In the index (pages ), reduce all page numbers by 2.

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1 Errata or Nilpotence and periodicity in stable homotopy theory (Annals O Mathematics Study No. 28, Princeton University Press, 992) by Douglas C. Ravenel, July 2, 997, edition. Most o these were ound by Peter Landweber. Major revisions: In the index (pages ), reduce all page numbers by 2. Page 9, insert between.5.4 and last paragraph: Some comments are in order about the deinition o a v n -map given in the theorem. First, X is not logically required to have type n, but that is the only case o interest. I X has type > n, then the trivial map satisies the deinition, and i X has type < n, it is not diicult to show that no map satisies it (3.3.). Second, it does not matter i we require K(m) () to be trivial or merely nilpotent or m > n. I it is nilpotent or each m > n, then some iterate o it will be trivial or all m > n. For d > 0 this ollows because some iterate o H () must be trivial or dimensional reasons, and K(m) () = K(m) H () or m 0. The case d = 0 occurs only when n = 0, or which the theorem is trivially true since the degree p map satisies the deinition. Pages 35 36, omit and the preceding two paragraphs, and replace the sketch o proo by the ollowing: Proo o Theorem Note that C p,0 = CΓ (p) by convention and we have a decreasing iltration CΓ (p) = C p,0 C p, C p,n with n 0 C p,n = {0} by Corollary 3.3.9(ii). Now suppose C CΓ (p) is thick. I C {0}, choose the largest n so that C p,n C. Then C C p,n+, and we want to show that C = C p,n, so we need to veriy that C C p,n. Let M be a comodule in C but not in C p,n+. Thus vm M = 0 or m < n M 0. Choosing a Landweber iltration o M in CΓ, but v n 0 = F 0 M F M F k M = M, all F s M are in C, hence so are all the subquotients Since vn M 0, we must have F s M/F s M = Σ d s MU /I p,ms. v n (MU /I p,ms ) 0 or some s, so some m s is no more than n. This m s must be n, since a smaller value would contradict the assumption that C C p,n. Hence we conclude that MU /I p,n C. (3.4.6)

2 Now let N be in C p,n ; we want to show that it is also in C. Then v n M = 0, so each subquotient o a Landweber iltration o N is a suspension o MU /I p,m or some m n. Since MU /I p,n C by (3.4.6), it ollows that MU /I p,n C or all m n. Hence the Landweber subquotients o N are all in C, so N itsel is in C. Page 50, replace Proo o 5..5 (the rest o this section) with: Proo o Corollary Let R = DW W and let e : R be the adjoint o the identity map. R is a ring spectrum (A.2.8) whose unit is e and whose multiplication is the composite R R = DW W DW W DW De W DW W = R. The map : X Y is adjoint to ˆ : DX Y, and W is adjoint to the composite ˆ DX Y e DX Y R DX Y = F, which we denote by g. The map W (i) is adjoint to the composite g (i) F (i) = R (i) DX (i) Y (i) R DX (i) Y (i), the latter map being induced by the multiplication in R. By 5..4 it suices to show that MU g (i) is null or large i. Let T i = R DX (i) Y (i) and let T be the direct limit o g T T ˆ T 2 T 2 ˆ T 3. The desired conclusion will ollow rom showing that M U T is contractible, and our hypothesis implies that K(n) T is contractible or each n. Now we need to use the methods o Chapter 7. Since we are in a p-local situation, it suices to show that BP T is contractible. Using and the act that K(n) T is contractible, it suices to show that P (m) T is contractible or large m. Now or large enough m, K(m) (W ) = K(m) H (W ) and P (m) (W ) = P (m) H (W ). Our hypothesis implies that both o these homomorphisms are trivial, so P (m) T is contractible as required. 2

3 Page 99, replace last sentence o paragraph ater 9.0. with: We claim that h( ˆ) is nilpotent i MU () is. To see this, observe that i MU () = 0, then MU X is contractible, where X denotes the homotopy direct limit o X Σ d X Σ 2d X. Since X is inite, this means that or large enough m, the composite Σ md X m X MU X is null. Then h( m ) = h( ˆ) m = 0, so h( ˆ) is nilpotent. 3

