In
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...
, a quasi-excellent ring is a
Noetherian commutative ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not ...
that behaves well with respect to the operation of
completion, and is called an excellent ring if it is also
universally catenary. Excellent rings are one answer to the problem of finding a natural class of "well-behaved"
rings containing most of the rings that occur in
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...
and
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
. At one time it seemed that the class of Noetherian rings might be an answer to this problem, but
Masayoshi Nagata
Masayoshi Nagata (Japanese: 永田 雅宜 ''Nagata Masayoshi''; February 9, 1927 – August 27, 2008) was a Japanese mathematician, known for his work in the field of commutative algebra.
Work
Nagata's compactification theorem shows that va ...
and others found several strange counterexamples showing that in general Noetherian rings need not be well-behaved: for example, a normal Noetherian
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 algebrai ...
need not be
analytically normal In algebra, an analytically normal ring is a local ring whose completion is a normal ring, in other words a domain that is integrally closed in its quotient field.
proved that if a local ring of an algebraic variety
Algebraic varieties ar ...
.
The class of excellent rings was defined by
Alexander Grothendieck (1965) as a candidate for such a class of well-behaved rings. Quasi-excellent rings are
conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in ...
d to be the base rings for which the problem of
resolution of singularities
In algebraic geometry, the problem of resolution of singularities asks whether every algebraic variety ''V'' has a resolution, a non-singular variety ''W'' with a Proper morphism, proper birational map ''W''→''V''. For varieties over fields ...
can be solved; showed this in
characteristic 0, but the positive characteristic case is (as of 2016) still a major open problem. Essentially all Noetherian rings that occur naturally in algebraic geometry or number theory are excellent; in fact it is quite hard to construct examples of Noetherian rings that are not excellent.
Definitions
The definition of excellent rings is quite involved, so we recall the definitions of the technical conditions it satisfies. Although it seems like a long list of conditions, most rings in practice are excellent, such as
fields,
polynomial ring
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...
s,
complete
Complete may refer to:
Logic
* Completeness (logic)
* Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
* The completeness of the real numbers, which implies t ...
Noetherian rings,
Dedekind domains over characteristic 0 (such as
), and
quotient and
localization rings of these rings.
Recalled definitions
*A ring
containing a field
is called
geometrically regular over
if for any finite extension
of
the ring
is
regular.
*A
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 ...
of rings from
is called regular if it is flat and for every
the fiber
is geometrically regular over the residue field
of
.
*A ring
is called a
G-ring (or Grothendieck ring) if it is Noetherian and its formal fibers are geometrically regular; this means that for any
, the map from the local ring
to its completion is regular in the sense above.
Finally, a ring is
J-2 if any finite type
-algebra
is J-1, meaning the regular subscheme
is open.
Definition of (quasi-)excellence
A ring
is called quasi-excellent if it is a G-ring and J-2 ring. It is called excellent
pg 214 if it is quasi-excellent and
universally catenary. In practice almost all Noetherian rings are universally catenary, so there is little difference between excellent and quasi-excellent rings.
A scheme is called excellent or quasi-excellent if it has a cover by open affine subschemes with the same property, which implies that every open affine subscheme has this property.
Properties
Because an excellent ring
is a G-ring,
it is
Noetherian by definition. Because it is universally catenary, every maximal chain of
prime ideal
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together wi ...
s has the same length. This is useful for studying the dimension theory of such rings because their dimension can be bounded by a fixed maximal chain. In practice, this means infinite-dimensional Noetherian rings
which have an inductive definition of maximal chains of prime ideals, giving an infinite-dimensional ring, cannot be constructed.
Schemes
Given an excellent scheme
and a locally finite type morphism
, then
is excellent
pg 217.
Quasi-excellence
Any quasi-excellent ring is a
Nagata ring.
Any quasi-excellent
reduced local ring is
analytically reduced.
Any quasi-excellent normal local ring is
analytically normal In algebra, an analytically normal ring is a local ring whose completion is a normal ring, in other words a domain that is integrally closed in its quotient field.
proved that if a local ring of an algebraic variety
Algebraic varieties ar ...
.
