Limit Behavior of the Rational Powers of Monomial Ideals
aa r X i v : . [ m a t h . A C ] S e p LIMIT BEHAVIOR OF THE RATIONAL POWERS OFMONOMIAL IDEALS
JAMES LEWIS
Abstract.
We investigate the rational powers of ideals. We find that in thecase of monomial ideals, the canonical indexing leads to a characterizationof the rational powers yielding that symbolic powers of squarefree monomialideals are indeed rational powers themselves. Using the connection with sym-bolic powers techniques, we use splittings to show the convergence of depthsand normalized CastelnuovoMumford regularities. We show the convergenceof Stanley depths for rational powers, and as a consequence of this we show thebefore-now unknown convergence of Stanley depths of integral closure powers.In addition, we show the finiteness of asymptotic associated primes, and wefind that the normalized lengths of local cohomology modules converge forrational powers, and hence for symbolic powers of squarefree monomial ideals. Introduction
In a commutative ring A with ideal I and ab ∈ Q + we define the ab -rational powerof I to be I ab = { x ∈ A | x b ∈ I a } . Despite having been formally defined in the literature nearly two decades ago,these simplistically defined powers are sparsely discussed with [2, 7, 11, 13, 18, 23]being some of the only articles which discuss rational powers. For instance, inthe Noetherian case we know that the family { I β } β ∈ Q + of rational powers can belinearly indexed by N , i.e. the rational powers form a filtration. However, nothinghas been said about the homological invariants of this filtration.In this article we show the canonical indexing of the rational powers – i.e. { I ne } n ∈ N where e is the least common multiple of the Rees valuations of I (whoseset we denote RV ( I )) evaluated at I – leads to a characterization of the rationalpowers which allows for use of the techniques often found in arguments involvingthe symbolic powers of squarefree monomial ideals (e.g. in [15, 19]). The character-ization – having a uniform bound in the Rees valuation inequalities – allows us, inthe case of squarefree monomial ideals, to find that the symbolic powers are in factrational powers of some ideal. In general, we can use hyperplanes to characterizerational powers of monomial ideals via the following theorem (see Theorem 3.2)where a hyperplane is given by, for X ∈ Q d , h ( X ) = a · X for coefficients a ∈ Q d : Theorem A.
Let hyperplanes h , . . . , h r with coefficients in Q d > define a familyof monomial ideals { I σ } σ ∈ Q + in R = K [ x , . . . , x d ] so that x α ∈ I σ if and only if h i ( α ) > σ for all i r . Then there exists a monomial ideal J and g ∈ N so thatfor any σ ∈ Q + we have I σ = J σg . Mathematics Subject Classification.
Primary 13A30; Secondary 13F55, 13B22, 05E40.
Key words and phrases.
Rational powers, Symbolic powers, Stanley depth, Local cohomology.
The connection with symbolic powers leads to a generalization of the splittingmaps method found in [14, 15] which guarantees the convergence of depths andnormalized CastelnuovoMumford regularities (see Theorem 4.6 and Theorem 5.1):
Theorem B. If I is any monomial ideal and e = lcm { v ( I ) | v ∈ RV ( I ) } , then (1) lim n →∞ reg( I ne ) n exists and is equal to e lim n →∞ reg( I n ) n . (2) lim n →∞ depth( R/I ne ) exists and is equal to d − ℓ ( I ) , where ℓ ( I ) is the analyticspread of I . This computation also yields that the symbolic analytic spread (as discussed in[6]) can be computed via the symbolic polyhedron (as discussed in [3]) in the caseof squarefree monomial ideals (see Remark 5.2).Furthermore, we show the convergence of Stanley depths (a measure of the(multi-)graded structure of the filtration) for rational powers using methods from[22]. As a consequence we answer the open question (from personal communications[21] and discussed in [20]) that the Stanley depths must also converge for integralclosure powers. (see Theorem 5.3):
Theorem C. If I is a monomial ideal and e = lcm { v ( I ) | v ∈ RV ( I ) } . Then thelimits lim n →∞ sdepth( R/I ne ) and lim n →∞ sdepth( I ne ) exist. In particular, these limitsmust exist for { I n } n ∈ N . Furthermore: lim n →∞ sdepth( R/I ne ) = min n sdepth( R/I n ), andlim n →∞ sdepth( I ne ) = min n sdepth( I n ) . Herzog conjectured the existence of the Stanley depth limits for regular powersin [8, Conjecture 59], hence we confirm the conjecture in the case of normal ideals.One of the common factors in proving the aforementioned limits is that theassociated Rees algebra of the rational powers is Noetherian. Combining this withthe fact that rational powers are integrally closed, we conclude that ∪ n ∈ N Ass(
R/I ne )is finite (see Corollary 5.5).Furthermore we use the methods of [5] to find that the lengths of local cohomol-ogy modules involving the rational powers of a monomial ideal also converge (seeTheorem 5.8): Theorem D.
