  Many properties of a module can be expressed in terms of the dimension of the
vector space obtained by applying a finitely presented functor to that module.
For example, the dimension of the kernel, image or cokernel of the
multiplication map given by an algebra element; or the number of summands of a
certain type when the module is considered a module over a subalgebra. When the
indecomposable modules over the algebra are arranged in the Auslander-Reiten
quiver, the support of the finitely presented functor typically has the shape
of a hammock, spanned between sources and sinks. There may also be tangents
which are meshes where the hammock function at the middle term exceeds the sum
of the values at the start and end terms. We describe how sources, sinks and
tangents of the hammock relate to the modules which define the projective
resolution of the finitely presented functor. The key tool is the Cokernel
Complex Lemma which links the values of the hammock function to the
Auslander-Reiten structure of the category. We are also interested in exact
subcategories of module categories which have Auslander-Reiten sequences. Our
examples include quiver representations and invariant subspaces of nilpotent
linear operators.
