THE INFLATION-RESTRICTION SEQUENCE : AN INTRODUCTION TO SPECTRAL SEQUENCES TOM WESTON. Example We egin with aelian groups for every p, q and maps : + (here, as in all of homological algera, all maps are indexed y the group where they originate) making the columns into complexes; that is, + = for all p and q. For convenience, if either p or q is negative we define =, and we let e the zero map. E, E, E, E 3, E, E, E, E 3, E, E, E, E 3, Now, since the columns are complexes, we can take their cohomology. Specifically, for all p and q we define = ker / im dp,q. (By our conventions efore we automatically have = if either p or q is negative.) Now, suppose that we somehow magically otain horizontal maps : E p+,q which make the rows into complexes; that is, d p+,q =
TOM WESTON for all p and q. E, E, E, E 3, E, E, E, E 3, E, E, E, Again, let us take the cohomology; that is, define = ker / im dp,q E 3, for all p and q. Further, let us suppose that we somehow otain maps : E p+,q satisfying d p+,q =. Here the terminal point of each map has shifted one to the right and one down from the terminal point at the previous step. E, E, E, E 3, E, E, E, E 3, E, E, E, E 3, Note that the maps leaving the ottow row E p, all must e, since they land in E p+, =. Similarly, the maps entering the first two columns E,q and E,q all originiate in groups, and thus are. In particular, for oth groups E, and E, oth the maps entering and leaving are ; we will return to this oservation in a moment. Now, continue our procedure another step. That is, define 3 = ker / im dp,q+ 3. Again, let us suppose that we somehow otain maps 3 : 3 E p+3,q 3, shifted one to the right and one down of the previous maps, such that d p+3,q 3 3 =. E, 3 E, 3 E, 3 E 3, 3 E, 3 E, 3 E, 3 E 3, 3 E, 3 E, 3 E, 3 E 3, 3
SPECTRAL SEQUENCES 3 Returning to our previous oservation, we see that since oth the maps d, and d, entering and leaving E, are zero, Similarly, E, 3 = E,. E, 3 = E,. Thus these two entries have stailized at the E step. We will denote these (now and forever) constant values y E, and E, respectively. We get more entries stailizing now at the E 3 step. This time, all of the maps leaving the ottom two rows and entering the first three columns are the zero map. 3 have stailized. As Thus all of the entries E, 3, E, 3, E, 3, E, 3, E, 3 and E, efore, we denote these constant values y for the appropriate p and q; this is of course consistent with our earlier definitions of E, and E,. Roughly speaking, a spectral sequence is all of the data in the aove construction. The general idea is that one starts with interesting groups Er p,q at some early stage r and that the stale values E p,q are also related to other interesting groups. Thus the spectral sequence would somehow codify a relationship etween these two families of groups. Specifically, the content of a spectral sequence is in three pieces of information : interesting interpretations of Er p,q for some small r (very often ); the construction of the new maps r at each stage; and interesting interpretations of the stale values E p,q. Of course, it is not clear how precisely the spectral sequence relates all of these groups; in general the relationship is so complicated as to e unusale. There are several important cases where precise information can e otained, however; we will give two examples elow.. Formal Definitions Definition. A a th -stage (first quadrant cohomological) spectral sequence is a collection of aelian groups Er p,q for all p, q and for all r a for some positive integer a, together with maps such that and r : r E p+r,q r+ r d p+r,q r+ r r = r+ = ker r / im d p r,q+r r for all p, q and r as aove. As we have seen aove, in any spectral sequence the (p, q) spot eventually stailizes; we denote this stale value of Er p,q y E p,q. As can e seen easily y continuing the construction of the first section, this will e achieved at least y the p + q + step, and often earlier. The definition aove makes no mention of these stale values, however. To relate these to the spectral sequence we need to introduce the notion of convergence. Definition. An a th -stage spectral sequence Er p,q H n, written Er p,q H p+q, if there is a filtration is said to converge to groups = H n n+ H n n H n n H n H n H n = H n
4 TOM WESTON such that for all p. E p,n p = H n p /H n p+ Hn n H n p+ H n p H n H n E n, E p,n p E,n Thus the stale values E p,q on the line p + q = n are the succesive quotients in a filtration of H n, so that knowledge of them gives a lot of information aout H n. The simplest such information are the edge maps and Ea n, E n, = Hn n H n H n H n /H n = E,n E,n a. (E n, is a quotient of Ea n, since all maps leaving the (n, )-entry are, so that at each stage the kernel is the whole entry. E,n is a sugroup of Ea,n for similar reasons.) There is even a very important special case where knowledge of the E p,q actually gives complete information aout H n. Definition 3. An a th -stage spectral sequence Er p,q is said to collapse at the th - stage if there is only one non-zero row or column in Let a H p+q e a convergent spectral sequence collapsing at the th -stage, and assume that To fix ideas let us suppose that only the q row is non-zero.. E,q E,q E,q E 3,q Two important things now happen. First, every map must e the zero-map. Thus we must have + = Ep,q for all p and q, and continuing in this way we see that = for all p and q. Second, the p + q = n diagonal only has the one non-zero term, so our aove filtration yields E n q,q H n = E n q,q. Thus for a collapsing spectral sequence there is no amiguity aout the H n.
