Grothendieck spectral sequence

Intuition Beginner

Many useful operations in algebra and geometry lose information when they are applied to an exact sequence. Derived functors record the lost information degree by degree.

The Grothendieck spectral sequence explains what happens when two such operations are performed in a row. First one operation produces its derived corrections. Then the second operation is applied to those corrections and produces a grid of further corrections. The spectral sequence is the bookkeeping device that turns this grid into the derived functors of the whole composite operation.

The slogan is:

derived functors of a composite are computed from derived functors of the two pieces, provided the first piece sends good replacements to objects that are good enough for the second.

This is the reason the theorem appears throughout sheaf cohomology. A map of spaces lets us first push a sheaf forward to the target, then take global sections there. Grothendieck's theorem says that the higher pushforwards and the cohomology of the target assemble into the cohomology of the original space.

Visual Beginner

The rows and columns record the two stages separately. The diagonals record the total degree of the composite.

Worked example Beginner

Let a continuous map carry a space (X) to a space (Y), and let a sheaf live on (X).

One way to understand the sheaf is to push it forward to (Y). This does not lose all information, but it may create higher pushforwards. Then one computes cohomology on (Y).

The Grothendieck spectral sequence says that those two layers of data fit together to recover the cohomology on (X). In sheaf language this becomes the Leray spectral sequence, one of the main computational tools in algebraic geometry and topology.

Check your understanding Beginner

Formal definition Intermediate+

Let $$ F:\mathcal A\to\mathcal B,\qquad G:\mathcal B\to\mathcal C $$ be left-exact additive functors between abelian categories. Assume (\mathcal A) and (\mathcal B) have enough injectives, and assume that (F) sends injective objects of (\mathcal A) to (G)-acyclic objects of (\mathcal B). Equivalently, for every injective (I\in\mathcal A), $$ R^pG(FI)=0\qquad p>0. $$

Then for every object (X\in\mathcal A) there is a first-quadrant cohomological spectral sequence $$ E_2^{p,q}=R^pG\bigl(R^qF(X)\bigr) ;\Rightarrow; R^{p+q}(G\circ F)(X). $$

The abutment is filtered: for (n=p+q), the final page gives the associated graded pieces $$ E_\infty^{p,q}\cong \frac{F^pR^n(GF)(X)}{F^{p+1}R^n(GF)(X)}. $$

The derived-category form is shorter. Under the same acyclicity hypothesis there is a natural isomorphism in (D^+(\mathcal C)): $$ R(G\circ F)(X)\cong RG(RF(X)). $$ The spectral sequence is the filtered-complex expression of this derived-category comparison.

How to read the indices

• (q) records the derived degree produced by (F).
• (p) records the derived degree produced by (G).
• (p+q) is the total degree of the derived functor of the composite.
• The (E_2)-page is already expressed in derived functors; earlier pages depend on a chosen filtered resolution.

Counterexamples to common slips

• The theorem does not say (R^n(GF)\cong R^nG(R^nF)). The total degree (n) receives contributions from all pairs (p+q=n).
• The acyclicity hypothesis is structural, not decorative. Without it, the displayed (E_2)-page may compute the wrong target.
• (E_\infty) is usually an associated graded object for a filtration, not an automatic direct-sum decomposition of (R^n(GF)(X)).

Key theorem with proof Intermediate+

Theorem (Grothendieck spectral sequence). Let (F:\mathcal A\to\mathcal B) and (G:\mathcal B\to\mathcal C) be left-exact additive functors between abelian categories with enough injectives. If (F) sends injectives of (\mathcal A) to (G)-acyclic objects of (\mathcal B), then $$ E_2^{p,q}=R^pG(R^qF(X)) \Rightarrow R^{p+q}(G\circ F)(X) $$ for every (X\in\mathcal A).

Proof. Choose an injective resolution $$ 0\to X\to I^0\to I^1\to I^2\to\cdots. $$ Apply (F). Since each (I^q) is injective in (\mathcal A), the object (F(I^q)) is (G)-acyclic by hypothesis. Thus the complex (F(I^\bullet)) is a model for (RF(X)) built from objects on which (G) may be derived by termwise application followed by an injective or Cartan-Eilenberg replacement.

Take a Cartan-Eilenberg injective resolution of the complex (F(I^\bullet)) in (\mathcal B), and apply (G). This produces a first-quadrant double complex in (\mathcal C). Filter its total complex by columns.

One filtration first computes cohomology in the direction resolving the (G)-part. Because the terms (F(I^q)) are (G)-acyclic, this identifies the total complex with a model for (R(GF)(X)).

The other filtration first computes the cohomology of (F(I^\bullet)). Those cohomology objects are precisely (R^qF(X)). Applying the right-derived functors of (G) to them gives $$ E_2^{p,q}=R^pG(R^qF(X)). $$ The convergence theorem for first-quadrant spectral sequences of filtered complexes identifies the abutment with the cohomology of the same total complex, namely (R^{p+q}(GF)(X)). This proves the spectral sequence.

