Dependencies

The synthetic Adams spectral sequence is a \(\mathbb {Z}[\lambda ]\)-module spectral sequence, with \(\lambda \) in tridegree \((0,0,-1)\).

In this project, \(\mathcal{S}\) refers to the homotopy category of spectra (in the sense of stable homotopy theory). It can be constructed as the homotopy category of a stable model category, but we take it as given and work axiomatically with its triangulated and closed symmetric monoidal structure.

Suppose \(X\) is of finite type. Then:

1. If additionally \(H_i(X; {\mathbb Q}) = 0\) for some \(i\), then the Adams filtration on \(\pi _i(X) / \{ \text{odd-order elements}\} \) is bounded.

2. All subquotients of the Adams filtration on \(\pi _i(X) / \{ \text{odd-order elements}\} \) are finite groups.

All elements in the \(E_2\)-page of the mod 2 Adams spectral sequence have order 2, i.e., \(2x = 0\) for all \(x \in E_2^{s,t}(X)\). This holds because \(E_2^{s,t}(X)\) is an \({\mathbb F}_2\)-vector space.

The \(\mathrm{H}{\mathbb F}_2\)-Adams spectral sequence is a functor from \(\mathcal{S}^{op} \times \mathcal{S}\) to the category of bigraded spectral sequences. For spectra \(X\) and \(Y\), it produces a spectral sequence \(\{ E_r^{s,t}(X, Y), d_r\} \) with \(r \ge 2\) and differentials

A map \(f: X' \to X\) induces maps \(f^*: E_r^{s,t}(X, Y) \to E_r^{s,t}(X', Y)\), and a map \(g: Y \to Y'\) induces maps \(g_*: E_r^{s,t}(X, Y) \to E_r^{s,t}(X, Y')\), both compatible with differentials. When \(X = \mathbb {S}\), we write \(E_r^{s,t}(Y) = E_r^{s,t}(\mathbb {S}, Y)\).

For \(X \in {\mathcal S}^{fin}\) and \(Y\) of finite type, the Adams spectral sequence \(Adams(X,Y)\) weakly converges to \([X, Y]\): there is a natural isomorphism between the graded pieces of the Adams filtration on \([X, Y]\) and the \(E_\infty \)-page of \(Adams(X,Y)\).

The bigraded suspension commutes with the tensor product: for all \((m,n) \in {\mathbb Z}^2\) and objects \(X, Y\) of \(h\mathrm{Syn}\),

If \(X\) is \(n\)-connected, then \(H_i(X; {\mathbb F}_2) = 0\) for all \(i \le n\).

The homotopy groups of the Eilenberg–MacLane spectrum are

A spectrum \(X\) is of finite type if and only if:

1. \(X\) is bounded below, and

2. for each \(i\), the homotopy group \(\pi _i(X)\) is finitely generated as an abelian group.

If \(X\) is of finite type, then \(H_i(X; {\mathbb F}_2)\) is a finite-dimensional \({\mathbb F}_2\)-vector space for each \(i\).

Let \(X \in \mathcal{S}^{\mathrm{fin}}\) be a finite spectrum. The synthetic Adams spectral sequence for \(\nu X\) is isomorphic to the \(\lambda \)-Bockstein spectral sequence:

1. \(E_2^{s,t}(X) \cong \pi _{t-s,t}(\nu X / \lambda )\).

2. If there is a classical Adams differential \(d_r x = y\) for \(x \in E_2^{s,t}(X) \cong \pi _{t-s,t}(\nu X / \lambda )\), then \(x\) admits a lift to \(\pi _{t-s,t}(\nu X / \lambda ^{r-1})\) whose image under the Bockstein \(\nu X / \lambda ^{r-1} \to \Sigma ^{1,-r+1} \nu X / \lambda \) equals \(d_r(x)\).

Let \(X, Y \in \mathcal{S}^{\mathrm{fin}}\) be finite spectra and let \(f: X \to Y\) be a map that induces the zero map on \(\mathrm{H}{\mathbb F}_2\)-homology: \(H_*(f; {\mathbb F}_2) = 0\). Let

be a functorial distinguished triangle (so \(W = \Sigma ^{-1}C(f)\) is the desuspended cofiber). By the \(\nu \)-functor applied to this triangle, there exists a distinguished triangle of synthetic spectra

where \(g\) denotes the boundary map. Then:

i.e., the boundary map \(g\) composed with \(\lambda \) equals the synthetic image of \(f\).

If \(f\) does not have odd order, then \(f\) does not have Adams filtration \(\infty \).

Suppose \(X \xrightarrow {f} Y \xrightarrow {g} Z\) is a cofiber sequence of spectra such that

is a short exact sequence of \(\mathrm{H}{\mathbb F}_2\)-homology. Then \(\nu X \xrightarrow {\nu f} \nu Y \xrightarrow {\nu g} \nu Z\) is a distinguished triangle in \(h\mathrm{Syn}\).

The functor \(\nu \) intertwines suspension with bigraded suspension:

The natural comparison map \(\Sigma (\nu X) \to \nu (\Sigma X)\) is induced by \(\lambda \).

Let \(X \in \mathcal{S}^{\mathrm{fin}}\) be a finite spectrum. The synthetic Adams spectral sequence for \(\nu X\) has \(E_2\)-page

where an element in \(E_2^{s,t}(X)\) has tridegree \((s,t,t)\) and \(\lambda \) has tridegree \((0,0,-1)\). Given a classical Adams differential \(d_r^{\mathrm{cl}}(x) = y\), the corresponding synthetic differential is \(d_r(x) = \lambda ^{r-1}y\), which is \(\lambda \)-linear, and all synthetic Adams differentials arise in this way.

The sphere spectrum \({\mathbb S}\) is \((-1)\)-connected.

