2. ^-ADIC DERIVED CATEGORIES

9

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

follows.

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

A/ln+1

by A

n

. 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

n

)

n G

^ in 21

such that Mn G

2ln

for every n G N. The categories 21 and 2P are abelian.

The n

t h

component functor an : 21 — 2P is exact.

By construction, for m n, the category

2ln

is naturally a subcategory

of 2lm. This inclusion has a left adjoint 2lm — 2ln, which we denote by

M n M ^ A

n

. We may define M ® A

n

= M/£n+1M.

We say that an object (M

n

)

n G

^ of 21 is £-adic, if for every m n the

morphism M

m

® A

n

— M

n

(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

— M

n

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