nLab multiplicative spectral sequence

Contents

Context

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Schanuel's lemma

Homology theories

Theorems

Higher algebra

Contents

Idea

A spectral sequence is called multiplicative or a spectral ring if there is a bi-graded algebra structure on each page such that the differentials act as graded derivations of total degree 1.

For example the Serre-Atiyah-Hirzebruch spectral sequence with coefficients in a ring spectrum.

Constructions

From spectral products on Cartan-Eilenberg systems

The following gives sufficient conditions for a Cartan-Eilenberg spectral sequence to be multiplicative. This is due to (Douady 58). The following is taken from (Goette 15a).

Definition

Let (H,η,)(H,\eta,\partial), (H,η,)(H',\eta',\partial') und (H,η,)(H'',\eta'',\partial'') be Cartan-Eilenberg systems. A spectral product μ:(H,)×(H,)(H,)\mu\colon(H',\partial')\times(H'',\partial'')\to(H,\partial) is a sequence of homomorphisms

μ r:H(m,m+r)H(n,n+r)H(m+n,m+n+r) \mu_r\colon H'(m,m+r)\otimes H''(n,n+r)\to H(m+n,m+n+r)

such that for all mm, nn, r1r\ge 1, the following two diagrams commute:

H(m,m+r)H(n,n+r) μ r H(m+n,m+n+r) ηη η H(m,m+1)H(n,n+1) μ 1 H(m+n,m+n+1) \array{ H'(m, m+r) \otimes H''(n, n+r) &\stackrel{\mu_r}{\longrightarrow}& H(m+n, m+n+r) \\ \downarrow^{\mathrlap{\eta' \oplus \eta''}} && \downarrow^{\mathrlap{\eta}} \\ H'(m, m+1) \otimes H''(n, n+1) &\stackrel{\mu_1}{\longrightarrow}& H(m+n, m+n+1) }

and

H(m,m+r)H(n,n+r) μ r H(m+n,m+n+r) ηη H(m+r,m+r+1)H(n,n+1) μ 1+μ 1 H p+q1(m+n+r,m+n+r+1) H(m,m+1)H(n+r,n+r+1). \array{ H'(m, m+r) \otimes H''(n, n+r) &\stackrel{\mu_r}{\longrightarrow}& H(m+n,m+n+r) \\ \downarrow^{\mathrlap{\partial' \otimes \eta'' \oplus \eta' \otimes \partial''}} && \downarrow^{\mathrlap{\partial}} \\ H'(m+r, m+r+1) \otimes H''(n,n+1) \\ \oplus &\stackrel{\mu_1 + \mu_1}{\longrightarrow}& H_{p+q-1}(m+n+r, m+n+r+1) \\ H'(m,m+1) \otimes H''(n+r, n+r+1) } \,.
Remark

The first diagram in def. is weaker than in (Douady 58). The second may be read as a Leibniz rule.

Write EE for the Cartan-Eilenberg spectral sequence induced from the Cartan-Eilenberg system HH.

Proposition

A spectral product μ:(H,)×(H,)(H,)\mu\colon(H',\partial')\times(H'',\partial'')\to(H,\partial) as in def. induces products

μ r:E m rE n rE m+n r, \mu^r\colon E^{\prime r}_m\otimes E^{\prime\prime r}_n\to E^r_{m+n}\;,

such that

  1. μ 1=μ 1\mu^1=\mu_1

  2. d m+n rμ r=μ r(d m rid)±μ r(idd n r)d^r_{m+n}\circ\mu^r=\mu^r\circ(d^{\prime r}_m\otimes\mathrm{id})\pm\mu^r\circ(\mathrm{id}\circ d^{\prime\prime r}_n),

  3. μ r+1\mu^{r+1} is induced by μ r\mu^r.

(Goette 15a, following Douady 58, theorem II).

Proof

Assume by induction that μ r\mu^r is induced by μ 1\mu_1. In particular,

Z m rZ n rμ 1Z m+n r,Z^{\prime r}_m\otimes Z^{\prime\prime r}_n\stackrel{\mu_1}\to Z^r_{m+n}\;,
B m rZ n rμ 1B m+n r,B^{\prime r}_m\otimes Z^{\prime\prime r}_n\stackrel{\mu_1}\to B^r_{m+n}\;,
Z m rB n rμ 1B m+n r.Z^{\prime r}_m\otimes B^{\prime\prime r}_n\stackrel{\mu_1}\to B^r_{m+n}\;.

