Nakayama closures, interior operations, and core-hull duality
aa r X i v : . [ m a t h . A C ] A ug NAKAYAMA CLOSURES, INTERIOR OPERATIONS, ANDCORE-HULL DUALITYwith applications to tight closure theory
NEIL EPSTEIN, REBECCA R.G., AND JANET VASSILEV
Abstract.
Exploiting the interior-closure duality developed by Epsteinand R.G. [ER19], we show that for the class of Matlis dualizable modules M over a Noetherian local ring, when cl is a Nakayama closure and i itsdual interior, there is a duality between cl-reductions and i-expansionsthat leads to a duality between the cl-core of modules in M and the i-hull of modules in M ∨ . We further show that many algebra and moduleclosures and interiors are Nakayama and describe a method to computethe interior of ideals using closures and colons. We use our methods togive a unified proof of the equivalence of F-rationality with F-regularity,and of F-injectivity with F-purity, in the complete Gorenstein local case.Additionally, we give a new characterization of the finitistic tight closuretest ideal in terms of maps from R / p e . Moreover, we show that theliftable integral spread of a module exists. Contents
1. Introduction 22. Background 43. The Artinistic interior of an ideal 74. Many module and algebra closures are Nakayama 125. Nakayama interiors are dual to Nakayama closures 146. i-Expansions, co-generating sets, and core-hull duality 167. Core and hull comparisons for known closure operations andtheir dual interiors 238. Computations of interiors and hulls 29Acknowledgments 36References 37
Date : August 4, 2020.1991
Mathematics Subject Classification.
Primary: 13J10, Secondary: 13A35, 13B22,13C60.
Key words and phrases. closure operation, test ideal, interior operation, Nakayamaclosure, tight closure, integral closure, Frobenius closure, core, Matlis duality. Introduction
The core of an ideal, attributed to Sally and Rees [RS88], was initiallystudied in part because of its relationship to the Brian¸con-Skoda Theorem.While the core is in general difficult to compute, Huneke and Swanson gavea formula for computing the core of an m -primary integrally closed ideal I ina 2-dimensional regular local ring with infinite residue field [HS94]. Papersby Mohan [Moh97], Corso, Polini, and Ulrich [CPU01, CPU02, CPU03],Hyry and Smith [HS04, HS03], Huneke and Trung [HT05], Polini and Ulrich[PU05], and Fouli [Fou08], and Fouli, Polini, and Ulrich [FPU08] furtherexplored formulas for cores of ideals and modules.Fouli and the third named author [FV10] extended the notion of core toNakayama closures cl, defining the cl-core of an ideal as the intersection of allcl-reductions of the ideal. They compared the original core to tight closureand Frobenius closure cores, showing that core ( I ) ⊆ ∗ -core ( I ) ⊆ F -core ( I ) .They also came up with conditions when the core and the ∗ -core are equal.Fouli, Vassilev and Vraciu [FVV11] determined ∗ -core ( I ) = ( J ∶ I ) J for aminimal reduction J over a normal local ring of characteristic p > ∗ -coreof an ideal is an intersection of finitely many ∗ -reductions of the ideal.The first named author and Schwede introduced the tight interior in[ES14] as a dual to tight closure. In [ER19], the first two named authorsshowed that this is an example of a more general duality, which assigns toeach closure operation a corresponding interior operation. The third namedauthor noted in [Vas20] that given an interior operation i, we can define thei-hull of an ideal in a way analogous but seemingly dual to the cl-core. Inthis paper we expand on this duality, ultimately proving that it arises fromthe closure-interior duality defined in [ER19]. More specifically, we have thefollowing result: Theorem A (Duality Theorem) . Let cl be a Nakayama closure on Noe-therian R -modules, where R is a complete Noetherian local ring. Then thedual interior operation to cl , i , is a Nakayama interior operation defined onArtinian R -modules (Proposition 5.5). Let A ⊆ B be Artinian R -modules.(1) There exists an order-reversing correspondence between i -expansionsof A ⊆ B and cl -reductions of ( B / A ) ∨ in B ∨ (Theorem 6.3).(2) Every i -expansion of A in B is contained in a maximal i -expansionof A in B (Proposition 6.4).(3) The i -hull of A in B exists and is dual to the cl -core of ( B / A ) ∨ in B ∨ (Theorem 6.17). In order to do this, we define Nakayama interiors (Definition 5.1) andprove that they are dual to Nakayama closures (see Section 5). We definei-expansions of an ideal, and prove that there are maximal i-expansions
ORE-HULL DUALITY 3 (see Section 6). This involves a discussion of co-generation of modules (seeDefinition 6.6) as originally discussed in [V´am68].The best known examples of Nakayama closures on ideals are Frobeniusclosure, tight closure, and integral closure [Eps05, Eps10]. These closures allextend to the module setting, integral closure by means of liftable integralclosure [EU], and our results apply to all of these. We give additional exam-ples of Nakayama closures and interiors in Section 4. For instance, under alarge variety of circumstances, module and algebra closures are Nakayama:
Theorem B (See Theorems 4.1 and 4.7, and Corollary 4.8) . Let ( R, m ) bea Noetherian local ring. Let S be an R -algebra with m S contained in theJacobson radical of S .(1) cl S is a Nakayama closure on finite R -modules. (Theorem 4.7)(2) If S is local and N a finitely generated S -module, then cl N is aNakayama closure on finitely generated R -modules. (Theorem 4.1)(3) In particular, Frobenius closure (in characteristic p > ) and plusclosure (when R is complete) are Nakayama closures. (Corollary 4.8)(4) If B is a local big Cohen-Macaulay R -algebra and R → B is a localhomomorphism, then cl B is a Nakayama closure. (Corollary 4.8) In addition to the duality theorem above, we show that interiors of idealscan be computed using closures and colons, particularly the
Artinistic ver-sion of the interior (analogous to the finitistic test ideal):
Theorem C (See Theorem 3.3) . Let ( R, m , k ) be a complete Noetherianlocal ring that is approximately Gorenstein and let α f denote the finitisticversion of a functorial submodule selector α . If { J t } is a decreasing nestedsequence of irreducible ideals cofinal with the powers of m , then for any ideal I , ( α f ) ⌣ ( I ) = ⋂ t ≥ ( J t ∶ ( J t ∶ I ) cl R ) . We also show that when R is Gorenstein, if R is cl-rational then R iscl-regular (Corollary 3.5). Further, we use liftable integral closure [EU] toextend the notion of analytic spread to modules: Theorem D (See Theorem 7.8) . Let ( R, m , k ) be a Noetherian local ringsuch that k is infinite and R is either of equal characteristic 0 or reduced.Then the liftable integral spread exists and agrees with the analytic spread inthe sense of Eisenbud-Huneke-Ulrich. Throughout the paper, we demonstrate the usefulness of our results byapplying them to the case of tight closure. One such result is the followingnew characterization of the finitistic tight closure test ideal:
Theorem E (See Theorem 6.20) . Let R be a complete Noetherian local F -finite reduced ring of prime characteristic p > and c be a big test elementfor R . The finitistic test ideal of R consists of those elements a ∈ R suchthat for every m -primary ideal J of R and every nonnegative integer e ≥ ,there is an R -linear map g ∶ R / p e → R / J with g ( c / p e ) = a + J . NEIL EPSTEIN, REBECCA R.G., AND JANET VASSILEV
As a consequence of Corollary 3.5, we get a unified proof of two resultson F -rational and F -injective rings: Theorem F (See Corollaries 3.6 and 3.7) . Let ( R, m , k ) be a complete Goren-stein Noetherian local ring. Let cl be a functorial and residual closure op-eration. If there exists a system of parameters x ∶= x , . . . , x d for which J t ∶= ( x t , . . . , x td ) are cl -closed for infinitely many t ∈ N , then every ideal I of R is cl -closed and every finitely generated R -module M is cl -closed in M (Corollary 3.6).As a consequence we recover the well known results (Corollary 3.7):(1) If R is equicharacteristic and F -rational, then it is F -regular.(2) If R is of prime characteristic p > and F -injective, then it is F -pure. In Theorem 8.1, we use results of Lyubeznik and Smith to describe caseswhere the tight interior and its Artinistic version agree, which enables us tocompute tight interiors and hulls using Theorem 3.3.
Theorem G (See Theorem 8.1) . Suppose that ( R, m ) is a complete reducedF-finite local ring and I is an ideal of R . If R has mild singularities or thepair R, I arises from a graded situation, then the tight interior of I and theArtinistic version of the tight interior of I coincide. The organization of the paper is as follows: Section 2 gives relevant back-ground. In Section 3 we prove that we can compute interiors of ideals inNoetherian local rings using the dual closure and colons and we show thatwhen our ring is Gorenstein, if parameter ideals are cl-closed then all idealsare cl-closed. In Section 4 we show that many module and algebra closuresare Nakayama. In Section 5 we define a Nakayma interior and show thatNakayama interiors are dual to Nakayama closures. In Section 6, we dis-cuss i-expansions, finite co-generation, minimal co-generating sets, maximalexpansions and the core-hull duality. In Section 7 we compare related clo-sures and interiors, define the co-spread as a dual to spread, and then showliftable integral spread exists. In Section 8 we prove that in many cases ofinterest the tight interior and its Artinistic version coincide, which allows usto compute some examples of tight and Frobenius interiors and hulls.All rings will be assumed to be commutative unless otherwise specified.2.
Background
We describe the duality between closure operations and interior opera-tions over a complete Noetherian local ring as first given by the first twonamed authors in [ER19]. We recall the definition of a Nakayama closure,cl-reductions, and the cl-core, and give some of their properties.
Definition 2.1 ([ER19]) . Let R be a ring, not necessarily commutative. Let M be a class of (left) R -modules that is closed under taking submodules ORE-HULL DUALITY 5 and quotient modules. Let
P ∶= P M denote the set of all pairs ( L, M ) where M ∈ M and L is a submodule of M in M .A submodule selector is a function α ∶ M → M such that ● α ( M ) ⊆ M for each M ∈ M , and ● for any isomorphic pair of modules M, N ∈ M and any isomorphism ϕ ∶ M → N , we have ϕ ( α ( M )) = α ( ϕ ( M )) .An interior operation is a submodule selector that is ● order-preserving , i.e. for any L ⊆ M ∈ M , α ( L ) ⊆ α ( M ) , and ● idempotent , i.e. for all M ∈ M , α ( α ( M )) = α ( M ) .A submodule selector α is functorial if for any g ∶ M → N in M , we have g ( α ( M )) ⊆ α ( N ) .A closure operation is an operation that sends each pair ( L, M ) ∈ P to amodule L ⊆ L cl M ⊆ M such that ● cl is idempotent, i.e. ( L cl M ) cl M = L cl M , and ● cl is order-preserving on submodules, i.e. if L ⊆ N ⊆ M , L cl M ⊆ N cl M .A closure operation is residual if for any surjective map q ∶ M ↠ P in M ,we have ( ker q ) cl M = q − ( cl P ) . Note that because q is a surjection, we alsohave q (( ker q ) cl M ) = cl P .A closure operation cl is finitistic if for any L ⊆ M for which L cl M is defined,and for any z ∈ L cl M , there is some module N with L ⊆ N ⊆ M and N / L finitely generated such that z ∈ L cl N .If cl is a closure operation, the finitistic version cl f of cl is given by L cl f M = ⋃{ L cl N ∶ L ⊆ N ⊆ M and N / L is finitely generated } . For the following definition and result, R is a complete Noetherian localring with maximal ideal m , residue field k , and E ∶= E R ( k ) the injective hull.We will use ∨ to denote the Matlis duality operation. M is a category of R -modules closed under taking submodule and quotient modules, and suchthat for all M ∈ M , M ∨∨ ≅ M . For example, M could be the category offinitely generated R -modules, or of Artinian R -modules. Definition 2.2 ([ER19]) . Let R be a complete Noetherian local ring. Let S (M) denote the set of all submodule selectors on M . Define ⌣∶ S (M) → S (M ∨ ) as follows: For α ∈ S (M) and M ∈ M ∨ , α ⌣ ( M ) ∶= ( M ∨ / α ( M ∨ )) ∨ , considered as a submodule of M in the usual way. Theorem 2.3 ([ER19]) . Let r be a complete Noetherian local ring and α asubmodule selector on M . Then:(1) ( α ⌣ ) ⌣ = α ,(2) If α ( M ) ∶= cl M for a residual closure operation cl , then α ⌣ is aninterior operation. Conversely, if α is an interior operation, then α ⌣ ( M ) = cl M for a uniquely determined residual closure operation cl . NEIL EPSTEIN, REBECCA R.G., AND JANET VASSILEV
Remark 2.4.
