one copy of the field in degree $0$ and one in degree $1$. The cohomology of the two-dimensional abelian quotient is $1,2,1$ across degrees $0,1,2$. Laying these out on the grid and reading the diagonals gives total dimensions $1,2,2,1$ in degrees $0$ through $3$.

What this tells us: the Betti numbers of the Heisenberg algebra fall straight out of two abelian pieces, with no need to wrestle with the full three-dimensional complex.

Check your understanding Beginner

Formal definition Intermediate+

Let $\mathfrak{g}$ be a finite-dimensional Lie algebra over a field $k$ of characteristic zero, let $\mathfrak{h}⊴\mathfrak{g}$ be an ideal with quotient $\mathfrak{q}=\mathfrak{g}/\mathfrak{h}$, and let $M$ be a $\mathfrak{g}$-module. Lie-algebra cohomology $H^{\bullet }(\mathfrak{g},M)$ is computed by the Chevalley-Eilenberg complex 07.06.23 $$C^n(\mathfrak{g},M)={\mathrm{Hom}}_k({\Lambda }^n\mathfrak{g},M),$$ with the standard differential $d$.

Definition (the $\mathfrak{h}$-filtration). Filter $C^n(\mathfrak{g},M)$ by $$F^p{C}^n=\{c\in C^n:c({\xi }_1,\dots ,{\xi }_n)=0\text{ whenever more than }n-p\text{ of the }{\xi }_i\text{ lie in }\mathfrak{h}\}.$$ A cochain sits in $F^p$ when it vanishes as soon as it is fed too many arguments from the ideal; equivalently $F^p{C}^{\bullet }$ consists of cochains that depend on at most $p$ arguments transverse to $\mathfrak{h}$. This is a decreasing, bounded, multiplicative filtration compatible with $d$.

Definition (Hochschild-Serre spectral sequence). The spectral sequence of the filtered complex $(C^{\bullet }(\mathfrak{g},M),F^{\bullet })$ 03.13.01 is the Hochschild-Serre spectral sequence of the pair $(\mathfrak{g},\mathfrak{h})$ with coefficients in $M$. Its $E_2$ page is $$E_2^{p,q}=H^p(\mathfrak{q},H^q(\mathfrak{h},M))⟹H^{p+q}(\mathfrak{g},M).$$ Here $H^q(\mathfrak{h},M)$ is a $\mathfrak{q}$-module: $\mathfrak{g}$ acts on $H^{\bullet }(\mathfrak{h},M)$ through the conjugation (adjoint) action twisted by the action on $M$, the inner derivations from $\mathfrak{h}$ act as zero on cohomology, and the residual action descends to $\mathfrak{q}=\mathfrak{g}/\mathfrak{h}$.

A non-example that fixes ideas: if $\mathfrak{h}$ is merely a subalgebra and not an ideal, $\mathfrak{g}/\mathfrak{h}$ is only a vector space, $H^p(\mathfrak{g}/\mathfrak{h},-)$ is undefined as Lie-algebra cohomology, and the construction collapses. The ideal hypothesis is exactly what makes both axes Lie-algebraic.

Key theorem with proof Intermediate+

Theorem (Hochschild-Serre, 1953). For an ideal $\mathfrak{h}⊴\mathfrak{g}$ over a field of characteristic zero and a $\mathfrak{g}$-module $M$, the $\mathfrak{h}$-filtration of the Chevalley-Eilenberg complex gives a first-quadrant spectral sequence with $E_2^{p,q}=H^p(\mathfrak{g}/\mathfrak{h},H^q(\mathfrak{h},M))$ converging to $H^{p+q}(\mathfrak{g},M)$. ^{[Hochschild-Serre 1953]}

Proof. The filtration $F^{\bullet }{C}^{\bullet }(\mathfrak{g},M)$ of the formal definition is decreasing and bounded: $F^0=C^{\bullet }$ and $F^p{C}^n=0$ for $p>n$, since a cochain in degree $n$ can be fed at most $n$ transverse arguments. It is compatible with the differential because $d$ is built from the bracket and the module action, and feeding $dc$ an extra transverse argument either feeds $c$ one more transverse argument or pairs two ideal arguments through a bracket landing back in $\mathfrak{h}$. Thus $d{F}^p\subseteq F^p$, and a bounded filtered complex has a convergent spectral sequence 03.13.01.

