# Friday, March 19 ## Spectral Sequence of a Filtration :::{.definition title="?"} A **filtration** of a chain complex $C$ is an ordered family of subcomplexes \[ F\da & \cdots \subseteq F_{p-1}C \subseteq F_p C \subseteq \cdots \subseteq C && p\in \ZZ \] such that there are commutative diagrams \begin{tikzcd} {F_pC_n} && {C_n} \\ \\ {F_pC_{n-1}} && {C_{n-1}} \arrow["d", from=1-1, to=3-1] \arrow["d", from=1-3, to=3-3] \arrow[hook, from=1-1, to=1-3] \arrow[hook, from=3-1, to=3-3] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJGX3BDX24iXSxbMiwwLCJDX24iXSxbMCwyLCJGX3BDX3tuLTF9Il0sWzIsMiwiQ197bi0xfSJdLFswLDIsImQiXSxbMSwzLCJkIl0sWzAsMSwiIiwxLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiaG9vayIsInNpZGUiOiJ0b3AifX19XSxbMiwzLCIiLDEseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJob29rIiwic2lkZSI6InRvcCJ9fX1dXQ==) A filtration is **exhaustive** if $\Union_{p\in \ZZ} F_p C_n = C_n$ for all $n$. ::: :::{.remark} The construction of the spectral sequence will show that $C$ and $\Union_p F_p C$ give rise to the same spectral sequence. So we will assume that all filtrations are exhaustive. ::: :::{.theorem title="Construction of the spectral sequence of a filtration"} A filtration $F$ of $C\in \Ch(\rmod)$ determines a spectral sequence starting with \[ E_{p, q}^0 { F_p C_{p+q} \over F_{p-1} C_{p+q} } && E_{p ,q}^1 = H_{p+q}(E^0_{p, *}) .\] Since $d$ preserves numerators and denominators, we get well-defined differentials $\bar{d}$ on the quotients: \begin{tikzcd} &&&&& \textcolor{rgb,255:red,92;green,92;blue,214}{E_{p-1, q+1}^0} \\ && {F_{p-1}C_{p+q+1}} & {} & {F_{p}C_{p+q+1}} && \textcolor{rgb,255:red,92;green,92;blue,214}{E_{p, q+1}^0} \\ &&&&&&& \ddots \\ \textcolor{rgb,255:red,92;green,92;blue,214}{F_{p-2}C_{p+q}} && \textcolor{rgb,255:red,92;green,92;blue,214}{F_{p-1}C_{p+q}} && {F_{p}C_{p+q}} && {E_{p, q}^0} \\ \\ && {F_{p-1}C_{p+q-1}} && {F_{p}C_{p+q-1}} && {E_{p, q-1}^0} \arrow[from=2-3, to=2-5] \arrow[from=2-5, to=2-7] \arrow[from=4-3, to=4-5] \arrow[from=4-5, to=4-7] \arrow[from=6-3, to=6-5] \arrow[from=6-5, to=6-7] \arrow["{\bar{d}}"', from=2-7, to=4-7] \arrow["{\bar{d}}"', from=4-7, to=6-7] \arrow["d"', from=2-5, to=4-5] \arrow["d"', from=4-5, to=6-5] \arrow["d"', from=2-3, to=4-3] \arrow["d"', from=4-3, to=6-3] \arrow[color={rgb,255:red,92;green,92;blue,214}, from=4-1, to=4-3] \arrow[color={rgb,255:red,92;green,92;blue,214}, from=1-6, to=2-7] \arrow[from=2-7, to=3-8] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsMTMsWzIsMSwiRl97cC0xfUNfe3ArcSsxfSJdLFszLDFdLFsyLDMsIkZfe3AtMX1DX3twK3F9IixbMjQwLDYwLDYwLDFdXSxbMiw1LCJGX3twLTF9Q197cCtxLTF9Il0sWzQsMSwiRl97cH1DX3twK3ErMX0iXSxbNCwzLCJGX3twfUNfe3ArcX0iXSxbNCw1LCJGX3twfUNfe3ArcS0xfSJdLFs2LDEsIkVfe3AsIHErMX1eMCIsWzI0MCw2MCw2MCwxXV0sWzYsMywiRV97cCwgcX1eMCJdLFs2LDUsIkVfe3AsIHEtMX1eMCJdLFswLDMsIkZfe3AtMn1DX3twK3F9IixbMjQwLDYwLDYwLDFdXSxbNSwwLCJFX3twLTEsIHErMX1eMCIsWzI0MCw2MCw2MCwxXV0sWzcsMiwiXFxkZG90cyJdLFswLDRdLFs0LDddLFsyLDVdLFs1LDhdLFszLDZdLFs2LDldLFs3LDgsIlxcYmFye2R9IiwyXSxbOCw5LCJcXGJhcntkfSIsMl0sWzQsNSwiZCIsMl0sWzUsNiwiZCIsMl0sWzAsMiwiZCIsMl0sWzIsMywiZCIsMl0sWzEwLDIsIiIsMix7ImNvbG91ciI6WzI0MCw2MCw2MF19XSxbMTEsNywiIiwyLHsiY29sb3VyIjpbMjQwLDYwLDYwXX1dLFs3LDEyXV0=) Taking vertical homology of the $E^0$ terms on the right yields $E_{p, q}^1$. Note that the blue terms contribute to the same diagonal $p+q=n$. ::: :::{.definition title="Bounded Filtrations"} A filtration $F$ on a chain complex $C$ is **bounded** if for each $n$ there are $sn, p-1\geq n). \end{cases} \] So $E$ becomes a first quadrant spectral sequence. ::: :::{.remark} Note that all elements on all pages are subquotients of $E^0$ elements, so they can only get smaller, and terms that become 0 on some page stay 0 for all remaining pages. ::: ## Construction of the Spectral Sequence of a Filtration :::{.remark} For ease of notation, we'll suppress the subscript $q$ since it can always be recovered as $q = n-p$. Define the canonical quotients \[ \eta_p: F_p C \to F_p C / F_{p-1}C = E_p^0 .