In consequence, if cl is a residual closure operation on M , itsdual interior operation can be expressed asi ( M ) = ( M ∨ cl M ∨ ) ∨ , where M ∈ M ∨ . Definition 2.5.
Let R be a ring, not necessarily commutative, and let L ⊆ M be R -submodules of N ∈ M . We say that L ⊆ N is a cl -reduction of N in M if L cl M = N cl M .Note that L ⊆ N ⊆ L cl M if and only if L is a cl-reduction of N in M . Remark 2.6. (1) If N = R and J ⊆ I are ideals of R with I cl = J cl , thisagrees with the notion of J being a cl-reduction of I .(2) When M = R is the ring, we will write (−) cl in place of (−) cl R .)However, for a general R -module M , the closure of a submodule N may change depending on the ambient R -module M . Hence, wewrite N cl M to emphasize that we are taking the closure of N in M . Definition 2.7.
Let R be a ring, not necessarily commutative, and cl aclosure operation defined on a class of R -modules M that is closed undersubmodules, quotient modules, and extensions. If M, N are elements of M ,with N ⊆ M , the cl -core of N with respect to M is the intersection of all clreductions of N in M , orcl -core M ( N ) ∶= ⋂ L ⊆ N ⊆ L cl M L. Definition 2.8.
Let ( R, m ) be a Noetherian local ring and cl be a closureoperation on the class of finitely generated R -modules M . We say that clis a Nakayama closure if for L ⊆ N ⊆ M ∈ M satisfying L ⊆ N ⊆ ( L + m N ) cl M implies that L cl M = N cl M .Note that this is consistent with the definition for ideals [Eps05] by letting M = R / J and L = I / J where J ⊆ I are ideals. Proposition 2.9. [Eps05, Eps10]
Let ( R, m ) be a Noetherian local ring and cl a Nakayama closure operation on the class of finitely generated R -modules M . Let N ⊆ M be elements of M . For any cl -reduction L of N in M , thereexists a minimal cl -reduction K of N in M such that K ⊆ L . Moreover, anyminimal generating set of K extends to one of L . Remark 2.10.
The above result is stated only for ideals in [Eps05, Lemma2.2], but the full proof of [Eps05, Lemma 2.2] applies mutatis mutandis tofinitely generated modules (as is mentioned for part of the result on page2210 of [Eps10]).The principal component of the above property holds for Artinian modulestoo, and in much greater generality.
ORE-HULL DUALITY 7
Proposition 2.11.
Let R be any associative ring (not necessarily commu-tative), let C be an R -module, and let cl be a closure operation on (left) R -submodules of C , and let A ⊆ B ⊆ C be R -modules with A Artinian. Sup-pose that A is a cl -reduction of B in C . Then there is a minimal cl -reduction K of B in C , such that K ⊆ A .Proof. Let S be the set of R -submodules D of A such that D is a cl-reductionof B in C . Note that A ∈ S , whence S ≠ ∅ . Since A is Artinian and S isa nonempty collection of submodules of A , S has a minimal element, say K . We claim that K is in fact a minimal cl-reduction of B in C , since if E is a cl-reduction of B in C and E ⊆ K , then in particular E ⊆ A , whence E ∈ S . By minimality of K , then, we have E = K . Hence K is a minimalcl-reduction of B in C , and K ⊆ A . (cid:3) The Artinistic interior of an ideal
In this section we prove a result on finitistic versions of closure opera-tions that allows us to describe their dual interior operations. All rings arecommutative unless otherwise specified.
Definition 3.1 ([ER19, Definition 5.1]) . Let cl be a closure operation on aclass M of R -modules. We define the finitistic version cl f of cl by L cl f M ∶= ⋃{ L cl N ∶ L ⊆ N ⊆ M ∈ M , N / L finitely-generated } . Let α be a submodule selector on a class of R -modules. The finitisticversion α f of α is defined by α f ( N ) ∶= ⋃{ α ( M ) ∶ M ⊆ N is finitely generated } . We give a lemma that may be well-known, but that we include in theinterest of keeping this paper self-contained.
Lemma 3.2.
Let cl be a functorial, residual closure operation. Let L ⊆ M be modules such that A cl B is defined whenever T ⊆ M and A ⊆ B ⊆ M / T .Then L cl f M = ⋃{( L ∩ U ) cl U ∶ U is a finitely generated submodule of M } . Proof.
For the forward containment, let z ∈ L cl f M . Then there is some module N with L ⊆ N ⊆ M , N / L finitely generated and z ∈ L cl N . Set x ∶= z ,and choose x , . . . , x t ∈ N such that the images of the x j generate N / L .That is, L + U = N , where U ∶= ∑ tj = Rx j . By functoriality, this means¯ z ∈ cl N / L = cl ( L + U )/ L ≅ cl U /( L ∩ U ) , by the Second Isomorphism Theorem. Underthis canonical isomorphism, the image of z in N / L maps to the image of z in U /( L ∩ U ) . So by residuality, we then have z ∈ ( L ∩ U ) cl U .For the reverse containment, suppose U is a finitely generated submoduleof M and z ∈ ( L ∩ U ) cl U . Then ¯ z ∈ cl U /( L ∩ U ) ≅ cl U + L / L , so by the argument inthe first paragraph, we have z ∈ L cl U + L . Additionally, U + L is a submodule NEIL EPSTEIN, REBECCA R.G., AND JANET VASSILEV of M containing L such that U + L / L ≅ U /( U ∩ L ) is finitely generated(as it is isomorphic a quotient of the finitely generated module U ). Hence z ∈ L cl f M . (cid:3) The following Theorem is analogous to (and for the most part a general-ization of) [ER19, Theorem 5.5].
Theorem 3.3.
Let ( R, m , k ) be a complete Noetherian local ring. Let α be a functorial submodule selector (i.e. a preradical) on A , the class ofArtinian R -modules. Let α f be its finitistic version. Let cl (resp. cl f ) bethe operation such that L cl M / L = α ( M / L ) (resp. L cl f M / L = α f ( M / L ) ), where M / L is Artinian. Assume that cl and cl f are closure operations. Let I beany ideal of R . Then: α ⌣ ( I ) = ann R (( ann E ( I )) cl E ) = ⋂ M ∈A ann R (( ann M ( I )) cl M )⊆ ( α f ) ⌣ ( I ) = ann R (( ann E ( I )) cl f E ) = ⋂ M ⊆ E f.g. ann R (( ann M ( I )) cl M )= ⋂ λ ( M )<∞ ann R (( ann M ( I )) cl M ) ⊆ ⋂ λ ( R / J )<∞ ann R (( ann R / J ( I )) cl R / J )= ⋂ λ ( R / J )<∞ ( J ∶ ( J ∶ I ) cl R ) . Moreover, the last containment is an equality when R is approximately Goren-stein. In that case if { J t } is a decreasing nested sequence of irreducible idealsthat is cofinal with the powers of m , then in fact we have ( α f ) ⌣ ( I ) = ⋂ t ≥ ( J t ∶ ( J t ∶ I ) cl R ) . Remark 3.4.
For examples of computations using the last part of the result,see Section 8. For an explanation of the term “Artinistic” to describe thephenomenon here, see the end of Section 6, in particular Definition 6.18 andTheorem 6.19.
Proof.
For the first equality, we have α ⌣ ( I ) = ( I ∨ α ( I ∨ ) ) ∨ = ( E / ann E I ( ann E ( I ) cl E / ann E ( I )) )= ( E ann E ( I ) cl E ) ∨ = ann R ( ann E ( I ) cl E ) . Note that this also establishes the third equality.For the second equality, we prove both containments. For the forwardcontainment, let M ∈ A . Then there is some positive integer t and someinclusion map j ∶ M ↪ E ⊕ t . Then j ( ann M ( I )) ⊆ ann E ⊕ t ( I ) . Thus since clis functorial (by the assumption on α ), we have j ( ann M ( I ) cl M ) ⊆ ann E ⊕ t ( I ) cl E ⊕ t = (( ann E ( I )) cl E ) ⊕ t . ORE-HULL DUALITY 9
Hence,ann R ( ann M ( I ) cl M ) = ann R ( j ( ann M ( I ) cl M ))⊇ ann R (( ann E ( I ) cl E ) ⊕ t ) = ann R ( ann E ( I ) cl E ) . For the reverse containment, we merely note that E ∈ A .The first displayed containment holds because α f ≤ α and ⌣ is order-reversing.For the fourth equality, we prove both containments. For the forwardcontainment, let M be a finitely generated submodule of E . Then byLemma 3.2, ( ann M I ) cl M = (( ann E I ) ∩ M ) cl M ⊆ ( ann E I ) cl f E , so ann R (( ann E I ) cl f E ) ⊆ ann R (( ann M I ) cl M ) for all such M . For the reversecontainment, let a ∈ ⋂ M ⊆ E f.g. ann R (( ann M ( I )) cl M ) and let z ∈ ( ann E I ) cl f E .Then by Lemma 3.2, there is some finitely generated submodule M ⊆ E with z ∈ ( M ∩ ann E I ) cl M = ( ann M I ) cl M . Hence az = N be an R -module of finite length. Then we have N = ⊕ tj = N j ,where each N j is indecomposable and of finite length. Hence, for each j thereis some injective R -linear map i j ∶ N j ↪ E . Set M j ∶= i j ( N j ) . Then each M j is a finitely generated submodule of E . Thus we haveann R (( ann N I ) cl N ) = ann R ⎛⎝ t ⊕ j = ( ann N j I ) cl N j ⎞⎠ = t ⋂ j = ann R (( ann N j I ) cl N j )= t ⋂ j = ann R (( ann M j I ) cl M j ) ⊇ ⋂ M ⊆ E f.g. ann R (( ann M ( I )) cl M ) . For the reverse containment, we merely recall that any finitely generatedsubmodule of E has finite length.For the sixth equality, let J be an arbitrary ideal of finite colength. Thenann R / J I = ( J ∶ I )/ J , so by residuality of the closure operation, we have ( ann R / J I ) cl R / J = ( J ∶ I ) cl R / J . Now, for any ideal K of R that contains J , wehave ann R ( K / J ) = ( J ∶ K ) . Hence in particular, we haveann R (( ann R / J ( I )) cl R / J ) = ann R (( J ∶ I ) cl R / J ) = ( J ∶ ( J ∶ I ) cl R ) . Now consider the case where R is approximately Gorenstein, with theideals J t as given in the statement of the theorem. Let N t ∶= ann E J t foreach t ∈ N . By the theory of approximately Gorenstein rings [Hoc07, Page157], we have N t ≅ R / J t as R -modules, N t ⊆ N t + for each t , and E = ⋃ t ∈ N N t .Now let M be a finitely generated submodule of E . Say M = ∑ mi = Rz i . Theneach z i ∈ N t i for some i . Let s = max { t i ∣ ≤ i ≤ m } . Then every z i ∈ N s ,whence M ⊆ N s . Since cl preserves containment in submodules and ambient modules, we have ( ann M I ) cl M ⊆ ( ann M I ) cl N s ⊆ ( ann N s I ) cl N s . Hence, ⋂ t ≥ ( J t ∶ ( J t ∶ I ) cl R ) ⊆ ( J s ∶ ( J s ∶ I ) cl R ) = ann R (( ann R / J s I ) cl R / J s ) = ann R (( ann N s I ) cl N s ) ⊆ ann R (( ann M I ) cl M ) . But the intersection of all such ideals has already been shown to be equalto ( α f ) ⌣ ( I ) .We have shown that ⋂ t ( J t ∶ ( J t ∶ I ) cR ) ⊆ ( α f ) ⌣ ( I ) . But the left handside contains ⋂ λ ( R / J )<∞ ( J ∶ ( J ∶ I ) cR ) since each J t has finite colength. Thisshows that the last containment of the first part of the theorem is an equalityfor approximately Gorenstein rings. (cid:3) Note that the above result already has consequences beyond that of [ER19,Theorem 5.5], even in the case where I = R . Corollary 3.5.
Let ( R, m , k ) be a complete approximately Gorenstein Noe-therian local ring. Let cl be a functorial and residual closure operation. Let { J t } t ∈ N be a sequence of irreducible m -primary ideals cofinal with the powersof m and cl -closed. Then for every ideal I of R , we have I cl = I , and forevery finitely generated R -module M , we have cl M = .Proof. Let α be the preradical associated to cl, and α f the preradical asso-ciated to cl f . Then by Theorem 3.3, we have ( α f ) ⌣ ( R ) = ⋂ t ( J t ∶ J cl t ) = ⋂ t ( J t ∶ J t ) = R = ⋂ λ ( M )<∞ ann R ( cl M ) = ⋂ λ ( R / J )<∞ ( J ∶ J cl ) . Hence, 0 is cl-closed in every finite-length module (since 1 ∈ R = the commonannihilator of their closures), and similarly every finite colength ideal is cl-closed.Now let M be an arbitrary finitely generated R -module. Then for anypositive integer s , we have that M / m s M is finite length. Hence by theresidual property, we have ( m s M ) cl M / m s M = cl M / m s M = . That is, m s M is a cl-closed submodule of M . But by the Krull intersectiontheorem, 0 = ⋂ s ∈ N m s M . Hence 0, being an intersection of cl-closed submod-ules of M , is itself cl-closed (cf. [Eps12, Proposition 2.1.3], where the resultis stated for ideals but whose proof extends immediately to modules).Finally, let I be an arbitrary ideal. Then R / I is a finitely generated R -module, so by the above, we have 0 = cl R / I = I cl / I , whence I cl = I . (cid:3) The above is especially interesting in the Gorenstein case, so we state itseparately as follows.
