aa r X i v : . [ m a t h . A C ] A p r GRADED INTEGRAL CLOSURES
FRED ROHRER
Abstract.
It is investigated how graded variants of integral and complete integralclosures behave under coarsening functors and under formation of group algebras.
Introduction
Let G be a group, let R be a G -graded ring, and let S be a G -graded R -algebra.(Throughout, monoids, groups and rings are understood to be commutative, and algebrasare understood to be commutative, unital and associative.) We study a graded variantof (complete) integral closure, defined as follows: We denote by Int( R, S ) (or CInt(
R, S ),resp.) the G -graded sub- R -algebra of S generated by the homogeneous elements of S thatare (almost) integral over R and call this the (complete) integral closure of R in S . If R isentire (as a G -graded ring, i.e., it has no homogeneous zero-divisors) we consider its gradedfield of fractions Q ( R ), i.e., the G -graded R -algebra obtained by inverting all nonzero homogeneous elements, and then Int( R ) = Int( R, Q ( R )) (or CInt( R ) = CInt( R, Q ( R )),resp.) is called the (complete) integral closure of R . These constructions behave similarto their ungraded relatives, as long as we stay in the category of G -graded rings. Butthe relation between these constructions and their ungraded relatives, and more generallytheir behaviour under coarsening functors, is less clear; it is the main object of study inthe following.For an epimorphism of groups ψ : G ։ H we denote by • [ ψ ] the ψ -coarsening functorfrom the category of G -graded rings to the category of H -graded rings. We ask forconditions ensuring that ψ -coarsening commutes with relative (complete) integral closure,i.e., Int( R, S ) [ ψ ] = Int( R [ ψ ] , S [ ψ ] ) or CInt( R, S ) [ ψ ] = CInt( R [ ψ ] , S [ ψ ] ), or – if R and R [ ψ ] areentire – that ψ -coarsening commutes with (complete) integral closure, i.e., Int( R ) [ ψ ] =Int( R [ ψ ] ) or CInt( R ) [ ψ ] = CInt( R [ ψ ] ). Complete integral closure being a more delicatenotion than integral closure, it is not astonishing that the questions concerning the formerare harder than the other ones. Furthermore, the case of integral closures of entire gradedrings in their graded fields of fractions turns out to be more complicated than the relativecase, because Q ( R ) [ ψ ] almost never equals Q ( R [ ψ ] ), hence in addition to the coarseningwe also change the overring in which we form the closure.The special case H = 0 of parts of these questions was already studied by severalauthors. Bourbaki ([3, V.1.8]) treats torsionfree groups G , Van Geel and Van Oystaeyen Mathematics Subject Classification.
Primary 13B22; Secondary 13A02, 16S34.
Key words and phrases.
Graded ring, coarsening, integral closure, integrally closed ring, completeintegral closure, completely integrally closed ring, group algebra.The author was supported by the Swiss National Science Foundation. ([13]) consider G = Z , and Swanson and Huneke ([11, 2.3]) discuss the case that G is offinite type. Our main results, generalising these partial results, are as follows. Theorem 1
Let ψ : G ։ H be an epimorphism of groups and let R be a G -graded ring.a) If Ker( ψ ) is contained in a torsionfree direct summand of G then ψ -coarseningcommutes with relative (complete) integral closure.b) Suppose that R is entire. If G is torsionfree, or if Ker( ψ ) is contained in a torsion-free direct summand of G and the degree support of R generates G , then ψ -coarseningcommutes with integral closure. The questions above are closely related to the question of how (complete) integral clo-sure behaves under formation of group algebras. If F is a group, there is a canonical G ⊕ F -graduation on the algebra of F over R ; we denote the resulting G ⊕ F -gradedring by R [ F ]. We ask for conditions ensuring that formation of graded group algebrascommutes with relative (complete) integral closure, i.e., Int( R, S )[ F ] = Int( R [ F ] , S [ F ])or CInt( R, S )[ F ] = CInt( R [ F ] , S [ F ]), or – if R is entire – that formation of gradedgroup algebras commutes with (complete) integral closure, i.e., Int( R )[ F ] = Int( R [ F ])or CInt( R )[ F ] = CInt( R [ F ]). Our main results are the following. Theorem 2
Let G be a group and let R be a G -graded ring. Formation of graded groupalgebras over R commutes with relative (complete) integral closure. If R is entire thenformation of graded group algebras over R commutes with (complete) integral closure. It is maybe more interesting to consider a coarser graduation on group algebras, namelythe G -graduation obtained from R [ F ] by coarsening with respect to the canonical projec-tion G ⊕ F ։ G ; we denote the resulting G -graded R -algebra by R [ F ] [ G ] and call it thecoarsely graded algebra of F over R . We ask for conditions ensuring that formation ofcoarsely graded group algebras commutes with relative (complete) integral closure, i.e.,Int( R, S )[ F ] [ G ] = Int( R [ F ] [ G ] , S [ F ] [ G ] ) or CInt( R, S )[ F ] [ G ] = CInt( R [ F ] [ G ] , S [ F ] [ G ] ), or – if R and R [ F ] [ G ] are entire – that formation of coarsely graded group algebras commuteswith (complete) integral closure, i.e., Int( R )[ F ] [ G ] = Int( R [ F ] [ G ] ) or CInt( R )[ F ] [ G ] =CInt( R [ F ] [ G ] ). Ungraded variants of these questions (i.e., for G = 0) for a torsionfreegroup F were studied extensively by Gilmer ([6, § Theorem 3
Let G and F be groups and let R be a G -graded ring. Formation of thecoarsely graded group algebra of F over R commutes with relative (complete) integralclosure if and only if F is torsionfree. If R is entire and F is torsionfree then formationof the coarsely graded group algebra of F over R commutes with integral closure. Some preliminaries on graded rings, coarsening functors, and algebras of groups arecollected in Section 1. Relative (complete) integral closures are treated in Section 2,and (complete) integral closures of entire graded rings in their graded fields of fractionsare treated in Section 3. Our notation and terminology follows Bourbaki’s ´El´ements demath´ematique.
