In algebraic geometry, the Ramanujam–Samuel theorem gives conditions for a
divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a '' multiple'' of m. An integer n is divisible or evenly divisibl ...
of a
local ring
In mathematics, more specifically in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on algebraic varieties or manifolds, or of ...
to be principal.
It was introduced independently by in answer to a question of
Grothendieck
Alexander Grothendieck, later Alexandre Grothendieck in French (; ; ; 28 March 1928 – 13 November 2014), was a German-born French mathematician who became the leading figure in the creation of modern algebraic geometry. His research ext ...
and by
C. P. Ramanujam in an appendix to a paper by , and was generalized by .
Statement
Grothendieck's version of the Ramanujam–Samuel theorem is as follows.
Suppose that ''A'' is a local
Noetherian ring
In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals. If the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noethe ...
with
maximal ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals ...
''m'', whose
completion is
integral
In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
and
integrally closed, and ρ is a local
homomorphism
In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
from ''A'' to a local Noetherian ring ''B'' of larger
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...
such that ''B'' is
formally smooth over ''A'' and the
residue field
In mathematics, the residue field is a basic construction in commutative algebra. If R is a commutative ring and \mathfrak is a maximal ideal, then the residue field is the quotient ring k=R/\mathfrak, which is a field. Frequently, R is a local ri ...
of ''B'' is
finite over that of ''A''. Then a
cycle of
codimension
In mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties.
For affine and projective algebraic varieties, the codimension equals ...
1 in
Spec
The Standard Performance Evaluation Corporation (SPEC) is a non-profit consortium that establishes and maintains standardized benchmarks and performance evaluation tools for new generations of computing systems. SPEC was founded in 1988 and i ...
(''B'') that is principal at the point ''mB'' is principal.
References
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{{DEFAULTSORT:Ramanujam-Samuel theorem
Theorems in algebraic geometry