ORE-HULL DUALITY 11
Corollary 3.6.
Let ( R, m , k ) be a complete Gorenstein Noetherian local ring.Let cl be a functorial and residual closure operation. Suppose there is somesystem of parameters x ∶= x , . . . , x d such that the ideals J t ∶= ( x t , . . . , x td ) are cl -closed for infinitely many t ∈ N . Then for every ideal I of R , we have I cl = I , and for every finitely generated R -module M , we have cl M = .Proof. Recall [Mat86, Theorem 18.1] that in this case, any ideal generatedby a full system of parameters is irreducible. Hence some subsequence ofthe ideals J t satisfies the conditions of Corollary 3.5. (cid:3) Hence, we obtain a unified proof of the following results, which previouslyseemed to require completely different proofs:
Corollary 3.7.
Let ( R, m , k ) be a complete Gorenstein Noetherian localring.(1) [EH08, Remark 3.8, in char p only] If R is equicharacteristic and F -rational, then it is F -regular.(2) [Fed83, Lemma 3.3] If R is of prime characteristic p > and F -injective, then it is F -pure.Proof. For the first item, note that the definition of F -rational is that idealsgenerated by systems of parameters are tightly closed. For the second item,if a ring is Cohen-Macaulay, it is F -injective if and only if any ideal generatedby a system of parameters is Frobenius closed [FW89, Remark 1.9].Both results now follow from Corollary 3.6. (cid:3) It happens that the assumption of Gorensteinness is crucial in Corol-lary 3.6. To see this, we look at the special case where R is Cohen-Macaulay,and the closure operation in question is cl ω R . In that case, every ideal gener-ated by a system of parameters is cl ω R -closed, yet cl ω R is never trivial unless R is Gorenstein: Lemma 3.8.
Let R be a complete Cohen-Macaulay Noetherian local ring,let ω R be its canonical modules, and let cl = cl ω R . Then for any ideal I generated by part of a system of parameters, we have I cl = I .Proof. Let x be a full system of parameters. Let J = ( x ) . Then by [BH97,Theorem 3.3.4(a)], ω R / J ω R ≅ E R / J ( k ) , which by [BH97, Proposition 3.2.12]is a faithful ( R / J ) -module. Let a ∈ J cl = ann R ( ω R / J ω R ) . Then in R / J , wehave ¯ a ∈ ann R / J ( ω R / J ω R ) = ann R / J ( E R / J ( k )) = ¯0 in R / J . Hence, a ∈ J .Now take an arbitrary parameter ideal I = ( x , . . . , x t ) . Complete to afull system of parameters x , . . . , x t , x t + , . . . , x d . Then for every positiveinteger s , we have that the ideal J s ∶= I + ( x st + , . . . , x sd ) is generated by a fullsystem of parameters. But I = ⋂ s J s , and every J s is cl-closed by the firstparagraph of the proof. Hence I , being an intersection of cl-closed ideals, isitself cl-closed. (cid:3) On the other hand, by [HHS19, Lemma 2.1] and the proof of [PR19,Corollary 3.18], we have that that cl ω R is trivial if and only if R is Gorenstein,even though by the above, this closure is always trivial on parameter ideals.4. Many module and algebra closures are Nakayama
In this section we prove that certain module and algebra closures areNakayama closures, so that the results of this paper apply to them.
Theorem 4.1.
Let ( R, m ) → ( S, n ) be a local homomorphism of Noetherianlocal rings, and let B be a finitely generated S -module. Let M be the classof finitely generated R -modules, and consider the closure operation cl ∶= cl B on M . Then cl is a Nakayama closure.In particular, this holds when B is either a finitely generated R -module(i.e. the case R = S ) or any Noetherian local R -algebra (i.e. the case S = B ).Proof. Since cl is residual, it is enough to show that for any L ⊆ M ∈ M , if L ⊆ ( m L ) cl M , we have L ⊆ cl M . Accordingly suppose L ⊆ ( m L ) cl M . Let H bethe image of the induced map L ⊗ R B → M ⊗ R B . By the assumption, H isin fact contained in the image of m L ⊗ R B → M ⊗ R B . But any element ofthe latter is of the form t ∑ j = ( m j x j ) ⊗ b j = t ∑ j = m j ( x j ⊗ b j ) , with m j ∈ m , x j ∈ L , and b j ∈ B . But the latter representation of such anelement is clearly in m H , since each x j ⊗ b j is in H . This in turn is containedin n H since m S ⊆ n and H is an S -module.Hence, H ⊆ n H . But H is a submodule of M ⊗ R B , which is a finitelygenerated S -module. Since S is Noetherian, it follows that H is itself afinitely generated S -module. But then the Nakayama lemma (applied to H as a finitely generated S -module) implies that H =
0. In other words, L ⊆ cl M . (cid:3) To go further, we recall the direct limit scaffolding from [ER19]. Let Γ bea directed set. Let { α i ∣ i ∈ Γ } be a directed system of submodule selectors.Then α ∶= lim Ð→ α i , defined by α ( M ) ∶= ⋃ i ∈ Γ α i ( M ) [ER19, Definition 7.1], isa submodule selector as well. Moreover, [ER19, Proposition 7.2] if each ofthe α i arises from a functorial residual closure operation, then so does α .In particular, we have the following for algebra closures, which is implicitlyused for example in [R.G16] and [PR19], but not explicitly stated there. Lemma 4.2.
Let { A i } i ∈ I be a direct limit system of R -algebras with R -algebra homomorphisms, with direct limit A . Then ∑ i ∈ I cl A i is a closureoperation and in fact is equal to cl A . Remark 4.3.
Using the notation of [ER19], we may also refer to this closureas lim Ð→ cl A i . ORE-HULL DUALITY 13
Proof.
We check equality, which is enough to show that cl ∶= ∑ i ∈ I cl A i is aclosure operation. First, note that each A i has an R -algebra map to A , socl A i ≤ cl A for all i ∈ I [R.G16, Proposition 3.6]. This gives the containmentcl ≤ cl A . For the other containment, let M be an R -module, and u ∈ cl A M .This means that 1 ⊗ u = A ⊗ R M . Since A is the direct limit of the A i ,there must exist i ∈ I such that 1 ⊗ u = A i ⊗ R M , which implies that u ∈ cl AI M ⊆ cl M . Since all of the closures in question are residual, this provesthe result. (cid:3) Proposition 4.4.
Let ( R, m ) be a Noetherian local ring. Let Γ be a directedposet. Let { cl j } j ∈ Γ be a directed set of residual functorial Nakayama closureoperations (that is, Γ is a directed set, and whenever j ≤ j ′ , we have cl j ≤ cl j ′ ).Then cl ∶= lim → j ∈ Γ cl j , provided it is idempotent (and hence a closure operation),is also residual, functorial, and Nakayama.Proof. Residual and functorial follow from [ER19, Proposition 7.2].To see the Nakayama property, let L ⊆ M be finitely generated R -modules,and assume L ⊆ ( m L ) cl M . Let z , . . . , z t be a generating set for L . Then thereexist j , . . . , j t ∈ Γ with z i ∈ ( m L ) cl ji M for each 1 ≤ i ≤ t . Choose j ∈ Γ with j ≥ j i for 1 ≤ i ≤ t (which exists since Γ is a directed set). Then each z i ∈ ( m L ) cl j M , whence we have L ⊆ ( m L ) cl j M since the z i generate L . But thensince cl j is a Nakayama closure, we have L ⊆ cl j M ⊆ cl M . (cid:3) Next we need a lemma that may be well known, but we don’t know areference so we include it and its proof for the reader’s convenience.
Lemma 4.5.
Let ( R, m ) and ( B, n ) be (not necessarily Noetherian) localrings, and let ϕ ∶ R → B be a local homomorphism. Then as an R -algebra, B is the direct limit, via a directed indexing set I , of local R -algebras B i thatare essentially of finite type over R , such that m B i ≠ B i for each i ∈ I .Proof. Let G be the collection of all finite subsets of B . Let Γ be an indexingset in bijective correspondence with G . For any X ∈ G , we write X = X i where i ∈ Γ is the index corresponding to X . We partially order Γ such that i ≤ j if and only if X i ⊆ X j . For each i ∈ Γ, set A i ∶= ϕ ( R )[ X i ] , i.e. thesubring of B generated by ϕ ( R ) and X i . Then it is clear that if i ≤ j , wehave a natural corresponding R -algebra map µ ij ∶ A i → A j , given by simpleinclusion. Now, for each i ∈ Γ, set n i ∶= A i ∩ n . Let B i ∶= ( A i ) n i , and whenever i ≤ j define ν ij ∶ B i → B j by ν ij ( a / s ) ∶= a / s .To see that ν ij is well-defined, note first that if s ∈ A i ∖ n i , we have s ∉ n ,whence s ∈ A j ∖ n = A j ∖ n j . Moreover, if a / s = b / t in B i , then there is some u ∈ A i ∖ n with uta = usb . But we have u, t, s ∈ A j ∖ n j , whence a / s = b / t in B j . Now for each i , define ν i ∶ B i → B in the same fashion. That is, ν i ( a / s ) = a / s , which is well-defined for the same reasons as above. Then it is elemen-tary that the B i , along with the maps ν ij and ν i , form a direct limit systemwith direct limit of B . (cid:3) Corollary 4.6.
Let ( R, m ) be a Noetherian local ring, and let ( B, n ) bea local R -algebra such that m B ≠ B . Then cl B is a Nakayama closure onfinitely generated R -modules.Proof. Construct the direct limit system of B i as in Lemma 4.5. Since each B i is essentially of finite type over the Noetherian ring R , it is itself Noe-therian. Moreover, by construction the homomorphisms ( R, m ) → ( B i , n i ) are local. Then by Theorem 4.1, each cl B i is a Nakayama closure over R .By Proposition 4.4, cl B is then Nakayama as well. (cid:3) Theorem 4.7.
Let ( R, m ) be a Noetherian local ring. Let B be an R -algebrasuch that m B is contained in the Jacobson radical of B . Then cl B is aNakayama closure on finitely generated R -modules.Proof. Let L ⊆ M be finitely generated R -modules. Suppose L ⊆ ( m L ) cl B M .Then for any maximal ideal P of B , we have cl B ≤ cl B P (since B P is a B -algebra), so L ⊆ ( m L ) cl BP M . Since cl B P is Nakayama by Corollary 4.6, wehave L ⊆ cl BP M . That is, for all maximal ideals P of B , we have, for all z ∈ L ,that z ⊗ = M ⊗ R B P = ( M ⊗ R B ) P . Since vanishing is a local property,it follows that z ⊗ = M ⊗ R B for all z ∈ L . That is, L ⊆ cl B M . (cid:3) Corollary 4.8.
Let ( R, m ) be a Noetherian local ring. Then Frobeniusclosure (in characteristic p > ) and plus closure (when R is complete) areNakayama closures, as is cl B for any big Cohen-Macaulay R -algebra B thatis local.Proof. Note that Frobenius closure is cl R / p ∞ and plus closure is cl R + . Eachof the algebras R / p ∞ and R + is local, with maximal ideal containing m , sowe can apply Theorem 4.7. (cid:3) Nakayama interiors are dual to Nakayama closures
In this section, we define a Nakayama interior and prove that the dual ofa Nakayama closure is a Nakayama interior. R will be a commutative ringwith additional hypotheses as specified. Definition 5.1.
Let i be an interior operation on the class of Artinian R -modules, where ( R, m ) is a Noetherian local ring. We say that i is Nakayama if whenever A ⊆ B are Artinian modules such that i ( A ∶ B m ) ⊆ A , we havei ( A ) = i ( B ) .We need the following presumably well-known fact: ORE-HULL DUALITY 15
Lemma 5.2. If R is any associative (not necessarily commutative) ringand A is a nonzero Artinian left R -module, then A contains a simple left R -module.Proof. If A doesn’t contain a simple module, then suppose there is a properdescending chain of length n in A : A ⊋ A ⊋ ⋯ ⊋ A n ≠
0. Since A n isnonzero and not simple, there is an A n + ≠ A n .Hence there is an infinite descending chain which is a contradiction since A is Artinian. (cid:3) This allows us to give an example of a Nakayama interior:
Lemma 5.3.