RADED INTEGRAL CLOSURES 3
Before we start, a remark on notation and terminology may be appropriate. Since we tryto never omit coarsening functors (and in particular forgetful functors) from our notationsit seems conceptually better and in accordance with the general yoga of coarsening to notfurnish names of properties of G -graded rings or symbols denoting objects constructedfrom G -graded rings with additional symbols that highlight the dependence on G or onthe graded structure. For example, if R is a G -graded ring then we will denote by Nzd( R )(instead of, e.g., Nzd G ( R )) the monoid of its homogeneous non-zerodivisors, and we call R entire (instead of, e.g., G -entire) if Nzd( R ) consists of all homogeneous elements of R different from 0. Keeping in mind that in this setting the symbol “ R ” always denotesa G -graded ring (and never, e.g., its underlying ungraded ring), this should not lead toconfusions (whereas mixing up different categories might do so). Throughout the following let G be a group. Preliminaries on graded rings
First we recall our terminology for graded rings and coarsening functors. (1.1)
By a G -graded ring we mean a pair ( R, ( R g ) g ∈ G ) consisting of a ring R and afamily ( R g ) g ∈ G of subgroups of the additive group of R whose direct sum equals theadditive group of R such that R g R h ⊆ R g + h for g, h ∈ G . If no confusion can arise wedenote a G -graded ring ( R, ( R g ) g ∈ G ) just by R . Accordingly, for a G -graded ring R and g ∈ G we denote by R g the component of degree g of R . We set R hom := S g ∈ G R g and calldegsupp( R ) := { g ∈ G | R g = 0 } the degree support of R . We say that R has full supportif degsupp( R ) = G and that R is trivially G -graded if degsupp( R ) = { } . Given G -gradedrings R and S , by a morphism of G -graded rings from R to S we mean a morphism ofrings u : R → S such that u ( R g ) ⊆ S g for g ∈ G . By a G -graded R -algebra we meana G -graded ring S together with a morphism of G -graded rings R → S . We denote by GrAnn G the category of G -graded rings with this notion of morphism. This category hasinductive and projective limits. In case G = 0 we canonically identify GrAnn G with thecategory of rings. (1.2) Let ψ : G ։ H be an epimorphism of groups. For a G -graded ring R we define an H -graded ring R [ ψ ] , called the ψ -coarsening of R ; its underlying ring is the ring underlying R , and its H -graduation is given by ( R [ ψ ] ) h = L g ∈ ψ − ( h ) R g for h ∈ H . A morphism u : R → S of G -graded rings can be considered as a morphism of H -graded rings R [ ψ ] → S [ ψ ] , and as such it is denoted by u [ ψ ] . This gives rise to a functor • [ ψ ] : GrAnn G → GrAnn H .This functor has a right adjoint, hence commutes with inductive limits, and it has a leftadjoint if and only if Ker( ψ ) is finite ([10, 1.6; 1.8]). For a further epimorphism of groups ϕ : H ։ K we have • [ ϕ ◦ ψ ] = • [ ϕ ] ◦ • [ ψ ] . (1.3) We denote by F G the set of subgroups of finite type of G , ordered by inclusion, sothat G = lim −→ U ∈ F G U . FRED ROHRER (1.4)
Let F ⊆ G be a subgroup. For a G -graded ring R we define an F -graded ring R ( F ) with underlying ring the subring L g ∈ F R g ⊆ R and with F -graduation ( R g ) g ∈ F . For an F -graded ring S we define a G -graded ring S ( G ) with underlying ring the ring underlying S and with G -graduation given by S ( G ) g = S g for g ∈ F and S ( G ) g = 0 for g ∈ G \ F . If R is a G -graded ring and F is a set of subgroups of G , ordered by inclusion, whose inductivelimit is G , then R = lim −→ F ∈ F (( R ( F ) ) ( G ) ).The next remark recalls the two different notions of graded group algebras and, moregeneral, of graded monoid algebras. (1.5) Let M be a cancellable monoid, let F be its group of differences, and let R be a G -graded ring. The algebra of M over R , furnished with its canonical G ⊕ F -graduation,is denoted by R [ M ] and called the finely graded algebra of M over R , and we denote by( e f ) f ∈ F its canonical basis. So, for ( g, f ) ∈ G ⊕ F we have R [ M ] ( g,f ) = R g e f . Denotingby π : G ⊕ F ։ G the canonical projection we set R [ M ] [ G ] := R [ M ] [ π ] and call this the coarsely graded algebra of M over R . If S is a G -graded R -algebra then S [ M ] isa G ⊕ F -graded R [ M ]-algebra, and S [ M ] [ G ] is a G -graded R [ M ] [ G ] -algebra. We have R [ F ] = lim −→ U ∈ F F R [ U ] ( G ⊕ F ) and R [ F ] [ G ] = lim −→ U ∈ F F R [ U ] [ G ] (1.3).We will need some facts about graded variants of simplicity (i.e., the property of “beinga field”) and entirety. Although they are probably well-known, we provide proofs for thereaders convenience. Following Lang we use the term “entire” instead of “integral” (toavoid confusion with the notion of integrality over some ring which is central in thisarticle) or “domain” (to avoid questions as whether a “graded domain” is the same asa “domain furnished with a graduation”), and we use the term “simple” (which is morecommon in noncommutative algebra) instead of “field” for similar reasons. (1.6) Let R be a G -graded ring. We denote by R ∗ the multiplicative group of invertiblehomogeneous elements of R and by Nzd( R ) the multiplicative monoid of homogeneousnon-zerodivisors of R . We call R simple if R ∗ = R hom \
0, and entire if Nzd( R ) = R hom \ R is entire then the G -graded ring of fractions Nzd( R ) − R is simple; we denote it by Q ( R ) and call it the (graded) field of fractions of R . If ψ : G ։ H is an epimorphism ofgroups and R [ ψ ] is simple or entire, then R is so. Let F ⊆ G be a subgroup. If R is simpleor entire, then R ( F ) is so, and an F -graded ring S is simple or entire if and only if S ( G ) isso. (1.7) Let I be a nonempty right filtering preordered set, and let (( R i ) i ∈ I , ( ϕ ij ) i ≤ j ) be aninductive system in GrAnn G over I . Analogously to [2, I.10.3 Proposition 3] we see that if R i is simple or entire for every i ∈ I , then lim −→ i ∈ I R i is simple or entire. If R i is entire forevery i ∈ I and ϕ ij is a monomorphism for all i, j ∈ I with i ≤ j , then by [8, 0.6.1.5] weget an inductive system ( Q ( R i )) i ∈ I in GrAnn G over I with lim −→ i ∈ I Q ( R i ) = Q (lim −→ i ∈ I R i ). (1.8) Let F ⊆ G be a subgroup and let ≤ be an ordering on F that is compatiblewith its structure of group. The relation “ g − h ∈ F ≤ ” is the finest ordering on G thatis compatible with its structure of group and induces ≤ on F ; we call it the canonical RADED INTEGRAL CLOSURES 5 extension of ≤ to G . If ≤ is a total ordering then its canonical extension to G induces atotal ordering on every equivalence class of G modulo F . (1.9) Lemma Let ψ : G ։ H be an epimorphism of groups such that Ker( ψ ) is tor-sionfree, let R be an entire G -graded ring, and let x, y ∈ R hom[ ψ ] \ with xy ∈ R hom . Then, x, y ∈ R hom and xy = 0 .Proof. (cf. [2, II.11.4 Proposition 8]) By [2, II.11.4 Lemme 1] we can choose a total orderingon Ker( ψ ) that is compatible with its structure of group. Let ≤ denote its canonicalextension to G (1.8). Let h := deg( x ) and h ′ := deg( y ). There exist strictly increasingfinite sequences ( g i ) ni =0 in ψ − ( h ) and ( g ′ j ) mj =0 in ψ − ( h ′ ), x i ∈ R g i \ i ∈ [0 , n ], and y j ∈ R g ′ j \ j ∈ [0 , m ] such that x = P ni =0 x i and y = P mj =0 y j . If k ∈ [0 , n ] and l ∈ [0 , m ] with g k + g ′ l = g n + g ′ m then k = n and l = m by [2, VI.1.1 Proposition 1]. Thisimplies that the component of xy of degree g n + g ′ m equals x n y m = 0, so that xy = 0. As x y = 0 and xy ∈ R hom it follows g + g ′ = g n + g ′ m , hence n = m = 0 and therefore x, y ∈ R hom . (cid:3) (1.10) Corollary Let ψ : G ։ H be an epimorphism of groups such that Ker( ψ ) istorsionfree, and let R be an entire G -graded ring. Then, R ∗ = R ∗ [ ψ ] .Proof. Clearly, R ∗ ⊆ ( R [ ψ ] ) ∗ . If x ∈ ( R [ ψ ] ) ∗ ⊆ R hom[ ψ ] \ y ∈ R hom[ ψ ] \ xy = 1 ∈ R hom , so 1.9 implies x ∈ R hom , hence x ∈ R ∗ . (cid:3) (1.11) Let R be a G -graded ring and let F be a group. It is readily checked that R [ F ]is simple or entire if and only if R is so. Analogously to [6, 8.1] it is seen that R [ F ] [ G ] isentire if and only if R is entire and F is torsionfree. Together with 1.10 it follows that R [ F ] [ G ] is simple if and only if R is simple and F = 0. (1.12) Proposition Let ψ : G ։ H be an epimorphism of groups.a) The following statements are equivalent: (i) ψ is an isomorphism; (ii) ψ -coarseningpreserves simplicity.b) The following statements are equivalent: (i) Ker( ψ ) is torsionfree; (ii) ψ -coarseningpreserves entirety; (iii) ψ -coarsening maps simple G -graded rings to entire H -gradedrings. Proof. If K is a field and R = K [Ker( ψ )] ( G ) , then R is simple and R [ ψ ] is trivially H -graded, hence R [ ψ ] is simple or entire if and only if R [0] is so (1.11, 1.6). If Ker( ψ ) = 0then R [0] is not simple, and if Ker( ψ ) is not torsionfree then R [0] is not entire (1.11). Thisproves a) and the implication (iii) ⇒ (i) in b). The remaining claims follow from 1.9. (cid:3) (1.13) A G -graded ring R is called reduced if 0 is its only nilpotent homogeneous element.With arguments similar to those above one can show that statements (i)–(iii) in 1.12 b) arealso equivalent to the following: (iv) ψ -coarsening preserves reducedness; (v) ψ -coarseningmaps simple G -graded rings to reduced H -graded rings. We will make no use of this fact. In case H = 0 the implication (i) ⇒ (ii) is [2, II.11.4 Proposition 8]. FRED ROHRER
Finally we make some remarks on a graded variant of noetherianness. (1.14)
Let R be a G -graded ring. We call R noetherian if ascending sequences of G -graded ideals of R are stationary, or – equivalently – if every G -graded ideal of R is offinite type. Analogously to the ungraded case one can prove a graded version of Hilbert’sBasissatz: If R is noetherian then so are G -graded R -algebras of finite type. If ψ : G ։ H is an epimorphism of groups and R [ ψ ] is noetherian, then R is noetherian. Let F ⊆ G bea subgroup. It follows from [4, 2.1] that if R is noetherian then so is R ( F ) . Moreover, an F -graded ring S is noetherian if and only if S ( G ) is so.If F is a group then it follows from [4, 2.1] and the fact that e f ∈ R [ F ] ∗ for f ∈ F that R [ F ] is noetherian if and only if R is so. Analogously to [6, 7.7] one sees that R [ F ] [ G ] isnoetherian if and only if R is noetherian and F is of finite type. More general, it followsreadily from a result by Goto and Yamagishi ([4, 1.1]) that G is of finite type if and only if ψ -coarsening preserves noetherianness for every epimorphism of groups ψ : G ։ H . (Thiswas proven again two years later by Nˇastˇasescu and Van Oystaeyen ([9, 2.1]).)2. Relative integral closures
We begin this section with basic definitions and first properties of relative (complete)integral closures. (2.1)
Let R be a G -graded ring and let S be a G -graded R -algebra. An element x ∈ S hom is called integral over R if it is a zero of a monic polynomial in one indeterminate withcoefficients in R hom . This is the case if and only if x , considered as an element of S [0] , isintegral over R [0] , as is seen analogously to the first paragraph of [3, V.1.8]. Hence, using[3, V.1.1 Th´eor`eme 1] we see that for x ∈ S hom the following statements are equivalent:(i) x is integral over R ; (ii) the G -graded R -module underlying the G -graded R -algebra R [ x ] is of finite type; (iii) there exists a G -graded sub- R -algebra of S containing R [ x ]whose underlying G -graded R -module is of finite type.An element x ∈ S hom is called almost integral over R if there exists a G -graded sub- R -module T ⊆ S of finite type containing R [ x ]. This is the case if and only if x , consideredas an element of S [0] , is almost integral over R [0] . Indeed, this condition is obviouslynecessary. It is also sufficient, for if T ⊆ S [0] is a sub- R [0] -module of finite type containing R [0] [ x ] then the G -graded sub- R -module T ′ ⊆ S generated by the set of homogeneouscomponents of elements of T is of finite type and contains T , hence R [ x ]. It follows fromthe first paragraph that if x ∈ S hom is integral over R then it is almost integral over R ;analogously to [3, V.1.1 Proposition 1 Corollaire] it is seen that the converse is true if R is noetherian (1.14). (2.2) Let R be a G -graded ring and let S be a G -graded R -algebra. The G -gradedsub- R -algebra of S generated by the set of elements of S hom that are (almost) integralover R is denoted by Int( R, S ) (or CInt(
R, S ), resp.) and is called the (complete) integralclosure of R in S . We have Int(
R, S ) ⊆ CInt(
R, S ), with equality if R is noetherian(2.1). For an epimorphism of groups ψ : G ։ H we have Int( R, S ) [ ψ ] ⊆ Int( R [ ψ ] , S [ ψ ] ) andCInt( R, S ) [ ψ ] ⊆ CInt( R [ ψ ] , S [ ψ ] ) (2.1). RADED INTEGRAL CLOSURES 7
Let R ′ denote the image of R in S . We say that R is (completely) integrally closed in S if R ′ = Int( R, S ) (or R ′ = CInt( R, S ), resp.), and that S is (almost) integral over R ifInt( R, S ) = S (or CInt( R, S ) = S , resp.). If R is completely integrally closed in S thenit is integrally closed in S , and if S is integral over R then it is almost integral over R ;the converse statements are true if R is noetherian. If ψ : G ։ H is an epimorphism ofgroups, then S is (almost) integral over R if and only if S [ ψ ] is (almost) integral over R [ ψ ] ,and if R [ ψ ] is (completely) integrally closed in S [ ψ ] then R is (completely) integrally closedin S . If G ⊆ F is a subgroup then Int( R, S ) ( F ) = Int( R ( F ) , S ( F ) ) and CInt( R, S ) ( F ) =CInt( R ( F ) , S ( F ) ), hence R is (completely) integrally closed in S if and only if R ( F ) is(completely) integrally closed in S ( F ) .From [3, V.1.1 Proposition 4 Corollaire 1] and [12, § we know that sums andproducts of elements of S [0] that are (almost) integral over R [0] are again (almost) integralover R [0] . Hence, Int( R, S ) hom (or CInt( R, S ) hom , resp.) equals the set of homogeneouselements of S that are (almost) integral over R , and thus Int( R, S ) (or CInt(
R, S ), resp.)is (almost) integral over R by the above. Moreover, Int( R, S ) is integrally closed in S by[3, V.1.2 Proposition 7]. One should note that CInt( R, S ) is not necessarily completelyintegrally closed in S , not even if R is entire and S = Q ( R ) ([7, Example 1]). (2.3) Suppose we have a commutative diagram of G -graded rings R / / (cid:15) (cid:15) S h (cid:15) (cid:15) R / / S. If x ∈ S hom is (almost) integral over R , then h ( x ) ∈ S hom is (almost) integral over R (2.1, [3, V.1.1 Proposition 2], [5, 13.5]). Hence, if the diagram above is cartesian and R is (completely) integrally closed in S , then R is (completely) integrally closed in S . (2.4) Let R be a G -graded ring, let S be a G -graded R -algebra, and let T ⊆ R hom be a subset. Analogously to [3, V.1.5 Proposition 16] one shows that T − Int(
R, S ) =Int( T − R, T − S ). Hence, if R is integrally closed in S then T − R is integrally closed in T − S .Note that there is no analogous statement for complete integral closures. Although T − CInt(
R, S ) ⊆ CInt( T − R, T − S ) by 2.3, this need not be an equality. In fact, by [3,V.1 Exercice 12] there exists an entire ring R that is completely integrally closed in Q ( R )and a subset T ⊆ R \ Q ( R ) is the complete integral closure of T − R . (2.5) Let I be a right filtering preordered set, and let ( u i ) i ∈ I : ( R i ) i ∈ I → ( S i ) i ∈ I be amorphism of inductive systems in GrAnn G over I . By 2.3 we have inductive systems(Int( R i , S i )) i ∈ I and (CInt( R i , S i )) i ∈ I in GrAnn G over I , and we can consider the sub-lim −→ i ∈ I R i -algebras lim −→ i ∈ I Int( R i , S i ) ⊆ lim −→ i ∈ I CInt( R i , S i ) ⊆ lim −→ i ∈ I S i Note that van der Waerden calls “integral” what we call “almost integral”.
