Multiplicative filtrations and graded products

PDF p. 3: CAlg(FilSp); PDF p. 10: Multiplication square

A decreasing filtered spectrum is a coherent diagram $F$ in spectra indexed by the integers, with a transition map $F^i\to F^k$ whenever $i\ge k$, as in Decreasing spectral filtrations and associated graded. Write $\mathrm{FilSp}$ for the infinity-category of such diagrams and coherent natural transformations. A multiplicative filtered spectrum, written $F\in \mathrm{CAlg}(\mathrm{FilSp})$, is such a diagram equipped with coherently compatible symmetric, associative and unital multiplication data: for every finite ordered list $(i_1,\dots ,i_n)$, including the empty list, there is a map

${\mu }_{i_1,\dots ,i_n}:F^{i_1}\wedge \cdots \wedge F^{i_n}⟶F^{i_1+\cdots +i_n}\mathrm{.}$

The empty smash product is the sphere spectrum $\mathbb{S}$, and the empty sum is $0$. The maps are compatible with all transition maps and permutations, with substitution of lists into lists, and with the unit, together with all higher homotopies required for a commutative algebra object. In particular, the one-input operation is the identity up to the specified coherent identification. Thus the notation specifies coherent data; unrelated binary pairings do not suffice.

A multiplicative filtration on a commutative ring spectrum $E$ additionally has a coherent multiplicative augmentation $F\to \mathrm{const}(E)$: its component maps $F^i\to E$ form an augmentation of the underlying tower, and the $n$-input products commute with these maps and the $n$-fold product of $E$, coherently. A commutative ring spectrum means a commutative algebra object in spectra under smash product; $\mathrm{const}(E)$ is its constant filtered diagram.

Put ${\mathrm{gr}}^{i}F=\mathrm{cofib}(F^{i+1}\to F^i)$ and write $q_i:F^i\to {\mathrm{gr}}^{i}F$ for the canonical cofiber map. The associated graded multiplication is the coherent family

${\bar{\mu }}_{i,j}:{\mathrm{gr}}^{i}F\wedge {\mathrm{gr}}^{j}F⟶{\mathrm{gr}}^{i+j}F$

induced by ${\mu }_{i,j}$. It satisfies ${\bar{\mu }}_{i,j}(q_i\wedge q_j)\simeq q_{i+j}{\mu }_{i,j}$. These are maps from filtration stages to graded pieces; the definition does not assert that there is a map from $E$ or from an image subgroup of ${\pi }_{\ast }E$ to a graded piece.

1.1 If $i_r\ge k_r$ for every $r$, then $\sum i_r\ge \sum k_r$, so the indicated target transition map exists. Addition is associative and commutative with identity $0$; consequently the stated compatibility, symmetry and substitution requirements have matching source and target indices. The augmentation is a coherent natural transformation to the constant tower, with additional compatibility with all products. These clauses specify structure and do not assert that every tower admits it. [F1, F2, given]

2.1 For fixed $i,j$, compatibility with transitions identifies the restriction of ${\mu }_{i,j}$ along $F^{i+1}\wedge F^j\to F^i\wedge F^j$ with a map factoring through $F^{i+j+1}\to F^{i+j}$; the same holds for $F^i\wedge F^{j+1}$. Hence their composites with $q_{i+j}$ have the canonical coherent nullhomotopies coming from the cofiber. Compatibility on their common corner follows from the coherent multiplication and transition data. [F1, F2, given, step 1.1]

3.1 Since smash preserves colimits separately, ${\mathrm{gr}}^{i}F\wedge {\mathrm{gr}}^{j}F$ is the iterated cofiber of the square with corners $F^{i+1}\wedge F^{j+1}$, $F^{i+1}\wedge F^j$, $F^i\wedge F^{j+1}$ and $F^i\wedge F^j$. The compatible nullhomotopies in step 2.1 therefore induce ${\bar{\mu }}_{i,j}$ by the universal property of that iterated cofiber, and its composite with $q_i\wedge q_j$ is $q_{i+j}{\mu }_{i,j}$. [F2, step 2.1, construct]

4.1 The same construction for any finite list uses the cube formed by replacing each $F^{i_r}$ by $F^{i_r+1}$ in every possible subset of inputs. Its iterated cofiber is the smash of the corresponding graded pieces, and every face with a raised index maps through $F^{\sum i_r+1}$. The coherent permutation, substitution and unit data of $F$ descend through these universal cofiber constructions. Thus the graded products are coherently commutative, associative and unital, with unit $\mathbb{S}\to F^0\stackrel{q_0}{\to }{\mathrm{gr}}^{0}F$. Together with step 3.1 this justifies the definition. [F1, F2, given, step 3.1] ∎