  We construct examples of bounded below, noncontractible, acyclic complexes of
finitely generated projective modules over some rings $S$, as well as bounded
above, noncontractible, acyclic complexes of injective modules. The rings $S$
are certain rings of infinite matrices with entries in the rings of commutative
polynomials or formal power series in infinitely many variables. In the world
of comodules or contramodules over coalgebras over fields, similar examples
exist over the cocommutative symmetric coalgebra of an infinite-dimensional
vector space. A simpler, universal example of a bounded below, noncontractible,
acyclic complex of free modules with one generator, communicated to the author
by Canonaco, is included at the end of the paper.