4 Misprints and minor corrections: Page xi, line 2 o second paragraph: algebraic patterns Page xi, line 4 o second paragraph: Northwestern Page xiv, line 6 o second paragraph: realizability Page xiv, line 8 o second paragraph: chosen Chapter Page, line 3: three sections Page, last line: [(X, x 0 ), (Y, y 0 )]. Page 4, line 9: point out the properties Page 5, (.3.2): E i+ (ΣX). Page 5, line 3 o.3.3: E () = 0 Page 6, paragraph ater.4.2: Actually this is the weakest o the three orms o the nilpotencetheorem; the other two (5..4 and 9.0.) are equivalent and imply this one. Page 6, line ater.4.: E to Page 6, line 3 ater.4.2: bordism theory, Page 6, line 3 above.5: Chapter 3 Page 8, line 5: I X is as above then Page 8, line 7: i X is simply connected and not contractible. Page 8, line 5 o.5.4: as a v n -map; see page 53. Page 9, lines 5 and 2: asymptotically Chapter 2 Page, line : In this chapter Page 2, line 3 ater 2..: It is easy to construct a inite CW- complex o dimension 2 Page 3, last line beore 2.2.6: was proved in [CMN79]. Page 5, line 2 above 2.3.4: three o the undamental Page 7, line o 2.4.2(ii): ollowing composite: Page 7, line 5 o 2.4.2(ii): o its suspensions.) Page 8, line 6 ater (2.4.3): so we can use 2.3.4(iii) Page 8, line above (2.4.4): and consider the diagram Page 8, line o (2.4.4): Σ td X e X k+e j k = X X k+e k+e = Sk+e Page 8, line 2 ater (2.4.4): i e = e. Page 9, line 2 o (2.4.5): Page 9, line 3 o (2.4.5): i Σ td X k+ k+ Σ td X k+e k+ i Σ td X k+2 k+2 Σ td X k+e k+2 4

5 Page 20, line 5 o second paragraph in 2.5: known to be isomorphic to H (Y ; Q). Page 20, line 3 above (2.5.): maps rom the inverse Page 22, line 4: For suiciently large Page 22, line 5 ater diagram: the orm g i Page 22, line 6 ater diagram: the coibre o i Page 23, line 3 o diagram: W (2) = C i Chapter 3 Page 25, line 3 o 3..: compact smooth maniold Page 25, line 3 above 3..2: e.g. a nonsingular complex algebraic variety Page 3, line 5 ater 3.3.6: initely generated module M Page 3, lines 5 and 4 above 3.3.7: Now L is not Noetherian, but it is a direct limit o Noetherian rings, so initely presented modules over it admit similar iltrations. Page 32, line 2 above 3.3.: Another consequence Page 34, line 2 o 3.4.2: p-local modules in CΓ Page 34, line ater 3.4.2: We will give the proo o this result below. Page 34, insert ater 3.4.3: The condition vn MU (X) = 0 is equivalent to K(n ) (X) = 0, in view o.5.2(v) and 7.3.2(d) below. Chapter 4 Page 42, line 3 o 4.3.4: elements o inite order Chapter 5 Page 45, irst sentence: In this chapter we will derive the thick subcategory theorem (3.4.3) rom a variant o the nilpotencetheorem (5..4 below) with the use o some standard tools rom homotopy theory, which we must introduce beore we can give the proo. Page 48, 5.2. (i): For any spectrum Y, the graded group [X, Y ] is isomorphic to π (DX Y ), and this isomorphism is natural in both X and Y. In particular, D =. Page 48, 5.2.(ii): This isomorphism is relected in Morava K- theory, namely (since K(n) (X) is ree over K(n) ) Hom K(n) (K(n) (X), K(n) (Y )) = K(n) (DX Y ). In particular or Y = X, K(n) (e) 0 when K(n) (X) 0. Similar statements hold or ordinary mod p homology. For X =, this isomorphism is the identity. Page 48, line 3: which we will prove at the end o Section 5.2, using some methods rom Chapter 7. Page 48, 5.2.(iii): DDX X and [X, Y ] = [DY, DX]. Page 48, insert at end o 5.2.: (vi) The unctor X DX is contravariant. Page 5, bottom o page, insert: Equivalently (by 7.3.2(d) and.5.2(v)), n is the smallest integer such that C contains an X with vn BP (X) 0. 5

