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A critical process in high-brightness photoinjectors is emittance compensation, which brings under control the correlated transverse emittance growth due to the linear space-charge force. Although emittance compensation has been used and studied for almost two decades, the exact criteria to achieve emittance compensation is not as clear as it should be. In this paper, a perturbative analysis of slice envelopes and emittance evolution close to any reference envelope is developed, via which space-charge and chromatic effects are investigated. A new criterion for emittance compensation is found, which is complementary to the well-known matching condition for the invariant envelope and agrees very well with simulations.