I am trying to extract a particular, more lightweight and more focussed at the same time, case of my recent question Which of the physics dualities are closest in essence to the Spanier-Whitehead duality (with a subquestion)?
From time to time I keep becoming fascinated anew by one of the deepest notions in algebraic geometry - divisors.
Specifically, I would like to understand better the interplay of analysis and elementary arithmetic of integers that occurs there.
At the first sight, forming the group of divisors is a purely "topological" move: you take formal linear combinations of codimension one subvarieties with integer coefficients. Topologists do such things all the time, and with coefficients in arbitrary abelian groups.
However as soon as you add the notion of principal divisor and linear equivalence, two things happen: first, integers become absolutely distinguished among the possible coefficient systems, since the coefficient starts to mean something very analytic and non-topological - order of vanishing/infinity of a function along the subvariety. Second, again from the purely topological point of view, some kind of duality becomes apparent, since involvement of functions on the variety suggests something cohomological, as opposed to homology presumed by considering subvarieties as cycles.
If a topologist is given an $n$-dimensional manifold, then something similar can be done relating $(n-1)$-cycles and $1$-cocycles. But the divisor class group is even more interestingly behaved in the nonsmooth case, and here I am not sure what a topologist would relate this to.
My question then is whether there exists a purely topological version of this subtle mixture of homology and cohomology. Maybe some version of Spanier-Whitehead-like duality or something like that, but the main point is that there must be a single group incorporating homology and cohomology simultaneously for non-manifolds, looking as basic and fundamental as the divisor class group, and the coefficients (in the generalized version, the spectrum) chosen being distinguished among all other possible choices of coefficients.
Turning the same question backwards - is there analog, in algebraic geometry, of taking coefficients other than $\mathbb Z$ for divisors? I am aware of the notion of $\mathbb Q$-divisor but more generally I mean, say, divisors with coefficients in the field of coefficients for varieties defined over that field, or, say, something like $\ell$-adic coefficients, or something similar. Does this have sense in the motivic context, for example? Also, from topological viewpoint, there must be higher versions of that, relating codimension $k$ cycles with zeros/poles of $(k-1)$-forms or some related gadgets (norm residue symbols? higher dimensional local fields? or what?)