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 divisible by ...
of a
local ring In abstract algebra, more specifically 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 varieties or manifolds, or of algebraic num ...
to be principal.
It was introduced independently by in answer to a question of
Grothendieck and by
C. P. Ramanujam
Chakravarthi Padmanabhan Ramanujam (9 January 1938 – 27 October 1974) was an Indian mathematician who worked in the fields of number theory and algebraic geometry. He was elected a fellow of the Indian Academy of Sciences in 1973.
Like his ...
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-Noether ...
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 cont ...
''m'', whose
completion is
integral
In mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
and
integrally closed, and ρ is a local
homomorphism
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
from ''A'' to a local Noetherian ring ''B'' of larger
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
such that ''B'' is
formally smooth In algebraic geometry, a morphism f:X \to S between schemes is said to be smooth if
*(i) it is locally of finite presentation
*(ii) it is flat, and
*(iii) for every geometric point \overline \to S the fiber X_ = X \times_S is regular.
(iii) mean ...
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 ''m'' is a maximal ideal, then the residue field is the quotient ring ''k'' = ''R''/''m'', which is a field. Frequently, ''R'' is a ...
of ''B'' is
finite over that of ''A''. Then a
cycle
Cycle, cycles, or cyclic may refer to:
Anthropology and social sciences
* Cyclic history, a theory of history
* Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr.
* Social cycle, various cycles in soc ...
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 the ...
1 in
Spec Spec may refer to:
*Specification (technical standard), an explicit set of requirements to be satisfied by a material, product, or service
**datasheet, or "spec sheet"
People
* Spec Harkness (1887-1952), American professional baseball pitcher
...
(''B'') that is principal at the point ''mB'' is principal.
References
*
*
*
{{DEFAULTSORT:Ramanujam-Samuel theorem
Theorems in algebraic geometry