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A local study of linear electrostatic modes, applicable to the dynamics perpendicular to the magnetic field in a plasma with crossed electric and magnetic fields, is presented. The analysis is based on a two-fluid model that takes into account the finite-electron-gyroradius effects through a rigorous gyroviscosity tensor and includes a dissipative friction force in the electron momentum equation. A comprehensive dispersion relation, valid for arbitrary electron temperature, is derived, which describes properly all the relevant waves for wavelengths longer than the electron Larmor radius. For a homogeneous and dissipationless plasma, such a fluid dispersion relation agrees with the long-wavelength limit of the kinetic electron–cyclotron-drift instability dispersion relation that extends the results to arbitrary short wavelengths. The general fluid dispersion relation covers different parametric regimes that depend on the relative ion-to-electron drift velocity and on the presence of equilibrium inhomogeneities and/or dissipation. Depending on such conditions, its roots yield as follows: two-stream instabilities, driven solely by the relative drift between species; drift-gradient instabilities, driven by the combination of the relative drift and equilibrium gradients; and drift-dissipative instabilities, driven by the combination of the relative drift and friction. Instability thresholds are determined and some distinctive unstable modes are described analytically.