The synthetic Adams spectral sequence is a functor from \(h\mathrm{Syn}\) to the category of 3-graded spectral sequences. For a synthetic spectrum \(X \in h\mathrm{Syn}\), it produces a spectral sequence \(\prescript {\mathrm{syn}\! }{}{E}_r^{s,t,w}(X)\) with differentials

For a finite spectrum \(X \in \mathcal{S}^{\mathrm{fin}}\), the synthetic Adams spectral sequence for \(\nu X\) converges to \(\pi _{*,*}(\nu X)\).

Let \(X, Y \in \mathcal{S}^{\mathrm{fin}}\) be finite spectra. If a map \(f: X \to Y\) has Adams filtration \(\mathrm{AF}(f) = k\), then there exists a factorization

with a dashed lift \(\Sigma ^{0,-k}\tilde{f}: \nu X \dashrightarrow \Sigma ^{0,-k}\nu Y\) such that \(\lambda ^k \circ \Sigma ^{0,-k}\tilde{f} = \nu f\), where \(\tilde{f}: \Sigma ^{0,k}\nu X \to \nu Y\) is called a synthetic lift of \(f\).

(Universal Coefficient Theorem) For any spectrum \(X\) and \(n \in {\mathbb Z}\), the mod 2 cohomology is canonically isomorphic to the \({\mathbb F}_2\)-linear dual of the mod 2 homology:

If \(\mathrm{AF}(f) = k\), then \(d_i^f = 0\) for \(i {\lt} k\) in the \(f\)-extension spectral sequence.

Consider a homotopy commutative diagram of converging spectral sequences \(V_1 \xrightarrow {p} V_3 \xrightarrow {g} V_4\) with \(q = g \circ p\). If \(d_m^p(x) = z\) and \(d_l^g(z) = w\) has no crossing, then \(d_{m+l}^q(x) = w\).

If \(g \circ f = 0\) (as morphisms of converging spectral sequences) and \(d_n^f(x) = y\), then \(y\) is a permanent cycle in the \(g\)-ESS: \(d_m^g(y) = 0\) for all \(m \ge 0\).

Consider a homotopy commutative diagram of converging spectral sequences

Assume \({}^{p}\! E_0 = {}^{p}\! E_r\) and \({}^{q}\! E_0 = {}^{q}\! E_r\) for some \(r \ge 0\) (i.e., the \(p\)-ESS and \(q\)-ESS have stationary pages up to \(r\)). Then \((d_r^p, d_r^q)\) induces a map from the \(f\)-ESS to the \(g\)-ESS.

In the setting of Theorem 0.52, if condition (4) is strengthened to “\(d_l^g(z) = w\) has no crossing” (and condition (5) is dropped), then \(d_{m+l-n}^q(y) = w\).

Consider a homotopy commutative diagram of converging spectral sequences \(V_1 \xrightarrow {f} V_2 \xrightarrow {q} V_3\) with \(p = q \circ f\). If \(d_n^f(x) = y\), \(d_m^p(x) = z\), and one of the two extensions has no crossing, then \(d_{m-n}^q(y) = z\).

The Adams spectral sequence induces a natural decreasing filtration on \(\pi _* X\), called the Adams filtration. An element \(\alpha \in \pi _n X\) has Adams filtration \(\ge s\) if it lies in \(F^s \pi _n X\).

A map \(f \in [X, Y]\) has Adams filtration \(\infty \) if \(\mathrm{AF}(f) \ge n\) for all \(n\).

The Adams filtration extends to maps between spectra. For \(X \in {\mathcal S}^{fin}\) and \(Y\) of finite type, a map \(f \in [X, Y]\) has Adams filtration \(\mathrm{AF}(f) \ge k\) if \(f\) factors as a composition

where each \(f_i\) induces the zero map on mod 2 homology: \((f_i)_* = 0: H_*(X_{i-1}; {\mathbb F}_2) \to H_*(X_i; {\mathbb F}_2)\). The Adams filtration \(\mathrm{AF}(f)\) is the largest such \(k\).

Given a filtered graded abelian group \((A, F)\), the associated graded is

Given a filtered chain complex \((C_*, d, F^s)\), the associated graded complex is the bigraded object \(\mathrm{gr}^s C_n\) equipped with the induced differential \(\bar{d}: \mathrm{gr}^s C_n \to \mathrm{gr}^s C_{n-1}\) (well-defined since \(d\) preserves the filtration).

The synthetic sphere \(S^{0,0}\) is defined as the monoidal unit \(\mathbf{1}\) of \(h\mathrm{Syn}\). The bigraded spheres are

The category \(h\mathrm{Syn}\) admits a bigraded suspension \(\Sigma ^{m,n}\) for each \((m,n) \in \mathbb {Z}^2\), which is an autoequivalence of \(h\mathrm{Syn}\) satisfying

The functor \(\Sigma ^{1,0}\) is identified with the triangulated suspension functor.

A spectrum \(X\) is bounded below if it is \(n\)-connected for some \(n \in {\mathbb Z}\).

A filtered chain complex \((C_*, d, F^s)\) is bounded if for each \(n\), there exist \(a \le b\) with \(F^a C_n = C_n\) and \(F^{b+1} C_n = 0\). A bounded filtration is both exhaustive and Hausdorff.

A closed symmetric monoidal category is a category \(\mathcal{C}\) equipped with:

1. A unit object \(S\).

2. A bifunctor \((X, Y) \mapsto X \wedge Y\) (the smash product or tensor product) from \(\mathcal{C} \times \mathcal{C}\) to \(\mathcal{C}\), which is associative and commutative up to coherent natural isomorphism, with \(S \wedge X \cong X\) up to coherent natural isomorphism.

3. Function objects \(F(X, Y)\), functorial contravariantly in \(X\) and covariantly in \(Y\), satisfying the tensor-hom adjunction

   \[ [X, F(Y, Z)] \cong [X \wedge Y, Z] \]

   naturally in all three variables.

A closed symmetric tensor triangulated category is a triangulated category \(\mathcal{C}\) equipped with a closed symmetric monoidal structure (Definition 0.62) that is compatible with the triangulation:

1. The smash product preserves suspensions: there is a natural equivalence \(e_{X,Y}: \Sigma X \wedge Y \xrightarrow {\sim } \Sigma (X \wedge Y)\), compatible with the unit and associativity isomorphisms.

