  This paper is an introduction to work motivated by the question "can
multipartite entanglement be detected by homological algebra?" We introduce
cochain complexes associated to multipartite density states whose cohomology
detects factorizability. The $k$th cohomology components of such cochain
complexes produce tuples of $(k+1)$-body operators that are non-locally
correlated due to the non-factorizability of the state. Associated Poincare
polynomials are invariant under local invertible linear transformations
(automorphisms that decompose as tensor products). These complexes can be
considered as a step toward realizing mutual information as an Euler
characteristic. We motivate the definition of the "state index" associated to a
multipartite state: a three-parameter function which is: invariant under local
invertible transformations, well-behaved under tensor products of states, and
interpolates between multipartite mutual information and the integer-valued
Euler characteristics of our complexes. We compute cohomologies and state
indices of multipartite W and GHZ states. The approach in this paper is
directed toward practitioners of finite-dimensional quantum mechanics, although
the machinery developed generalizes far beyond. Some results are applicable in
infinite dimensions and should be generalizable to the context of quantum field
theory. In order to compensate for the long length, a detailed summary is
provided in the introduction section.
