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  "body": "\n## Statement\n\nLet $p$ be a prime, let $\\mathcal{A}$ be the mod $p$ [[Steenrod algebra|JWTTXSGD0E]], and write $H^*(-)$ for mod $p$ cohomology of [[spectra|1GHF4EKBXS]].\nLet $X$ be a finite CW spectrum, and let $Y$ be a bounded-below spectrum whose homotopy groups are finitely generated in each degree.\nFor $n \\in \\Z$, let $[X, Y]_n$ be the group of maps $\\Sigma^n X \\to Y$ in the [[stable homotopy category|26GTNMAF7B]].\nThen there is a spectral sequence $(E_r^{s,t}, d_r)_{r \\ge 2}$ with $s \\ge 0$.\nIts differentials are $d_r \\colon E_r^{s,t} \\to E_r^{s+r, t+r-1}$.\nIts second page is\n$$E_2^{s,t} = \\operatorname{Ext}_{\\mathcal{A}}^{s,t}\\big(H^*(Y), H^*(X)\\big).$$\nHere $\\operatorname{Ext}^{s,t}$ is the [[Ext group|4P7GBWJ4RR]] of $\\mathcal{A}$-modules in homological degree $s$ and internal degree $t$.\nThe spectral sequence converges strongly to $[X, Y]_{t-s} \\otimes \\Z_p$, where $\\Z_p$ is the ring of $p$-adic integers.\n\n## Remarks\n\nThe construction resolves $Y$ by a tower of spectra built from mod $p$ Eilenberg–MacLane spectra, the Adams resolution, and applies $[X, -]_*$.\nThe filtration degree $s$ counts how many maps in the tower a class passes through.\n\nThe main case is $X = Y = S$, the [[sphere spectrum|14CGGHA85T]].\nThen $E_2^{s,t} = \\operatorname{Ext}_{\\mathcal{A}}^{s,t}(\\F_p, \\F_p)$ and the abutment is the $p$-primary part of the [[stable homotopy groups of spheres|HGZAYPQBXW]] $\\pi_{t-s}^s$ for $t - s \u003e 0$.\nFor $p = 2$, the group $\\operatorname{Ext}^{1,t}$ has a basis of classes $h_j$ with $t = 2^j$, one for each $j \\ge 0$, coming from the indecomposable squares $Sq^{2^j}$.\nThe classes $h_0, h_1, h_2, h_3$ detect $2$, $\\eta$, $\\nu$ and $\\sigma$, while $d_2(h_j) = h_0 h_{j-1}^2 \\neq 0$ for $j \\ge 4$; this is Adams' theorem on Hopf invariant one.\n\nWithout the finiteness hypotheses the spectral sequence still exists, but it may converge only to the homotopy of a $p$-completion of $Y$, or not converge at all.\n\n## Explanations\n\n### L1 human\n\nStable homotopy groups of spheres are very hard to compute directly.\nThe Adams spectral sequence starts from a purely algebraic object, built from the Steenrod operations on mod $p$ cohomology.\nAfter a series of corrections, it gives the $p$-primary part of these groups.\n\n### L2 human\n\nThe $E_2$ term can be computed by machine with minimal resolutions over the Steenrod algebra.\nIn a range of degrees the differentials are known, and this gives the stable stems far beyond what other methods reach.\nIts algebraic structure, such as Massey products and the May spectral sequence for $E_2$, controls much of the computation.\nThe Adams–Novikov spectral sequence replaces mod $p$ cohomology by complex cobordism and is better suited for periodic phenomena.\n",
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