L-COMPLETE HOPF ALGEBROIDS 21

Corollary B.3. The module L1(

k

M/N) is non-zero.

Proof. For each s 1, the natural short exact sequence of (1.1) and the fact

that M/N is L-complete and so LsM/N = 0, together yield

lim

n

Tor2

R(R/mn,K)

= 0 =

lim1

n

Tor2

R(R/mn,K).

This gives one of the hypotheses of Lemma B.1, and Lemma B.2 gives the other.

Therefore

lim1

n

Tor2

R(R/mn,M/N)

= 0.

Now applying (1.1) to M/N gives L1(

k

M/N) = 0.

Lemma B.4. If the sequence p, u acts regularly on the R-module K, then LsK =

0 for s 0.

Proof. Using the exact sequence (1.1), it suﬃces to show that for all s 0,

Tors

R(R/mn,K)

= 0. This can be deduced from the case n = 1 since R/mn has a

composition series with simple quotient terms isomorphic R/m. This case n = 1

can be directly veriﬁed using the Koszul resolution.

Here is the main result of this Appendix which complements an example of [6].

Theorem B.5. The natural map L0(

k

N) −→ L0(

k

M) is not injective.

Proof. The short exact sequence

0 →

k

N −→

k

M −→

k

M/N → 0

induces an exact sequence

L1(

k

M) −→ L1(

k

M/N) −→ L0(

k

N) −→ L0(

k

M) → 0.

The sequence p, u acts regularly on

k

M, so Lemma B.4 shows that L1(

k

M) =

0, while Corollary B.3 shows that L1(

k

M/N) = 0.

References

[1] A. Baker, A version of the Landweber ﬁltration theorem for vn-periodic Hopf algebroids,

Osaka J. Math. 32 (1995), 689–99.

[2] A. Baker & B. Richter, Galois extensions of Lubin-Tate spectra, Homology, Homotopy and

Appl. 10 (2008), 27–43.

[3] S. U. Chase, D. K. Harrison & A. Rosenberg, Galois theory and Galois cohomology of

commutative rings, Mem. Amer. Math. Soc. 52 (1965), 15–33.

[4] L. N. Childs, G. Garﬁnkel & M. Orzech, The Brauer group of graded Azumaya algebras,

Trans. Amer. Math. Soc. 175 (1973), 299–326.

[5] J. P. C. Greenlees & J. P. May, Derived functors of I-adic completion and local homology,

J. Alg. 149 (1992), 438–453.

[6] M. Hovey, Morava E-theory of ﬁltered colimits, Trans. Amer. Math. Soc. 360 (2008), 369–

382.

[7] M. Hovey & N. P. Strickland, Morava K-theories and localisation, Mem. Amer. Math. Soc.

139 no. 666 (1999).

[8] , Comodules and Landweber exact homology theories, Adv. Math. 192 (2005),

427–456.

[9] , Local cohomology of BP∗BP -comodules, Proc. Lond. Math. Soc. (3) 90

(2005), 521–544.

[10] M. K¨ unzer, On representations of twisted group rings, J. Group Theory 7 (2004), 197–229.

21

Corollary B.3. The module L1(

k

M/N) is non-zero.

Proof. For each s 1, the natural short exact sequence of (1.1) and the fact

that M/N is L-complete and so LsM/N = 0, together yield

lim

n

Tor2

R(R/mn,K)

= 0 =

lim1

n

Tor2

R(R/mn,K).

This gives one of the hypotheses of Lemma B.1, and Lemma B.2 gives the other.

Therefore

lim1

n

Tor2

R(R/mn,M/N)

= 0.

Now applying (1.1) to M/N gives L1(

k

M/N) = 0.

Lemma B.4. If the sequence p, u acts regularly on the R-module K, then LsK =

0 for s 0.

Proof. Using the exact sequence (1.1), it suﬃces to show that for all s 0,

Tors

R(R/mn,K)

= 0. This can be deduced from the case n = 1 since R/mn has a

composition series with simple quotient terms isomorphic R/m. This case n = 1

can be directly veriﬁed using the Koszul resolution.

Here is the main result of this Appendix which complements an example of [6].

Theorem B.5. The natural map L0(

k

N) −→ L0(

k

M) is not injective.

Proof. The short exact sequence

0 →

k

N −→

k

M −→

k

M/N → 0

induces an exact sequence

L1(

k

M) −→ L1(

k

M/N) −→ L0(

k

N) −→ L0(

k

M) → 0.

The sequence p, u acts regularly on

k

M, so Lemma B.4 shows that L1(

k

M) =

0, while Corollary B.3 shows that L1(

k

M/N) = 0.

References

[1] A. Baker, A version of the Landweber ﬁltration theorem for vn-periodic Hopf algebroids,

Osaka J. Math. 32 (1995), 689–99.

[2] A. Baker & B. Richter, Galois extensions of Lubin-Tate spectra, Homology, Homotopy and

Appl. 10 (2008), 27–43.

[3] S. U. Chase, D. K. Harrison & A. Rosenberg, Galois theory and Galois cohomology of

commutative rings, Mem. Amer. Math. Soc. 52 (1965), 15–33.

[4] L. N. Childs, G. Garﬁnkel & M. Orzech, The Brauer group of graded Azumaya algebras,

Trans. Amer. Math. Soc. 175 (1973), 299–326.

[5] J. P. C. Greenlees & J. P. May, Derived functors of I-adic completion and local homology,

J. Alg. 149 (1992), 438–453.

[6] M. Hovey, Morava E-theory of ﬁltered colimits, Trans. Amer. Math. Soc. 360 (2008), 369–

382.

[7] M. Hovey & N. P. Strickland, Morava K-theories and localisation, Mem. Amer. Math. Soc.

139 no. 666 (1999).

[8] , Comodules and Landweber exact homology theories, Adv. Math. 192 (2005),

427–456.

[9] , Local cohomology of BP∗BP -comodules, Proc. Lond. Math. Soc. (3) 90

(2005), 521–544.

[10] M. K¨ unzer, On representations of twisted group rings, J. Group Theory 7 (2004), 197–229.

21