FRED ROHRER and compare them with the sub-lim −→ i ∈ I R i -algebrasInt(lim −→ i ∈ I R i , lim −→ i ∈ I S i ) ⊆ CInt(lim −→ i ∈ I R i , lim −→ i ∈ I S i ) ⊆ lim −→ i ∈ I S i . Analogously to [8, 0.6.5.12] it follows lim −→ i ∈ I Int( R i , S i ) = Int(lim −→ i ∈ I R i , lim −→ i ∈ I S i ), hence if R i is integrally closed in S i for every i ∈ I then lim −→ i ∈ I R i is integrally closed in lim −→ i ∈ I S i .Note that there is no analogous statement for complete integral closures. Althoughlim −→ i ∈ I CInt( R i , S i ) ⊆ CInt(lim −→ i ∈ I R i , lim −→ i ∈ I S i ) by 2.3, this need not be an equality (butcf. 3.2). In fact, by [3, V.1 Exercice 11 b)] there exist a field K and an increasing family( R n ) n ∈ N of subrings of K such that R n is completely integrally closed in Q ( R n ) = K forevery n ∈ N and that lim −→ n ∈ N R n is not completely integrally closed in Q (lim −→ n ∈ N R n ) = K .Now we turn to the behaviour of finely and coarsely graded group algebras with respectto relative (complete) integral closures. (2.6) Theorem Let R be a G -graded ring.a) Formation of finely graded group algebras over R commutes with relative (complete)integral closure.b) Let S be a G -graded R -algebra, and let F be a group. Then, R is (completely)integrally closed in S if and only if R [ F ] is (completely) integrally closed in S [ F ] .Proof. a) Let F be a group, and let S be a G -graded R -algebra. Let x ∈ S [ F ] hom . Thereare s ∈ S hom and f ∈ F with x = se f . If x ∈ Int(
R, S )[ F ] hom then s ∈ Int(
R, S ) hom ,hence s ∈ Int( R [ F ] , S [ F ]) (2.3), and as e f ∈ Int( R [ F ] , S [ F ]) it follows x = se f ∈ Int( R [ F ] , S [ F ]). This shows Int( R, S )[ F ] ⊆ Int( R [ F ] , S [ F ]). Conversely, suppose that x ∈ Int( R [ F ] , S [ F ]) hom . As e f ∈ R [ F ] ∗ it follows s ∈ Int( R [ F ] , S [ F ]) hom . So, there is afinite subset E ⊆ S [ F ] hom such that the G ⊕ F -graded sub- R [ F ]-algebra of S [ F ] generatedby E contains R [ F ][ s ]. As e h ∈ R [ F ] ∗ for every h ∈ F we can suppose E ⊆ S . If n ∈ N then s n is an R [ F ]-linear combination of products in E , and comparing the coefficientsof e shows that s n is an R -linear combination of products in E . Thus, R [ s ] is containedin the G -graded sub- R -algebra of S generated by E , hence s ∈ Int(
R, S ), and therefore x ∈ Int(
R, S )[ F ]. This shows Int( R [ F ] , S [ F ]) ⊆ Int(
R, S )[ F ]. The claim for completeintegral closures follows analogously. b) follows immediately from a). (cid:3) (2.7) Theorem Let F be a group. The following statements are equivalent: (i) Formation of coarsely graded algebras of F over G -graded rings commutes with rel-ative integral closure;(i’) Formation of coarsely graded algebras of F over G -graded rings commutes with rel-ative complete integral closure;(ii) If R is a G -graded ring and S is a G -graded R -algebra, then R is integrally closedin S if and only if R [ F ] [ G ] is integrally closed in S [ F ] [ G ] ;(ii’) If R is a G -graded ring and S is a G -graded R -algebra, then R is completely integrallyclosed in S if and only if R [ F ] [ G ] is completely integrally closed in S [ F ] [ G ] ; In case G = 0 the implication (iii) ⇒ (i) is [3, V.1 Exercice 24]. RADED INTEGRAL CLOSURES 9 (iii) F is torsionfree.Proof. “(i) ⇒ (ii)” and “(i’) ⇒ (ii’)”: Immediately from 2.3.“(ii) ⇒ (iii)” and “(ii’) ⇒ (iii)”: Suppose that F is not torsionfree. It suffices to find anoetherian ring R and an R -algebra S such that R is integrally closed in S and that R [ F ] [0] is not integrally closed in S [ F ] [0] , for then furnishing R and S with trivial G -graduationsit follows that R [ F ] [ G ] is not integrally closed in S [ F ] [ G ] (2.2). The ring Z is noetherianand integrally closed in Q . By hypothesis there exist g ∈ F \ n ∈ N > with ng = 0,so that e ng = 1 ∈ Q [ F ] [0] . It is readily checked that f := P n − i =0 1 n e ig ∈ Q [ F ] [0] \ Z [ F ] [0] isidempotent. Setting c := 1 + ( n − e n − g ∈ Z [ F ] [0] we get d := f c = f + ( n − f e n − g = f + P n − i =0 n − n e i + n − g = f + P n − i =0 n − n e i − g = f + n − n e − g + P n − i =0 n − n e ig + n − n e n − g − n − n e n − g = f + ( n − f = nf ∈ Z [ F ] [0] . Therefore, f + ( c − f − d = f + d − f − d = 0 yields an integral equation for f over Z [ F ] [0] . Thus, Z [ F ] [0] is not integrally closed in Q [ F ] [0] .“(iii) ⇒ (i)” and “(iii) ⇒ (i’)”: Without loss of generality suppose G = 0 (2.2). Sup-pose that F is torsionfree, let R be a ring, and let S be an R -algebra. If n ∈ N thenInt( R, S )[ N n ] [0] = Int( R [ N n ] [0] , S [ N n ] [0] ) ([3, V.1.3 Proposition 12]), henceInt( R [ Z n ] [0] , S [ Z n ] [0] ) = Int( R, S )[ Z n ] [0] (2.4). This proves (i) in case F is of finite type, and so we get (i) in general by 1.5, 2.2and 2.5. It remains to show (i’). The inclusion CInt( R, S )[ F ] [0] ⊆ CInt( R [ F ] [0] , S [ F ] [0] )follows immediately from 2.2 and 2.3. We prove the converse inclusion analogously to [7,Proposition 1]. Since F is torsionfree we can choose a total ordering ≤ on F that is com-patible with its structure of group ([2, II.11.4 Lemme 1]). Let x ∈ CInt( S [ F ] [0] , R [ F ] [0] ).There are n ∈ N , a family ( x i ) ni =1 in S \
0, and a strictly increasing family ( f i ) ni =1 in F with x = P ni =1 x i e f i . We prove by induction on n that x ∈ CInt(
R, S )[ F ] [0] . If n = 0 thisis clear. Suppose that n > n .There exists a finite subset P ⊆ S [ F ] [0] with R [ F ] [0] [ x ] ⊆ h P i R [ F ] . Let Q denote the finiteset of coefficients of elements of P . Let k ∈ N . There exists a family ( s p ) p ∈ P in R [ F ] with x k = P p ∈ P s p p . By means of the ordering of F we see that x kn is the coefficient of e f kn in x k , hence the coefficient of e f kn in P p ∈ P s p p . This latter being an R -linear combinationof Q we get x kn ∈ h Q i R . It follows R [ x n ] ⊆ h Q i R , and thus x n ∈ CInt(
