In the context of schemes, the importance of ideal sheaves lies mainly in the correspondence between closed subschemes and "quasi-coherent" ideal sheaves. Consider a scheme "X" and a quasi-coherent ideal sheaf "J" in O"X". Then, the support "Z" of O"X"/"J" is a closed subspace of "X", and ("Z", O"X"/"J") is a scheme (both assertions can be checked locally). It is called the closed subscheme of "X" defined by "J". Conversely, let "i": "Z" → "X" be a closed immersion, i.e., a morphism which is a homeomorphism onto a closed subspace such that the associated map