Let I be a monomial ideal and e = lcm { v ( I ) | v ∈ RV ( I ) } . Assumethat λ (H i m ( R/I ne )) < ∞ for n ≫ . Then the limit lim n →∞ λ (H i m ( R/I ne )) n d exists and is rational. In particular, this limit must exist for { I ( n ) } n ∈ N when I issquarefree. Furthermore lim n →∞ λ (H i m ( R/I ne )) n d = e d lim n →∞ λ (H i m ( R/I n )) n d . Preliminaries
We now outline preliminaries of integral closure and valuation theory that weuse throughout the article. For full details and proofs we refer the reader to [13,Chapters 1,6, and 10] and [12].
IMIT BEHAVIOR OF THE RATIONAL POWERS OF MONOMIAL IDEALS 3
For an ideal I in a commutative Noetherian ring A , we define the integral closure of I to be the ideal I = { x ∈ A | x n + a x n − + · · · + a n − x + a n = 0 for some n ∈ N , a i ∈ I i for 1 i n } . Using the theory of valuations, integral closure becomes easier to compute and workwith. To begin with valuations:Let K be a field, K ∗ = K − { } the multiplicative group. A discrete (rank one)valuation on K is a group homomorphism v : K ∗ → Z with the added property that v ( x + y ) > min { v ( x ) , v ( y ) } for any x, y ∈ K ∗ . We can associate to each valuation v a valuation ring R v = { x ∈ K | v ( x ) > } ∪ { } which is a local domain. We call avaluation, v , normalized if v ( K ∗ ) ⊆ Z has greatest common divisor one.For an ideal I in a domain A and a discrete (rank one) valuation, v , on thefraction field of A , Q ( A ), we define v ( I ) = min { v ( x ) | x ∈ I } . We may consider only the set of generators of I to compute v ( I ). One may showfurther that v ( I n ) = nv ( I ).There is a powerful connection between integral closures and valuations whichsays that, if A is a domain, for an ideal I and for any n ∈ N we have the valuativecriterion of integral closure x ∈ I n if and only if v ( x ) > nv ( I )for all discrete valuations of rank one v with R v between A and Q ( A ) and suchthat the maximal ideal of R v contracts to a maximal ideal of A . This connectionalso shows that v ( I ) = v ( I ).From a construction of Rees, for a given ideal I there exist a finite set of unique(up to equivalence of valuations) normalized discrete rank one valuations for whichwe need to check the valuative inequality for integral closure powers. We call thesevaluations the Rees valuations of I and denote the set of them by RV ( I ). That is, x ∈ I n if and only if v ( x ) > nv ( I ) for all v ∈ RV ( I )where, critically, RV ( I ) is finite. Definition 2.1.
In a commutative ring A with ideal I and ab ∈ Q + we define the ab rational power of I to be I ab = { x ∈ A | x b ∈ I a } . Surprisingly this set turns out to be an ideal. The fact of it being an ideal followsfrom the properties of valuations and the valuative criterion of integral closureabove. The following theorem outlines some basic facts about rational powers in aNoetherian ring.