R, S ). So, weget x n e f n ∈ CInt( R [ F ] [0] , S [ F ] [0] ) (2.2, 2.3), hence x − x n e f n ∈ CInt( R [ F ] [0] , S [ F ] [0] ), thus x − x n e f n ∈ CInt(
R, S )[ F ] [0] by our hypothesis, and therefore x ∈ CInt(
R, S )[ F ] [0] asdesired. (cid:3) (2.8) The proof above shows that 2.7 remains true if we replace “If R is a G -gradedring” by “If R is a noetherian G -graded ring” in (ii) and (ii’).The rest of this section is devoted to the study of the behaviour of relative (complete)integral closures under arbitrary coarsening functors. Although we are not able to charac-terise those coarsenings with good behaviour, we identify two properties of the coarsening, one that implies good behaviour of (complete) integral closures, and one that is impliedby good behaviour of (complete) integral closures. (2.9) Let ψ : G ։ H be an epimorphism of groups. We say that ψ -coarsening commuteswith relative (complete) integral closure if Int( R, S ) [ ψ ] = Int( R [ ψ ] , S [ ψ ] ) (or CInt( R, S ) [ ψ ] =CInt( R [ ψ ] , S [ ψ ] ), resp.) for every G -graded ring R and every G -graded R -algebra S . (2.10) Proposition Let ψ : G ։ H be an epimorphism of groups. We consider thefollowing statements:(1) ψ -coarsening commutes with relative integral closure;(1’) ψ -coarsening commutes with relative complete integral closure;(2) If R is a G -graded ring, S is a G -graded R -algebra, and x ∈ S hom[ ψ ] , then x is in-tegral over R [ ψ ] if and only if all its homogeneous components (with respect to the G -graduation) are integral over R ;(2’) If R is a G -graded ring, S is a G -graded R -algebra, and x ∈ S hom[ ψ ] , then x is almostintegral over R [ ψ ] if and only if all its homogeneous components (with respect to the G -graduation) are almost integral over R ;(3) If R is a G -graded ring and S is a G -graded R -algebra, then R is integrally closed in S if and only if R [ ψ ] is integrally closed in S [ ψ ] .(3’) If R is a G -graded ring and S is a G -graded R -algebra, then R is completely integrallyclosed in S if and only if R [ ψ ] is completely integrally closed in S [ ψ ] .Then, we have (1) ⇔ (2) ⇔ (3) and (1’) ⇔ (2’) ⇒ (3’).Proof. The implications “(1) ⇔ (2) ⇒ (3)” and “(1’) ⇔ (2’) ⇒ (3’)” follow immediately fromthe definitions. Suppose (3) is true, let R be a G -graded ring R , and let S be a G -graded R -algebra. As Int( R, S ) is integrally closed in S (2.2) it follows that Int( R, S ) [ ψ ] is integrally closed in S [ ψ ] , implyingInt( R [ ψ ] , S [ ψ ] ) ⊆ Int(Int(
R, S ) [ ψ ] , S [ ψ ] ) = Int( R, S ) [ ψ ] ⊆ Int( R [ ψ ] , S [ ψ ] )(2.2) and thus the claim. (cid:3) The argument above showing that (3) implies (1) cannot be used to show that (3’)implies (1’), as CInt(
R, S ) is not necessarily completely integrally closed in S (2.2). (2.11) Let ψ : G ։ H be an epimorphism of groups, suppose that there exists a section π : H → G of ψ in the category of groups, and let R be a G -graded ring. For g ∈ G thereis a morphism of groups j πR,g : R g → R [0] [Ker( ψ )] , x xe g + π ( ψ ( g )) . The family ( j πR,g ) g ∈ G induces a morphism of groups j πR : L g ∈ G R g → R [0] [Ker( ψ )] [0] thatis readily checked to be a morphism j πR : R [ ψ ] → R [ ψ ] [Ker( ψ )] [ H ] of H -graded rings. (2.12) Theorem Let ψ : G ։ H be an epimorphism of groups.a) If Ker( ψ ) is contained in a torsionfree direct summand of G then ψ -coarseningcommutes with relative (complete) integral closure. RADED INTEGRAL CLOSURES 11 b) If ψ -coarsening commutes with relative (complete) integral closure then Ker( ψ ) istorsionfree.Proof. a) First, we consider the case that Ker( ψ ) itself is a torsionfree direct summandof G . Let R be a G -graded ring, let S be a G -graded R -algebra, and let x ∈ S hom[ ψ ] be(almost) integral over R [ ψ ] . As Ker( ψ ) is a direct summand of G there exists a section π of ψ in the category of groups. So, we have a commutative diagram R [ ψ ] j πR (cid:15) (cid:15) / / S [ ψ ] j πS (cid:15) (cid:15) R [ ψ ] [Ker( ψ )] [ H ] / / S [ ψ ] [Ker( ψ )] [ H ] of H -graded rings (2.11). Since Ker( ψ ) is torsionfree it follows j πS ( x ) ∈ Int( R [ ψ ] [Ker( ψ )] [ H ] , S [ ψ ] [Ker( ψ )] [ H ] ) = Int( R [ ψ ] , S [ ψ ] )[Ker( ψ )] [ H ] (and similarly for complete integral closures) by 2.3 and 2.7. By the construction of j πS this implies x g ∈ Int( R [ ψ ] , S [ ψ ] ) (or x g ∈ CInt( R [ ψ ] , S [ ψ ] ), resp.) for every g ∈ G , and thusthe claim (2.10).Next, we consider the general case. Let F be a torsionfree direct summand of G containing Ker( ψ ), let χ : G ։ G/F be the canonical projection and let λ : H ։ G/F bethe induced epimorphism of groups, so that λ ◦ ψ = χ . Let R be a G -graded ring, and let S be a G -graded R -algebra. By 2.9 and the first paragraph,Int( R [ χ ] , S [ χ ] ) = Int( R, S ) [ χ ] = (Int( R, S ) [ ψ ] ) [ λ ] ⊆ Int( R [ ψ ] , S [ ψ ] ) [ λ ] ⊆ Int(( R [ ψ ] ) [ λ ] , ( S [ ψ ] ) [ λ ] ) = Int( R [ χ ] , S [ χ ] ) , hence (Int( R, S ) [ ψ ] ) [ λ ] = Int( R [ ψ ] , S [ ψ ] ) [ λ ] and therefore Int( R, S ) [ ψ ] = Int( R [ ψ ] , S [ ψ ] ) (or theanalogous statement for complete integral closures) as desired.b) Suppose K := Ker( ψ ) is not torsionfree. By 2.7 and 2.8 there exist a noetherian ring R and an R -algebra S such that R is integrally closed in S (hence completely integrallyclosed in S ) and that R [ K ] [0] is not integrally closed in S [ K ] [0] (hence not completelyintegrally closed in S [ K ] [0] (2.2)). Then, R [ K ] is completely integrally closed in S [ K ](2.6). Extending the K -graduations of R and S trivially to G -graduations it follows that R [ K ] [ G ] is completely integrally closed in S [ K ] [ G ] , while ( R [ K ] [ G ] ) [ ψ ] is not integrally closedin ( S [ K ] [ G ] ) [ ψ ] . This proves the claim. (cid:3) (2.13) Corollary Let ψ : G ։ H be an epimorphism of groups. If G is torsionfree then ψ -coarsening commutes with relative (complete) integral closure. Proof.
