10 1. THE ^-ADIC FORMALISM
denote the full subcategory of 21 consisting of those projective systems each
of whose components is in 2tc. It is clear that 2tc is closed in 21. The symbols
2l^c and
21AR-^,C
denote the categories of ^-adic and AR-£-adic systems in
2tc, respectively.
L E M MA 1.9. Let a : F M be an AR-isomorphism in 2t7 where F is
i-adic. If M is in 2lc? then so is F.
P R O O F .
Choose r as in Lemma 1.8. Then if Mn+r is in 2lc, so is M
n + r
®
A
n
, as the cokernel of multiplication by
£n+1.
Then Fn is also in 2lc as a
direct summand.
P R O P O S I T I O N 1.10. The category %AR-I,C is closed in 21. In particular,
it is an abelian category.
P R O O F .
Let a : M N be a homomorphism in the category
21AR-^,C-
Then kero; and cok a are in 2tc. We would like to show that they are AR-^-
adic. But this follows from Proposition 5.2.1 of [13, Exp. V] applied to the
category C = 2tc.
Now let M and N be objects of
21AR-^,C
and let
0 M E N 0
be an extension of N by M in 21. Since E is then also in 2lc, applying
Corollaire 5.2.5 of [loc. cit.] to C = 2lc, we see that E is in fact AR-[-
adic.
We may now carry out the construction of Proposition 1.3, using the
subcategory
21AR-^,C
of 21- We get a sub-t-category of the derived category
D (21), which we shall denote by
Z?AR-^,C(21).
We shall now construct a certain quotient of
AAR-^,C(21).
P R O P O S I T I O N
1.11. The objects of%AR-i,c that are AR-null form a thick
subcategory.
P R O O F .
This is clear. See also [13, Exp. V, Proposition 2.2.2].
Thus we may apply Proposition 1.4 to our situation. We denote the quo-
tient of
21AR-^,C
modulo the AR-null sheaves by
AR-21AR-^,C-
Thus
AR-21AR-^,
C
is obtained by inverting the AR-isomorphisms in
21AR-^,C-
We call a mor-
phism M N in
Z?AR-^,C(21)
a quasi-AR-isomorphism if the induced map
hlM
»
hlN
is an AR-isomorphism for every i G Z. The triangulated cate-
gory obtained from
-DAR-£,C(21)
by inverting the quasi-AR-isomorphisms will
be denoted by
AR-DAR-^,C(21).
P R O P O S I T I O N
1.12. The natural functor 2l^c
AR-^LAR-^C
is
an
equiv-
alence of categories.
P R O O F .
This is easily proved using Lemma 1.7 and Lemma 1.9.
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