Let ( R, m ) be a Noetherian local ring. Then the identity op-eration is a Nakayama interior on the class of Artinian modules.Proof. Let A ⊆ B be Artinian modules such that ( A ∶ B m ) ⊆ A . If A ≠ B ,then B / A is a nonzero Artinian module, so it contains a nonzero simplesubmodule S . Let 0 ≠ x ∈ S . Then x = y + A ∈ B / A for some y ∈ B , and since S ≅ R / m as R -modules, we have m x =
0, which means that m y ⊆ A . That is, y ∈ ( A ∶ B m ) ∖ A , which contradicts the assumption. Hence A = B . (cid:3) As promised, we prove that the dual of a Nakayama closure is a Nakayamainterior. First we prove a lemma.
Lemma 5.4.
Let R be a complete Noetherian local ring. Let L ⊆ M beMatlis-dualizable R -modules (i.e. R -modules isomorphic to their doubleMatlis duals), let B ∶= M ∨ , and let A ⊆ B be the R -submodule of B suchthat A = ( M / L ) ∨ = { f ∈ M ∨ ∣ L ⊆ ker f } . Let I be an ideal of R . Then whenthought of as a submodule of M ∨ , we have ( M / IL ) ∨ = ( A ∶ B I ) .Proof. Let f ∈ M ∨ = Hom R ( M, E ) . We need to show that IL ⊆ ker f if andonly if for all g ∈ If , we have L ⊆ ker g .Accordingly, suppose that IL ⊆ ker f . Let g ∈ If . Then g = µf for some µ ∈ I . Then for any z ∈ L , we have g ( z ) = ( µf )( z ) = µ ⋅ f ( z ) = f ( µz ) = µz ∈ IL . Hence L ⊆ ker g .Conversely, suppose that L ⊆ ker g for all g ∈ If . Let z ∈ IL . Then wehave z = ∑ tj = µ j y j , where µ j ∈ I and y j ∈ L . Let g j = µ j ⋅ f for 1 ≤ j ≤ t . Notethat g j ∈ If , so that L ⊆ ker g j for all 1 ≤ j ≤ t . Then we have f ( z ) = f ( ∑ j µ j y j ) = ∑ j µ j ⋅ f ( y j ) = ∑ j ( µ j f )( y j ) = ∑ j g j ( y j ) = . Hence IL ⊆ ker f . (cid:3) Proposition 5.5.
Let ( R, m ) be a complete Noetherian local ring. Let cl bea residual closure operation on the category of finitely generated R -modules,and let i be the interior operation on the category of Artinian R -modulesgiven by i ( A ) = ( A ∨ cl A ∨ ) ∨ , i.e. the interior operation dual to cl . Then cl is aNakayama closure if and only if i is a Nakayama interior. Proof.
First assume cl is a Nakayama closure operation. Let A ⊆ B beArtinian R -modules such that i ( A ∶ B m ) ⊆ A . Let M = B ∨ , and set L ∶ = { f ∈ B ∨ ∣ A ⊆ ker f } ≅ ( B / A ) ∨ . Then by Lemma 5.4, ( A ∶ B m ) = ( M / m L ) ∨ . The fact that i ( A ∶ B m ) ⊆ A means then that ( M / L ) ∨ = A ⊇ i ( A ∶ B m ) = i (( M / m L ) ∨ ) ≅ ⎛⎝ M / m L cl M / m L ⎞⎠ ∨ = ( M /( m L ) cl M ) ∨ . Applying Matlis duality, we obtain M / L ↠ M /( m L ) cl M , which is to say that L ⊆ ( m L ) cl M . Then by the Nakayama closure property, we obtain L ⊆ cl M .This then implies M / L ↠ M / cl M . Applying Matlis duality, we have i ( B ) = ( M / cl M ) ∨ ⊆ ( M / L ) ∨ = A , as was to be shown.Conversely, assume i is a Nakayama interior operation, and let L ⊆ N ⊆ M be finitely generated R -modules such that N ⊆ ( L + m N ) cl M . Without lossof generality (by the residual property of cl), we may assume L =
0. Nowlet B = M ∨ and A = ( M / N ) ∨ ⊆ B . The fact that N ⊆ ( m N ) cl M means that M / N ↠ M /( m N ) cl M , which translates toi ( A ∶ B m ) = i (( M / m N ) ∨ ) = ⎛⎝ M / m N cl M / m N ⎞⎠ ∨ = ( M /( m N ) cl M ) ∨ ⊆ ( M / N ) ∨ = A. Then since i is a Nakayama interior, it follows that ( M cl M ) ∨ = ( B ∨ cl B ∨ ) ∨ = i ( B ) ⊆ A = ( M / N ) ∨ . Thus, we have M / N ↠ M / cl M , whence N ⊆ cl M . (cid:3) i-Expansions, co-generating sets, and core-hull duality We define i-expansions, discuss the duality between cl-reductions and i-expansions, define the i-hull, and incorporate the notion of co-generationin order to prove that the i-hull is dual to the cl-core. We will close thesection by applying these tools to define the Artinistic version of a submoduleselector; in turn, we use the definition to obtain a new characterization ofthe finitistic (i.e. classical) test ideal.
Definition 6.1.
Suppose R is an associative (not necessarily commutative)ring and A ⊆ B are left R -modules. Suppose i is an interior operation thatoperates on at least the submodules of B that contain A . We say C with A ⊆ C ⊆ B is an i -expansion of A in B if i ( A ) = i ( C ) . Setup:
If i is an interior operation on Artinian R -modules, set α = i ⌣ ,and cl to be the corresponding residual closure operation. ORE-HULL DUALITY 17
Lemma 6.2.
Let R be a complete Noetherian local ring. Let C ⊆ B be R -modules, and j ∶ C ↪ B the inclusion. Let π = j ∨ ∶ B ∨ → C ∨ and L ⊆ C .Then π − (( C / L ) ∨ ) = ( B / L ) ∨ . Proof.
Let g ∈ π − (( C / L ) ∨ ) , so that g ∈ Hom R ( B, E ) . Then π ( g ) ∈ ( C / L ) ∨ ⊆ C ∨ , i.e., g ○ j ∈ ( C / L ) ∨ . This implies that g ○ j kills L . Hence L ⊆ ker ( g ○ j ) , so L = L ∩ C ⊆ ker ( g ) . This implies that g ∈ ( B / L ) ∨ .Now suppose that g ∈ ( B / L ) ∨ . Then g ∈ Hom R ( B / L, E ) , so L ⊆ ker ( g ) .Then π ( g ) = g ○ j ∶ C → E must also kill L , so g ∈ π − (( C / L ) ∨ ) . (cid:3) Theorem 6.3.
Let R be a Noetherian local ring and A ⊆ B Matlis-dualizable R -modules. Let i be an interior operation, let α ∶ = i ⌣ , and let cl be thecorresponding residual closure operation. There exists an order reversingone-to-one correspondence between the poset of i -expansions of A in B andthe poset of cl -reductions of ( B / A ) ∨ in B ∨ . Under this correspondence, an i -expansion C of A in B maps to ( B / C ) ∨ , a cl -reduction of ( B / A ) ∨ in B ∨ .Proof. Let C be an i-expansion of A in B ; in other words, i ( C ) ⊆ A ⊆ C . Let π ∶ B ∨ → B ∨ /( B / C ) ∨ = C ∨ be the quotient map. Let α ∶ = i ⌣ , let cl be thecorresponding residual closure operation. We have: (( B / C ) ∨ ) cl B ∨ = π − ( α ( B ∨ /( B / C ) ∨ )) = π − ( α ( C ∨ )) = π − (( C / i ( C )) ∨ ) , where the last equality follows because α = i ⌣ . By Lemma 6.2, we have π − (( C / i ( C )) ∨ ) = ( B / i ( C )) ∨ . Since i ( C ) = i ( A ) , this is equal to ( B / i ( A )) ∨ , and the latter is equal to (( B / A ) ∨ ) cl B ∨ by the same argument used for C .Hence (( B / C ) ∨ ) cl B ∨ = (( B / A ) ∨ ) cl B ∨ . This implies that ( B / C ) ∨ is a cl-reduction of ( B / A ) ∨ ; in other words, ( B / C ) ∨ ⊆ ( B / A ) ∨ ⊆ (( B / C ) ∨ ) cl B ∨ , establishing one direction of the corre-spondence.Now let N ∶ = ( B / A ) ∨ ⊆ B ∨ = ∶ M and let L be a cl-reduction of N in M ;in other words, L ⊆ N ⊆ L cl M . We need to show ( M / L ) ∨ is an i-expansionof ( M / N ) ∨ in M ∨ . Note that the natural surjections M ↠ M / L ↠ M / N yield inclusions ( M / N ) ∨ ↪ ( M / L ) ∨ ↪ M ∨ , so that the above make sense. Accordingly, since L cl M = N cl M , we havei (( M / L ) ∨ ) = α ⌣ (( M / L ) ∨ ) = ( M / Lα ( M / L ) ) ∨ = ( M / LL cl M / L ) ∨ = ( M / L cl M ) ∨ = ( M / N cl M ) ∨ = ( M / Nα ( M / N ) ) ∨ = α ⌣ (( M / N ) ∨ ) = i (( M / N ) ∨ ) (cid:3) Proposition 6.4.
Let ( R, m , k ) be a complete Noetherian local ring. Let A ⊆ B be Artinian R -modules and i a Nakayama interior defined on Artinian R -modules. Maximal i -expansions of A exist in B . In fact, if C is an i -expansion of A in B , then there is some maximal expansion D of A in B such that A ⊆ C ⊆ D ⊆ B .Proof. Recall that the dual statement (Proposition 2.9) holds for minimalcl-reductions and finitely generated R -modules.By Theorem 6.3, ( B / C ) ∨ is a cl-reduction of ( B / A ) ∨ in B ∨ . Hence byProposition 2.9, there is some minimal cl-reduction U of ( B / A ) ∨ in B ∨ suchthat U ⊆ ( B / C ) ∨ . Let D ∶ = ( B ∨ / U ) ∨ . Then by Theorem 6.3, D is ani-expansion of A in B , and clearly C ⊆ D .Now suppose that D ′ is an i-expansion of A in B with D ⊆ D ′ . ByTheorem 6.3, we then have that ( B / D ′ ) ∨ is a cl-reduction of ( B / A ) ∨ in B ∨ ,and we have ( B / D ′ ) ∨ ⊆ ( B / D ) ∨ = U . By minimality of U , we therefore have ( B / D ′ ) ∨ = U = ( B / D ) ∨ , whence D = D ′ . Thus, D is maximal. (cid:3) We now show that Proposition 6.4 also holds for Noetherian R -modules,and in far greater generality. Proposition 6.5.
Let R be an associative (i.e. not necessarily commutative)ring with identity. Let L ⊆ M be (left) R -modules, and let i be an interioroperation on submodules of M . Let U be an i -expansion of L in M . Assume M / U is Noetherian. Then there is an R -module N with U ⊆ N ⊆ M , suchthat N is a maximal i -expansion of L in M .Proof. Let S be the set of all R -modules D such that U ⊆ D ⊆ M and D is ani-expansion of L . Since U ∈ S , we have S ≠ ∅ . Since M / U is Noetherian and S corresponds to a nonempty collection of submodules of M / U , S containsa maximal element N . Moreover, if N ′ is an i-expansion of L in M such that N ⊆ N ′ , then in particular U ⊆ N ′ ⊆ M , so that N ′ ∈ S . Then by maximalityof N , we have N = N ′ . Hence N is a maximal i-expansion of L in M . (cid:3) Next we describe a dual to the notion of a generating set. This will enableus to dualize a property of minimal reductions, namely that if L is a minimalreduction of N in M , a minimal generating set for L extends to a minimalgenerating set for N . Definition 6.6.
Let R be a Noetherian local ring, L an R -module, and g , . . . , g t ∈ L ∨ . We say that the quotient of L co-generated by g , . . . , g t is L / ( ⋂ i ker ( g i )) . ORE-HULL DUALITY 19
We say that L is co-generated by g , . . . , g t if ⋂ i ker ( g i ) = L is minimal if it is irredundant, i.e.,for all 1 ≤ j ≤ t , ⋂ i ≠ j ker ( g i ) ≠ Lemma 6.7.
Let ( R, m , k ) be a Noetherian local ring and E = E R ( k ) . Let V be an R -module such that m V = and g ∶ V → E an R -linear map. Then im ( g ) is contained in the unique copy of k in E .Proof. We have m ⋅ im ( g ) = , so im ( g ) ⊆ soc E . Since the socle of E is equalto this copy of k inside of E , we get the desired result. (cid:3) Lemma 6.8.