It remains to identify $E_2$. The associated graded ${\mathrm{gr}}^p{C}^n=F^p/F^{p+1}$ consists of cochains depending on exactly $p$ transverse arguments and $n-p$ ideal arguments, so $${\mathrm{gr}}^p{C}^{p+q}\cong {\mathrm{Hom}}_k({\Lambda }^p\mathfrak{q},{\mathrm{Hom}}_k({\Lambda }^q\mathfrak{h},M))=C^p(\mathfrak{q},C^q(\mathfrak{h},M)),$$ where ${\Lambda }^p\mathfrak{q}$ arises because the transverse arguments are read modulo $\mathfrak{h}$. The piece of $d$ that preserves filtration degree (the $d_0$ differential of the spectral sequence) is precisely the Chevalley-Eilenberg differential of $\mathfrak{h}$ acting in the inner slot, with the transverse arguments held fixed. Therefore $$E_1^{p,q}=H^q(C^{\bullet }(\mathfrak{h},M))\text{-valued cochains}=C^p(\mathfrak{q},H^q(\mathfrak{h},M)).$$ The induced $d_1$ on $E_1$ is the part of $d$ that raises transverse degree by one while preserving $\mathfrak{h}$-degree; a direct bracket computation shows it is the Chevalley-Eilenberg differential of $\mathfrak{q}$ acting on the $\mathfrak{q}$-module $H^q(\mathfrak{h},M)$, where the $\mathfrak{q}$-action is the conjugation action descended from $\mathfrak{g}$. Taking $d_1$-cohomology gives $$E_2^{p,q}=H^p(\mathfrak{q},H^q(\mathfrak{h},M)),$$ as claimed. Boundedness forces convergence to the associated graded of $H^{p+q}(\mathfrak{g},M)$ under the induced filtration. $\square$

Bridge. This construction builds toward every later dévissage of solvable cohomology, and the same filtration appears again in the Lyndon-Hochschild-Serre sequence for a group extension, where conjugation by the quotient plays the identical role. The foundational reason the $E_2$ page splits into a quotient-of-ideal form is that the $\mathfrak{h}$-degree filtration is multiplicative and bounded; this is exactly the mechanism of the spectral sequence of a filtered complex 03.13.01 specialised to the Chevalley-Eilenberg complex 07.06.23. Putting these together, the Lie-algebra case generalises the abelian-extension computation and is dual, under the de Rham comparison, to the Leray-Serre sequence of a fibration. The central insight is that an algebraic filtration by "how many directions point out of the ideal" reproduces the geometric filtration of a fibration by base and fibre.

Exercises Intermediate+

Advanced results Master

Inflation-restriction and transgression. The low-degree exact sequence $$0\to H^1(\mathfrak{q},M^{\mathfrak{h}})\stackrel{\inf }{\to }{H}^1(\mathfrak{g},M)\stackrel{\mathrm{res}}{\to }{H}^1(\mathfrak{h},M{)}^{\mathfrak{q}}\stackrel{\tau }{\to }{H}^2(\mathfrak{q},M^{\mathfrak{h}})\stackrel{\inf }{\to }{H}^2(\mathfrak{g},M)$$ is the five-term sequence extracted from the corner of the spectral sequence, with $\tau =d_2$ the transgression. Inflation is the pullback along $\mathfrak{g}\twoheadrightarrow \mathfrak{q}$; restriction is the pullback along $\mathfrak{h}\hookrightarrow \mathfrak{g}$. This is the Lie-algebra incarnation of the inflation-restriction sequence familiar from group cohomology, and it is the universal tool for relating an extension's cohomology to that of its kernel and quotient.

Dévissage of solvable algebras. A solvable Lie algebra admits a chain $\mathfrak{g}={\mathfrak{g}}_0\supseteq {\mathfrak{g}}_1\supseteq \cdots \supseteq {\mathfrak{g}}_r=0$ with each ${\mathfrak{g}}_{i+1}$ an ideal of ${\mathfrak{g}}_i$ and each successive quotient abelian (the derived series, or for nilpotent algebras the lower central series). Iterating Hochschild-Serre along this chain computes $H^{\bullet }(\mathfrak{g},M)$ from the cohomology of abelian algebras, where the Chevalley-Eilenberg complex is an exterior algebra and everything is explicit. Each step contributes one abelian layer and one transgression.