Immediately from 2.12. (cid:3) (2.14)
Supposing that the torsion subgroup T of G is a direct summand of G , it isreadily checked that a subgroup F ⊆ G is contained in a torsionfree direct summand of G if and only if the composition of canonical morphisms T ֒ → G ։ G/F has a retraction. In case H = 0 the statement about relative integral closures is [3, V.1 Exercice 25]. A torsionfree subgroup F ⊆ G is not necessarily contained in a torsionfree directsummand of G , not even if G is of finite type. Using the criterion above one checks thata counterexample is provided by G = Z ⊕ Z /n Z for n ∈ N > and F = h ( n, i Z ⊆ G . (2.15) Questions Let ψ : G ։ H be an epimorphism of groups. The above result givesrise to the following questions:a) If Ker( ψ ) is torsionfree, does ψ -coarsening commute with (complete) integral closure? b) If ψ -coarsening commutes with (complete) integral closure, is then Ker( ψ ) containedin a torsionfree direct summand of G ? Note that, by 2.14, at most one of these questions has a positive answer.3.
Integral closures of entire graded rings
In this section we consider (complete) integral closures of entire graded rings in theirgraded fields of fractions. We start with the relevant definitions and basic properties. (3.1)
Let R be an entire G -graded ring. The (complete) integral closure of R in Q ( R )is denoted by Int( R ) (or CInt( R ), resp.) and is called the (complete) integral closure of R . We say that R is (completely) integrally closed if it is (completely) integrally closed in Q ( R ). Keep in mind that Int( R ) is integrally closed, but that CInt( R ) is not necessarilycompletely integrally closed (2.2). If ψ : G ։ H is an epimorphism of groups and R is a G -graded ring such that R [ ψ ] (and hence R ) is entire, then Q ( R ) [ ψ ] is entire and R [ ψ ] ⊆ Q ( R ) [ ψ ] ⊆ Q ( Q ( R ) [ ψ ] ) = Q ( R [ ψ ] ) . From 2.9 it follows Int( R ) [ ψ ] ⊆ Int( R [ ψ ] ) and CInt( R ) [ ψ ] ⊆ CInt( R [ ψ ] ). Hence, if R [ ψ ] is(completely) integrally closed then so is R . We say that ψ -coarsening commutes with(complete) integral closure if whenever R is an entire G -graded ring then R [ ψ ] is entireand Int( R ) [ ψ ] = Int( R [ ψ ] ) (or CInt( R ) [ ψ ] = CInt( R [ ψ ] ), resp.). Clearly, if ψ -coarseningcommutes with (complete) integral closure then Ker( ψ ) is torsionfree (1.12 b)). Let F ⊆ G be a subgroup. An F -graded ring S is entire and (completely) integrally closed ifand only if S ( G ) is so. (3.2) Let I be a nonempty right filtering preordered set, and let (( R i ) i ∈ I , ( ϕ ij ) i ≤ j ) be aninductive system in GrAnn G over I such that R i is entire for every i ∈ I and that ϕ ij isa monomorphism for all i, j ∈ I with i ≤ j . By 2.5 and 1.7 we have inductive systems(Int( R i )) i ∈ I and (CInt( R i )) i ∈ I in GrAnn G over I , and we can consider the sub-lim −→ i ∈ I R i -algebras lim −→ i ∈ I Int( R i ) ⊆ lim −→ i ∈ I CInt( R i ) ⊆ Q (lim −→ i ∈ I R i )and compare them with the sub-lim −→ i ∈ I R i -algebrasInt(lim −→ i ∈ I R i ) ⊆ CInt(lim −→ i ∈ I R i ) ⊆ Q (lim −→ i ∈ I R i ) . RADED INTEGRAL CLOSURES 13
It follows immediately from 2.5 and 1.7 that lim −→ i ∈ I Int( R i ) = Int(lim −→ i ∈ I R i ). Hence, if R i is integrally closed for every i ∈ I then lim −→ i ∈ I R i is integrally closed, and lim −→ i ∈ I CInt( R i ) ⊆ CInt(lim −→ i ∈ I R i ).Suppose now in addition that (lim −→ i ∈ I R i ) ∩ Q ( R i ) = R i for every i ∈ I . Then, analo-gously to [1, 2.1] one sees that lim −→ i ∈ I CInt( R i ) = CInt(lim −→ i ∈ I R i ), hence if R i is completelyintegrally closed for every i ∈ I then so is lim −→ i ∈ I R i . This additional hypothesis is fulfilledin case R is a G -graded ring, F is a group, I = F F (1.3), and R U equals R [ U ] ( G ⊕ F ) or R [ U ] [ G ] for U ∈ F F . (3.3) Let R , S and T be G -graded rings such that R ⊆ S ⊆ T as graded subrings.Clearly, CInt( R, S ) ⊆ CInt(
R, T ) ∩ S . Gilmer and Heinzer found an (ungraded) exampleshowing that this is not necessarily an equality ([7, Example 2]), not even if R , S and T are entire and have the same field of fractions. In [7, Proposition 2] they also presentedthe following criterion for this inclusion to be an equality, whose graded variant is provenanalogously: If for every G -graded sub- S -module of finite type M ⊆ T with R ⊆ M thereexists a G -graded S -module N containing M such that R is a direct summand of N , thenCInt( R, S ) = CInt(
R, T ) ∩ S .In [7, Remark 2] Gilmer and Heinzer claim (again in the ungraded case) that thiscriterion applies if S is principal. As this would be helpful to us later (3.6, 3.10) we takethe opportunity to point out that it is wrong. Namely, suppose that S is not simple, that T = Q ( S ), and that the hypothesis of the criterion is fulfilled. Let x ∈ S \
0. Thereexists an S -module N containing h x − i S ⊆ T such that S is a direct summand of N . Thetensor product with S/ h x i S of the canonical injection S ֒ → N has a retraction, but it alsofactors over the zero morphism S/ h x i S → h x − i S /S . This implies x ∈ S ∗ , yielding thecontradiction that S is simple.Now we consider graded group algebras. We will show that both variants behave wellwith integral closures and that the finely graded variant behaves also well with completeintegral closure. (3.4) Theorem a) Formation of finely graded group algebras over entire G -graded ringscommutes with (complete) integral closure.b) If F is a group, then an entire G -graded ring R is (completely) integrally closed ifand only if R [ F ] is so.Proof. Keeping in mind that Q ( R )[ F ] = Q ( R [ F ]) (1.11) this follows immediately from2.6. (cid:3) (3.5) Lemma If R is a simple G -graded ring then R [ Z ] [ G ] is entire and completelyintegrally closed.Proof. First we note that S := R [ Z ] [ G ] is entire (1.11). The argument in [2, IV.1.6 Propo-sition 10] shows that S allows a graded version of euclidean division, i.e., for f, g ∈ S hom with f = 0 there exist unique u, v ∈ S hom with g = uf + v and deg Z ( v ) < deg Z ( f ), wheredeg Z denotes the usual Z -degree of polynomials over R . Using this we see analogously to [2, IV.1.7 Proposition 11] that every G -graded ideal of S has a homogeneous generatingset of cardinality 1. Next, developing a graded version of the theory of divisibility in entirerings along the line of [2, VI.1], it follows analogously to [2, VI.1.11 Proposition 9 (DIV);VII.1.2 Proposition 1] that for every x ∈ Q ( S ) hom there exist coprime a, b ∈ S hom with x = ab . So, the argument in [3, V.1.3 Proposition 10] shows that S is integrally closed.As it is noetherian the claim is proven (2.2). (cid:3) (3.6) Theorem a) Formation of coarsely graded algebras of torsionfree groups overentire G -graded rings commutes with integral closure.b) If F is a torsionfree group, then an entire G -graded ring R is integrally closed if andonly if R [ F ] [ G ] is so. Proof.
