This composition for forming an extreme-ultraviolet (EUV) or electron-beam upper-layer resist film including (a) a polymer (P) and (b) a solvent, the solvent containing 1 to 13 mass % of a C4-12 ketone compound with respect to the entire solvent, is used in the lithography process of a procedure for manufacturing a semiconductor device. Without needing to be intermixed with a resist, and particularly on the occasion of EUV exposure, the composition for forming an EUV or electron-beam upper-layer resist film blocks undesirable exposure light, e.g., ultraviolet (UV) or deep ultraviolet (DUV) rays, and selectively transmits only the EUV rays, and can be developed using a developing solution after exposure.