2. The smash product is exact: if \(X \xrightarrow {f} Y \xrightarrow {g} Z \xrightarrow {h} \Sigma X\) is an exact triangle, then for any object \(W\),

   \[ X \wedge W \xrightarrow {f \wedge 1} Y \wedge W \xrightarrow {g \wedge 1} Z \wedge W \xrightarrow {h \wedge 1} \Sigma (X \wedge W) \]

   is exact.

3. The functor \(F(X, Y)\) is exact in \(Y\); it is exact in \(X\) up to sign.

4. The smash product interacts with suspension in a graded-commutative manner: the twist map \(T: S^r \wedge S^s \to S^s \wedge S^r\) and the equivalence \(S^r \wedge S^s \simeq S^{r+s}\) satisfy \(T = (-1)^{rs}\).

This definition follows Hovey–Palmieri–Strickland  [ , Definition A.2.1.

In a triangulated category, a distinguished triangle \(X \xrightarrow {f} Y \to C_f \to \Sigma X\) is called a cofiber sequence, and \(C_f\) is the cofiber (or cone) of \(f\). By (TR1), every morphism admits a cofiber sequence.

A spectrum \(X\) is \(n\)-connected if \(\pi _i(X) = 0\) for all \(i \le n\).

Let \(E\) be a spectral sequence with \(n\)-grading and let \(A\) be an \((n-1)\)-graded object equipped with a decreasing filtration (Definition 0.14). We say \(E\) converges to \(A\), written \(E_{r_0} \Longrightarrow A\), if there exists an isomorphism of \(n\)-graded objects

where the \(n\)-grading on \(\mathrm{gr}\, A\) is obtained from the \((n-1)\)-grading of \(A\) together with the filtration index, possibly composed with a fixed reindexing map.

Given a reindexing rule, the category of converging spectral sequences has objects \((E, A, F, \varphi )\) where \(E\) is a spectral sequence, \(A\) is a filtered graded object, and \(\varphi : E_\infty \xrightarrow {\sim } \mathrm{gr}\, A\) is the convergence isomorphism. A morphism \((E_1, A_1) \to (E_2, A_2)\) consists of:

1. A morphism of spectral sequences \(E_1 \to E_2\) (Definition 0.5), inducing maps on \(E_\infty \)-pages.

2. A filtration-preserving map \(A_1 \to A_2\).

3. Compatibility: the induced \(E_\infty \)-map and the associated graded map commute with the convergence isomorphisms.

Let \(f: V_1 \to V_2\) be a morphism of converging spectral sequences, and suppose \(d_n^f(x) = y\) is an \(f\)-extension differential with \(x \in E_\infty ^{s}(V_1)\) and \(y \in E_\infty ^{s+n}(V_2)\).

1. We say that \(d_n^f(x) = y\) has a crossing that hits filtration \(p\) for some \(p \le s+n\), if there exists \(x' \in E_\infty ^{s+a}(V_1)\) with \(a {\gt} 0\) and an essential differential \(d_{n-a}^f(x') = y'\) for \(0 \ne y' \in E_\infty ^{s+a+(n-a)}(V_2)\) such that

   \[ p \le \mathrm{Fil}(y') \le \mathrm{Fil}(y) = s + n. \]

2. We say that \(d_n^f(x) = y\) has no crossing that hits the range \(\mathrm{Fil} \ge p\) if there does not exist such \(x'\) and \(y'\) with \(p \le \mathrm{Fil}(y') \le \mathrm{Fil}(y)\).

3. We say that \(d_n^f(x) = y\) has no crossing if it has no crossing that hits the range \(\mathrm{Fil} \ge \mathrm{Fil}(x) + 1 = s + 1\).

Suppose \(E\) converges to \(A\). Let \(x \in A\) be an element with filtration at least \(r\), i.e., \(x \in F^r A\). Let \(\bar{x}\) denote the image of \(x\) in \(\mathrm{gr}^r A = F^r A / F^{r+1} A\), and let \(y \in E_\infty \) be the element corresponding to \(\bar{x}\) under the convergence isomorphism. We say \(y\) detects \(x\).

A differential datum in a spectral sequence \(E\) is a triple \((r, x, y)\) where \(r \ge r_0\), \(x \in E_r^{\mathbf{k}}\), and \(y \in E_r^{\mathbf{k}+\mathbf{d}(r)}\) with \(d_r(x) = y\). We say the differential datum is essential if \(y \ne 0\), and denote it \(d_r(x) = y\).

The Eilenberg–MacLane spectrum \(\mathrm{H}{\mathbb F}_2\) is the spectrum representing mod 2 ordinary cohomology. It is a commutative ring spectrum in \(\mathcal{S}\).

The \(E_\infty \)-page of a spectral sequence \(E\) at index \(\mathbf{k}\) is given by the nested subspace data: \(E_\infty ^{\mathbf{k}} = Z_\infty ^{\mathbf{k}} / B_\infty ^{\mathbf{k}}\). For bounded or degenerate spectral sequences, \(E_\infty = E_r\) for all sufficiently large \(r\).

The \(f\)-extension spectral sequence (\(f\)-ESS) is the spectral sequence

obtained by applying the filtered complex construction (Theorem 0.25) to the underlying filtered chain complex of Definition 0.34. By Theorem 0.26, if the filtrations on \(A_1\) and \(A_2\) are exhaustive and Hausdorff (which holds when they are bounded), then the \(f\)-ESS converges to the homology of the underlying complex, i.e., to \(\ker (f_A) \oplus \mathrm{coker}(f_A)\).

Let \(f: V_1 \to V_2\) be a morphism of converging spectral sequences, with underlying map \(f_A: A_1 \to A_2\) on the target groups. The underlying filtered chain complex of \(f\) is the two-term chain complex

equipped with the filtrations on \(A_1\) and \(A_2\) inherited from the convergence data: \(F^s C_1 = F^s A_1\) and \(F^s C_0 = F^s A_2\). Since \(f_A\) preserves the filtration, this is a filtered chain complex in the sense of Definition 0.16.