Theorem 2.2. [13, Section 10.5]
For a commutative Noetherian ring A , ideal I ,and α, β ∈ Q + , set e = lcm { v ( I ) | v ∈ RV ( I ) } where lcm denotes least commonmultiple, then: (1) If a, b, c, d ∈ N , then I ab is a well-defined ideal, that is, if ab = cd then I ab = I cd (2) I n = I n (3) if α β then I β ⊆ I α (4) I α is integrally closed (5) x ∈ I α if and only if v ( x ) > αv ( I ) for every Rees valuation v of I (6) I α I β ⊆ I α + β JAMES LEWIS (7) for all α ∈ Q + , I α = I ne where n = (cid:6) eab (cid:7) Of particular interest is the second property that will allow us to move backand forth between rational and integral closure powers, so that some features ofrational powers will be able to pull back to the integral closure powers. We alsonote that the last property allows us to describe the rational powers of an ideal asan N -indexed filtration of ideals. By filtration we mean a decreasing chain of ideals { I n } n ∈ N such that I = A and I n · I k ⊆ I n + k for any n, k ∈ N .From now on let R = K [ x , . . . , x d ] for some field K and d ∈ N and I be amonomial ideal of R . We will denote a general monomial of R by x α = x α . . . x α d d where α = ( α a , · · · , α d ) ∈ N d . Then we say the exponent set of I is G ( I ) = { α ∈ N d | x α ∈ I } and we call the Newton Polyhedron the convex hull of the exponent set of I , thatis NP( I ) = Conv( G ( I )) . The Newton Polyhedron connects convex geometry to the integral closure via [13,Proposition 1.4.6] so that for any monomial ideal I we have G ( I ) = NP( I ) ∩ N d . That is, we can read-off the integral closure of an ideal via the lattice points of theNewton Polyhedron.The connection between convex geometry and integral closure can also be ex-tended to the valuation theory of integral closures. To set this up: a valuation, v ,on the field of fractions of R is called monomial if for any polynomial f we havethat v ( f ) = min { v ( x α ) | x α is a monomial supporting f } . Then we have that theRees valuations of a monomial ideal are monomial (see [13, Theorem 10.3.4]).We say a hyperplane in Q d is a function h on Q d defined by h ( X ) = X · α forcoefficients α ∈ Q d Remark 2.3.
Now, let v be a monomial valuation on R = K [ x , . . . , x d ]. Thevaluation is determined by how it behaves on monomials and so, since valuationsare multiplicative group homomorphisms, v ( x a . . . x a d d ) = a v ( x ) + · · · + a d v ( x d )shows the valuation is given by the hyperplane v ( x ) X + · · · + v ( x d ) X d in Q d > .Hence it may be no surprise that the Rees valuations of a monomial ideal aregiven by the bounding faces of the Newton polyhedron (see [13, Theorem 10.3.5]).That is, we can read-off the Rees valuations of a monomial ideal by the hyperplaneswhich make up the Newton polyhedron, and furthermore the valuations given bythe hyperplanes are normalized from [12, Corollary 3.3].For n ∈ N we define the nth-symbolic power of an ideal I to be I ( n ) = \ p ∈ Min( I ) I n R p ∩ R where Min( I ) denotes the minimal primes of I .For any graded R -module M set a i ( M ) = max { j | H i m ( M ) j = 0 } IMIT BEHAVIOR OF THE RATIONAL POWERS OF MONOMIAL IDEALS 5 for 0 i dim M , we call a i the i th a -invariant . Then the CastelnuovoMumfordregularity of M is reg( M ) = max { a i ( M ) + i | i dim M } . Another measure of the graded structure of a module is the Stanley Depth. Fora survey of Stanley depth, we refer the reader to [16]. Let M be a finitely generated Z d -graded R -module, Z a subset of the variables (i.e. Z ⊆ { x , . . . , x d } ), and u ∈ M homogeneous. If u · K [ Z ] is a free K [ Z ]-module, then we call u · K [ Z ] a Stanley space of dimension | Z | . A k -vector spaces decomposition, D , of M into a direct sum ofStanley spaces is called a Stanley decomposition of M . We call the Stanley depth of the decomposition the minimum dimension of a Stanley space appearing in D and is notated sdepth( D ). That is, if we can write D as the direct sum of k -vectorspaces: M = u · K [ Z ] ⊕ · · · ⊕ u r · K [ Z r ]where each u i · K [ Z i ] is a free K [ Z i ]-module then we saysdepth( D ) = min {| Z i | | i r } . The maximum of the Stanley depths over all Stanley decompositions of M is calledthe Stanley depth of M . That is,sdepth( M ) := max { sdepth( D ) | D is a Stanley decomposition of M } . Characterizing Rational Powers
In this section we extend the valuative criterion of integral closures to helpcharacterize rational powers. Notice that we have a version of the valuative criterionin Theorem 2.2 using rational numbers. Recall that R = K [ x , . . . , x d ] for some field K and d ∈ N , and we have I a monomial ideal of R . Remark 3.1.
Using the e = lcm { v ( I ) | v ∈ RV ( I ) } indexing of the rational powers,we have that u ∈ I ne if and only if v ( u ) > ne v ( I ) for each v ∈ RV ( I ). As before, wecan rearrange this to u ∈ I ne if and only if ev ( I ) v ( u ) > n for each v ∈ RV ( I ) . