Let's stipulate that

1. Connective -- i.e. nonnegatively-(homologically)-graded -- chain complexes have a very natural geometric interpretation: by the Dold-Kan theorem, they are a way of thinking about simplicial abelian groups.

2. Dually, coconnective -- i.e. nonpositively-graded -- chain complexes have a very natural geometric interpretation as complexes of functions on spaces.

The motivations for considering the category of all (unbounded) chain complexes, then, are quite good -- this category provides a home for both the connective and coconnective chain complexes, and has excellent formal properties like stability and a good duality theory.

However, these motivations are quite formal in nature -- they don't provide a geometric interpretation in line with (1) or (2) above. For the most part, these motivations operate "one category level higher", discussing properties of the category of chain complexes. I'm specifically looking for something which gives a geometric, natural way to think about an individual chain complex.

Question 1: What is a good geometric interpretation of nonconnective, noncoconnective chain complexes?

Notes:

• A similar discussion would more generally stipulate that grouplike $E_\infty$-spaces have a natural geometric interpretation, and ask for a similarly "geometric" interpretation of more general spectra. I'd be equally happy with a discussion in this setting.

• Similarly, I'd be happy with a discussion in the context of complexes of sheaves of various flavors.

Guess: Here's a guess of a picture which might be appropriate, based on my understanding of Tyler and Adeel's answers to this question, and inspired by Sanath's comment below.

• A general $E_\infty$ ring spectrum $A$ can be thought of as the global sections $A = \Gamma(X, \mathcal O_X)$ of a spectral scheme $X$ (note that the structure sheaf $\mathcal O_X$ takes values $\Gamma(U,\mathcal O_X)$ in connective $E_\infty$ ring spectra when $U$ is affine open, but its global sections can be nonconnective).

  • Thus the negative-dimensional homotopy groups of $A$ can be thought of as measuring the cohomology of $X$, i.e. the global structure of how $A$ is glued together from affines. The positive-dimensional homotopy groups of $A$ can be thought of as some sort of nilpotent thickening / infinitesimal structure of $X$.

• A spectrum $M$ is a module over an $E_\infty$-ring spectrum $A = \Gamma(X, \mathcal O_X)$, and so we should think of $M = \Gamma(X,\mathcal F)$ as the global sections of a quasicoherent sheaf on $X$ (where the values $\Gamma(U,\mathcal F)$ of $\mathcal F$ on affine opens $U$ are connective, but its global sections need not be).

  • Thus the negative-dimensional homotopy groups of $M$ can be thought of as measuring the cohomology of $\mathcal F$, and the positive-dimensional homotopy groups of $M$ can be thought of as the infinitesimal structure of $\mathcal F$.

Question 2: Is this a good picture to have in mind when trying to think about nonconnective, noncoconnective chain complexes / spectra geometrically?

Question 3: For example, do we have $KU = \Gamma(X, \mathcal O_X)$ for a natural (or even canonical) spectral scheme $X$?