It suffices to prove the first claim. We can without loss of generality supposethat F is of finite type, hence free of finite rank (1.3, 1.5, 3.2). By induction on therank of F we can furthermore suppose F = Z . We have Int( R [ Z ] [ G ] ) ∩ Q ( R )[ Z ] [ G ] =Int( R [ Z ] [ G ] , Q ( R )[ Z ] [ G ] ). Since Q ( R )[ Z ] [ G ] is integrally closed (3.5) we getInt( R [ Z ] [ G ] ) ⊆ Int( Q ( R )[ Z ] [ G ] , Q ( R [ Z ] [ G ] )) = Q ( R )[ Z ] [ G ] . It followsInt( R [ Z ] [ G ] ) = Int( R [ Z ] [ G ] ) ∩ Q ( R )[ Z ] [ G ] = Int( R [ Z ] [ G ] , Q ( R )[ Z ] [ G ] ) = Int( R )[ Z ] [ G ] (2.7) and thus the claim. (cid:3) (3.7) Let F be a torsionfree group, let R be an entire G -graded ring, and suppose thatCInt( R [ Z ] [ G ] ) ∩ Q ( R )[ Z ] [ G ] = CInt( R [ Z ] [ G ] , Q ( R )[ Z ] [ G ] ). Then, the same argument as in3.6 (keeping in mind 3.2) yields CInt( R )[ F ] [ G ] = CInt( R [ F ] [ G ] ), hence R is completelyintegrally closed if and only if R [ F ] [ G ] is so. However, although R [ Z ] [ G ] is principal by theproof of 3.5, we have seen in 3.3 that it is unclear whether CInt( R [ Z ] [ G ] ) ∩ Q ( R )[ Z ] [ G ] andCInt( R [ Z ] [ G ] , Q ( R )[ Z ] [ G ] ) are equal in general. (3.8) Corollary a) Formation of coarsely graded algebras of torsionfree groups overnoetherian entire G -graded rings commutes with complete integral closure.b) If F is a torsionfree group, then a noetherian entire G -graded ring R is completelyintegrally closed if and only if R [ F ] [ G ] is so.Proof. We can without loss of generality suppose that F is of finite type (1.3, 1.5, 3.2).Then, R [ F ] [ G ] is noetherian (1.14), and the claim follows from 3.6 and 2.2. (cid:3) In the rest of this section we study the behaviour of (complete) integral closures underarbitrary coarsening functors, also using the results from Section 2. (3.9) Proposition
Let ψ : G ։ H be an epimorphism of groups.a) ψ -coarsening commutes with integral closure if and only if a G -graded ring R isentire and integrally closed if and only if R [ ψ ] is so. In case G = 0 the statement that integral closedness of R implies integral closedness of R [ F ] [ G ] is [3,V.1 Exercice 24]. RADED INTEGRAL CLOSURES 15 b) If ψ -coarsening commutes with complete integral closure, then a G -graded ring R isentire and completely integrally closed if and only if R [ ψ ] is so.Proof. If ψ -coarsening commutes with (complete) integral closure then it is clear thata G -graded ring R is entire and (completely) integrally closed if and only if R [ ψ ] is so.Conversely, suppose that ψ -coarsening preserves the property of being entire and integrallyclosed. Let R be an entire G -graded ring. Since simple G -graded rings are entire andintegrally closed, R [ ψ ] is entire (1.12 b)). As Int( R ) is integrally closed (2.2) the same istrue for Int( R ) [ ψ ] , implyingInt( R [ ψ ] ) = Int( R [ ψ ] , Q ( R [ ψ ] )) ⊆ Int(Int( R ) [ ψ ] , Q ( R [ ψ ] )) =Int(Int( R ) [ ψ ] ) = Int( R ) [ ψ ] ⊆ Int( R [ ψ ] )(3.1) and thus the claim. (cid:3) The argument used in a) cannot be used to prove the converse of b), as CInt( R ) is notnecessarily completely integrally closed (3.1). (3.10) Proposition Let ψ : G ։ H be an epimorphism of groups. Suppose that ψ -coarsening commutes with relative integral closure and maps simple G -graded rings toentire and integrally closed H -graded rings. Then, ψ -coarsening commutes with integralclosure.Proof. If R is an entire G -graded ring, then R [ ψ ] is entire (2.12 b), 1.12 b)) and Q ( R ) [ ψ ] is integrally closed, and as Q ( Q ( R ) [ ψ ] ) = Q ( R [ ψ ] ) (3.1) it followsInt( R [ ψ ] ) = Int( R [ ψ ] , Q ( R [ ψ ] )) ⊆ Int( Q ( R ) [ ψ ] , Q ( R [ ψ ] )) = Int( Q ( R ) [ ψ ] ) = Q ( R ) [ ψ ] , hence Int( R [ ψ ] ) = Int( R [ ψ ] , Q ( R ) [ ψ ] ) = Int( R, Q ( R )) [ ψ ] = Int( R ) [ ψ ] . (cid:3) (3.11) We have seen in 3.3 that it is (in the notations of the proof of 3.10) not clear thatCInt( R [ ψ ] ) ⊆ Q ( R ) [ ψ ] implies CInt( R [ ψ ] ) = CInt( R [ ψ ] , Q ( R ) [ ψ ] ). Therefore, the argumentfrom that proof cannot be used to get an analogous result for complete integral closures. (3.12) Lemma Let F be a free direct summand of G , let H be a complement of F in G ,let ψ : G ։ H be the canonical projection, let R be a simple G -graded ring, and supposethat ψ (degsupp( R )) ⊆ degsupp( R ) . Then, R [ ψ ] ∼ = R ( H ) [degsupp( R ) ∩ F ] [ H ] in GrAnn H .Proof. We set D := degsupp( R ). As F is free the same is true for D ∩ F . Let E be a basisof D ∩ F . If e ∈ E then R e = 0, so that we can choose y e ∈ R e \ ⊆ R ∗ . For f ∈ D ∩ F there exists a unique family ( r e ) e ∈ E of finite support in Z with f = P e ∈ E r e e , and we set y f := Q e ∈ E y r e e ∈ R f \
