2. ALGEBRAIC DISCRETE MORSE THEORY 9
Sometimes it is useful to consider the same construction for matchings which
are not acyclic. Clearly, Theorem 2.2 does not hold anymore for C• M if M is not
acyclic. In general, there is not even a definition for a map replacing the differentials
∂i M. But for calculating invariants it is nonetheless sometimes useful to consider
C• M for matchings that are not acyclic. Thus, if M is a not necessarily acyclic
matching we denote by C• M the corresponding sequences of free R-modules. For
applications of this viewpoint see also [31].
Finally. we would like to generalize the construction of the Morse complex to
infinite acyclic matchings:
Note, that the definition of an acyclic matching makes perfect sense also for infinite
sets of edges. But if M is an infinite acyclic matching then Γ(c, c ) may no longer
be well defined in case the set of paths from c to c is infinite. Moreover, we indeed
use finiteness in our proof of Theorem 2.2 since we use induction on the cardinality
of the acyclic matching.
In order to be able to formulate a result similar to Theorem 2.2 for infinite
acyclic matchings we have to introduce an additional finiteness condition.
Let C• be a complex and M an infinite acyclic matching. Clearly the matching
M induces a finite acyclic matching on each finite subcomplex C•
f
of C•. Therefore,
we make the following definition:
Definition 2.3 (Finiteness). Let C• be a complex of free R-modules and let
M be an infinite acyclic matching. We say that M defines a Morse matching if
there exists a sequence of finite subcomplexes Di := (D•)i, i 0 of C• such that:
(1) Di is a subcomplex of Di+1, for all i 0.
(2) C• = colimi≥0 Di.
(3)
(Di)M
is a subcomplex of
(Di+1)M.
Note that the last condition implies that Γ(c, c ) is well defined and thus the con-
clusion of Theorem 2.2 still holds for those infinite acyclic matchings.
In our applications all complexes are multigraded by α
Nn
and the α-graded
part of C• is finite. Therefore, the subcomplexes Di, defined by
Di :=
i
j=0
α∈Nn
|α|=j
(C•)α
are finite subcomplexes. It is easy to see that for multigraded complexes whose
graded parts are of finite rank any acyclic matching fulfills the additional finiteness
condition on the sequence Di. This indeed holds for all complexes in our applica-
tions.
Therefore, we get:
Theorem 2.4. Let C• be a Nn-graded complex of free R-modules such that
(C•)α is a finite subcomplex for all α Nn. Then the conclusion of Theorem 2.2
still holds for infinite acyclic matchings M.
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