The invention relates to an air cooling and air dehumidifying module comprising plastic capillary tube mats, which are formed by folding and/or winding into a compact assembly with a virtually cuboidal outer shape, which cools and dehumidifies the air stream passed through the mat assembly when cold water is conducted through the capillary tubes. Furthermore, the invention relates to a method of operating the air cooling and air dehumidifying module in combination with cooling ceilings or suspended cooling panels. Such solutions serve for the decentralized cooling of rooms and for dehumidifying the air in a room.