Whitehead's lemmas re-derived. For $\mathfrak{g}$ reductive, write $\mathfrak{g}=\mathfrak{z}\oplus \mathfrak{s}$ with $\mathfrak{z}$ the center and $\mathfrak{s}$ semisimple, and take $\mathfrak{h}=\mathfrak{s}$. Complete reducibility 07.06.22 gives $H^q(\mathfrak{s},M)=0$ for $q=1,2$ and finite-dimensional $M$, so the spectral sequence collapses onto its bottom row $E_2^{p,0}=H^p(\mathfrak{z},M^{\mathfrak{s}})$. The vanishing of $E_2^{0,1}$ and $E_2^{0,2}$ recovers Whitehead's first and second lemmas, and the collapse identifies $H^{\bullet }(\mathfrak{g},M)\cong H^{\bullet }(\mathfrak{z},M^{\mathfrak{s}})={\Lambda }^{\bullet }{\mathfrak{z}}^{\ast }\otimes M^{\mathfrak{g}}$, the Hopf-algebra structure of reductive cohomology.

Borel and Heisenberg subalgebras. For a Borel subalgebra $\mathfrak{b}=\mathfrak{h}\oplus \mathfrak{n}$ with $\mathfrak{n}$ the nilradical (an ideal) and $\mathfrak{h}$ a Cartan, the sequence computes $H^{\bullet }(\mathfrak{b},M)$ from $H^{\bullet }(\mathfrak{n},M)$ as an $\mathfrak{h}$-module — the input to Kostant's theorem on $\mathfrak{n}$-cohomology. For the $(2m+1)$-dimensional Heisenberg algebra with center $\mathfrak{h}=⟨Z⟩$, the two-row sequence ($H^q(\mathfrak{h},k)=0$ for $q\ge 2$) reduces the computation to the exterior algebra of the $2m$-dimensional abelian quotient modulo the image of the single transgression $d_2=\omega \wedge (-)$, where $\omega$ is the symplectic bracket form.

Deformation theory. In Lie-algebra deformation theory, infinitesimal deformations are classified by $H^2(\mathfrak{g},\mathfrak{g})$ with the adjoint coefficients and obstructions by $H^3(\mathfrak{g},\mathfrak{g})$. For $\mathfrak{g}$ built as an extension, the spectral sequence organises these adjoint-cohomology groups by ideal and quotient, isolating which deformations move the ideal, which move the quotient, and which move the gluing cocycle.

Synthesis. The Hochschild-Serre sequence is the foundational reason that the cohomology of an extension is governed by the cohomology of its kernel and quotient, and this is exactly the algebraic shadow of the Leray-Serre sequence of a fibration. The construction generalises the inflation-restriction sequence of group cohomology to the Lie-algebra setting, and it is dual, through Chevalley-Eilenberg-de Rham comparison, to the cohomology of the corresponding homogeneous space. The central insight is that a single filtration by ideal-degree simultaneously yields the five-term sequence, the dévissage of solvable algebras, and the cohomological proof of Whitehead's lemmas; putting these together, one filtration underwrites the entire computational apparatus of Lie-algebra cohomology, and the bridge to topology is that the same bookkeeping computes both algebraic and geometric extensions.

Full proof set Master

Proposition (the five-term exact sequence). For an ideal $\mathfrak{h}⊴\mathfrak{g}$ and a $\mathfrak{g}$-module $M$ there is an exact sequence $$0\to H^1(\mathfrak{q},M^{\mathfrak{h}})\to H^1(\mathfrak{g},M)\to H^1(\mathfrak{h},M{)}^{\mathfrak{q}}\stackrel{\tau }{\to }{H}^2(\mathfrak{q},M^{\mathfrak{h}})\to H^2(\mathfrak{g},M).$$

Proof. The spectral sequence is first-quadrant, so $E_2^{p,q}=0$ for $p<0$ or $q<0$, and convergence equips $H^n(\mathfrak{g},M)$ with a finite filtration whose graded pieces are the $E_{\infty }^{p,q}$ with $p+q=n$. In total degree $0$ the only entry is $E_2^{0,0}=M^{\mathfrak{g}}$, and no differential touches it, so $H^0(\mathfrak{g},M)=M^{\mathfrak{g}}$.

