Decreasing spectral filtrations and associated graded

PDF p. 3: FilSp and the image filtration

Let $E$ be a spectrum. A decreasing filtration on $E$ consists of spectra $F^{j}E$, indexed by $j\in \mathbb{Z}$, and a coherent diagram whose transition maps $F^{j}E\to F^{k}E$ are defined whenever $j\ge k$, together with a coherent compatible family of maps $F^{j}E\to E$. Thus the composite $F^{j+1}E\to F^{j}E\to E$ is identified with the given map $F^{j+1}E\to E$. Coherent means that the maps, their composition identifications, and all higher compatibility data form a diagram and a cone in the infinity-category of spectra.

The associated graded spectrum in degree $j$ is ${\mathrm{gr}}^{j}E=\mathrm{cofib}(F^{j+1}E\to F^{j}E)$. For $i\in \mathbb{Z}$, its image filtration on the homotopy group ${\pi }_{i}E$ is the family of subgroups

$F^j{\pi }_{i}E=\mathrm{im}({\pi }_i{F}^{j}E⟶{\pi }_{i}E)\mathrm{.}$

The filtration is exhaustive if the canonical map ${\mathrm{colim}}_{j\to -\mathrm{\infty }}{F}^{j}E\to E$ is an equivalence. Here the colimit is over the maps from larger indices to smaller indices. It is complete, in the tower-vanishing convention used here, if ${\lim }_{j\to +\mathrm{\infty }}{F}^{j}E$ is equivalent to the zero spectrum; the limit is taken over the same transition maps. It is nonnegative if every structure map $F^{j}E\to E$ is an equivalence for $j\le 0$. Completeness imposes no injectivity condition on the maps ${\pi }_i{F}^{j}E\to {\pi }_{i}E$.

1.1 The index category has one object for each integer and one arrow $j\to k$ when $j\ge k$. It is small, so the tower has a limit and a colimit. The compatible maps to $E$ form a cone and induce the displayed canonical colimit map. Each transition map has a cofiber, so every ${\mathrm{gr}}^{j}E$ is defined. These constructions are invariant under equivalence of the coherent input diagrams. [F1, given]

2.1 Applying ${\pi }_i$ gives group homomorphisms. The image of each structure map is therefore a subgroup of ${\pi }_{i}E$, and compatibility with $F^{j+1}E\to F^{j}E$ gives $F^{j+1}{\pi }_{i}E\subseteq F^j{\pi }_{i}E$. Equivalence to $E$ or to the zero spectrum is a well-defined condition on the specified maps or limit. None of the three conditions includes injectivity of the homomorphisms defining the image filtration; in particular, completeness is the stated condition on the tower limit. [F1, given, step 1.1] ∎