0; in case f ∈ E we recover the element y f defined above.As ( R ( H ) ) [0] is a subring of R [0] there exists a unique morphism of ( R ( H ) ) [0] -algebras p : R ( H ) [ D ∩ F ] [0] → R [0] with p ( e f ) = y f for f ∈ D ∩ F . If h ∈ H , then for f ∈ D ∩ F and x ∈ R h we have p ( xe f ) = xy f ∈ R h + f ⊆ ( R [ ψ ] ) h , so that p (( R ( H ) [ D ∩ F ] [ H ] ) h ) ⊆ ( R [ ψ ] ) h ,and therefore we have a morphism p : R ( H ) [ D ∩ F ] [ H ] → R [ ψ ] in GrAnn H . Let χ : G ։ F denote the canonical projection. For g ∈ G with χ ( g ) ∈ D there is amorphism of groups q g : R g → R ( H ) [ D ∩ F ] , x xy χ ( g ) e χ ( g ) , and for g ∈ G with χ ( g ) / ∈ D we denote by q g the zero morphism of groups R g → R ( H ) [ D ∩ F ]. For h ∈ H the morphisms q g with g ∈ ψ − ( h ) induce a morphism of groups q h : ( R [ ψ ] ) h → R ( H ) [ D ∩ F ]. So, we get a morphism of groups q := M h ∈ H q h : R [ ψ ] → R ( H ) [ D ∩ F ] . Let g ∈ G and x ∈ R g . If χ ( g ) / ∈ D then g / ∈ D , hence x = 0, and therefore p ( q ( x )) = x . Otherwise, p ( q ( x )) = p ( xy χ ( g ) e χ ( g ) ) = xy χ ( g ) p ( e χ ( g ) ) = xy χ ( g ) y χ ( g ) = x . Thisshows that q is a right inverse of p . If x ∈ R ( H ) then q ( p ( x )) = x , and if f ∈ D ∩ F then q ( p ( e f )) = q ( y f ) = y f y f e f = e f , hence q is a left inverse of p . Therefore, q is an inverse of p , and thus p is an isomorphism. (cid:3) (3.13) Proposition Let ψ : G ։ H be an epimorphism of groups, let R be a simple G -graded ring, and suppose that one of the following conditions is fulfilled:i) G is torsionfree;ii) Ker( ψ ) is contained in a torsionfree direct summand of G and R has full support.Then, R [ ψ ] is entire and completely integrally closed.Proof. First, we note that R [ ψ ] is entire (1.12 b)). In case i) it suffices to show that R [0] isintegrally closed, so we can replace H with 0 and hence suppose Ker( ψ ) = G . In case ii),by the same argument as in the proof of 2.12 (and keeping in mind 3.1) we can supposewithout loss of generality that K := Ker( ψ ) itself is a torsionfree direct summand of G and hence consider H as a complement of K in G . In both cases, as K = lim −→ L ∈ F K L (1.3)we have G = K ⊕ H = lim −→ L ∈ F K ( L ⊕ H ), hence R = lim −→ L ∈ F K (( R ( U ⊕ H ) ) ( G ) ). Setting ψ L := ψ ↾ L ⊕ H : L ⊕ H ։ H we get R [ ψ ] = lim −→ L ∈ F K (( R ( L ⊕ H ) ) [ ψ L ] ) (1.2). Hence, if ( R ( L ⊕ H ) ) [ ψ ] is integrally closed for every L ∈ F K then R [ ψ ] is integrally closed (3.2). Therefore, as R ( L ⊕ H ) is simple for every L ∈ F K (1.6) we can suppose that K is of finite type, hencefree. As R is simple it is clear that D := degsupp( R ) ⊆ G is a subgroup, hence D ∩ K ⊆ K is a subgroup, and thus D ∩ K is free. In both cases, our hypotheses ensure ψ ( D ) ⊆ D ,so that 3.12 implies R [ ψ ] ∼ = R ( H ) [ D ∩ K ] [ H ] . As R is simple it is completely integrallyclosed, hence R ( H ) is completely integrally closed (3.1), thus R ( H ) [ D ∩ K ] [ H ] is completelyintegrally closed (3.6), and so the claim is proven. (cid:3) (3.14) Theorem Let ψ : G ։ H be an epimorphism of groups, let R be an entire G -graded ring, and suppose that one of the following conditions is fulfilled:i) G is torsionfree;ii) Ker( ψ ) is contained in a torsionfree direct summand of G and h degsupp( R ) i Z = G .Then, Int( R ) [ ψ ] = Int( R [ ψ ] ) , and R is integrally closed if and only if R [ ψ ] is so. In case i) and H = 0 this is [3, V.1 Exercice 25]. RADED INTEGRAL CLOSURES 17
Proof.
As degsupp( Q ( R )) = h degsupp( R ) i Z this follows immediately from 3.10, 3.13 and2.12. (cid:3) (3.15) Questions Let R be an entire G -graded ring. The above, especially 3.7 and 3.11,gives rise to the following questions:a) Let ψ : G ։ H be an epimorphism of groups such that Ker( ψ ) is torsionfree. Do wehave CInt( R [ ψ ] ) ∩ Q ( R ) [ ψ ] = CInt( R [ ψ ] , Q ( R ) [ ψ ] ) ? b) Do we have
CInt( R [ Z ] [ G ] ) ∩ Q ( R )[ Z ] [ G ] = CInt( R [ Z ] [ G ] , Q ( R )[ Z ] [ G ] ) ? If both these questions could be answered positively, then the same arguments as abovewould yield statements for complete integral closures analogous to 3.6, 3.10, and 3.14.
Acknowledgement:
I thank Benjamin Bechtold and the reviewer for their commentsand suggestions. The remarks in 2.14 were suggested by Micha Kapovich and Will Sawinon http://mathoverflow.net/questions/108354 . The counterexample in 3.3 is due toan anonymous user on http://mathoverflow.net/questions/110998 . References [1] D. F. Anderson,
Graded Krull domains.
Comm. Algebra 7 (1979), 79–106.[2] N. Bourbaki, ´El´ements de math´ematique. Alg`ebre. Chapitres 1 `a 3.
Masson, 1970;
Chapitres 4 `a 7.
Masson, 1981.[3] N. Bourbaki, ´El´ements de math´ematique. Alg`ebre commutative. Chapitres 5 `a 7.
Hermann, 1975.[4] S. Goto, K. Yamagishi,
Finite generation of noetherian graded rings.
Proc. Amer. Math. Soc. 89(1983), 41–44.[5] R. Gilmer,
Multiplicative ideal theory.
Pure Appl. Math. 12, Marcel Dekker, 1972.[6] R. Gilmer,
Commutative semigroup rings.
Chicago Lectures in Math., Univ. Chicago Press, 1984.[7] R. Gilmer, W. J. Heinzer,
On the complete integral closure of an integral domain.
J. Aust. Math.Soc. 6 (1966), 351–361.[8] A. Grothendieck, J. A. Dieudonn´e, ´El´ements de g´eom´etrie alg´ebrique. I: Le langage des sch´emas(Seconde ´edition).
Grundlehren Math. Wiss. 166, Springer, 1971.[9] C. Nˇastˇasescu, F. Van Oystaeyen,
Graded rings with finiteness conditions II.
Comm. Algebra 13(1985), 605–618.[10] F. Rohrer,
Coarsenings, injectives and Hom functors.
Rev. Roumaine Math. Pures Appl. 57 (2012),275–287.[11] I. Swanson, C. Huneke,
Integral closure of ideals, rings, and modules.
London Math. Soc. LectureNote Ser. 336, Cambridge Univ. Press, 2006.[12] B. L. van der Waerden,
Algebra. Zweiter Teil. (F¨unfte Auflage).
Heidelb. Taschenb. 23, Springer,1967.[13] J. Van Geel, F. Van Oystaeyen,
About graded fields.
Indag. Math. (N.S.) 84 (1981), 273–286.
Universit¨at T¨ubingen, Fachbereich Mathematik, Auf der Morgenstelle 10, 72076 T¨u-bingen, Germany
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