Introduction

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,

respectively.

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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