Let \(E\) be a spectral sequence. A differential \(d_r(x) = y\) is called essential if \(y \ne 0\) on the \(E_r\)-page. Equivalently, \(d_r(x) = y\) is essential if \(x\) is not a boundary and \(y\) is not zero in \(E_r\). A differential is inessential if \(y = 0\) on the \(E_r\)-page.

A filtered chain complex \((C_*, d, F^s)\) is exhaustive if \(\bigcup _s F^s C_n = C_n\) for all \(n\).

The \(f\)-extension differential \(d_n^f: E_\infty ^{s}(V_1) \to E_\infty ^{s+n}(V_2)\) is the unique nonzero component of the \(n\)-th differential of the \(f\)-ESS (Proposition 0.37). We say there is an \(f\)-extension from \(x \in E_\infty ^{s}(V_1)\) to \(y \in E_\infty ^{s+n}(V_2)\) if \(d_n^f(x) = y\). The extension is essential if \(y\) is nontrivial on the \(E_n\)-page; otherwise it is inessential.

A filtered chain complex is a chain complex \((C_*, d)\) in the category of filtered graded abelian groups: the differential \(d: C_n \to C_{n-1}\) preserves the filtration, i.e., \(d(F^s C_n) \subseteq F^s C_{n-1}\) for all \(s\) and \(n\).

A morphism of filtered chain complexes \(f: (C_*, F) \to (D_*, G)\) is a chain map \(f: C_* \to D_*\) preserving the filtration: \(f(F^s C_n) \subseteq G^s D_n\) for all \(s, n\).

Given a filtered chain complex \((C_*, d, F^s)\), we construct an SSData structure. The spectral sequence starts at \(E_0\) with \((n+1)\)-grading.

1. Define \(V^{s,t} = \mathrm{gr}^s C_{s+t}\), the associated graded.

2. Define \(Z_r^{s,t}\) as the image in \(\mathrm{gr}^s C_{s+t}\) of those \(x \in F^s C_{s+t}\) with \(dx \in F^{s+r} C_{s+t-1}\).

3. Define \(B_r^{s,t}\) as the image in \(\mathrm{gr}^s C_{s+t}\) of those elements of the form \(dy\) with \(y \in F^{s-r} C_{s+t+1}\).

The differential \(d_r: E_r^{s,t} \to E_r^{s+r,t-r+1}\) is induced by \(d\): for \([x] \in Z_r^{s,t}\), define \(d_r[x] = [dx] \in E_r^{s+r,t-r+1}\).

A morphism of filtered graded abelian groups is a graded homomorphism \(f: A \to B\) that preserves the filtration: \(f(F^s A^{\mathbf{k}}) \subseteq F^s B^{\mathbf{k}}\) for all \(s\) and \(\mathbf{k}\). Filtered graded abelian groups and their morphisms form a category.

A filtered \(n\)-graded abelian group is an \(n\)-graded abelian group \(A\) equipped with a decreasing filtration: for each index \(\mathbf{k} \in \mathbb {Z}^n\), a family of subgroups

Let \((E, A, F)\) be a converging spectral sequence with filtration bounded below. The completion of \(A\) with respect to the filtration \(F\) is the inverse limit

There is a natural map \(\iota : A \to \widehat{A}\) induced by the projections \(A \to A/F^{s+1}A\). The filtration is called complete if \(\iota \) is an isomorphism. The Hausdorff condition (Definition 0.21) ensures \(\iota \) is injective; completeness further requires \(\iota \) to be surjective.

The component of the \(n\)-grading on \(E\) that corresponds to the filtration on \(A\) is called the filtration degree.

The full subcategory \({\mathcal S}^{fin} \subset {\mathcal S}\) of finite spectra is the smallest full subcategory containing the sphere spectrum \({\mathbb S}\) and closed under suspension, desuspension, and taking cofibers.

A spectrum \(X\) is of finite type if for each \(n\), there exists a finite spectrum \(X_n \in {\mathcal S}^{fin}\) and a map \(X_n \to X\) whose cofiber is \(n\)-connected.

A functorial cofiber on a triangulated category \(\mathcal{T}\) is an assignment that to each morphism \(f: X \to Y\) in \(\mathcal{T}\) associates a distinguished triangle \(X \xrightarrow {f} Y \to Cf \to \Sigma X\) that is functorial: given a commutative square

there is an induced map \(C\alpha \beta : Cf \to Cf'\) making the diagram of distinguished triangles commute.

A filtered chain complex \((C_*, d, F^s)\) is Hausdorff (or separated) if \(\bigcap _s F^s C_n = 0\) for all \(n\).

For a spectrum \(X \in \mathcal{S}\), the \(n\)-th homotopy group of \(X\) is

There exists a natural transformation

in \(h\mathrm{Syn}\). For a synthetic spectrum \(X\), \(\lambda _X\) denotes the component \(\Sigma ^{0,-1}X \to X\).

The \(\lambda \)-action on synthetic homotopy groups is the map

defined as follows: given \(f: S^{m,n} \to X\), the element \(\lambda \cdot f \in \pi _{m,n-1}(X)\) is the composite

where the first arrow is the biShift composition isomorphism \(S^{m,n-1} = \Sigma ^{m,n-1}\mathbf{1} \cong \Sigma ^{0,-1}\Sigma ^{m,n}\mathbf{1} = \Sigma ^{0,-1}S^{m,n}\), and \(\lambda _{S^{m,n}}: \Sigma ^{0,-1}S^{m,n} \to S^{m,n}\) is the component of \(\lambda \) at \(S^{m,n}\).

For \(n \ge 0\), define \(\lambda ^n_X: \Sigma ^{0,-n}X \to X\) by induction:

1. \(\lambda ^0_X = \mathrm{id}_X: X \to X\).

