8 ANDREW BAKER
Let K = ker f. Tensoring the exact sequence
0 K −→ Fm −→ P 0
with R/m, by flatness of P we obtain the exact sequence
0 K/mK −→ F/mF

=
−→ P/mP 0
so K/mK = 0. Hence K = 0 by Nakayama’s Lemma.
In general, tensoring with a pro-free module need not be left exact on M as is
shown by an example in Appendix B. In particular, when n 1, infinitely generated
pro-free modules may not always be flat. Instead we can restrict attention to L-
flatness on restricted classes of L-complete modules. We say that P is weakly L-flat
if the functor
Mbt −→ M ; M P ⊗M
is exact on the subcategory Mbt of bounded m-torsion modules. Then if Q is a flat
module, L0Q is weakly L-flat since for any N Mbt,
L0Q⊗N

=
L0(Q N)

=
Q⊗N.
2. L-complete Hopf algebroids
To ease notation, from now on we assume that (R, m) is a commutative Noe-
therian regular local ring which is m-adically complete, i.e., R = R = Rm. We
assume that R is an algebra over some chosen local subring (k0, m0) so that the
inclusion map is local, i.e., m0 = k0 m. We write k = R/m for the residue field.
Let Γ Mk0 . We need to assume extra structure on Γ to define the notion of
an L-complete Hopf algebroid. Unfortunately this is quite complicated to describe.
A (non-unital) ring object A Mk0 is equipped with a product morphism
ϕ: A ⊗k0 A −→ A which is associative, i.e., the following diagram commutes.
A ⊗k0 A ⊗k0 A
id⊗ϕ
id⊗ϕ
A ⊗k0 A
ϕ
A ⊗k0 A
ϕ
A
It is commutative if
A ⊗k0 A
switch
ϕ
A ⊗k0 A
ϕ
A
also commutes. An R-unit for ϕ is a k0-algebra homomorphism η : R −→ A.
Definition 2.1. A ring object is R-biunital if it has two units ηL,ηR : R −→ A
which extend to give a morphism ηL ηR : R ⊗k0 R −→ A.
To distinguish between the two R-module structures on A, we will write RA and
AR. When discussing AR we will emphasise the use of the right module structure
whenever it occurs. In particular, from now on tensor products over R are to be
interpreted as bimodule tensor products R⊗R, even though we often write ⊗.
Definition 2.2. An R-biunital ring object A is L-complete if A is L-complete
as both a left and a right R-module.
8
Previous Page Next Page