Spectral sequence: Difference between revisions

Revision as of 09:56, 1 January 2008

Spectral sequences were invented by Jean Leray as an approach to computing sheaf cohomology.

A (cohomology) spectral sequence (starting at $E_{a}$ ) in an abelian category $A$ consists of the following data:

A family $\{E_{r}^{pq}\}$ of objects of $A$ defined for all integers $p,q$ and $r\geq a$
Morphisms $d_{r}^{pq}:E_{r}^{pq}\to E_{r}^{p+r,q-r+1}$ that are differentials in the sense that $d_{r}\circ d_{r}=0$ , so that the lines of "slope" $-r/(r+1)$ in the lattice $E_{r}^{**}$ form chain complexes (we say the differentials "go to the right")
Isomorphisms between $E_{r+1}^{pq}$ and the homology of $E_{r}^{**}$ at the spot $E_{r}^{pq}$ :

E_{r+1}^{pq}\simeq \ker(d_{r}^{pq})/{\text{image}}(d_{r}^{p-r,q+r+1})

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 ==Definition==
+A (cohomology) spectral sequence (starting at <math>E_a</math>) in an abelian category <math>A</math> consists of the following data:
+#A family <math>\{E_r^{pq}\}</math> of objects of <math>A</math> defined for all integers <math>p,q</math> and <math>r\geq a</math>
+#Morphisms <math>d_r^{pq}:E_r^{pq}\to E_r^{p+r,q-r+1}</math> that are differentials in the sense that <math>d_r\circ d_r=0</math>, so that the lines of "slope" <math>-r/(r+1)</math> in the lattice <math>E_r^{**}</math> form chain complexes (we say the differentials "go to the right")
+#Isomorphisms between <math>E_{r+1}^{pq}</math> and the homology of <math>E_r^{**}</math> at the spot <math>E_r^{pq}</math>:
+:<math>E_{r+1}^{pq}\simeq \ker(d_r^{pq})/\text{image}(d_r^{p-r,q+r+1})</math>
 ==Convergence==
-==Examples
+==Examples==
 #The Leray spectral sequence
 #The Grothendieck spectral sequence