2. ^-ADIC DERIVED CATEGORIES
If F has a left adjoint G : T2 — V\ mapping V2 to V[, then G : T2/V2 —
Vi/V^ is a left adjoint of F.
R E M A R K
1.6. Clearly, the constructions of Propositions 1.3 and 1.4 com-
mute with passage to D + , T~ or P 6 .
2. £-Adic Derived Categories
Using results from [13, Exp. V] we will develop an ^-adic formalism as
Let A be a discrete valuation ring and [ the maximal ideal of A. Let £
be a generator of I. For n E N w e denote
. For an abelian A-
category 21 we denote by 2ln the category of (An, 2l)-modules. In other words,
2P is the full subcategory of 21 consisting of those objects on which £n+1 acts
as zero. By 21 we denote the category of projective systems (M
^ in 21
such that Mn G
for every n G N. The categories 21 and 2P are abelian.
component functor an : 21 — 2P is exact.
By construction, for m n, the category
is naturally a subcategory
of 2lm. This inclusion has a left adjoint 2lm — 2ln, which we denote by
M n M ^ A
. We may define M ® A
We say that an object (M
^ of 21 is £-adic, if for every m n the
(which is given by adjunction) is an isomorphism.
We denote the full subcategory of £-adic projective systems in 21 by 21^.
An object M = (Mn)ne^ of 21 is called AR-null if there exists an integer
r such that M
n + r
is the zero map for every n G N. We say that a
homomorphism 0 : M —• A^ in 21 is an AR-isomorphism if ker / and cok 0
are AR-null. An object M of 21 is called AR-£-adic1 if there exists an ^-adic
object F and an AR-isomorphism (j) : F — M . The symbol 21AR-^ denotes
the category of AR-£-adic objects in 21.
Some important properties of ^-adic objects with respect to AR-
isomorphisms are the following:
LEMMA 1.7. Let a : M' -^ M be an AR-isomorphism in 2l; where M is
£-adic. Then a has a section.
L E M M A
1.8. Let a : F — » M be an AR-isomorphism in 2l; with F being
£-adic. Then there exists an r 0 such that Fn is a direct summand of
Mn+r (g) An for every n.
PROOF. This follows from the fact that an AR-isomorphism gives rise
to an isomorphism in the category of projective systems modulo translation.
Now let 2lc C 21 be a finite subcategory, closed in 2t. This means (Defi-
nition 1.2) that 2lc is a full subcategory closed under kernels, cokernels and
extensions in 21, such that every object of 2tc is noetherian and artinian.
For n G N, we let 21J? be the intersection of 2ln and 2lc. Moreover, 2lc will