SpectralSequence 2003-08-21 22:07:41 2005-03-02 06:50:52 Definition % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics \usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \newtheorem{thm}{Theorem} \newtheorem{prop}{Proposition} \newcommand{\ab}[1]{{#1}_{\mathrm{ab}}} \newcommand{\Ad}{\mathrm{Ad}} \newcommand{\ad}{\mathrm{ad}} \newcommand{\Aut}{\mathrm{Aut}\,} \newcommand{\Aff}[2]{\mathrm{Aff}_{#1} #2} \newcommand{\aff}[2]{\mathfrak{aff}_{#1} #2} \newcommand{\mcB}{\mathcal{B}} \newcommand{\bb}[1]{\mathbb{#1}} \newcommand{\bfrac}[2]{\left[\frac{#1}{#2}\\right]} \newcommand{\bkh}{\backslash} \newcommand{\Cyc}[2]{\mathcal{C}^{#1}_{#2}} \newcommand{\Cbar}[2]{\overline{\C{#1}{#2}}} %\newcommand{\CD}{\R[\Delta]} \newcommand{\C}{\mathbb{C}} \newcommand{\CF}[2]{\ensuremath{\mathfrak{C}(#1,#2)}} \newcommand{\Cinf}{\EuScript{C}^{\infty}} \newcommand{\cmp}{cyclic mod $p$\xspace} \newcommand{\cp}{\mathrm{c.p.}} \newcommand{\CS}{\EuScript{CS}} \newcommand{\deck}{\EuScript{D}} \newcommand{\defl}[1]{\mathfrak{def}_{#1}} \newcommand{\Der}{\mathrm{Der}\,} \newcommand{\eH}{[X_H]-[Y_H]} \newcommand{\EL}{\mathcal{EL}} \newcommand{\End}{\mathrm{End}} \newcommand{\ES}[1]{\EuScript{#1}} \newcommand{\Ext}{\mathrm{Ext}} \newcommand{\Fix}{\mathrm{Fix}} \newcommand{\fr}[1]{\mathfrak{#1}} \newcommand{\Frat}{\mathrm{Frat}\,} \newcommand{\Gal}[1]{\Gamma(#1 |\Q)} \newcommand{\GL}[2]{\mathrm{GL}_{#1} #2} \newcommand{\gl}[2]{\mathfrak{gl}_{#1} #2} \newcommand{\GrR}[1]{a(#1 G)} \newcommand{\Gr}{\mathrm{Gr}\,} \newcommand{\mcH}{\mathcal{H}} \renewcommand{\H}{\mathbb{H}} \newcommand{\Hom}[2]{\mathrm{Hom}(#1,#2)} \newcommand{\id}{\mathrm{id}} \newcommand{\im}{\mathrm{im}} \newcommand{\ind}[2]{\mathrm{ind}^{#1}_{#2}} \newcommand{\indp}[2]{\mathfrak{ind}^{#1}_{#2}} \renewcommand{\inf}[1]{\mathfrak{inf}_{#1}} \newcommand{\inn}[1]{\langle #1\rangle} \renewcommand{\int}{\mathrm{int}} \newcommand{\Iso}{\mathrm{Iso}} \newcommand{\K}{\mathcal{K}} \renewcommand{\ker}{\mathrm{ker}\,} \renewcommand{\L}[1]{\mathfrak{L}(#1)} \newcommand{\lap}[1]{\Delta_{#1}} \newcommand{\lapM}{\Delta_M} \newcommand{\Lie}{\mathrm{Lie}} \newcommand{\lineq}{linearly equivalent\xspace} \newcommand{\mc}[1]{\mathcal{#1}} \newcommand{\mG}{m_G} \newcommand{\mK}{m_{\K}} \newcommand{\mindeg}[1]{\fr{md}(#1)} \newcommand{\N}{\mathbb{N}} \renewcommand{\O}{\mathcal{O}} \newcommand{\Om}{\Omega} \newcommand{\om}{\omega} \newcommand{\Orb}{\mathrm{Orb}} \newcommand{\pad}{\hat{\Z}_p} \newcommand{\pder}[2]{\frac{\partial #1}{\partial #2}} \newcommand{\pderw}[1]{\frac{\partial}{\partial #1}} \newcommand{\pdersec}[2]{\frac{\partial^2 #1}{\partial {#2}^2}} \newcommand{\perm}[1]{\pi_{#1}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\R}{\mathbb{R}} \newcommand{\rad}{\mathrm{rad}\,} \newcommand{\res}[2]{\mathrm{res}^{#1}_{#2}} \newcommand{\resp}[2]{\mathfrak{res}^{#1}_{#2}} \newcommand{\RG}{\EuScript{R}_G} \newcommand{\rk}{\mathrm{rk}\,} \newcommand{\V}[1]{\mathbf{#1}} \newcommand{\vp}{\varphi} \newcommand{\Stab}{\mathrm{Stab}} \newcommand{\SL}[2]{\mathrm{SL}_{#1} #2} \renewcommand{\sl}[2]{\fr{sl}_{#1} #2} \newcommand{\SO}[2]{\mathrm{SO}_{#1} #2} \newcommand{\Sp}[2]{\mathrm{Sp}_{#1} #2} \renewcommand{\sp}[2]{\fr{sp}_{#1} #2} \newcommand{\SU}[1]{\mathrm{SU}( #1)} \newcommand{\su}[1]{\fr{su}_{#1}} \newcommand{\Sym}{\mathrm{Sym}} \newcommand{\sym}{\mathrm{sym}} \newcommand{\Tg}{\mc{T}(\fr g)} \newcommand{\tom}{\tilde{\omega}} \newcommand{\ghtghp}{\fr g/\fr h\oplus(\fr g/\fr h^\perp)^*} \newcommand{\ghps}{(\fr g/\fr h^\perp)^*} \newcommand{\Tr}{\mathrm{Tr}} \newcommand{\tr}{\mathrm{tr}} %\renewcommand{\thechapter}{\Roman{chapter}} %\renewcommand{\thesection}{\thechapter.\arabic{section}} %\renewcommand{\thethm}{\thechapter.\arabic{thm}} \newcommand{\Ug}{\mc{U}(\fr g)} \newcommand{\Uh}{\mc{U}(\fr h)} \renewcommand{\V}[1]{\mathbf{#1}} \newcommand{\Z}{\mathbb{Z}} \newcommand{\Zp}{\Z/p} \PMlinkescapeword{arrow} \PMlinkescapeword{arrows} \PMlinkescapeword{collection} \PMlinkescapeword{group} \PMlinkescapeword{information} \PMlinkescapeword{interpretation} \PMlinkescapeword{source} A \emph{spectral sequence} is a collection of $R$-modules (or more generally, objects of an abelian category) $\{E^r_{p,q}\}$ for all $r\in\mathbb{N}$, $p$, $q\in\Z$, equipped with maps $d^r_{pq}:E^r_{p,q}\to E^r_{p-r,q+r-1}$ such that \[\xymatrix{ \cdots & E^r_{p-r,q+r-1}\ar[l] & E_{p,q}\ar[l]_(0.35){d^r_{p,q}} & & E^r_{p+r,q-r+1}\ar[ll]_(0.575){d^r_{p+r,q-r+1}} & \ar[l]\cdots }\] is a chain complex, and the $E^{r+1}$'s are its homology, that is, \[ E^{r+1}_{p,q}\cong \mathrm{ker}(d^r_{p,q})/\mathrm{im}({d^r_{p+r,q-r+1}}). \] (Note: what I have defined above is a homology spectral sequence. Cohomology spectral sequences are identical, except that all the arrows go in the other direction.) Most interesting spectral sequences are upper right quadrant, meaning that $E^r_{p,q}=0$ if $p$ or $q<0$. If this is the case then for any $p,q$, both $d^r_{pq}$ and $d^r_{p+r,q-r+1}$ are 0 for sufficiently large $r$ since the target or source is out of the upper right quadrant, so that for all $r>r_0$ $E^r_{p,q}=E^{r+1}_{p,q}\cdots$. This \PMlinkname{group}{Group} is called $E^{\infty}_{p,q}$. A upper right quadrant spectral sequence $\{E^r_{p,q}\}$ is said to converge to a sequence $F_n$ of $R$-modules if there is an exhaustive filtration $F_{n,0}=0\subset F_{n,1}\subset\cdots\subset$ of each $F_n$ such that \[ F_{p+q,q+1}/F_{p+q,q}\cong E^\infty_{p,q}. \] This is typically written $E^r_{p,q}\Rightarrow F_{p+q}$. Typically spectral sequences are used in the following manner: we find an interpretation of $E^r$ for a small value of $r$, typically 1, and of $E^\infty$, and then in cases where enough groups and differentials are $0$, we can obtain information about one from the other.