Using up to nine different ways to represent homogeneous technologies with decreasing returns to scale, this article presents and proves identities between those different representations of such technologies, outlining the homogeneity properties of each representation. These identities, which allow to shift from one representation of a technology to another -- and which are summarized in a matrix of identities -- can be useful since they provide a tool to obtain explicit functional forms for homogeneous technologies. They can also be useful to simplify computational procedures when different representations of a technology are needed. Finally, the article also refers explicitly to some aspects of producer theory that are often neglected or treated in a marginal way in the literature, such as the inverse supply, the non conditional cost and the inverse input demands functions.