Ringed space
A topological space with a sheaf of rings
. The sheaf
is called the structure sheaf of the ringed space
. It is usually understood that
is a sheaf of associative and commutative rings with a unit element. A pair
is called a morphism from a ringed space
into a ringed space
if
is a continuous mapping and
is a homomorphism of sheaves of rings over
which transfers units in the stalks to units. Ringed spaces and their morphisms constitute a category. Giving a homomorphism
is equivalent to giving a homomorphism
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which transfers unit elements to unit elements.
A ringed space is called a local ringed space if
is a sheaf of local rings (cf. Local ring). In defining a morphism
between local ringed spaces
it is further assumed that for any
, the homomorphism
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is local. Local ringed spaces form a subcategory in the category of all ringed spaces. Another important subcategory is that of ringed spaces over a (fixed) field , i.e. ringed spaces
where
is a sheaf of algebras over
, while the morphisms are compatible with the structure of the algebras.
Examples of ringed spaces.
1) For each topological space there is a corresponding ringed space
, where
is the sheaf of germs of continuous functions on
.
2) For each differentiable manifold (e.g. of class
) there is a corresponding ringed space
, where
is the sheaf of germs of functions of class
on
; moreover, the category of differentiable manifolds is a full subcategory of the category of ringed spaces over
.
3) The analytic manifolds (cf. Analytic manifold) and analytic spaces (cf. Analytic space) over a field constitute full subcategories of the category of ringed spaces over
.
4) Schemes (cf. Scheme) constitute a full subcategory of the category of local ringed spaces.
References
[1] | I.R. Shafarevich, "Basic algebraic geometry" , Springer (1977) (Translated from Russian) MR0447223 Zbl 0362.14001 |
[2] | R. Hartshorne, "Algebraic geometry" , Springer (1977) MR0463157 Zbl 0367.14001 |
Comments
If is a sheaf over a topological space
and
is a mapping of topological spaces, then the induced sheaf
over
is the sheaf defined by
for all open
.
Ringed space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ringed_space&oldid=23966