Photon absorption requirements

If I blast a material with photons at a frequency strictly greater than the difference between any two energy levels of all electrons existing in that material, will any of these electrons jump to a higher energy state? Some say the energy match must be exact. Some say the balance moves on as a lower energy photon, which might be eternal if lower than any bottom level Intuitively, the material should heat up. How is this heat manifested? Thank you