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In mathematics, the inverse image functor is a covariant construction of sheaves. The direct image functor is the primary operation on sheaves, with the simplest definition. The inverse image exhibits some relatively subtle features. ==Definition== Suppose given a sheaf on and that we want to transport to using a continuous map . We will call the result the ''inverse image'' or pullback sheaf . If we try to imitate the direct image by setting : for each open set of , we immediately run into a problem: is not necessarily open. The best we could do is to approximate it by open sets, and even then we will get a presheaf and not a sheaf. Consequently, we define to be the sheaf associated to the presheaf: : (Here is an open subset of and the colimit runs over all open subsets of containing .) For example, if is just the inclusion of a point of , then is just the stalk of at this point. The restriction maps, as well as the functoriality of the inverse image follows from the universal property of direct limits. When dealing with morphisms of locally ringed spaces, for example schemes in algebraic geometry, one often works with sheaves of -modules, where is the structure sheaf of . Then the functor is inappropriate, because in general it does not even give sheaves of -modules. In order to remedy this, one defines in this situation for a sheaf of -modules its inverse image by :. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「inverse image functor」の詳細全文を読む スポンサード リンク
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