Let R be a Noetherian local ring, and A an R -module. Assume A is Matlis-dualizable (e.g. if A is Artinian). The elements g , . . . , g t ∈ A ∨ co-generate A if and only if they generate A ∨ .Proof. Let ψ ∶ R ⊕ t → A ∨ be the map given by the row matrix [ g g ⋯ g t ] .This induces a dual map ψ ∨ ∶ A ∨∨ → E ⊕ t . Let j ∶ A → A ∨∨ be the bidual-ity map, which is an isomorphism by the hypotheses of the lemma. Then ϕ ∶ = ψ ∨ ○ j ∶ A → E ⊕ t is given by the column matrix ⎡⎢⎢⎢⎢⎢⎢⎢⎣ g g ⋮ g t ⎤⎥⎥⎥⎥⎥⎥⎥⎦ , sending each a ∈ A to ( g ( a ) , . . . , g t ( a )) . We have ker ϕ = ⋂ tj = ker g j . Hence, we have g , . . . , g t generate A ∨ ⇐⇒ ψ is surjective ⇐⇒ ψ ∨ is injective ⇐⇒ ϕ = ψ ∨ ○ j is injective ⇐⇒ ker ϕ = ⇐⇒ ⋂ j ker g j = ⇐⇒ g , . . . , g t co-generate A. (cid:3) Remark 6.9.
One could obtain as a corollary that a Matlis-dualizable mod-ule is finitely co-generated if and only if it is Artinian. However, in [V´am68]it is shown that over a Noetherian ring, this equivalence holds for any mod-ule. Note that V´amos uses the term “finitely embedded” for what we andothers (see e.g. [Lam99]) call “finitely co-generated”.
Lemma 6.10.
Let ( R, m , k ) be a Noetherian local ring. Let A be an Artinian R -module, and let g , . . . , g t ∈ A ∨ . The following are equivalent:(1) g , . . . , g t is a co-generating set for A ,(2) The restrictions of the g i to soc A span Hom R ( soc A, k ) as a k -vectorspace.Proof. (1) ⇒ (2): Let h ∈ Hom R ( soc A, k ) , and let j ∶ k → E be the naturalinclusion. Since E is injective, there exists a map ˜ h ∶ A → E extending j ○ h ∶ soc A → E . By Lemma 6.8, since the g , . . . , g t co-generate A and since A is Artinian, we have that g , . . . , g t generate A ∨ . Hence ˜ h = ∑ ti = r i g i forsome r i ∈ R . Restricting to soc A , we get j ○ h = ˜ h ∣ soc A = t ∑ i = ( r i g i )∣ soc A = t ∑ i = r i ( g i ∣ soc A ) . By Lemma 6.7, the images of the g i ∣ soc A are all inside of the copy of k insideof E , and so h = ∑ ti = ¯ r i ( g i ∣ soc A ) , as desired.(2) ⇒ (1): Set B = ⋂ i ker ( g i ) ⊆ A . Let x ∈ soc B ⊆ soc A . Then g i ∣ soc A ( x ) = ≤ i ≤ t . Since the g i ∣ soc A span Hom R ( soc A, k ) asa k -vector space, this implies that x =
0. Hence soc B =
0, which by Re-mark 5.2 means that B =
0, since B is Artinian. Therefore, g , . . . , g t is aco-generating set for A . (cid:3) Proposition 6.11.
Let R be a Noetherian local ring, and let g , . . . , g t be aco-generating set of an Artinian R -module A . The following are equivalent:(1) g , . . . , g t is a minimal co-generating set for A ,(2) The restrictions of the g i to soc A form a basis for Hom R ( soc A, k ) as a k -vector space.Proof. First we prove that (1) implies that the g i ∣ soc A are linearly indepen-dent. We already know that they are a spanning set for Hom R ( soc A, k ) from Lemma 6.10. Suppose ∑ ti = ¯ r i ( g i ∣ soc A ) =
0, with the ¯ r i ∈ k and at leastone ¯ r j ≠
0. Without loss of generality, suppose that ¯ r t ≠
0, and by multiply-ing by an appropriate unit that ¯ r t = − ¯1. Then we can rewrite our equationas g t ∣ soc A = t − ∑ i = ¯ r i ( g i ∣ soc A ) . We have ker ( g t ∣ soc A ) ⊆ ⋂ t − i = ker ( g i ∣ soc A ) . Hence t − ⋂ i = ker ( g i ∣ soc A ) = t ⋂ i = ker ( g i ∣ soc A ) = , which contradicts our hypothesis that g , . . . , g t is a minimal co-generatingset for A .Next we prove that (2) implies that g , . . . , g t are a minimal co-generatingset for A . We already know that they are a co-generating set for A byLemma 6.10, so it suffices to prove minimality. Suppose without loss ofgenerality that ⋂ t − i = ker ( g i ) =
0. Then g , . . . , g t − also form a co-generatingset for A , so they generate A ∨ . Hence g t = ∑ t − i = r i g i for some r i ∈ R . Thisimplies that g t ∣ soc A = ∑ t − i = ¯ r i ( g i ∣ soc A ) , which contradicts the hypothesis thatthe g i ∣ soc A are linearly independent. (cid:3) Lemma 6.12. [HRR02, Theorem 2.3]
Let ( R, m ) be a Noetherian local ringand L ⊆ M be finitely generated R -modules. The following are equivalent.(1) L ∩ m M = m L .(2) Any minimal generating set of L extends to a minimal generatingset for M . ORE-HULL DUALITY 21
Remark 6.13.
We have LL ∩ m M ≅ L + m M m M , the second of which is the image of the vector space L / m L in M / m M . So(1) means the map L / m L → M / m M is injective. Proposition 6.14.
Let ( R, m ) be a Noetherian local ring and i a Nakayamainterior on Artinian R -modules. Let A ⊆ B Artinian R -modules. Supposethat C ⊆ D are i -expansions of A in B , with D a maximal i -expansion. Thenany minimal co-generating set of B / D extends to a minimal co-generatingset for B / C .Proof. Given the setup of the statement of the proposition, we have: ( B / D ) ∨ ⊆ ( B / C ) ∨ ⊆ ( B / A ) ∨ ⊆ B ∨ , with ( B / D ) ∨ a minimal cl-reduction of ( B / A ) ∨ in B ∨ by Theorem 6.3. ByProposition 2.9, any minimal set of generators of ( B / D ) ∨ extends to a min-imal set of generators for ( B / C ) ∨ , where said modules are being consideredover ˆ R . Given that a minimal set of generators for ( B / D ) ∨ is a minimalcogenerating set for B / D in B by Lemma 6.8, and the same holds with D replaced by C , this gives the desired result. (cid:3) Lemma 6.15.
Let R be a complete Noetherian local ring. Let B be an R -module such that it and all of its quotient modules are Matlis-dualizable. Let { C i } i ∈ I a collection of submodules of B . Then ( B ∑ i C i ) ∨ ≅ ⋂ i ( B / C i ) ∨ and ( B ⋂ i C i ) ∨ ≅ ∑ i ( B / C i ) ∨ , where all the dualized modules are considered as submodules of B ∨ .Proof. Note that elements of the left hand side of the first isomorphism aremaps from B → E R ( k ) whose kernel contains ∑ i C i . Elements of the righthand side are maps from B → E R ( k ) whose kernel contains C i for all i ∈ I ,which proves the first isomorphism.For the second isomorphism, we will apply Matlis duality to the first iso-morphism. Namely, let M = B ∨ , and L i ∶ = ( B / C i ) ∨ for each i ∈ I , consideredas a submodule of B ∨ . Then we have B ⋂ i C i = M ∨ ⋂ i ( M / L i ) ∨ = M ∨ ( M / ∑ i L i ) ∨ = ( ∑ i L i ) ∨ = ( ∑ i ( B / C i ) ∨ ) ∨ . One more application of Matlis duality to the start and end of the chain ofequalities then yields the second isomorphism. (cid:3)
Definition 6.16.
Let R be an associative (not necessarily commutative)ring and i an interior operation defined on a class of (left) R -modules M . If A ⊆ B are elements of M , the i -hull of a submodule A with respect to B isthe sum of all i-expansions of A in B , ori -hull B ( A ) ∶ = ∑ i ( C )⊆ A ⊆ C ⊆ B C. Theorem 6.17.
Let R be a complete Noetherian local ring. Let A ⊆ B beArtinian R -modules, and let i be a Nakayama interior defined on Artinian R -modules. Then the i -hull of A in B is dual to the cl -core of ( B / A ) ∨ in B ∨ , where cl is the closure operation dual to i .Proof. Let M = B ∨ and N = ( B / A ) ∨ . We need to show that ( M / cl -core M ( N )) ∨ = i -hull B ( A ) . This follows from the definition of cl -core, i -hull, and Lemma 6.15. (cid:3)
We conclude the section by exploring the dual to the concept of the finitelygenerated version of a submodule selector (see Definition 3.1). Moreover, weshow how this dual notion relates to Theorem 3.3.
Definition 6.18.
Let R be a complete Noetherian local ring. Let α bea submodule selector on a class of R -modules that is closed under takingsubmodules and quotient modules. For any fixed R -module M ∈ M , if N is a submodule of N , for now we denote π N ∶ M ↠ M / N to be the naturalsurjection. The Artinistic version α f of α is defined as α f ( M ) ∶ = ⋂ { π − N ( α ( M / N )) ∶ M / N is Artinian } . Recall (cf. Remark 6.9) that it is equivalent to take the intersection overall finitely co-generated quotients.
Theorem 6.19.
Let R be a complete Noetherian local ring. Let M be aclass of Matlis-dualizable R -modules that is closed under taking submodulesand quotient modules. Let α be a preradical on M . Then ( α f ) ⌣ = ( α ⌣ ) f .Proof. Let A ∈ M ∨ . We have ( α f ) ⌣ ( A ) = ( A ∨ α f ( A ∨ ) ) ∨ = ⎛⎜⎜⎜⎝ A ∨ ∑ N ⊆ A ∨ , N f.g. α ( N ) ⎞⎟⎟⎟⎠ ∨ . By Lemma 6.15 and Matlis duality, the above is equal to ⋂ N ⊆ A ∨ , N f.g. ( A ∨ / α ( N )) ∨ = ⋂ N ⊆ A ∨ , N f.g. π − ( A ∨ / N ) ∨ (( N / α ( N )) ∨ ) . By the usual one-to-one correspondence between submodules of A and quo-tient modules of A ∨ , this is equal to ⋂ B ⊆ A, ( A / B ) ∨ f.g. π − B (( ( A / B ) ∨ α (( A / B ) ∨ ) ) ∨ ) = ⋂ B ⊆ A, ( A / B ) ∨ f.g. π − B ( α ⌣ ( A / B )) . ORE-HULL DUALITY 23
But by Definition 6.18 and Lemma 6.8, the latter equals ( α ⌣ ) f ( A ) . (cid:3) Hence, Theorem 3.3 can be reinterpreted as a statement about the Ar-tinistic version of the interior operation dual to a closure operation. In theparticular case of tight closure, it allows us to extend the interpretation ofthe big test ideal in terms of maps from R / p e developed in [ES14] to acomparable interpretation of the finitistic tight closure test ideal, as follows: Theorem 6.20.
Let R be a complete Noetherian local F -finite reduced ringof prime characteristic p > . Let c be a big test element for R . Then thefinitistic test ideal of R consists of those elements a ∈ R such that for every m -primary ideal J of R and every nonnegative integer e ≥ , there is an R -linear map g ∶ R / p e → R / J with g ( c / p e ) = a + J .Proof. Let α ( M ) = ∗ M for any R -module M . Let c ∈ R be a big test elementfor R . We have the following sequence of equalities, which is justified below: τ fg ( R ) = ( α f ) ⌣ ( R ) = ( α ⌣ ) f ( R ) = ⋂ { π − I ( α ⌣ ( R / I )) ∶ R / I finitely co-generated } = ⋂ { π − I ( α ⌣ ( R / I )) ∶ λ ( R / I ) < ∞ } = ⋂ { π − I (( R / I ) ∗ R ) ∶ λ ( R / I ) < ∞ } = { a ∈ R ∶ a + I ∈ ( R / I ) ∗ R whenever λ ( R / I ) < ∞ } = { a ∈ R ∶ a + I ∈ tr c / q ,R / q ( R / I ) , ∀ finite colength I, ∀ q } . We justify the steps of this proof one by one. The first equality is by The-orem 3.3 (or [ER19, Theorem 5.5]). The second equality is by Theorem 6.19.The equality on the second line is by definition.To see the equality on the third line, note that R / I is finitely co-generatedif and only if ( R / I ) ∨ is finitely generated (by Lemma 6.8). But ( R / I ) ∨ ≅ ann E ( I ) is finitely generated if and only if it is of finite length, since itis already Artinian. But of course ( R / I ) ∨ has finite length if and only if λ ( R / I ) < ∞ .The equality on the fourth line follows from [ES14, Corollary 3.6], whichin our terminology says that tight interior (in the sense given in [ES14]) issmile-dual to tight closure. The equality on the fifth line is by definition of π I . The equality on the final line then follows from [ES14, Theorem 2.5]. (cid:3) Core and hull comparisons for known closure operationsand their dual interiors
In this section we extend many of the results of [FV10] on cl-spread tothe module setting, proving that liftable integral spread exists along the way,and then prove dual results for i-hulls.