SPECTRAL SEQUENCES 5 3. The Hochschild-Serre Spectral Sequence It is well past time for an example. We will give the most important general family of examples. For details see [, Chapter 5]; the construction is fairly involved. So, suppose that we have two aelian categories A and B. Further suppose that we have functors G : A B and F : B A, where A is the category of aelian groups. (There isn t really any need to require that G takes values in A; it could just as well e any aelian category. One merely has to replace the words aelian group with oject in an aelian category throughout the previous two sections.) A F G G F A We suppose further that oth A and B have enough injectives, and that F and G are left exact, so that they have right derived functions R r F and R r G for all r. The Grothendieck spectral sequence compares the composition of the derived functors with the derived functors of the composition. Theorem 3. (Grothendieck Spectral Sequence). Given the aove setup, suppose further that for any injective oject I of A, R r F (G(I)) = for all r >. Then for any oject A of A, there exists a second stage (first-quadrant cohomological) spectral sequence The edge maps and are the natural maps. = (R p F )(R q G)(A) R p+q (F G)(A). B (R p F )(GA) R p (F G)(A) R q (F G)(A) F (R q G(A)) As a particular example, we have the Hochschild-Serre spectral sequence for the cohomology of groups. Theorem 3. (Hochschild-Serre Spectral Sequence). Let G e a group, H a normal sugroup and A a G-module. Then there is a second stage (first-quadrant, cohomological) spectral sequence The edge maps and = H p (G/H; H q (H; A)) H p+q (G; A). H n (G/H; A H ) H n (G; A) H n (G; A) H n (H; A) G/H are induced from inflation and restriction respectively. Proof. This is a special case of the Grothendieck spectral sequence; for details, see [, Chapter 6, Section 8].
6 TOM WESTON H (G/H; H (H; M)) H (G/H; H (H; M)) H (G/H; H (H; M)) H (G/H; H (H; M)) H (G/H; H (H; M)) H (G/H; H (H; M)) H (G/H; H (H; M)) H (G/H; H (H; M)) H (G/H; H (H; M)) 4. The Inflation-Restriction Sequence Even when a spectral sequence does not collapse, one can often still otain information aout some of the low-degree terms. In this section we will construct an exact sequence of fundamental importance in group cohomology. Let H p+q e a convergent second stage spectral sequence. E, E, E, E, E, E, E, E, E, Let us follow through the computation of 3 a it. Straight from the definitions, we get the following picture for the third stage, in which the oxed entries have already stailized to their final values. ker E, E, ker E, E 3, mess ker E, E, ker E, E 3, mess E, E, E, / im E, Now, let us comine this with our knowledge of convergence. First, directly from the filtration of H, we get an exact sequence E, H (ker E, E, ). Next, we know that we have an edge map E, / im E, H. Thus these two exact sequences splice together to give a five term exact sequence E, H E, E, H.
SPECTRAL SEQUENCES 7 Further, every map except for E, E, is just an edge map. We state the special case of this for the Hochschild-Serre spectral sequence as a theorem. Theorem 4. (Inflation-Restriction). If H is a normal sugroup of G and M is a G-module, then there is an exact sequence H (G/H; M H ) H (G/H; M H ) inf H (G; M) inf H (G; M) where inf and res are the inflation and restriction maps. res H (H; M) G/H Now, suppose further that the q = row of our spectral sequence vanishes. That is, suppose that E p, = for all p. Since every Er p, is a suquotient of E p,, it follows that this row is zero for all r. (In general, any entry in the spectral sequence which is ever zero is always and forever zero.) In particular, our aove exact sequence now ecomes an isomorphism E, = H and an injection E, H. In this case we can actually extend this second exact sequence farther to the right. So, we egin with a second stage spectral sequence with zero q = row. E, E, E, E 3, E, E, E, E 3, We again follow along to the third stage. This time we get E, E, mess mess E, E, E, E 3,
8 TOM WESTON Now, let us go to the fourth stage. This time we get ker E, E 3, ker E, E 3, mess mess E, E, E, E 3, / im E, where all of the entires pictured aove have stailized to their values. As efore, the filtration of H is reflected y the exact sequence E, H (ker E, E 3, ). We also again get an edge map injection E 3, / im E, H 3. These splice together to give the five term exact sequence E, H E, E 3, H 3, in which all ut one of the maps is simply an edge map. It is easy to generalize the aove construction to the case where each of the rows q = up to q = q vanishes. We state the group cohomology case as a theorem. Theorem 4. (Higher Inflation-Restriction). Let H e a normal sugroup of G and M a G-module. Suppose that H q (H; M) = for q < q. Then for q < q, inflation induces isomorphisms H q (G/H; M H ) = H q (G; M) and there is an exact sequence H q (G/H; M H ) H q+ (G/H; M H ) inf H q (G; M) inf H q+ (G; M) where inf and res are the inflation and restriction maps. References res H q (H; M) G/H [] Charles Weiel, An introduction to homological algera. Camridge; Camridge University Press, 994.