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  "body": "\n## Definition\n\nLet $\\mathcal{A}$ be an [[abelian category|Y9CZ3MTNF1]] and let $a \\ge 0$ be an integer.\nA *cohomological spectral sequence* in $\\mathcal{A}$ starting at page $a$ consists of:\n\n1. for every $r \\ge a$ and all integers $p, q$, an object $E_r^{p,q}$ of $\\mathcal{A}$;\n2. for every $r \\ge a$ and all $p, q$, a morphism $d_r^{p,q} \\colon E_r^{p,q} \\to E_r^{p+r, q-r+1}$ such that $d_r^{p+r, q-r+1} \\circ d_r^{p,q} = 0$;\n3. for every $r \\ge a$ and all $p, q$, an isomorphism of $E_{r+1}^{p,q}$ with the [[homology|2J4JJJYP22]] $\\ker d_r^{p,q} / \\im d_r^{p-r, q+r-1}$.\n\nThe family $(E_r^{p,q})_{p,q}$ is the *$r$-th page*.\n\n## Remarks\n\nA spectral sequence is *first quadrant* if $E_a^{p,q} = 0$ unless $p \\ge 0$ and $q \\ge 0$.\nThen for fixed $(p, q)$ the differentials into and out of $E_r^{p,q}$ vanish for $r$ large, so $E_r^{p,q}$ is eventually constant; its value is $E_\\infty^{p,q}$.\nThe sequence *converges* to objects $H^n$, written $E_a^{p,q} \\Rightarrow H^{p+q}$, if each $H^n$ has a finite [[filtration|HNK1Y62G7V]] $H^n = F^0 H^n \\supseteq F^1 H^n \\supseteq \\cdots \\supseteq F^{n+1} H^n = 0$ with $F^p H^n / F^{p+1} H^n \\cong E_\\infty^{p, n-p}$.\nExample: if $E_2^{p,q} = 0$ for $q \\ne 0$, then every $d_r$ with $r \\ge 2$ starts or ends at a zero object, because it changes $q$ by $1 - r \\ne 0$; so $E_2 = E_\\infty$ and $H^n \\cong E_2^{n,0}$.\nNon-example: let $E_2^{0,1} = E_2^{2,0} = \\Z/2\\Z$, all other $E_2^{p,q} = 0$, let $d_2^{0,1}$ be the identity of $\\Z/2\\Z$, and put $E_3 = E_2$.\nClause 3 fails, since the homology of $d_2$ is zero at $(0,1)$ and at $(2,0)$, while $E_3^{0,1} = \\Z/2\\Z$.\nHomological spectral sequences are defined in the same way with $d^r \\colon E^r_{p,q} \\to E^r_{p-r, q+r-1}$.\n",
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