In total degree $1$ the entries are $E_2^{1,0}$ and $E_2^{0,1}$. The differential $d_2:E_2^{0,1}\to E_2^{2,0}$ leaves $E_2^{0,1}$ on its source side; the entry $E_2^{1,0}$ receives no differential and emits none into the first quadrant. Hence $E_{\infty }^{1,0}=E_2^{1,0}=H^1(\mathfrak{q},M^{\mathfrak{h}})$ and $E_{\infty }^{0,1}=\ker (d_2:E_2^{0,1}\to E_2^{2,0})=\ker (\tau :H^1(\mathfrak{h},M{)}^{\mathfrak{q}}\to H^2(\mathfrak{q},M^{\mathfrak{h}}))$. The filtration on $H^1(\mathfrak{g},M)$ has graded pieces $E_{\infty }^{1,0}$ and $E_{\infty }^{0,1}$, giving the short exact sequence $$0\to E_{\infty }^{1,0}\to H^1(\mathfrak{g},M)\to E_{\infty }^{0,1}\to 0.$$ This realises the first three nonzero terms: inflation injects $H^1(\mathfrak{q},M^{\mathfrak{h}})$, restriction maps onto $\ker \tau$.

For the connecting map, $E_3^{2,0}=\mathrm{coker}(d_2:E_2^{0,1}\to E_2^{2,0})$, and since no later differential hits the bottom row in this range, $E_{\infty }^{2,0}=E_3^{2,0}$. The edge map $E_{\infty }^{2,0}\hookrightarrow H^2(\mathfrak{g},M)$ is inflation in degree $2$. Splicing the cokernel description of $E_3^{2,0}$ to the previous sequence yields $$H^1(\mathfrak{h},M{)}^{\mathfrak{q}}\stackrel{\tau }{\to }{H}^2(\mathfrak{q},M^{\mathfrak{h}})\to H^2(\mathfrak{g},M),$$ exact at $H^2(\mathfrak{q},M^{\mathfrak{h}})$ because the kernel of inflation there is the image of $\tau$. Concatenating the two segments gives the stated five-term sequence, exact throughout. $\square$

Proposition (collapse for semisimple ideal). If $\mathfrak{h}$ is semisimple and $M$ is finite-dimensional, then $H^n(\mathfrak{g},M)\cong H^n(\mathfrak{q},M^{\mathfrak{h}})$ for all $n$.

Proof. By Whitehead's lemmas, recovered as in the worked dévissage, $H^q(\mathfrak{h},M)=0$ for $q\ge 1$ when $\mathfrak{h}$ is semisimple and $M$ finite-dimensional, since every finite-dimensional $\mathfrak{h}$-module is completely reducible 07.06.22 and the higher cohomology of $\mathfrak{h}$ with completely reducible coefficients vanishes. Thus $E_2^{p,q}=0$ for $q\ge 1$ and the spectral sequence is concentrated in the bottom row $E_2^{p,0}=H^p(\mathfrak{q},M^{\mathfrak{h}})$. A spectral sequence concentrated in one row degenerates: every differential has source or target zero, so $E_2=E_{\infty }$, and the abutment filtration has a single graded piece in each total degree. Therefore $H^n(\mathfrak{g},M)\cong E_{\infty }^{n,0}=H^n(\mathfrak{q},M^{\mathfrak{h}})$. $\square$

Connections Master

Cohomology of a Lie algebra and Whitehead's lemmas 07.06.23 supplies the input to both axes: the Chevalley-Eilenberg complex whose $\mathfrak{h}$-filtration produces the spectral sequence, and the low-degree dictionary $H^0=M^{\mathfrak{g}}$, $H^1=$ derivations modulo inner, $H^2=$ central extensions that the five-term sequence relates across $\mathfrak{h}$, $\mathfrak{g}$, and $\mathfrak{q}$.

The spectral sequence of a filtered complex 03.13.01 is the general mechanism specialised here: the $\mathfrak{h}$-degree filtration is bounded and multiplicative, so the abstract convergence theorem applies verbatim, and the only Lie-specific content is the identification of the $E_2$ page and the conjugation action of $\mathfrak{q}$ on $H^q(\mathfrak{h},M)$.

Weyl complete reducibility 07.06.22 is what makes the semisimple-ideal collapse work: vanishing of $H^1$ and $H^2$ of a semisimple algebra with finite-dimensional coefficients forces the spectral sequence onto its bottom row, re-deriving Whitehead's lemmas and the reductive-cohomology product structure from the Casimir-based complete reducibility rather than from a direct cocycle argument.

The Lyndon-Hochschild-Serre spectral sequence for a group extension is the lateral analogue: $H^p(G/N,H^q(N,M))\Rightarrow H^{p+q}(G,M)$ uses conjugation of $G/N$ on $H^q(N,M)$ in exactly the way $\mathfrak{q}$ acts here, and the inflation-restriction sequences match term for term.

The Leray-Serre spectral sequence of a fibration is the topological counterpart: $H^p(B,H^q(F))\Rightarrow H^{p+q}(E)$ filters by base degree as Hochschild-Serre filters by quotient degree, and the de Rham comparison turns the fibration $F\to E\to B$ into the ideal extension $\mathfrak{h}\to \mathfrak{g}\to \mathfrak{q}$ on cohomology.

Historical & philosophical context Master

Gerhard Hochschild and Jean-Pierre Serre introduced the spectral sequence for Lie algebras and, in a companion paper, for group extensions in 1953 ^{[Hochschild-Serre 1953]}, building directly on the cohomology theory of Lie algebras that Claude Chevalley and Samuel Eilenberg had founded in 1948 ^{[Chevalley-Eilenberg 1948]} as the algebraic model for the de Rham cohomology of a compact Lie group. The filtration by ideal-degree is the algebraic transcription of the filtration Jean Leray had introduced for fibre spaces during his wartime captivity, and the parallel between the two constructions was understood by Henri Cartan's seminar as one structure viewed algebraically and topologically.

The inflation-restriction sequence the construction produces had appeared earlier in the group-cohomological work of Hochschild and others on class field theory, where it controls the relation between the cohomology of a Galois group and that of a normal subgroup. Serre incorporated the Lie-algebra version into his lecture notes on Lie algebras and Lie groups ^{[Serre LALG]}, and Charles Weibel's textbook ^{[Weibel Ch. 7]} gives the modern homological-algebra treatment as a special case of the spectral sequence of a filtered complex.

Bibliography Master

@article{hochschild-serre1953,
  author = {Hochschild, Gerhard and Serre, Jean-Pierre},
  title = {Cohomology of Lie Algebras},
  journal = {Annals of Mathematics},
  volume = {57},
  number = {3},
  pages = {591--603},
  year = {1953}
}

@article{hochschild-serre1953group,
  author = {Hochschild, Gerhard and Serre, Jean-Pierre},
  title = {Cohomology of Group Extensions},
  journal = {Transactions of the American Mathematical Society},
  volume = {74},
  pages = {110--134},
  year = {1953}
}

@article{chevalley-eilenberg1948,
  author = {Chevalley, Claude and Eilenberg, Samuel},
  title = {Cohomology Theory of Lie Groups and Lie Algebras},
  journal = {Transactions of the American Mathematical Society},
  volume = {63},
  pages = {85--124},
  year = {1948}
}

@book{weibel1994,
  author = {Weibel, Charles A.},
  title = {An Introduction to Homological Algebra},
  publisher = {Cambridge University Press},
  series = {Cambridge Studies in Advanced Mathematics 38},
  year = {1994}
}

@book{serre-lalg,
  author = {Serre, Jean-Pierre},
  title = {Lie Algebras and Lie Groups},
  publisher = {Springer},
  series = {Lecture Notes in Mathematics 1500},
  year = {1992}
}

@article{kostant1961,
  author = {Kostant, Bertram},
  title = {Lie Algebra Cohomology and the Generalized Borel-Weil Theorem},
  journal = {Annals of Mathematics},
  volume = {74},
  number = {2},
  pages = {329--387},
  year = {1961}
}