Definition 7.1.
Let R be an associative, not necessarily commutative ring.If cl and cl are both closure operations on a class of left R -modules M , we say that cl ≤ cl if N cl M ⊆ N cl M for all R -modules M ∈ M and all submodules N ⊆ M .The following proposition generalizes [FV10, Lemma 3.3] to the modulesetting. Proposition 7.2.
Let R be a local ring. If cl ≤ cl are closure operationsdefined on the class of finitely generated R -modules M with cl Nakayama,and if L is a cl -reduction of N in M , then there exists a minimal cl -reduction K of N with K ⊆ L .Proof. Notice for all cl -reductions L of N in M , L ⊆ N ⊆ L cl M . Since cl ≤ cl , L cl M ⊆ L cl M for all submodules L ⊆ M . Hence L ⊆ N ⊆ L cl M ⊆ L cl M . So L is acl -reduction of N . Now by Proposition 2.9, there is a minimal cl -reduction K ⊆ L of N in M. (cid:3) As [FV10, Proposition 3.4] does for ideals, we use this result to establisha containment between the cl -core and the cl -core of N in M . Proposition 7.3.
Let R be a local ring and cl ≤ cl be closure operationsdefined on the class of finitely generated R -modules M with cl Nakayama.If N ⊆ M are R -modules in M , then cl -core M ( N ) ⊆ cl -core M ( N ) .Proof. For any submodule N ⊆ M in M , cl -core M ( N ) = ⋂ L ⊆ N ⊆( L ) cl1 M L .By Proposition 7.2, for every cl -reduction of N in M , there is a minimalcl -reduction L of N in M such that L ⊆ L . Nowcl -core M ( N ) ⊆ ⋂ L ⊆ L ⊆ N ⊆( L ) cl2 M L , and ⋂ L ⊆ L ⊆ N ⊆( L ) cl2 M L ⊆ ⋂ L ⊆ N ⊆( L ) cl1 M L = cl -core M ( N ) . (cid:3) The next corollary extends Corollary 3.5 of [FV10] to the module setting.
Corollary 7.4.
Let R be a Noetherian local ring of characteristic p , and let N ⊆ M be R -modules. Then z -core M ( N ) ⊆ ∗ -core M ( N ) ⊆ F -core M ( N ) . Proof.
It is clear from the framework where Frobenius closure is first in-troduced [HH90, Section 10] that the Frobenius closure of a submodule isalways contained in its tight closure. That the tight closure of a submoduleis always contained in its liftable integral closure follows from [EU, Proposi-tion 2.4 (5)]. Now the result follows directly from Proposition 7.3 since N FM ⊆ N ∗ M ⊆ N z M for all R -submodules N ⊆ M . (cid:3) ORE-HULL DUALITY 25
Definition 7.5.
Let R be a Noetherian local ring and M a finitely generated R -module such that cl is defined on submodules of M . A submodule N ⊆ M is said to have cl-spread if all the minimal cl-reductions of N in M have thesame minimal number of generators. In this case, we denote this commonnumber by ℓ cl M ( N ) and call it the cl-spread of N in M .Next we extend [FV10, Proposition 3.7] to the module setting. Proposition 7.6.
Let ( R, m ) be a Noetherian local ring. Let cl ≤ cl be Nakayama closure operations defined on a class of finitely generated R -modules M . If N ⊆ M ∈ M and the cl - and cl -spread of N in M exist,then ℓ cl M ( N ) ≥ ℓ cl M ( N ) .Proof. Let L be a minimal cl -reduction of N in M . Then µ ( L ) = ℓ cl M ( N ) .Since L ⊆ N ⊆ L cl M ⊆ L cl M , then L is a cl -reduction of N in M (but notnecessarily a minimal cl -reduction of N ). By Proposition 2.9, ℓ cl M = µ ( L ) ≥ ℓ cl M ( N ) . (cid:3) Remark 7.7.
This leads to the question: when does the cl-spread exist?The first named author showed in [Eps10] that whenever R is an excellentand analytically irreducible local domain of prime characteristic p > N ⊆ M with M finitely generated, both the ∗ -spread ℓ ∗ M ( N ) and the F -spread ℓ FM ( N ) exist. In the next result we prove that the liftable integralspread typically exists and agrees with the analytic spread, the integralclosure spread originally defined for ideals. Theorem 7.8.
Let ( R, m , k ) be a Noetherian local ring such that k is infinite.Assume that either R is Z -torsion free (e.g. if it is of equal characteristic 0)or that it is unmixed and generically Gorenstein (e.g. if it is reduced). Thenthe liftable integral spread exists.In particular, if L ⊆ M are finitely generated R -modules and π ∶ F ↠ M is a minimal surjection from a finitely generated free module F , then theliftable integral spread of L in M is the analytic spread of π − ( L ) in thesense of Eisenbud-Huneke-Ulrich.Proof. First assume that M is itself a finitely generated free module, sothat π is the identity map. By [EHU03, Theorem 0.3], integrality in theirsense coincides with integrality within the symmetric algebra of M underour hypotheses. That is, say M = ∑ ni = Rx i , where the x i ∈ M are linearlyindependent over R . Then any submodule U of M is generated by R -linearcombinations of the x i . Denote by R [ U ] the R -subalgebra of the polynomialring R [ x , . . . , x n ] generated by a generating set for U . Then for submodules U ⊆ L ⊆ M , we have L ⊆ U z M if and only if the induced map R [ U ] → R [ L ] ismodule-finite.Let ℓ = dim ( R [ L ]/ m R [ L ]) . We claim that every minimal z -reductionof L in M is ℓ -generated. Since we know minimal z -reductions exist (seeProposition 2.9), it will be enough to show the following two things: (1) No z -reduction of L in M can be generated by fewer than ℓ elements,and(2) Every z -reduction of L in M contains a z -reduction that is ℓ -generated.First suppose that L contains a z -reduction U of M with µ ( U ) = t < ℓ .Then R [ L ] is module-finite over R [ U ] (which is t -generated as an R -algebra),whence R [ L ]/ m R [ L ] is module-finite over R [ U ]/( m R [ L ] ∩ R [ U ]) . But thelatter is at most t -generated as a k -algebra, so dim ( R [ U ]/( m R [ L ] ∩ R [ U ])) ≤ t . On the other hand, the module-finiteness shows that dim ( R [ U ]/( m R [ L ] ∩ R [ U ])) = ℓ > t , and we have a contradiction that proves (1).For (2), let U be a z -reduction of L in M . We have that A ∶ = R [ U ]/( m R [ L ] ∩ R [ U ]) is a standard graded k -algebra, with k an infinite field, and by theabove reasoning about module-finiteness, we have dim A = ℓ . Then by thegraded Noether normalization theorem [BH97, Theorem 1.5.17], there existalgebraically independent degree one elements a , . . . , a ℓ ∈ A such that A is module-finite over k [ a , . . . , a ℓ ] . Choose elements u j ∈ U whose residueclasses mod m R [ L ] ∩ R [ U ] are the a j . Let V ∶ = ∑ ℓj = Ru j . By the module-finiteness condition, we have A s = k [ a , . . . , a ℓ ] s for s ≫
0, thought of asfinite dimensional vector spaces over k . By Nakayamas lemma, it followsthat R [ V ] s = R [ U ] s for s ≫
0, whence R [ U ] is module-finite over R [ V ] .Since module-finiteness is transitive, R [ L ] is also module-finite over R [ V ] .Hence L has an ℓ -generated z -reduction contained in U , namely V .Now we examine the general case. Let π ∶ F ↠ M be as in the statementof the theorem, so that ker π ⊆ m F . Let ˜ L = π − ( L ) , and let U be a minimal z -reduction of L in M . Then ˜ U = π − ( U ) is a minimal z -reduction of ˜ L in F . Hence by the above, we have ˜ U ∩ m F = m ˜ U , so since ker π ⊆ m F , we have˜ U ∩ ker π ⊆ m ˜ U . Thus, since U ≅ ˜ U /( ker π ∩ ˜ U ) , we have µ ( U ) = dim k ( U / m U ) = dim k ( ˜ U /( ker π ∩ ˜ U ) m ( ˜ U /( ker π ∩ ˜ U )) ) = dim k ( ˜ U /( m ˜ U + ( ker π ∩ ˜ U ))) = dim k ( ˜ U / m ˜ U ) = µ ( ˜ U ) . (cid:3) Now we can extend Corollary 3.8 of [FV10] to modules, using liftableintegral closure as our integral closure on modules.
Corollary 7.9.
Let ( R, m ) be an excellent, analytically irreducible domainof characteristic p > , with infinite residue field. Then for all finitely gen-erated modules N ⊆ M , ℓ z M ( N ) ≤ ℓ ∗ M ( N ) ≤ ℓ FM ( N ) . Now we explore similar results on expansions and hulls, using the dualitybuilt up in Section 6. In particular, we discuss relationships between ex-pansions and hulls for the interior operations dual to Frobenius, tight, andliftable integral closure.
ORE-HULL DUALITY 27
Definition 7.10.
Let R be an associative but not necessarily commutativering. Let i and i be interior operations defined on a class M of R -modules.We say that i ≤ i if for all M ∈ M , i ( M ) ⊆ i ( M ) .Using the notion of Nakayama interior, we derive similar statements toPropositions 7.2 and 7.3 for the containments of i-expansions and i-hulls. Proposition 7.11.
Let ( R, m ) be a Noetherian local ring and i ≤ i be in-terior operations on the class of Artinian R -modules M with i a Nakayamainterior then any i -expansion of A in B is contained in a maximal i -expansion of A in B .Proof. Suppose A ⊆ C ⊆ B and C is an i -expansion of A in B . Thusi ( C ) ⊆ A ⊆ C . Since i ( C ) ⊆ i ( C ) ⊆ A ⊆ C , C is also an i -expansion of A in B . Finally, C must be contained in a maximal i -expansion of A. (cid:3) Proposition 7.12.
Let R be an associative (i.e. not necessarily commuta-tive) ring and i ≤ i interior operations on a class M of (left) R -modules.Let A ⊆ B be R -modules such that i and i are defined on all R -modulesbetween A and B . Then i -hull B ( A ) ⊆ i -hull B ( A ) .Proof. Let C be an i -expansion of A in B . Then we havei ( C ) ⊆ i ( C ) ⊆ A ⊆ C, whence C is an i -expansion of A in B . Hencei -hull B ( A ) = ∑ i ( C )⊆ A ⊆ C ⊆ B C ⊆ ∑ i ( D )⊆ A ⊆ D ⊆ B D = i -hull B ( A ) . (cid:3) Corollary 7.13.
Let ( R, m ) be a Noetherian local F -finite ring of charac-teristic p and A ⊆ B Artinian R -modules, then F -hull B ( A ) ⊆ ∗ -hull B ( A ) ⊆ z -hull B ( A ) . Proof.
First note that the first named author and Ulrich showed that ∗ ≤ z [EU] as closure operations. Now by [ER19, Proposition 7.5], z ⌣ ≤ ∗ ⌣ ≤ F ⌣ or in other words, the liftable integral interior is less than or equal to thestar interior, which is less than or equal to the Frobenius interior. Hence byProposition 7.12 we obtain F -hull B ( A ) ⊆ ∗ -hull B ( A ) ⊆ z -hull B ( A ) . (cid:3) We discuss cases where we can say more about the integral and ∗ -hull. Proposition 7.14.
Let ( R, m ) be a complete Noetherian local equidimen-sional ring having no embedded primes. Assume dim R ≥ . Then anyfinitely generated free module F has liftable integral interior equal to zero.Hence, for any finitely generated free module F and any submodule L ⊆ F ,the liftable integral hull of L in F is F . In particular, the liftable integralhull of any ideal is the unit ideal. Proof.
Let α ( M ) denote the liftable integral closure of 0 in M , and β beits dual interior operation. Let F = R t be a free module of rank t . By [EU,Theorem 5.1], we have 0 z E = E . Then β ( F ) = α ⌣ ( F ) = ( F ∨ α ( F ∨ ) ) ∨ = ( E t α ( E ) t ) ∨ = ∨ = . (cid:3) Next, we note the following general fact about the interior of a local ring.