Examples
Excellent rings
Most naturally occurring commutative rings in number theory or algebraic geometry are excellent. In particular:
*All complete Noetherian local rings, for instance all fields and the ring of
-adic integers, are excellent.
*All Dedekind domains of characteristic are excellent. In particular the ring of
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s is excellent. Dedekind domains over fields of characteristic greater than need not be excellent.
*The rings of
convergent power series
In mathematics, a power series (in one variable) is an infinite series of the form
\sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots
where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...
in a finite number of variables over or are excellent.
*Any localization of an excellent ring is excellent.
*Any finitely generated algebra over an excellent ring is excellent. This includes all polynomial algebras
with
excellent. This means most rings considered in algebraic geometry are excellent.
A J-2 ring that is not a G-ring
Here is an example of a
discrete valuation ring of dimension and characteristic which is but not a -ring and so is not quasi-excellent. If is any field of characteristic with and is the ring of power series such that is finite then the formal fibers of are not all geometrically regular so is not a -ring. It is a ring as all Noetherian local rings of dimension at most are rings. It is also universally catenary as it is a Dedekind domain. Here denotes the image of under the
Frobenius morphism
In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphi ...
.
A G-ring that is not a J-2 ring
Here is an example of a ring that is a G-ring but not a J-2 ring and so not quasi-excellent. If is the
subring
In mathematics, a subring of ''R'' is a subset of a ring that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and which shares the same multiplicative identity as ''R''. For those ...
of the polynomial ring in infinitely many generators generated by the squares and cubes of all generators, and is obtained from by adjoining inverses to all elements not in any of the
ideals generated by some , then is a 1-dimensional Noetherian domain that is not a ring as has a cusp singularity at every closed point, so the set of singular points is not closed, though it is a G-ring.
This ring is also universally catenary, as its localization at every prime ideal is a quotient of a regular ring.
A quasi-excellent ring that is not excellent
Nagata's example of a 2-dimensional Noetherian local ring that is catenary but not universally catenary is a G-ring, and is also a J-2 ring as any local G-ring is a J-2 ring . So it is a quasi-excellent catenary local ring that is not excellent.
Resolution of singularities
Quasi-excellent rings are closely related to the problem of
resolution of singularities
In algebraic geometry, the problem of resolution of singularities asks whether every algebraic variety ''V'' has a resolution, a non-singular variety ''W'' with a Proper morphism, proper birational map ''W''→''V''. For varieties over fields ...
, and this seems to have been Grothendieck's motivation
pg 218 for defining them. Grothendieck (1965) observed that if it is possible to resolve singularities of all complete integral local Noetherian rings, then it is possible to resolve the singularities of all reduced quasi-excellent rings. Hironaka (1964)
proved this for all complete integral Noetherian local rings over a field of characteristic 0, which implies his theorem that all singularities of excellent schemes over a field of characteristic 0 can be resolved. Conversely if it is possible to resolve all singularities of the spectra of all integral finite algebras over a Noetherian ring ''R'' then the ring ''R'' is quasi-excellent.
See also
*
Resolution of singularities
In algebraic geometry, the problem of resolution of singularities asks whether every algebraic variety ''V'' has a resolution, a non-singular variety ''W'' with a Proper morphism, proper birational map ''W''→''V''. For varieties over fields ...
References
*
Alexandre Grothendieck,
Jean Dieudonné
Jean Alexandre Eugène Dieudonné (; 1 July 1906 – 29 November 1992) was a French mathematician, notable for research in abstract algebra, algebraic geometry, and functional analysis, for close involvement with the Nicolas Bourbaki pseudonym ...
''Eléments de géométrie algébrique IV'' Publications Mathématiques de l'IHÉS 24 (1965), section 7
*
*
Heisuke Hironaka ''Resolution of singularities of an algebraic variety over a field of characteristic zero.'' III Annals of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study.
History
The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as th ...
(2) 79 (1964), 109-203; ibid. (2) 79 1964 205-326.
*Hideyuki Matsumura, ''Commutative algebra'' {{ISBN, 0-8053-7026-9, chapter 13.
Algebraic geometry
Commutative algebra