This is clear for r=1r=1 if we put μ 1=μ 1\mu^1=\mu_1 because E p 1=Z p 1=H(p,p+1)E^1_p=Z^1_p=H(p,p+1) and B p 1=0B^1_p=0.

Let [a]Z m r[a]\in Z^{\prime r}_m, [b]Z n r[b]\in Z^{\prime\prime r}_n be represented by a=η(a 0)H(m,m+1)a=\eta'(a_0)\in H'(m,m+1), b=η(b 0)H(n,n+1)b=\eta''(b_0)\in H''(n,n+1) with a 0H(m,m+r)a_0\in H'(m,m+r), b 0H(n,n+r)b_0\in H''(n,n+r). Using the first diagram and the construction of d m+n rd^r_{m+n}, we conclude that

(d m+n rμ r)([a][b])=d m+n r[μ 1(ab)]=d m+n r[η(μ r(a 0b 0))]=(μ r)(a 0b 0). (d^r_{m+n}\circ\mu^r)([a]\otimes[b])=d^r_{m+n}[\mu_1(a\otimes b)]=d^r_{m+n}[\eta(\mu_r(a_0\otimes b_0))]=(\partial\circ\mu_r)(a_0\otimes b_0) \;.

From the second diagram, we get

(μ r)(a 0b 0)=μ 1(a 0ηb 0)±μ 1(ηa 0b 0)=μ r(d m r[a][b])±μ r([a]d n r[b]). (\partial\circ\mu_r)(a_0\otimes b_0)=\mu_1(\partial'a_0\otimes\eta''b_0)\pm\mu_1(\eta'a_0\otimes\partial''b_0)=\mu^r(d^{\prime r}_m[a]\otimes[b])\pm\mu^r([a]\otimes d^{\prime\prime r}_n[b]) \;.

This proves the Leibniz rule (2).

From the Leibniz rule and the facts that ker(d p r)=Z p r+1/B p r\ker(d^r_p)=Z^{r+1}_p/B^r_p and im(d p r)=B p r+1/B p r\mathrm{im}(d^r_p)=B^{r+1}_p/B^r_p, we conclude that μ r\mu^r induces a product on E p r+1ker(d p r)/im(d p r)E^{r+1}_p\cong\ker(d^r_p)/\mathrm{im}(d^r_p), which proves (3). Because μ r\mu^r is induced by μ 1\mu_1, so is μ r+1\mu^{r+1}, and we can continue the induction.

Examples

AHSS for multiplicative cohomology

We discuss that the multiplicative structure on the cohomology Serre-Atiyah-Hirzebruch spectral sequence for multiplicative generalized cohomology. This is taken from (Goette 15b).

Definition

For π:XB\pi\colon X\to B a Serre fibration over a CW-complex BB. And for (h˜ ,δ,)(\tilde h^\bullet,\delta,\wedge) a multiplicative reduced generalized (Eilenberg-Steenrod) cohomology theory, define a Cartan-Eilenberg system (H,η,)(H,\eta,\partial) by

H(p,q)=h˜ (X q1/X p1) H(p,q)=\tilde h^\bullet(X^{q-1}/X^{p-1})

(where X k=π 1(B k)X^k=\pi^{-1}(B^k)) for pqp\le q with the obvious maps η:H(p,q)H(p,q)\eta\colon H(p',q')\to H(p,q) for ppp\le p', qqq\le q'.

The Cartan-Eilenberg spectral sequence of this Cartan-Eilenberg system is the Serre-Atiyah-Hirzebruch spectral sequence.