6 Chapter 6 Page 56, sentence beginning on line 8: Hence 5..5 tells us that ad( ˆ) is nilpotent and the argument above applies to give the desired result. Page 58, insert at end o 6.: The ollowing will be used in the proo o Lemma Let X be a inite complex with a v n -map and let /sp X denote the telescope associated with X. Then C /sp X = /pnt. Proo: X and X are both v n -maps on the inite complex X X. Thus by 6..4 we can replace by a suitable interate so that X and X are homotopic. It ollows that /sp X is an equivalence so C /sp X = /pnt. Page 64, Lemma 6.3.3: x Ext AN (Z/(p), Z/(p)) Chapter 7 Page 7, diagram: η W W E X η E X E X E W E E X E X. E m X Page 73, 7.2.6(iii): Omit the second statement. Page 74, 7.2.9, line 4: Then FBA (p) is the ree Page 75, irst line o proo o 7.2.9: 7.2.6(iii) and 6..7 give Page 76, bottom paragraph: Replace [X, Y ] and [X, cy ] with [X, Y ] and [X, cy ]. Page 80, bottom line: X lim L n X. Chapter 8 Page 85, line 3 ater 8.2.3: E(2) Page 85, sentence beginning at line 7: A.6. is actually a characterization; or a countable ring spectrum E, all spectra are E-prenilpotent i and only the conbergence condition is satisied. Page 87, line 9: L n Y Page 95, (8.6.2): lim π i (C n X) Page 95, De 8.6.3: A spectrum Y is E-convergent i the E-based Adams spectralsequence or Y converges to π (Y ) and there is a nondecreasing unction s(i) such that, Page 95, line 4, put a space ater theorem Page 95, line ater (8.6.2) and page 96, line 5: lim Chapter 9 Page 99, line 3 ater 9.0.: let R = DX X 6

7 Page 00, replace irst sentence with: Proo that 9.0. implies 5..4: (In the ollowing argument, MU could be replaced by any ring spectrum or which 9.0. holds.) Page 00, third displayed ormula: X Page 00, ourth displayed ormula: η X MU X η X α X α MU X α Page 0, lines 4 and 3: interested in the localization o F p j at p, which we denote by G j. These spectra interpolate between X(n) (p) and X(n + ) (p). Page 02, Lemma 9..6:... then or each prime p... Page 02, Lemma 9..7: X(n) (p). Page 02, Replace last next to last sentence o Proo o 9.. with: By 9..7 this means that α R X(n) (p) is contractible or each prime p, so α R X(n) is contractible. Y s+ = G s /im i 0 im i im i s Page 05, last paragraph o proo: Projection onto the irst coordinate gives compatible maps o the G s to E, and hence a stable vector bundle over each o them. This means that we can Thomiy the entire construction. To each o the quotients Y s and L s we associate a reduced Thom spectrum, which is deined as ollows. Given a space A with a vector bundle and a subspace B A, the reduced Thom space or A/B is the space D A /(S A D B ) where D X and S X denote disk and sphere bundles over the space X. Thus we get the coibre sequences (9.2.5) deining the desired Adams resolution by Thomiying (9.2.4). Page 08, Prop : For p > 2, For p = 2, (x 2p i+ m )β = y 2p i+ m 2 or i 0 (y 2p i+ m 2)β = 0 or i 0 (x 2p i m )P j = 0 or i, j 0 (y 2p i+ m 2)P = y p 2p i m 2 or i 0 (y 2p i m 2)P j = 0 or i 0, j > (x 2 i+ m )Sq = x 2 2 i m or i 0 (x 2 i+ m )Sq j = 0 or i 0, j >. Appendix A Page 2, omit A..6 and preceding sentence. Page 24, line 6 o A.2.8: are each homotopic to the identity on E 7

8 Page 24, bottom o page: such that the ollowing diagram commutes up to homotopy m E E E M E M E µ µ E M µ M and the composite M = M η M M M µ M is homotopic to the identity. Page 26, line 4 above A.3: There is a Hurewicz theorem or connective spectra Page 30, irst sentence o bottom paragraph: This theorem can used to construct a spectrum by constructing the cohomology theory it represents. Page 4, line 8: E (E X) = E (E) π (E) E (X). Appendix B Page 48, line ater B..7: For a paracompact base space X Page 55, line : E (E) and the map ψ Page 62, last line o B.4.7: with coeicients in R Page 63, line 2 o Proo o B.4.0: w(x, y) = log F (F (x, y)) log F (x) log F (y). Page 7, line 2 o part (c) o Proo: at most one ormal summand Page 7, line o B.5.6: an ideal J S Page 73, third displayed ormula: 0 Tor BP (M, BP /(p)) M p M M/pM 0. Page 73, three lines above B.6.2: 0 Tor BP (M, BP /I n M) M/I n M p M/I n M M/I n M 0. Thus we see that Tor BP (M, BP /I n ) = 0 provided that M/I n M is v n -torsion ree, i.e., that multiplication by v n in M/I n M is monic. Page 74, line 2: or each positive integer n Page 74, line 9 ater B.6.2: In all three cases Page 75, bottom line: conjugate o τ i 8