commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideal (ring theory), ideals, and module (mathematics), modules over such rings. Both algebraic geometry and algebraic number theo ...

, a G-ring or Grothendieck ring is a

Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite leng ...

ring (The) Ring(s) may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell Arts, entertainment, and media Film and TV * ''The Ring'' (franchise), a ...

such that the map of any of its

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 ...

s to the completion is regular (defined below). Almost all Noetherian rings that occur naturally in

algebraic geometry Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...

number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

are G-rings, and it is quite hard to construct examples of Noetherian rings that are not G-rings. The concept is named after

Alexander 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 ...

. A ring that is both a G-ring and a

J-2 ring In commutative algebra, a J-0 ring is a ring R such that the set of regular points, that is, points p of the spectrum at which the localization R_p is a regular local ring, contains a non-empty open subset, a J-1 ring is a ring such that the set of ...

is called a

quasi-excellent ring In commutative algebra, a quasi-excellent ring is a Noetherian commutative ring 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 th ...

, and if in addition it is

universally catenary In mathematics, a commutative ring ''R'' is catenary if for any pair of prime ideals ''p'', ''q'', any two strictly increasing chains :''p'' = ''p''0 ⊂ ''p''1 ⊂ ... ⊂ ''p'n'' = ''q'' of prime ideals are contained in maximal strictly ...

it is called an

excellent ring In commutative algebra, a quasi-excellent ring is a Noetherian ring, Noetherian commutative ring that behaves well with respect to the operation of completion of a ring, completion, and is called an excellent ring if it is also universally catenary ...

Definitions

*A (Noetherian) ring ''R'' containing a field ''k'' is called

geometrically regular In algebraic geometry, a geometrically regular ring is a Noetherian ring over a field that remains a regular ring after any finite extension of the base field. Geometrically regular schemes are defined in a similar way. In older terminology, point ...

over ''k'' if for any

finite extension In mathematics, more specifically field theory, the degree of a field extension is a rough measure of the "size" of the field extension. The concept plays an important role in many parts of mathematics, including algebra and number theory—in ...

''K'' of ''k'' the ring ''R'' ⊗_''k'' ''K'' is a

regular ring In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let A be any Noetherian local ring with unique maxi ...

. *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 ''R'' to ''S'' is called regular if it is flat and for every ''p'' ∈ Spec(''R'') the fiber ''S'' ⊗_''R'' ''k''(''p'') is geometrically regular over 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 ...

''k''(''p'') of ''p''. (see also Popescu's theorem.) *A ring is called a local G-ring if it is a Noetherian local ring and the map to its completion (with respect to its

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 ...

) is regular. *A ring is called a G-ring if it is Noetherian and all its localizations at

prime ideal In algebra, a prime ideal is a subset of a ring (mathematics), ring that shares many important properties of a prime number in the ring of Integer#Algebraic properties, integers. The prime ideals for the integers are the sets that contain all th ...

s are local G-rings. (It is enough to check this just for the maximal ideals, so in particular local G-rings are G-rings.)

Examples

*Every field is a G-ring *Every complete Noetherian local ring is a G-ring *Every ring 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 ''a_n'' represents the coefficient of the ''n''th term and ''c'' is a co ...

in a finite number of variables over R or C is a G-ring. *Every

Dedekind domain In mathematics, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily un ...

in characteristic 0, and in particular the ring of

integers An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

, is a G-ring, but in positive characteristic there are Dedekind domains (and even

discrete valuation ring In abstract algebra, a discrete valuation ring (DVR) is a principal ideal domain (PID) with exactly one non-zero maximal ideal. This means a DVR is an integral domain ''R'' that satisfies any and all of the following equivalent conditions: # '' ...

s) that are not G-rings. *Every localization of a G-ring is a G-ring *Every finitely generated

algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...

over a G-ring is a G-ring. This is a

theorem In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to esta ...

due to Grothendieck. Here is an example of a discrete valuation ring ''A'' of characteristic ''p''>0 which is not a G-ring. If ''k'' is any field of characteristic ''p'' with /nowiki>''k'' : ''k''^''p''/nowiki> = ∞ and ''R'' = ''k'' ''x'' and ''A'' is the

subring In mathematics, a subring of a ring is a subset of that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and that shares the same multiplicative identity as .In general, not all s ...

of power series Σ''a_ixⁱ'' such that /nowiki>''k''^''p''(''a''₀,''a''₁,...) : ''k''^''p''/nowiki> is finite then the formal fiber of ''A'' over the generic point is not geometrically regular so ''A'' is not a G-ring. Here ''k''^''p'' denotes the image of ''k'' under the Frobenius morphism ''a''→''a''^''p''.

References

*A. Grothendieck, J. Dieudonné
''Eléments de géométrie algébrique IV''
Publ. Math. IHÉS 24 (1965), section 7 *H. Matsumura, ''Commutative algebra'' , chapter 13. Commutative algebra {{commutative-algebra-stub