2. \(\lambda ^{n+1}_X = \lambda _X \circ \Sigma ^{0,-1}(\lambda ^n_X): \Sigma ^{0,-n-1}X \to \Sigma ^{0,-1}X \to X\), using the composition isomorphism \(\Sigma ^{0,-n-1} \cong \Sigma ^{0,-1}\Sigma ^{0,-n}\).

We define \(X/\lambda ^n\) as the cofiber of \(\lambda ^n_X\). In particular, \(X/\lambda ^0 = 0\) and \(X/\lambda ^1 = X/\lambda \). The cofiber construction gives a distinguished triangle

For spectra \(X, Y \in \mathcal{S}\), the mapping spectrum \(F(X, Y)\) is the internal hom (function object) from the closed symmetric monoidal structure (Definition 0.63), adjoint to the smash product:

In particular, \(\pi _n F(X, Y) \cong [\Sigma ^n X, Y]\).

The mod 2 cohomology of a spectrum \(X\) is

The mod 2 homology of a spectrum \(X\) is

There exists an additive functor

from the homotopy category of spectra to the homotopy category of synthetic spectra.

Given morphisms \(f: X \to Y\) and \(g: X \to Z\) in \(\mathcal{S}\), the pushout of the diagram \(Y \xleftarrow {f} X \xrightarrow {g} Z\) is the cofiber \(C(f, g)\) of the canonical morphism \((f, g): X \to Y \vee Z\), where \(Y \vee Z\) denotes the coproduct (wedge sum) in \(\mathcal{S}\).

The smash product \(\wedge : \mathcal{S} \times \mathcal{S} \to \mathcal{S}\) is the symmetric monoidal product from the closed symmetric monoidal structure of \(\mathcal{S}\) (Definition 0.63), with unit \(\mathbb {S}\). It satisfies \(S^m \wedge S^n \simeq S^{m+n}\) and induces a pairing on homotopy groups:

A spectral sequence in an abelian category \(\mathcal{C}\) with index type \(\iota \) (typically \(\mathbb {Z}^n\)) consists of:

1. A starting page index \(r_0 \in \mathbb {Z}\).

2. For each \(r \ge r_0\) and index \(\mathbf{k} \in \iota \), an object \(E_r^{\mathbf{k}} \in \mathcal{C}\) (the \(E_r\)-page at index \(\mathbf{k}\)).

3. A differential degree function \(\mathbf{d}: \mathbb {Z} \to \iota \), and differentials

   \[ d_r^{\mathbf{k}}: E_r^{\mathbf{k}} \to E_r^{\mathbf{k} + \mathbf{d}(r)} \]

   satisfying \(d_r \circ d_r = 0\) componentwise.

4. For each index \(\mathbf{k}\), nested subspace data \((V^{\mathbf{k}}, Z_r^{\mathbf{k}}, B_r^{\mathbf{k}})\) such that \(E_r^{\mathbf{k}} = Z_r^{\mathbf{k}} / B_r^{\mathbf{k}}\) for \(r \ge r_0\).

5. An isomorphism \(H(E_r, d_r) \cong E_{r+1}\) for each \(r\). By the nesting \(B_r \subseteq B_{r+1} \subseteq Z_{r+1} \subseteq Z_r\), the differential \(d_r\) has kernel \(Z_{r+1}/B_r\) and image \(B_{r+1}/B_r\), so

   \[ H(E_r, d_r) = \frac{Z_{r+1}/B_r}{B_{r+1}/B_r} \cong Z_{r+1}/B_{r+1} = E_{r+1} \]

   by the third isomorphism theorem.

For an integer \(n \in \mathbb {Z}\), define

the \(n\)-fold suspension of the sphere spectrum.

The sphere spectrum \(\mathbb {S} \in \mathcal{S}\) is the unit object for the smash product.

Let \(\mathcal{C}\) be an abelian category. The nested subspace data (or SS data) for a spectral sequence at a single grading index consists of:

1. An ambient object \(V \in \mathcal{C}\).

2. A decreasing family of subobjects \(\{ Z_r\} _{r \in \mathbb {N} \cup \{ \infty \} }\) of \(V\), called \(r\)-cycles:

   \[ V = Z_0 \supseteq Z_1 \supseteq Z_2 \supseteq \cdots \supseteq Z_\infty . \]

3. An increasing family of subobjects \(\{ B_r\} _{r \in \mathbb {N} \cup \{ \infty \} }\) of \(V\), called \(r\)-boundaries:

   \[ B_0 \subseteq B_1 \subseteq B_2 \subseteq \cdots \subseteq B_\infty . \]

4. The containment \(B_r \subseteq Z_r\) for all \(r\).

5. \(Z_\infty \) is the greatest lower bound of the finite cycles: for any subobject \(X \subseteq V\), if \(X \subseteq Z_i\) for all \(i \in \mathbb {N}\), then \(X \subseteq Z_\infty \). Together with the monotonicity of \(Z\), this gives \(Z_\infty = \bigcap _{i \in \mathbb {N}} Z_i\).

6. \(B_\infty \) is the least upper bound of the finite boundaries: for any subobject \(X \subseteq V\), if \(B_i \subseteq X\) for all \(i \in \mathbb {N}\), then \(B_\infty \subseteq X\). Together with the monotonicity of \(B\), this gives \(B_\infty = \bigcup _{i \in \mathbb {N}} B_i\).

Together these form the nested chain

The \(r\)-th page at this index is defined as \(E_r = Z_r / B_r\). The \(E_\infty \)-page is \(E_\infty = Z_\infty / B_\infty \).

A morphism of spectral sequences \(f: E \to E'\) consists of a family of maps \(\varphi _{\mathbf{k}}: V^{\mathbf{k}} \to V'^{\mathbf{k}}\) on the underlying objects at each index, preserving the cycle and boundary subobjects:

for all \(r\), and commuting with differentials on the induced page maps. A morphism induces maps on all pages \(E_r \to E'_r\) and on the \(E_\infty \)-page.

The stable homotopy category \(\mathcal{S}\) is a closed symmetric tensor triangulated category with functorial cofiber (Definition 0.64). Its objects are called spectra.