Lemma 7.15.
Let ( R, m ) be a local (not necessarily Noetherian) commu-tative ring and let i be an interior operation defined at least on ideals of R .Let J be an ideal of R . Then i ( R ) ⊆ J ⇐⇒ i -hull ( J ) = R .Proof. First, if i ( R ) ⊆ J , then R is an i-expansion of J , whence i -hull ( J ) ⊇ R .Conversely, suppose i -hull ( J ) = R . Then 1 ∈ i -hull ( J ) , the sum of thei-expansions of J , whence there are i-expansions K , . . . , K t of J such that1 ∈ ∑ tj = K j . Say 1 = ∑ tj = a j , a j ∈ K j . Then there is some i with a i ∉ m (otherwise the sum is in m , but the sum is 1), whence a i is a unit and K i = R . That is, R is an i-expansion of J , which means that i ( R ) ⊆ J . (cid:3) In dimension 1, we have the following consequence:
Proposition 7.16.
Let ( R, m ) be a 1-dimensional Cohen-Macaulay ap-proximately Gorenstein complete Noetherian local ring, with infinite residuefield. Let J be an ideal of R . Let C R be the conductor of R . Then C R ⊆ J ⇐⇒ z -hull ( J ) = R .Proof. By [EU, Theorem 4.4], C R is the uniform annihilator of the modules I − / I for all ideals I of R . Since any integrally closed ideal is the intersectionof integrally closed m -primary ideals [HS06, Corollary 6.8.5], C R is thenalso equal to the uniform annihilator of the modules I − / I for finite colengthideals I of R by [ER19, Theorem 5.5]. Since R is approximately Gorenstein,an appeal to Theorem 3.3, along with the fact that liftable integral closureis finitistic [EU, Lemma 2.3], shows that C R is the liftable integral interiorof R . An application of Lemma 7.15 finishes the proof. (cid:3) In the next result, τ will denote the tight closure test ideal of R . Proposition 7.17.
Let ( R, m ) be an F -finite ring of characteristic p > ,then ∗ -hull ( I ) = R if and only if τ ⊆ I .Proof. By [ES14, Proposition 2.3], the tight interior of R is τ . Then theresult follows from Lemma‘7.15.. (cid:3) Definition 7.18.
Let ( R, m ) be a Noetherian local ring. Let i be an interioroperation defined on a class of Artinian R -modules M . Let A ⊆ B beArtinian R -modules. We define the i -co-spread ℓ B i ( A ) of A to be the minimalnumber of cogenerators of B / C of any maximal i-expansion C of A , if thisnumber exists. ORE-HULL DUALITY 29
Proposition 7.19.
Let ( R, m ) be a Noetherian local ring and i a Nakayamainterior operation defined on a class of Artinian R -modules. Let cl be theclosure operation dual to i . Let A ⊆ B be Artinian R -modules, M = B ∨ ,and N = ( B / A ) ∨ . If the cl -spread ℓ cl M ( N ) of N in M exists, then the i -co-spread ℓ Bi ( A ) of A in B exists. In particular, the tight interior co-spread andFrobenius interior co-spread exist under the hypotheses of Remark 7.7 andthe liftable integral interior co-spread exists under the hypotheses of Theorem7.8.Proof. Let C be a maximal i-expansion of A in B . By the proof of Proposi-tion 6.4, ( B / C ) ∨ is a minimal cl-reduction of N in M . Since the cl-spreadof N in M exists, ( B / C ) ∨ is minimally generated by ℓ cl M ( N ) elements. ByLemma 6.8, B / C is minimally co-generated by ℓ cl M ( N ) elements. Since thisholds for every maximal i-expansion C of A in B , the i-co-spread ℓ Bi ( A ) exists.The last sentence of the Proposition follows immediately. (cid:3) Proposition 7.20.
Let ( R, m ) be a Noetherian local ring and i ≤ i beNakayama interior operations on the class of Artinian R -modules M suchthat ℓ Bi ( A ) and ℓ Bi ( A ) exist. Then ℓ B i ( A ) ≤ ℓ B i ( A ) .Proof. Suppose A ⊆ C ⊆ B and C is a maximal i -expansion of A in B . Then ℓ B i ( A ) is the minimal number of cogenerators of B / C . By Proposition 7.11there exists A ⊆ C ⊆ D ⊆ B with D a maximal i -expansion of A . Thus anyminimal cogenerating set of B / D extends to a minimal cogenerating set of B / C by Proposition 6.14. Hence ℓ B i ( A ) ≥ ℓ B i ( A ) . (cid:3) Corollary 7.21.
Let ( R, m ) be a Noetherian local ring of characteristic p > satisfying the hypotheses of both Remark 7.7 and Theorem 7.8 and A ⊆ B Artinian R -modules, then ℓ BF ( A ) ≤ ℓ B ∗ ( A ) ≤ ℓ B z ( A ) . Computations of interiors and hulls
To illustrate Theorem 3.3 in action, we construct the tight and Frobeniusinteriors of some ideals in certain nice rings of prime characteristic. Havingdone this, we then compute the hulls of some of these ideals.In order to use the theorem to compute tight interiors, we need a resulttelling us when the tight interior equals its Artinistic version. To that end,we repurpose work from two papers of Lyubeznik and Smith from aroundthe turn of the century.
Theorem 8.1.
Let ( R, m ) be a complete reduced F -finite local ring, and I an ideal of R . Suppose either(1) There is a positively graded N -graded algebra A over a field K , withgraded maximal ideal n , and a homogeneous ideal J of A , such that R = ̂ A n and I = J R , or(2) R is an isolated singularity. Then ann E ( I ) ∗ E = ann E ( I ) ∗ fgE . Hence the tight interior of I and the Artin-istic version of the tight interior of I coincide.Proof. First we show that in both cases (1) and (2), we have ( ann E I ) ∗ E = ( ann E I ) ∗ fgE .First assume we are in case (1). Clearly K must be F -finite. By [LS99,Theorem 3.3], we have ( ann E A ( A / n ) J ) ∗ E A ( A / n ) = ( ann E A ( A / n ) J ) ∗ fgE A ( A / n ) (us-ing here also the fact that ann E A ( A / n ) J must be a graded submodule of E A ( A / n ) ). But E = E A ( A / n ) , and by Hom-tensor adjointness we haveann E I = ann E J . Set D to be this module. The tight closure and finitistictight closure of 0 in G as A -modules coincide. By persistence of tight closurethis tight closure is contained in the finitistic tight closure of 0 in G as an R -module. In turn, this is contained in the ordinary tight closure of 0 in G as an R -module. Hence it is enough to show that the latter is equal (asa subset of G ) to the tight closure of 0 in G as an A -module. But since G ∶ = E / D is Artinian, every element of A ∖ n acts as a unit on it, and its A n -module structure coincides with its R -module structure. Hence, we haveour result.On the other hand, suppose we are in case (2). Let D ∶ = ann E I . By [LS01,Theorem 8.12], 0 ∗ E / D = ∗ fgE / D . Since both ∗ and ∗ f g are residual closures, itfollows that D ∗ E = D ∗ fgE , as desired.Now we can prove the last sentence of the Theorem. In either case, setting α ( − ) ∶ = ∗− , we have I ∗ = α ⌣ ( I ) = ann R (( ann E I ) ∗ E ) = ann R (( ann E I ) ∗ fgE ) = ( α f ) ⌣ ( I ) = ( α ⌣ ) f ( I ) , where the first equality comes from [ES14, Corollary 3.6]. The second andfourth equalities follow from Theorem 3.3. The third equality follows fromthe earlier parts of the proof. Finally, the last equality follows from Theo-rem 6.19. (cid:3) Our first few examples are in numerical semigroup rings, which are approx-imately Gorenstein. Following [Hoc07, Page 177], in the following exampleswe construct irreducible ideals J n cofinal with the maximal ideal in order tocompute the double colons needed to determine the finitistic interior relatedto a residual closure operation. Example 8.2.
Let k be an infinite F -finite field of characteristic p > R = k [[ t , t ]] with maximal ideal m = ( t , t ) . We use Theorem 3.3 tocompute the ∗ -interior of all ideals of R and then compute the ∗ -hulls of allideals of R .The nonzero, non-unital ideals of R are either of the form ( t m , t m + ) or ( t m + at m + ) where m ≥ a ∈ k . The lattice of ideals includes thefollowing: ORE-HULL DUALITY 31 ( t m , t m + ) ♥♥♥♥♥♥♥♥♥♥♥♥♥♥ ( t m + , t m + ) ❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖ ( t m + at m + )( t m + + at m + ) ( t m + , t m + ) where the boxed ideals contain ∣ k ∣ ideals.Let J n = ( t n + at n + ) for n ≥
1. Then the J n are a decreasing sequenceof irreducible ideals cofinal with the powers of the maximal ideal. They areirreducible because R is Gorenstein and t n + at n + is a regular element,whence R /( t n + at n + ) is Gorenstein. By Theorem 3.3, this implies thatfor any ideal I of R , the Artinistic tight interior of I is equal to ⋂ n ≥ J n ∶ ( J n ∶ I ) ∗ R . Since R is an isolated singularity, Theorem 8.1 guarantees that Artinisticand ordinary tight interior agree on ideals of R .For the following computations, we note that ( t n + at n + ) ∶ ( t m , t m + ) = ( t n + at n + ) ∶ ( t m ) ∩ ( t n + at n + ) ∶ ( t m + ) = ( t n − m + at n − m + ) ∩ ( t n − m − + at n − m ) = ( t n − m + , t n − m + ) . Since R is a 1-dimensional domain with infinite residue field, it is known[Hun96, Example 1.6.2] that ( t m ) ∗ = ( t m ) − = ( t m , t m + ) for any m ≥
2. Forany a ∈ k , we compute ( t m + at m + ) ∗ using Theorems 3.3 and 8.1. ( t m + at m + ) ∗ = ⋂ n ≥ J n ∶ ( J n ∶ ( t m + at m + )) ∗ = ⋂ n ≥ ( t n + at n + ) ∶ (( t n + at n + ) ∶ ( t m + at m + )) ∗ . For 2 ≤ m < n − ( t n + at n + ) ∶ ( t m + at m + ) = ( t n − m ) , and given that theintersection is over ideals that decrease as n increases the above is equal to ⋂ n ≥ ( t n + at n + ) ∶ ( t n − m ) ∗ = ⋂ n ≥ ( t n + at n + ) ∶ ( t n − m , t n − m + ) = ⋂ n ≥ m ( t n + at n + ) ∶ ( t n − m , t n − m + ) = ⋂ n ≥ m ( t m + , t m + ) = ( t m + , t m + ) , which agrees with the the computations in [Vas20].We similarly compute ( t m , t m + ) ∗ with J n = ( t n ) . ( t m , t m + ) ∗ = ⋂ n ≥ J n ∶ ( J n ∶ ( t m , t m + )) ∗ = ⋂ n ≥ ( t n ) ∶ (( t n ) ∶ ( t m , t m + )) ∗ = ⋂ n ≥ ( t n ) ∶ ( t n − m + , t n − m + ) = ⋂ n ≥ ( t m , t m + ) = ( t m , t m + ) . Next we compute the ∗ -hull of the ideals ( t m + , t m + ) for m ≥
2. Observ-ing the lattice of ideals and using the computations above, ( t m + at m + ) ∗ = ( t m + , t m + ) for all a ∈ k . Hence ( t m + at m + ) is a maximal ∗ -expansion of ( t m + , t m + ) since ( t m , t m + ) ∗ = ( t m , t m + ) . This implies that ∗ -hull ( t m + , t m + ) = ∑ a ∈ k ( t m + at m + ) = ( t m , t m + ) for m ≥
2. As the test ideal of R is ( t , t ) , Proposition 7.17 implies that the ∗ -hull ( t , t ) = R . Note that ∗ -hull ( t , t ) = ( t , t ) and ∗ -hull ( t m + at m + ) = ( t m + at m + ) for m ≥ ∗ -interior. Example 8.3.
