4
INTRODUCTION
cohomology of BGm is a polynomial ring in one variable, thus represented
by an unbounded complex. This is why we have to use a different approach.
(Another -£-adic formalism is constructed in [11]. But although a tri-
angulated category of unbounded complexes is defined, boundedness is as-
sumed to prove the existence of a t-structure on this triangulated category.)
Overview. We describe our £-adic formalism in Section 1. It uses only
results from [13, Exp. V]. The main idea is to construct our derived category
D
c
(X
sm
, A) as an inductive limit over the categories D(5£)(X
sm
, A). Here the
subscript (S, £) denotes what we call an L-stratification (Definition 2.8) of
X&- Here S is a stratification of X& and C associates to every stratum 23 G S
a finite set £(53) of locally constant constructible sheaves of A-modules
on 03^- We require objects of D^^(XsmiA) to have (S,£)-constructible
cohomology sheaves. The key fact is that the category Mod^^(X^t^ A) of
(S, £)-constructible sheaves of A-modules on X&t is finite, i.e. noetherian and
artinian. (To be precise, this would be true for 5-constructibility, already.
The £-part is introduced to deal with higher direct images.)
Section 2 introduces the formalism needed to deal with L-stratifications.
We introduce what we call a d-structure. This just formalizes the situation
of a noetherian topological space X and its locally closed subspaces, where
one has the functors i*, i*, v and i\ between the derived categories on the
various locally closed subspaces of X. A cd-structure gives the additional
data required to introduce L-stratifications.
In Section 3 we apply our £-adic formalism to c-topoi. We reach the
central Definition 3.46 of the category of constructible A-complexes on a c-
topos X in Section 4. An important role is also played by Proposition 3.59,
where d-structures, c-topoi and our ^-adic formalism come together, giving
rise to an ^-adic d-structure on a noetherian c-topos.
In Section 4 we apply our results to algebraic stacks. Our central object
of study, Dc(Xsm,^4) is introduced in Remark 4.38. The technical heart of
this work is contained in Sections 5, 6 and 7. Our main result is summarized
in Remark 4.75. In Section 8 we go slightly beyond the two operations /*
and R/*, by defining R/
-
for certain kinds of representable morphisms of
algebraic stacks (essentially smooth morphisms and closed immersions).
Finally, in Section 5, we consider the case of finite ground field ¥q. Our
goal is to generalize the triangulated category D^(X§t, Q£) of bounded mixed
complexes on a scheme X to the case of algebraic stacks. We introduce (Def-
inition 5.16) the category D+(£
s m
, Q^), which is obtained form 0+(3£sm, Qg)
by requiring cohomology objects to be mixed. This concept behaves well
with respect to the three operations /*, R/* and R / \
But to define the trace of Frobenius on mixed objects we need a fur-
ther restriction. We pass to B+(5tsm, Q^), the subcategory of D+(X
s m
,Q^)
consisting of absolutely convergent objects. Roughly speaking, an object
M G obD+(X
s m
, Q£) is absolutely convergent, if for every finite field ¥qn
Previous Page Next Page