In the setting of Definition 0.6, the component of the \(n\)-grading on \(E\) that corresponds to the original \((n-1)\)-grading of \(A\) is called the stem (or topological degree).

The category \(h\mathrm{Syn}\) of \(\mathrm{H}{\mathbb F}_2\)-synthetic spectra is a closed symmetric monoidal triangulated category with functorial cofiber (it arises as the homotopy category of a stable model category \(\mathrm{Syn}_{\mathrm{H}{\mathbb F}_2}\)). We denote its tensor product by \(\otimes \) and its unit object by \(\mathbf{1}\).

The synthetic homotopy groups of a synthetic spectrum \(X\) are

These form a bigraded abelian group with a natural \(\mathbb {Z}[\lambda ]\)-module structure (Proposition 0.106).

A closed symmetric tensor triangulated category with functorial cofiber is a closed symmetric tensor triangulated category \(\mathcal{T}\) (Definition 0.63) equipped with:

1. A functorial cofiber (Definition 0.60).

2. A tensor-cofiber exchange: for any morphism \(f: X \to Y\) and any object \(W\), a natural isomorphism

   \[ C(f \wedge W) \cong C(f) \wedge W, \]

   where \(f \wedge W : X \wedge W \to Y \wedge W\) denotes the right-tensoring of \(f\) with \(W\) (i.e., \(f \otimes \mathrm{id}_W\)), compatible with the triangulated structure.

A triangulated category is an additive category \(\mathcal{T}\) equipped with:

1. An additive autoequivalence \(\Sigma : \mathcal{T} \to \mathcal{T}\), called the shift (or suspension) functor.

2. A class of distinguished triangles \(X \to Y \to Z \to \Sigma X\), satisfying the axioms (TR1)–(TR4):

   • Every morphism \(f: X \to Y\) extends to a distinguished triangle \(X \xrightarrow {f} Y \to Z \to \Sigma X\). The triangle \(X \xrightarrow {\mathrm{id}} X \to 0 \to \Sigma X\) is distinguished.

   • A triangle \(X \to Y \to Z \to \Sigma X\) is distinguished if and only if \(Y \to Z \to \Sigma X \to \Sigma Y\) is distinguished.

   • Given a morphism of the first two terms of two distinguished triangles, there exists a (not necessarily unique) morphism of the third terms making the diagram commute.

   • The octahedral axiom.

Let \((E, A, F)\) be a converging spectral sequence with filtration \(F^\bullet A\) bounded below. For an integer \(s\), the \(s\)-truncation of \((E, A, F)\) is the converging spectral sequence \((E^{[\le s]}, A^{[\le s]}, F^{[\le s]})\) defined as follows:

1. The truncated spectral sequence: for each filtration degree \(p\), set

   \[ V^{[\le s],p} = \begin{cases} V^p & \text{if } p \le s, \\ 0 & \text{otherwise,} \end{cases} \]

   so that \(E_r^{[\le s],p} = E_r^p\) for \(p \le s\) and \(E_r^{[\le s],p} = 0\) for \(p {\gt} s\).

2. The truncated abutment is the quotient

   \[ A^{[\le s]} = A \big/ F^{s+1} A, \]

   equipped with the induced filtration

   \[ F^{[\le s],p} A^{[\le s]} = \begin{cases} \big(F^p A + F^{s+1} A\big) \big/ F^{s+1} A & \text{if } p \le s, \\ 0 & \text{if } p {\gt} s. \end{cases} \]

   When \(F^p A = A\) (which holds for all sufficiently negative \(p\) by the bounded below assumption), this reduces to \(F^{[\le s],p} A^{[\le s]} = A^{[\le s]}\).

The truncation \((E^{[\le s]}, A^{[\le s]}, F^{[\le s]})\) is a converging spectral sequence with bounded (hence exhaustive and Hausdorff) filtration.

Let \(f: (E_1, A_1) \to (E_2, A_2)\) be a morphism of converging spectral sequences with filtrations bounded below (not necessarily bounded above). For each integer \(s\), let \(f^{[\le s]}: (E_1^{[\le s]}, A_1^{[\le s]}) \to (E_2^{[\le s]}, A_2^{[\le s]})\) denote the induced morphism of truncated converging spectral sequences. The unbounded \(f\)-extension spectral sequence is the spectral sequence (without converging data) defined as the inverse limit in the category of spectral sequences:

where each \(\mathrm{ESS}(f^{[\le s]})\) is the (bounded) extension spectral sequence of Definition 0.35. The inverse limit is well-defined as a spectral sequence by the stabilization property (Proposition 0.49).

For a map \(f: X \to Y\) which is part of a distinguished triangle \(X \xrightarrow {f} Y \xrightarrow {g} Cf \to \Sigma X\), define

When \(\mathrm{AF}(f) = 0\), we also denote \(\nu f\) by \(\hat{f}\). In both cases, we have \(\hat{f}: \Sigma ^{0,e(f)}\nu X \to \nu Y\) and \(\nu f = \lambda ^{e(f)}\hat{f}\). Furthermore, \(C\hat{f} \simeq \Sigma ^{0,-e(g)}\nu Cf\) (by Axiom 0.124).

For \(x \in E_\infty ^{s}(V_1)\), let \(\{ x\} \subseteq A_1\) denote the set of elements detected by \(x\):

For \([x] \in \{ x\} \), we call \([x]\) a representative of \(x\). Note that \(\{ x\} \) is a coset of \(F^{s+1}A_1\) in \(F^s A_1\).

A map \(f: X \to Y\) with \(\mathrm{AF}(f) = k\) maps \(F^s \pi _* X\) into \(F^{s+k} \pi _* Y\) for all \(s\).

For any synthetic spectrum \(X\),

Two elements \(x, x' \in F^r A\) are both detected by the same \(y \in E_\infty \) if and only if \(x - x' \in F^{r+1} A\).

An element \(x \in F^r A\) is detected by \(0 \in E_\infty \) if and only if \(x \in F^{r+1} A\).

