A vanishing line in the BP〈1〉-Adams spectral sequence

Using techniques of relative homological algebra, for an odd prime p, we describe a vanishing line of slope (p2−p−1)−1 in the second term of the BP〈1〉-Adams spectral sequence for the sphere spectrum. As a consequence, the E∞ term of the classical Adams spectral sequence is shown to have a similar line of slope (2p−1)/[(2p−2) (p2−p−1)], above which only the image of the stable J-homomorphism lies. This produces upper bounds for the exponent at p of the stable homotopy groups of spheres.