information:filtering
Ringed spaces and sheaves of modules

In several geometrical disciplines, including algebraic geometry and differential geometry, the spaces come along with a natural sheaf of rings, often called the structure sheaf and denoted by O X {\displaystyle {\mathcal {O}}_{X}}. Such a pair (X, O X) {\displaystyle (X,{\mathcal {O}}_{X})} is called a ringed space. Many types of spaces can be defined as certain types of ringed spaces. Commonly, all the stalks O X, x {\displaystyle {\mathcal {O}}_{X,x}} of the structure sheaf are local rings, in which case the pair is called a locally ringed space.

For example, an n {\displaystyle n} -dimensional C k {\displaystyle C^{k}} manifold M {\displaystyle M} is a locally ringed space whose structure sheaf consists of C k {\displaystyle C^{k}} -functions on the open subsets of M {\displaystyle M}. The property of being a locally ringed space translates into the fact that such a function, which is nonzero at a point x {\displaystyle x}, is also non-zero on a sufficiently small open neighborhood of x {\displaystyle x}. Some authors actually define real (or complex) manifolds to be locally ringed spaces that are locally isomorphic to the pair consisting of an open subset of R n {\displaystyle \mathbb {R} ^{n}} (respectively C n {\displaystyle \mathbb {C} ^{n}}) together with the sheaf of C k {\displaystyle C^{k}} (respectively holomorphic) functions. Similarly, schemes, the foundational notion of spaces in algebraic geometry, are locally ringed spaces that are locally isomorphic to the spectrum of a ring.

Given a ringed space, a sheaf of modules is a sheaf M {\displaystyle {\mathcal {M}}} such that on every open set U {\displaystyle U} of X {\displaystyle X}, M (U) {\displaystyle {\mathcal {M}}(U)} is an O X (U) {\displaystyle {\mathcal {O}}_{X}(U)} -module and for every inclusion of open sets V ⊆ U {\displaystyle V\subseteq U}, the restriction map M (U) → M (V) {\displaystyle {\mathcal {M}}(U)\to {\mathcal {M}}(V)} is compatible with the restriction map O (U) → O (V) {\displaystyle {\mathcal {O}}(U)\to {\mathcal {O}}(V)}: the restriction of fs is the restriction of f {\displaystyle f} times that of s {\displaystyle s} for any f {\displaystyle f} in O (U) {\displaystyle {\mathcal {O}}(U)} and s {\displaystyle s} in M (U) {\displaystyle {\mathcal {M}}(U)}.

Most important geometric objects are sheaves of modules. For example, there is a one-to-one correspondence between vector bundles and locally free sheaves of O X {\displaystyle {\mathcal {O}}_{X}} -modules. This paradigm applies to real vector bundles, complex vector bundles, or vector bundles in algebraic geometry (where O {\displaystyle {\mathcal {O}}} consists of smooth functions, holomorphic functions, or regular functions, respectively). Sheaves of solutions to differential equations are D {\displaystyle D} -modules, that is, modules over the sheaf of differential operators. On any topological space, modules over the constant sheaf Z _ {\displaystyle {\underline {\mathbf {Z} }}} are the same as sheaves of abelian groups in the sense above.

There is a different inverse image functor for sheaves of modules over sheaves of rings. This functor is usually denoted f ∗ {\displaystyle f^{*}} and it is distinct from f − 1 {\displaystyle f^{-1}}. See inverse image functor.