The zeroth extension differential \(d_0^f\) equals the map induced by \(f\) on the associated graded:

Under the identification \(\mathrm{gr}^s A_i \cong E_\infty ^s(V_i)\), this is the map \(E_\infty ^s(V_1) \to E_\infty ^s(V_2)\) induced by \(f\) on \(E_\infty \)-pages.

Under the decomposition \(\mathrm{ESS}(f)_0^{s} \cong E_\infty ^{s}(V_1) \oplus E_\infty ^{s}(V_2)\), the differentials of the \(f\)-ESS have the form

and only the component \(E_\infty ^{s}(V_1) \to E_\infty ^{s+n}(V_2)\) can be nonzero; the other three components (source-to-source, target-to-source, target-to-target) vanish. This follows from the chain complex structure: the differential of \(C_*\) maps \(C_1 = A_1\) to \(C_0 = A_2\) and is zero in the reverse direction.

The \(E_0\)-page of the \(f\)-ESS is canonically isomorphic to the direct sum of the \(E_\infty \)-pages of the source and target:

This follows from \(E_0^s = \mathrm{gr}^s C_* = \mathrm{gr}^s A_1 \oplus \mathrm{gr}^s A_2 \cong E_\infty ^s(V_1) \oplus E_\infty ^s(V_2)\).

Suppose the sequence of converging spectral sequences \(V_1 \xrightarrow {f} V_2 \xrightarrow {g} V_3\) induces an exact sequence on target groups \(A_1 \xrightarrow {f_A} A_2 \xrightarrow {g_A} A_3\) that is exact at \(A_2\). Then all permanent \(d^g\)-cycles in \(E_\infty (V_2)\) are boundaries in the \(f\)-ESS.

An \(f\)-extension \(d_n^f(x) = y\) holds if and only if there exists \([x] \in \{ x\} \) such that \(f_A[x] \in \{ y\} \).

Consider \(f: V_1 \to V_2\), \(x \in E_\infty ^{s}(V_1)\), \(y \in E_\infty ^{s+n}(V_2)\) and \(y' \in E_\infty ^{s+m}(V_2)\) for \(m, n \ge 0\).

1. An \(f\)-extension from \(x\) to \(y\) is inessential, i.e., \(y\) is trivial on the \(E_n\)-page of the \(f\)-ESS, if and only if there exists \(x' \in E_\infty ^{s+a}(V_1)\) for some \(0 {\lt} a \le n\) with an essential differential \(d_{n-a}^f(x') = y\). Equivalently, there exists \([x'] \in \{ x'\} \subseteq A_1\) with \(\mathrm{Fil}(x') {\gt} \mathrm{Fil}(x)\) such that \(f_A[x'] \in \{ y\} \).

2. Suppose \(d_n^f(x) = y\) and \(d_m^f(x) = y'\).

   1. If \(m = n\), then \(y - y' \in {}^f\! B_{n-1}^{s+n}(V_2)\): there exists \(x' \in E_\infty ^{s+a}(V_1)\) for some \(0 {\lt} a \le n\) with an essential differential \(d_{n-a}^f(x') = y - y'\).

   2. If \(m {\gt} n\), then the \(f\)-extension from \(x\) to \(y\) is inessential.

Let \(X, Y, Z \in \mathcal{S}^{\mathrm{fin}}\) be finite spectra. Suppose that \(X \xrightarrow {f} Y \xrightarrow {g} Z \xrightarrow {h} \Sigma X\) is a distinguished triangle with \(e(f) + e(g) + e(h) = 1\). Then there is a distinguished triangle of synthetic spectra

The construction uses Axiom 0.124 to establish the \(\lambda \)-relations between the classical and synthetic connecting maps.

A morphism of filtered chain complexes induces a morphism of the associated spectral sequences. This construction is functorial.

The differential \(d_r\) in Definition 0.23 is well-defined and satisfies \(d_r \circ d_r = 0\).

A filtered chain complex \((C_*, d, F^s)\) induces a natural filtration on its homology:

For any spectrum \(X\), the collection \(\pi _* X = \{ \pi _n X\} _{n \in \mathbb {Z}}\) is a graded abelian group.

Let \((E, A, F)\) be a converging spectral sequence with filtration bounded below. If the filtration on \(A\) is complete (Definition 0.50), then the converging spectral sequence \((E, A)\) is the inverse limit of the inverse system of truncations in the category of converging spectral sequences:

Let \((E, A, F)\) be a converging spectral sequence with filtration bounded below. Then the spectral sequence \(E\) is the inverse limit of the spectral sequences of the truncations: at each filtration degree \(p\) and page \(r\) one has

For a distinguished triangle \(X \to Y \to Z \to \Sigma X\) and any object \(W\), the long exact sequence on hom-sets

holds (covariant), and dually for the contravariant case.

Let \(X \xrightarrow {f} Y \to C_f \to \Sigma X\) be a cofiber sequence. Then there is a long exact sequence of homotopy groups:

This follows from Proposition 0.61 and the representability \(\pi _n(-) = [S^n, -]\).

An \(f\)-extension from \(x\) to \(y\) has no crossing in the range \(\mathrm{Fil} \ge p\) if and only if for all \([x] \in \{ x\} \) such that \(\mathrm{Fil}(f_A[x]) \ge p\) we have \(f_A[x] \in \{ y\} \). In particular:

1. An \(f\)-extension from \(x\) to \(y\) has no crossing if and only if for all \([x] \in \{ x\} \) we have \(f_A[x] \in \{ y\} \).

2. An \(f\)-extension \(d_n^f(x) = 0\) has no crossing if and only if for all \([x] \in \{ x\} \), \(\mathrm{Fil}(f_A[x]) {\gt} \mathrm{Fil}(x) + n\).