\] Define \[ A^r_p \da \ts{ c\in F_p C \st d(c) \in F_{p-r}(C) } ,\] which are elements of $F_p C$ which are cycles modulo $F_{p-r} C$, the **approximate cycles**. Note that any actual cycle is in all $A^r$. This differential takes things $r$ columns to the left, so we'll want to define a differential that associates the following terms \begin{tikzcd} &&& {F_{p-1}C_{n+1}} & {} & {F_{p}C_{n+1}} \\ \\ &&& \textcolor{rgb,255:red,153;green,92;blue,214}{F_{p-1}C_{n}} && {F_{p}C_{n}} & c \\ \\ \textcolor{rgb,255:red,153;green,92;blue,214}{F_{p-r}C} & \cdots && {F_{p-1}C_{n-1}} && {F_{p}C_{n-1}} & dc \arrow[hook, from=1-4, to=1-6] \arrow[hook, from=3-4, to=3-6] \arrow[hook, from=5-4, to=5-6] \arrow["d"', from=1-6, to=3-6] \arrow["d"', from=3-6, to=5-6] \arrow["d"', from=1-4, to=3-4] \arrow["d"', from=3-4, to=5-4] \arrow[hook, from=5-1, to=5-2] \arrow[maps to, from=3-7, to=5-7] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=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) Similarly, define \[ Z_p^r &\da \eta_p(A_p^r \subseteq E_p^0 \\ B_p^r &\da \eta_p(d A_{p+r-1}^{r-1}) \subseteq \eta_p(F_p C) \subseteq E_p^0 .\] ::: :::{.observation} Some key observations: 1. $F_p C = A_p^0 = A_p^{-1} = A_p^{-2} = \cdots$ 2. $A_p^{r+1} \subseteq A_p^r$ 3. $A_p^r \intersect F_{p-1} C = A_{p-1}^{r-1}$. ::: :::{.exercise title="?"} Work through these facts using the diagram above. ::: :::{.remark} Some consequences: $(1) \implies Z_p^0 = E_p^0$ (taking $r=0$ in the quotient map $\eta_p$). $(2) \implies Z_p^{r+q} \subseteq Z_p^r$, since these are images of subgroups $(3) \implies A_{p+r-1}^{r-1} \subseteq A_{p+r}^r$, replacing $p\mapsto p+r$ in the intersection formula. Then applying $d$ yields $B_p^r \subseteq D_p^{r+1}$. $(1) \implies B_p^0 = \eta_p(d A_{p-1}^{-1}) \subseteq \eta_p(F_{p-1} C) = 0$, since this occurs in the denominator for $\eta_p$ and $d$ preserves filtration degree. So the $Z_p$ get smaller and the $B_p$ get bigger. What happens in the middle? ::: :::{.proposition title="All boundaries are contained in all cycles in a spectral sequence"} $B_p^r \subseteq Z_p^s$ for all $r, s\geq 0$. ::: :::{.proof title="?"} A sequence of implications: \[ B_p^r \ni x = \eta_p(dc) \text{ for some }c &\implies d(dc) = 0 \in F_{p-s}C \, \forall s \\ &\implies dc \in A_p^s \\ &\implies \eta_p(dc) \in Z_p^s .\] ::: :::{.remark} Set $B_p^{\infty } \da \union_{r\geq 1} B_p^r \subseteq Z_p^{\infty } \da \Intersect_{s\geq 1} Z_p^s$, which follows from a set theory exercise. ::: :::{.remark} Combining and summarizing these results: for every $p\geq 0$, we have a tower of groups: \begin{tikzcd} {0 = B_p^0} & {B_p^1} & \cdots & {B_p^r} & \cdots & {B_p^\infty} & {Z_p^{\infty}} & \cdots & {Z_p^{r}} & \cdots & {Z_p^{1}} & {Z_p^{0} = E_p^0} \arrow[hook, from=1-1, to=1-2] \arrow[hook, from=1-2, to=1-3] \arrow[hook, from=1-3, to=1-4] \arrow[hook, from=1-4, to=1-5] \arrow[hook, from=1-5, to=1-6] \arrow[hook, from=1-6, to=1-7] \arrow[hook, from=1-7, to=1-8] \arrow[hook, from=1-8, to=1-9] \arrow[hook, from=1-9, to=1-10] \arrow[hook, from=1-10, to=1-11] \arrow[hook, from=1-11, to=1-12] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=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) ::: :::{.remark} Note that using standard isomorphism theorems, we have \[ Z_p^r \cong {A_p^r \over A_p^r \intersect F_{p-1}C C} \equalsbecause{(3)} {A_p^r \over A_{p-1}^{r-1}} .\] So set \[ E_p^r \da Z_p^r/B_p^r \cong {A_p^r + F_{p-1} C \over d A_{p+r-1}^{r-1} + F_{p-1}C } \cong {A_p^r \over d A_{p+r-q}^{r-1} + A_{p-1}^{r-1}} ,\] making $E_p^r$ a quotient of $A_p^r$. Using a similar calculation, one can show \[ {Z_p^{r+1} \over B_p^r} \cong { A_p^{r+1} + A_{p-1}^{r-1} \over dA_{p+r-1}^{r-1} + A_{p-1}^{r-1} } .\] ::: :::{.remark} There will be an induced differential on this quotient, which will follow from checking that the different preserves the numerator and denominator. :::