Five-term exact sequence Intermediate+

The low-degree part is often the most useful computational output. A first-quadrant cohomological spectral sequence gives the exact sequence $$ 0\to E_2^{1,0}\to R^1(GF)(X)\to E_2^{0,1} \to E_2^{2,0}\to R^2(GF)(X). $$

For Grothendieck's spectral sequence this becomes $$ 0\to R^1G(FX)\to R^1(GF)(X)\to G(R^1F(X))\to R^2G(FX)\to R^2(GF)(X), $$ where (R^0F=F) and (R^0G=G). This sequence measures the first failure of the derived functor of the composite to be read from the two first derived functors separately.

Leray as the model example Intermediate+

Let $$ f\to Y $$ be a continuous map, and let (\mathcal F) be a sheaf of abelian groups on (X). Take $$ F=f_*,\qquad G=\Gamma(Y,-). $$ The composite (G\circ F) is (\Gamma(X,-)). The Grothendieck spectral sequence gives $$ E_2^{p,q}=H^p\bigl(Y,R^qf_*\mathcal F\bigr) \Rightarrow H^{p+q}(X,\mathcal F). $$ This is the Leray spectral sequence.

The hypothesis that (f_*) sends injective sheaves on (X) to (\Gamma(Y,-))-acyclic sheaves is satisfied in the usual sheaf categories. Hence Leray is not a separate trick; it is the sheaf cohomology instance of Grothendieck's composition theorem.

Derived-category interpretation Master

The modern proof compresses the construction into one statement: $$ R(GF)\simeq RG\circ RF. $$ This statement is not formal for arbitrary functors; it depends on the existence of a class of replacements adapted to the composite. The condition that (F) send injectives to (G)-acyclic objects gives such an adapted class.

Once the isomorphism is available in (D^+(\mathcal C)), the spectral sequence arises from filtering a representative complex for (RG(RF(X))). The (E_2)-page records the derived functors of (G) applied to the cohomology objects of (RF(X)), while the abutment is the cohomology of the total object (R(GF)(X)).

This is why the theorem functions as a bridge between two languages:

• the classical language of injective resolutions and derived functor groups;
• the derived-category language where total derived functors compose as objects and functors.

Convergence and filtrations Master

For a first-quadrant double complex, the filtration by columns is bounded in each total degree. Therefore the associated spectral sequence converges to the cohomology of the total complex without requiring extra completeness hypotheses.

The output is a finite filtration $$ 0=F^{n+1}H^n\subseteq F^nH^n\subseteq\cdots\subseteq F^0H^n=H^n $$ on $$ H^n=R^n(GF)(X), $$ with $$ F^pH^n/F^{p+1}H^n\cong E_\infty^{p,n-p}. $$ In applications, extension problems may remain after (E_\infty) is known. A spectral sequence gives the associated graded pieces; separate input may be needed to reconstruct the filtered object itself.

Variants and applications Master

The same pattern appears in many forms:

• Leray spectral sequence: (H^p(Y,R^qf_*\mathcal F)\Rightarrow H^{p+q}(X,\mathcal F)).
• Local-to-global Ext: sheaf Ext groups first, then sheaf cohomology, abutting to global Ext.
• Group extensions: invariants for a normal subgroup followed by invariants for the quotient give the Hochschild-Serre spectral sequence.
• Change of rings: derived functors for restriction, extension, or Hom across a ring map produce spectral sequences for Ext and Tor.
• Hypercohomology: global sections after resolving a complex of sheaves leads to spectral sequences whose pages mix sheaf cohomology and cohomology sheaves.

In each case the visible formula is less important than the adapted replacement condition. One verifies that the first operation sends chosen replacements to objects on which the second operation has no higher derived obstruction.

Historical note Master

Grothendieck's 1957 Tohoku paper introduced the systematic derived functor viewpoint for abelian categories and made spectral sequences a central organizing tool for sheaf cohomology. Cartan-Eilenberg supplied the resolution technology; Verdier and later Gelfand-Manin recast the same theorem in derived-category language.

The theorem's lasting value is conceptual economy: a large family of named spectral sequences becomes one composition principle for derived functors.

Connections Master

• 04.03.12 supplies total derived functors (RF) and (LF).
• 04.03.15 specializes this theorem to the Leray spectral sequence.
• 04.03.16 uses derived functor composition as part of the six-functor formalism.
• 03.13.01 gives the general spectral-sequence mechanics from filtered complexes and double complexes.

Exercises Intermediate+

Bibliography Master

• A. Grothendieck, "Sur quelques points d'algèbre homologique," Tohoku Mathematical Journal 9 (1957), especially §2.4.
• H. Cartan and S. Eilenberg, Homological Algebra, Princeton University Press, 1956.
• S. I. Gelfand and Y. I. Manin, Methods of Homological Algebra, Springer, Ch. III §6.
• C. A. Weibel, An Introduction to Homological Algebra, Cambridge University Press, Ch. 5 §8.
• J. McCleary, A User's Guide to Spectral Sequences, Cambridge University Press, Ch. 12.