3. If \(\mathrm{AF}(f) = n\), then all \(d_n^f\)-differentials have no crossing.

Let \(X, Y, Z \in \mathcal{S}^{\mathrm{fin}}\) be finite spectra. Suppose that \(X \xrightarrow {f} Y \xrightarrow {g} Z \xrightarrow {h} \Sigma X\) is a distinguished triangle of spectra with \(\mathrm{AF}(h) {\gt} 0\), and consequently a short exact sequence on \(\mathrm{H}{\mathbb F}_2\)-homology

Then there exists a distinguished triangle of synthetic spectra

such that \(\nu h = \lambda \hat{h}\). The last relation \(\nu h = \lambda \hat{h}\) follows from Axiom 0.124.

Let \(X \in \mathcal{S}^{\mathrm{fin}}\) be a finite spectrum. For \(r \ge 2\), the \(E_\infty \)-page of the synthetic Adams spectral sequence for \(\nu X / \lambda ^r\) is

Let \(X \in \mathcal{S}^{\mathrm{fin}}\) be a finite spectrum. The \(E_\infty \)-page of the synthetic Adams spectral sequence for \(\nu X\) is

The collection \(\{ (E^{[\le s]}, A^{[\le s]})\} _{s \in {\mathbb Z}}\) forms an inverse system in the category of converging spectral sequences, indexed by \(s\) with order \(s \ge s'\). For \(s \ge s'\), the transition morphism is the natural projection

which is compatible with the truncated filtrations and with the spectral sequence page maps.

Let \(f: (E_1, A_1) \to (E_2, A_2)\) be a morphism of converging spectral sequences with filtrations bounded below. Fix a filtration degree \(p \in {\mathbb Z}\) and a page index \(r \ge 1\). Then for all integers \(s \ge p + r\), the component of the \(s\)-truncated extension spectral sequence at filtration degree \(p\) and page \(r\) is canonically isomorphic to the corresponding component of the \(s'\)-truncated extension spectral sequence for any \(s' \ge s\):

In other words, once the truncation level \(s \ge p + r\) (so that the truncated ESS retains both the source filtration degree \(p\) and the target filtration degree \(p + r\)), the \(d_r^f\)-differential at filtration degree \(p\) is fully determined and independent of the truncation level.

The homotopy category \(h\mathrm{Syn}\) is enriched over \({\mathbb Z}[\lambda ]\)-modules.

All filtrations considered in this project are decreasing.

The following are equivalent:

• An \(f\)-extension \(d_n^f(x) = y\) has no crossing that hits the range \(\mathrm{Fil} \ge p\).

• For any \(a {\gt} 0\), if there is an \(f\)-extension from \(x' \in E_\infty ^{s+a}(V_1)\) to some nontrivial \(y'\), then \(\mathrm{Fil}(y') {\lt} p\) or \(\mathrm{Fil}(y') {\gt} \mathrm{Fil}(y)\).

The cohomological variant also holds: if \(X \to Y \to Z\) is a cofiber sequence with

short exact in mod 2 cohomology, then \(\nu \) sends it to a distinguished triangle. This follows from Axiom 0.117 and the universal coefficient theorem (Axiom 0.79).

This is convergence in the weak sense: we only require an isomorphism between \(E_\infty \) and \(\mathrm{gr}\, A\). Conditional convergence and strong convergence (in the sense of Boardman  [ ) are not considered in this project.

The quotient \([X, Y] / \{ \text{odd-order elements}\} \) carries an induced Adams filtration that is separated (Hausdorff).

If the filtration on a filtered chain complex is exhaustive and Hausdorff, then the associated spectral sequence converges to the homology:

The convergence isomorphism is \(E_\infty ^{s,t} \cong \mathrm{gr}^s H_{s+t}(C_*)\) with respect to the induced filtration from Proposition 0.17.

The nested subspace data and differentials from Definition 0.23 satisfy the axioms of a spectral sequence. Specifically:

1. \(B_r \subseteq B_{r+1} \subseteq Z_{r+1} \subseteq Z_r\) for all \(r\).

2. \(E_r^{s,t} = Z_r^{s,t} / B_r^{s,t}\).

3. \(H(E_r, d_r) \cong E_{r+1}\).

The spectral sequence starts at \(E_0^{s,t} = F^s C_{s+t} / F^{s+1} C_{s+t}\) with differentials \(d_r: E_r^{s,t} \to E_r^{s+r,t-r+1}\).

\(Z \in {\mathcal S}^{fin}\) if and only if \(Z\) can be obtained from \({\mathbb S}\) by finitely many operations of the following two types:

1. suspension or desuspension,

2. taking the cofiber of a map from a sphere spectrum \(S^n\).

Consider a homotopy commutative diagram of converging spectral sequences

Suppose \(m, n, l \ge 0\), \(0 {\lt} k \le m + l - n\),

and

1. \(d_n^f(x) = y\),

2. \(d_m^p(x) = z\),

3. the differential in (1) or (2) has no crossing,

4. \(d_l^g(z) = w\), and has no crossing that hits \(\mathrm{Fil} \ge s + n + k\),

5. \(d_{k-1}^q y = 0\), and has no crossing.

Then \(d_{m+l-n}^q(y) = w\).

For all \(n \in {\mathbb Z}\) and spectra \(X\),

If \(f\) has odd order in \([X,Y]\), then \(f\) has Adams filtration \(\infty \).

Let \(f: (E_1, A_1) \to (E_2, A_2)\) be a morphism of converging spectral sequences with filtrations bounded below. Suppose that the inverse systems \(\{ A_i^{[\le s]}\} _{s \in {\mathbb Z}}\) (for \(i = 1, 2\)) satisfy the Mittag-Leffler condition: for each \(s\), the images of the transition maps \(A_i^{[\le s']} \to A_i^{[\le s]}\) stabilize as \(s' \to \infty \). Then the unbounded \(f\)-extension spectral sequence \(\mathrm{ESS}^{\mathrm{unbd}}(f)\) converges to the bigraded associated-graded homology of the completions:

where \(\hat{f}_A: \widehat{A}_1 \to \widehat{A}_2\) is the completion of \(f_A\), and the right-hand side denotes the bigraded associated-graded homology of the two-term complex equipped with the filtrations induced on \(\widehat{A}_1\) and \(\widehat{A}_2\).