Definition

The spectral product μ:(H,η,)×(H,η,)(H,η,)\mu\colon(H,\eta,\partial)\times(H,\eta,\partial)\to(H,\eta,\partial), def. , on the Cartan-Eilenberg system of def. is that given by the following morphism

F m,n,r :(XX) m+n+r1/(XX) m+n1 a+b=m+n+r1(X aX b)/ c+d=m+n1(X cX d) a+b=m+n+r1(X aX b)/( a=0 m(X a1X m+n+ra) b=0 n(X m+n+rbX b1) a=m+1 m+r(X a1X m+n+ra)/(X m+r1X n1X m1X n+r1) X m+r1X n+r1/(X m+r1X n1X m1X n+r1) (X m+r1/X m1)(X n+r1/X n1). \begin{aligned} F_{m,n,r} & \colon (X\wedge X)^{m+n+r-1}/(X\wedge X)^{m+n-1} \cong \bigcup_{a+b=m+n+r-1}(X^a\wedge X^b) / \bigcup_{c+d=m+n-1}(X^c\wedge X^d) \\ & \twoheadrightarrow \bigcup_{a+b=m+n+r-1} (X^a\wedge X^b)/(\bigcup_{a=0}^m(X^{a-1}\wedge X^{m+n+r-a}) \cup\bigcup_{b=0}^n(X^{m+n+r-b}\wedge X^{b-1}) \\ & \cong \bigcup_{a=m+1}^{m+r}(X^{a-1}\wedge X^{m+n+r-a}) / \bigl(X^{m+r-1}\wedge X^{n-1}\cup X^{m-1}\wedge X^{n+r-1}\bigr) \\ & \hookrightarrow X^{m+r-1}\wedge X^{n+r-1}/(X^{m+r-1}\wedge X^{n-1}\cup X^{m-1}\wedge X^{n+r-1}) \\ & \cong (X^{m+r-1}/X^{m-1})\wedge(X^{n+r-1}/X^{n-1}) \;. \end{aligned}

Together with the diagonal map Δ\Delta, for r1r\ge 1, we define

μ r :H(m,m+r)H(n,n+r) h˜(X m+r1/X m1)h˜(X n+r1/X n1) h˜((X m+r1/X m1)(X n+r1/X n1)) F m,n,r *h˜((XX) m+n+r1/(XX) m+n1) Δ X *h˜(X m+n+r1/X m+n1)=H(m+n,m+n+r). \begin{aligned} \mu_r & \colon H(m,m+r)\otimes H(n,n+r) \\ & \cong\tilde h(X^{m+r-1}/X^{m-1})\otimes\tilde h(X^{n+r-1}/X^{n-1}) \\ &\stackrel\wedge\longrightarrow\tilde h\bigl((X^{m+r-1}/X^{m-1})\wedge(X^{n+r-1}/X^{n-1})\bigr) \\ &\stackrel{F_{m,n,r}^*}\longrightarrow\tilde h\bigl((X\wedge X)^{m+n+r-1}/(X\wedge X)^{m+n-1}\bigr) \\ &\stackrel{\Delta_X^*}\longrightarrow\tilde h(X^{m+n+r-1}/X^{m+n-1})=H(m+n,m+n+r)\;. \end{aligned}
Proposition

With def. 3, then for all mm, nn, r1r\ge 1, the following [[commuting diagram|diagram commutes]]

H(m,m+1)H(n,n+1) μ 1 H(m+n,m+n+1) ηη η H(m,m+r)H(n,n+r) μ r H(m+n,m+n+r) ηη H(m+r,m+r+1)H(n,n+1) μ 1±μ 1 H(m+n+r,m+n+r+1) H(m,m+1)H(n+r,n+r+1). \array{ H(m,m+1) \otimes H(n,n+1) &\stackrel{\mu_1}{\longrightarrow}& H(m+n, m+n+1) \\ \uparrow^{\mathrlap{\eta \oplus \eta}} && \uparrow^{\mathrlap{\eta}} \\ H(m,m+r) \otimes H(n,n+r) &\stackrel{\mu_r}{\longrightarrow}& H(m+n,m+n+r) \\ \downarrow^{\mathrlap{\partial \otimes \eta \oplus \eta \otimes \partial}} && \downarrow^{\mathrlap{\partial}} \\ H(m+r, m+r+1) \otimes H(n,n+1) \\ \oplus &\stackrel{\mu_1 \pm \mu_1}{\longrightarrow}& H(m+n+r, m+n+r+1) \\ H(m,m+1) \otimes H(n+r, n+r+1) } \,.

