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
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