0.1. A common wisdom says that difficulties arise in Koszul duality because
important spectral sequences diverge. What really happens here is that one con-
siders the spectral sequence of a complex endowed with, typically, a decreasing
filtration which is not complete. Indeed, the spectral sequence of a complete and
cocomplete filtered complex always converges in the relevant sense [14]. The solu-
tion to the problem, therefore, is to either replace the complex with its completion,
or choose a different filtration. In this paper, we mostly follow the second path.
This involves elaboration of the distinction between two kinds of derived categories,
as we will see below.
The first conclusion is that one has to pay attention to completions if one
wants one’s spectral sequences to converge. What this means in the case of the
spectral sequence related to a bicomplex is that the familiar picture of two spectral
sequences converging to the same limit splits in two halves when the bicomplex
becomes infinite enough. The two spectral sequences essentially converge to the
cohomology of two different total complexes. To obtains those, one takes infinite
products in the “positive” direction along the diagonals and infinite direct sums
in the “negative” direction (like in Laurent series). The two possible choices of
the “positive” and “negative” directions give rise to the two completions. The
word “essentially” here is to be understood as “ignoring the delicate, but often
manageable issues related to nonexactness of the inverse limit”.
0.2. This alternative between taking infinite direct sums and infinite products
when constructing the total complex leads to the classical distinction between dif-
ferential derived functors of the first and the second kind [25, section I.4]. Roughly
speaking, one can consider a DG-module either as a deformation of its cohomology
or as a deformation of itself considered with zero differential; the spectral sequences
related to the former and the latter kind of deformations essentially converge to
the cohomology of the differential derived functors of the first and the second kind,
Derived categories of the first and the second kind are intended to serve as the
domains of the differential derived functors of the first and the second kind. This
does not always work as smoothly as one wishes; one discovers that, for technical
reasons, it is better to consider derived categories of the first kind for algebras and
derived categories of the second kind for coalgebras. The distinction between the
derived functors/categories of the first and the second kind is only relevant when
certain finiteness conditions no longer hold; this happens when one considers either
unbounded complexes, or differential graded modules.
Let us discuss the story of two derived categories in more detail. When the
finiteness conditions do hold, the derived category can be represented in two simple
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