Hence by prop. 1 the spectral product of def. 3 defines a mutliplicative structure on the Serre-WhiteheadAtiyah-Hirzebruch spectral sequence for multiplicative generalizted cohomology.

Proof

The upper square commutes because the maps F m,n,rF_{m,n,r} are [[natural transformations]]. For the lower square, we consider the boundary morphism δ\delta of the triple

( X m+rX n+r1X m+r1X n+r, X m+rX n1X m+r1X n+r1X m1X n+r, X m+rX n1X m1X n+r). \begin{aligned} (& X^{m+r}\wedge X^{n+r-1}\cup X^{m+r-1}\wedge X^{n+r}, \\ & X^{m+r}\wedge X^{n-1}\cup X^{m+r-1}\wedge X^{n+r-1}\cup X^{m-1}\wedge X^{n+r}, \\ & X^{m+r}\wedge X^{n-1}\cup X^{m-1}\wedge X^{n+r}) \end{aligned} \;.

The following diagram commutes:

h˜ p(X m+r1/X m1)h˜ q(X n+r1/X n1) h˜ pq((X m+r1/X m1)(X n+r1/X n1) δididδ δ h˜(1p)(X m+r/X m+r1)h˜ q(X n+r1/X n1) h˜ 1pq((X m+r/X m+r1)(X n+r1/X n1)) h˜ p(X m+r1/X m1)h˜ 1q(X n+r/X n+r1) h˜ 1pq((X m+r1/X m1)(X n+r/X n+r1)). \array{ \tilde h^{-p}(X^{m+r-1}/ X^{m-1}) \otimes \tilde h^{-q}(X^{n+r-1}/X^{n-1}) &\stackrel{\wedge}{\longrightarrow}& \tilde h^{-p-q}((X^{m+r-1}/X^{m-1}) \wedge (X^{n+r-1}/X^{n-1}) \\ \downarrow^{\mathrlap{\delta \wedge id \oplus id \wedge \delta}} && \downarrow^{\mathrlap{\delta}} \\ \tilde h(1-p)(X^{m+r}/X^{m+r-1}) \otimes \tilde h^{-q}(X^{n+r-1}/X^{n-1}) && \tilde h^{1-p-q}((X^{m+r}/X^{m+r-1}) \wedge (X^{n+r-1}/X^{n-1})) \\ \oplus &\stackrel{\wedge \oplus \wedge}{\longrightarrow}& \\ \tilde h^{-p}(X^{m+r-1}/X^{m-1}) \otimes \tilde h^{1-q}(X^{n+r}/X^{n+r-1}) && \tilde h^{1-p-q}( (X^{m+r-1}/ X^{m-1}) \wedge (X^{n+r}/ X^{n+r-1}) ) } \,.

By extend this diagram to the right using the maps F m,n,rF_{m,n,r} once concludes that the lower square above also commutes.

  • [[multiplicative cohomology theory]]

References

  • [[Adrien Douady]], La suite spectrale d’Adams : structure multiplicative Séminaire Henri Cartan, 11 no. 2 (1958-1959), Exp. No. 19, 13 p (Numdam)

  • Brayton Gray, Products in the Atiyah-Hirzebruch spectral sequence and the calculation of MSO *M SO_\ast, Trans. Amer. Math. Soc. 260 (1980), 475-483 (web)

  • [[Stanley Kochmann]], prop. 4.2.9 of [[Bordism, Stable Homotopy and Adams Spectral Sequences]], AMS 1996

  • [[John McCleary]], section 2.3 in A User’s Guide to Spectral Sequences, Cambridge University Press (2000)

  • [[Daniel Dugger]], Multiplicative structures on homotopy spectral sequences I (arXiv:math/0305173)

  • [[Daniel Dugger]], Multiplicative structures on homotopy spectral sequences II (arXiv:math/0305187)

  • [[Sebastian Goette]], MO comment a, MO comment b Feb 15, 2015

[[!redirects multiplicative spectral sequences]]

Last revised on May 6, 2016 at 16:23:53. See the history of this page for a list of all contributions to it.