Let k be an F -finite field of characteristic p > R = k [[ t , t , t ]] with maximal ideal m = ( t , t , t ) . Since R is an isolated singu-larity, Theorem 8.1 guarantees that Artinistic and ordinary tight interior ofideals agree in R . We use Theorem 3.3 to compute the ∗ -interior of principalideals of R and then compute the ∗ -hulls of certain ideals of R .Unlike in the previous example, R is not Gorenstein, but ω ≅ ( t , t ) andtaking w = t in Hochster’s construction [Hoc07, Page 177], we have R / J n ≅ ( t )/(( t n + at n + + bt n + )( t , t ) ∩ ( t ) ≅ ( t )/( t n + + at n + + bt n + , t n + + at n + + bt n + ) Thus J n = ( t n + at n + + bt n + , t n + + at n + + bt n + ) are irreducible idealscofinal with powers of the maximal ideal. By Theorem 3.3, this implies thatfor any ideal I of R , the tight interior of I is equal to ⋂ n ≥ J n ∶ ( J n ∶ I ) ∗ R . Since R is a 1-dimensional domain with infinite residue field, it is known[Hun96, Example 1.6.2] that ( t m , t m + , t m + ) = ( t m ) − = ( t m ) ∗ = ( t m , t m + ) ∗ for m ≥
3. Using Theorem 3.3, ( t m + at m + + bt m + ) ∗ = ⋂ n ≥ J n ∶ ( J n ∶ ( t m + at m + + bt m + )) ∗ . ORE-HULL DUALITY 33
We note that J n ∶ ( t m , t m + , t m + ) = ( J n ∶ t m ) ∩ ( J n ∶ t m + ) ∩ ( J n ∶ t m + ) = J n − m ∩ J n − m − ∩ J n − m − = ( t n − m + , t n − m + , t n − m + ) . For 3 ≤ m < n − J n ∶ ( t m + at m + + bt m + ) = ( t n − m , t n − m + ) , and given that the intersection is over ideals that decrease as n increases theabove is equal to ⋂ n ≥ J n ∶ ( t n − m , t n − m + ) ∗ = ⋂ n ≥ J n ∶ ( t n − m , t n − m + , t n − m + ) = ⋂ n ≥ ( t m + , t m + , t m + ) = ( t m + , t m + , t m + ) . Thus, ( t m + at m + + bt m + ) ∗ = ( t m + , t m + , t m + ) which agrees with the thecomputations in [Vas20].Similarly we can compute the ∗ -interior of ( t m , t m + , t m + ) for m ≥ J n = ( t n , t n + ) . ( t m , t m + , t m + ) ∗ = ⋂ n ≥ J n ∶ ( J n ∶ ( t m , t m + , t m + )) ∗ = ⋂ n ≥ J n ∶ ( t n − m + , t n − m + , t n − m + ) = ⋂ n ≥ ( t m , t m + , t m + ) = ( t m , t m + , t m + ) . The nonzero, non-unital ideals of R are of the form ( t m , t m + , t m + ) ,generated by two binomials whose degrees differ by at most 2, or ( t m + at m + + bt m + ) where m ≥ a, b ∈ k . Hence ( t m + at m + + bt m + ) is a ∗ -expansion of ( t m + , t m + , t m + ) . Note that ( t m , t m + , t m + ) ⊇ ( t m + at m + + bt m + ) , but ( t m , t m + , t m + ) ∗ = ( t m , t m + , t m + ) . A maximal ∗ -expansion of ( t m + , t m + , t m + ) is at most an ideal I generated by two binomials satisfying ( t m + at m + + bt m + ) ⊆ I ⊆ ( t m , t m + , t m + ) . We have ∑ a,b ∈ k ( t m + at m + + bt k + ) = ( t m , t m + , t m + ) , so summing over such I , ∑ I I = ( t m , t m + , t m + ) , which implies ∗ -hull ( t m + , t m + , t m + ) = ( t m , t m + , t m + ) . Example 8.4.
Suppose k is an F -finite field of characteristic p >
0. Let R = k [[ x, y ]]/( xy ) . The nonzero, non-unital ideals in R are of the form ( x n ) , ( y m ) , ( x n + ay m ) for some nonzero a ∈ k , or ( x n , y m ) . Note that forvarious choices of positive gradings of x and y in k [ x, y ] , each of these idealsis extended from a homogeneous ideal of k [ x, y ] . Hence, by Theorem 8.1(1),the tight interior and the Artinistic tight interior of any ideal are the same,so this example could also be computed using Theorem 3.3.We compute the ∗ -hulls of the ideals ( x n , y m ) . Note that ( x n ) ∗ = ( x n ) = ( x n ) ∗ , ( y m ) ∗ = ( y m ) = ( y m ) ∗ . and ( x n , y m ) ∗ = ( x n , y m ) = ( x n , y m ) ∗ by[Hun96, Theorem 1.3(c)] and [ES14, Proposition 2.8]. However, ( x n + ay m ) ∗ = ( x n , y m ) and ( x n + ay m ) ∗ = ( x n + , y m + ) again by [Hun96, Theorem 1.3(c)] and [ES14, Proposition 2.8]. So ( x n + ay m ) is a ∗ -expansion of ( x n + , y m + ) for all nonzero a ∈ k . Part of the lattice ofideals for k [[ x, y ]]/( xy ) includes: ( x n , y m ) ♣♣♣♣♣♣♣♣♣♣♣♣ ◆◆◆◆◆◆◆◆◆◆◆◆ ( x n , y m + ) ◆◆◆◆◆◆◆◆◆◆◆◆ ( x n + ay m ) ( x n + , y m ) ♣♣♣♣♣♣♣♣♣♣♣♣ ( x n + , y m + ) where the boxed node contains ∣ k ∣ − ( x n , y m ) ∗ = ( x n , y m ) , we see that the ideals ( x n + ay m ) are maximal ∗ -expansions of ( x n + , y m + ) . Thus, ∗ -hull ( x n + , y m + ) = ∑ a ∈ k /{ } ( x n + ay m ) = ( x n , y m ) . Example 8.5.
Let R = k [[ x, y, z ]]/( x + y + z ) , where k is an F -finitefield of characteristic p >
3. The goal of this example is to show that the F -interior and F -hull of an ideal can vary depending on the characteristicof k . By Fedder’s F -purity criterion [Fed83], R is F -pure if and only if ( x + y + z ) p − ∉ m [ p ] which is true if and only if p ≡ F -interior of ( y, z ) . Note that ( y s , z t ) F = ⎧⎪⎪⎨⎪⎪⎩( y s , z t ) if p ≡ ( x y s − z t − , y s , z t ) = ( y s , z t ) ∗ if p ≡ ( y, z ) F = ⎧⎪⎪⎨⎪⎪⎩( y, z ) if p ≡ ( xy, xz, y , yz, z ) if p ≡ . ORE-HULL DUALITY 35
We compute the F -interior for the second case above using the methodsfrom Section 3. Since R is Gorenstein any system of parameters is irreducible.Let J t = ( y t , z t ) and I = ( y, z ) . Then J t ∶ ( J t ∶ I ) F = J t ∶ ( y t , z t , ( yz ) t − ) F . Note that ( y t , z t , y t − z t − ) = ( y t , z t − ) ∩ ( y t − , z t ) . The test ideal of R isthe maximal ideal [McD00, Proposition 1.4]. Now by [Vas14, Proposition2.4] ( y t , z t , ( yz ) t − ) ∗ = ( y t , z t , ( yz ) t − ) ∶ m = ( y t , z t , ( yz ) t − , x y t − z t − , x y t − z t − ) . When p ≡ ( y t , z t , ( yz ) t − ) F = ( y t , z t , ( yz ) t − ) ∗ by [McD00, Pro-postion 2.1]. Hence, ( y, z ) F = J t ∶ ( y t , z t , ( yz ) t − ) F = J t ∶ ( y t , z t , ( yz ) t − , x y t − z t − , x y t − z t − ) = ( xy, xz, y , yz, z ) when p ≡ . Next we compute the tight interior of the parameter ideal ( y + x , z ) for p ≡ R is an isolated singularity (as follows easily from theJacobian criterion on the uncompleted affine ring, k [ x, y, z ]/( x + y + z ) )Theorem 8.1(2) guarantees that for any ideal I , the tight interior and theArtinistic tight interior of I coincide. Using the same argument above with J t = (( y + x ) t , z t ) and I = ( y + x , z ) we obtain ( y + x , z ) ∗ = J t ∶ (( y + x ) t , z t , ( y + x ) t − z t − ) ∗ = J t ∶ (( y + x ) t , z t , ( y + x ) t − z t − , x ( y + x ) t − z t − , x ( y + x ) t − z t − )) = ( x ( y + x ) , xz, ( y + x ) , ( y + x ) z, z ) where the second equality is by [Vas14, Proposition 2.4]. Clearly yz ∈ ( y + x , z ) ∗ . Note that x + z = − y . We will write y times a unit as an elementof ( y + x , z ) ∗ = ( x ( y + x ) , xz, ( y + x ) , z ( y + x ) , z ) . First we will take acombination of ( y + x ) , x ( y + x ) and z and simplify algebraically. ( y + x ) − x ( y + x ) − xz = y + x y + x − x y − x − xz = y − x − xz = y + xy = y ( + xy ) . Since 1 + xy is a unit in R , y and hence xy are in ( y + x , z ) ∗ . Thus ( y + x , z ) ∗ = ( xy, xz, y , yz, z ) . Note now that ( y, z ) and ( y + x , z ) are both ∗ -expansions of the ideal ( xy, xz, y , yz, z ) , hence ( x , y, z ) = ( y, z ) + ( y + x , z ) ⊆ ∗ -hull ( xy, xz, y , yz, z ) . We will show that x ∉ ∗ -hull ( xy, xz, y , yz, z ) . First, we compute thetight closures of the ideals ( y − cx, z ) with c ≠ −
1. (A similar argument canbe used to compute the tight closure of the ideals ( y, z − dx ) with d ≠ − ( y − cx, z − dx ) with c + d ≠ − ( y t , ( z − cx ) t , y t − ( z − cx ) t − ) using [Vas14, Proposition 2.4]Note first that ( y − cx )( y + cxy + c x ) + z = y − c x + z = ( − − c ) x Since c ≠ − − c − x ∈ ( y − cx, z ) ∶ m . Asa consequence, since the socle of R /( y − cx, z ) is generated by one element, ( y − cx, z ) ∗ = ( y − cx, z ) ∶ m = ( x , y − cx, z ) . By [McD00, Proposition 5.2], x ∈ ( y − cx, z ) F and ( y − cx, z ) F = ( y − cx, z ) ∗ .As above, we compute the tight interior of ( y − cx, z ) . Let J t = (( y − cx ) t , z t ) and I = ( y − cx, z ) . Then ( y − cx, z ) ∗ = J t ∶ (( y − cx ) t , z t , ( y − cx ) t − z t − ) ∗ = J t ∶ (( y − cx ) t , z t , ( y − cx ) t − z t − , x ( y − cx ) t − z t − , x ( y − cx ) t − z t − ) = ( x ( y − cx ) , xz, ( y − cx ) , ( y − cx ) z, z ) where the second equality is by [Vas14, Proposition 2.4].Although yz ∈ ( y − cx, z ) ∗ , xy, y ∉ ( y − cx, z ) ∗ . Thus ( y − cx, z ) is not a ∗ -extension nor a F -extension of ( xy, xz, y , yz, z ) . Similarly ( y, z − dx ) and ( y − cx, z − dx ) are not ∗ -extensions nor F -extensions of ( xy, xz, y , yz, z ) . Note that ( y, z ) ⊆ F -hull ( xy, xz, y , yz, z ) ⊆ ∗ -hull ( xy, xz, y , yz, z ) = ( x , y, z ) for p ≡ m -primary ideal I , if I F ≠ I ∗ , then there is a Z -graded module M andirreducible submodule N of M with N ∗ ≠ N F . Hence it is not knownthat the tight closure and Frobenius closure agree on all ideals when p ≡ p ≡ (( y + x ) t , z t ) F = (( y + x ) t , z t ) ∗ for t >> F -hull ( xy, xz, y , yz, z ) = ( x , y, z ) .When the characteristic is p ≡ R is F -pure and F -hull ( xy, xz, y , yz, z ) = ( xy, xz, y , yz, z ) . Even if (( y + cx ) t , z t ) F ≠ (( y + cx ) t , z t ) ∗ for some c ≠
0, the F -hull of ( xy, xz, y , yz, z ) will depend on whether the characteristic is congruent to1 or 2 mod 3. Acknowledgments
The authors would like to thank Eleonore Faber for suggestions of ringswith known MCM modules and Jooyoun Hong and Craig Huneke for discus-sions of conormal modules. We are grateful to Karl Schwede for commentsthat improved the presentation of the paper. We also gratefully acknowledge
ORE-HULL DUALITY 37 the AMS-Simons Travel Grant program; the grant awarded to Rebecca R.G.subsidized the travel of Janet Vassilev to George Mason University wherethis collaboration commenced.
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Department of Mathematical Sciences, George Mason University, Fairfax,VA 22030
E-mail address : [email protected] Department of Mathematical Sciences, George Mason University, Fairfax,VA 22030
E-mail address : [email protected] Department of Mathematics and Statistics, University of New Mexico, Al-buquerque, NM 87131
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