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The rigorous construction and study of the BRST formalism for the Schwinger model is performed. The BRST transformation and BRST algebra are defined and the kernel of the BRST charge Q is studied. It is proved that our BRST quantization is physically correct, i.e., ker Q is a non-negative subspace (‘‘no-ghost theorem’’); moreover, the space of the BRST cohomologies ker Q/ran Q is one dimensional up to factors on which Q acts trivially. It is found that the well-known Lowenstein–Swieca solution cannot produce the correct BRST quantization (Theorem 1) and so the formalism is based on the Capri–Ferrari solution [Capri and Ferrari, Nuovo Cimento A 62, 273 (1981)] valid in the arbitrary local covariant gauge (although only the Lorentz gauge is considered here for simplicity reasons). The important by-product of our study is the general construction of the BRST–Fock moduli satisfying the no-ghost theorem. The construction is done with the aid of a special kind of the finite-dimensional approximation, the ‘‘N-cell technique.’’