Normal scheme - Citizendia

In mathematics, in the field of algebraic geometry, a normal scheme is a scheme X for which every stalk (local ring)

O_X,x

of its structure sheaf O_X is an integrally closed local ring; that is, each stalk is an integral domain such that its integral closure in its field of fractions is equal to itself. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with In Mathematics, a scheme is an important concept connecting the fields of Algebraic geometry, Commutative algebra and Number theory. In Mathematics, a ringed space is intuitively speaking a space together with a collection of Commutative rings the elements of which are "functions" on In Mathematics, more specifically in Abstract algebra, the concept of integrally closed has two meanings one for groups and one for rings. In Mathematics, more particularly in Abstract algebra, local rings are certain rings that are comparatively simple and serve to describe what is called In Abstract algebra, a branch of Mathematics, an integral domain is a Commutative ring with an additive identity 0 and a multiplicative identity 1 such In Commutative algebra, the notions of an element integral over a ring (also called an algebraic integer over the ring and of an integral extension of In Mathematics, every Integral domain can be embedded in a field; the smallest field which can be used is the field of fractions or field of quotients

Any reduced scheme has a normalization, whose construction we first give for irreducible reduced schemes. This is a glossary of scheme theory. For an introduction to the theory of schemes in Algebraic geometry, see Affine scheme, Projective space, sheaf

An irreducible and reduced scheme $X$ has the property that every affine chart is a domain. In Abstract algebra, a branch of Mathematics, an integral domain is a Commutative ring with an additive identity 0 and a multiplicative identity 1 such Choose an affine cover corresponding to rings $A i$ . Compute the integral closure of each of these in its fraction field, denote them by $\overline{A_i}$ . It is not hard to see that one can construct a new scheme $\overline X$ by gluing together the affine schemes Spec $\overline{A_i}$ .

If the initial scheme is not irreducible, one can define the normalization as the disjoint union of the normalizations of the irreducible components. An alternate, equivalent, definition uses integral closures in rings of fractions where any nonzero divisor is allowed in the denominator.

References

In Commutative algebra, the notions of an element integral over a ring (also called an algebraic integer over the ring and of an integral extension of

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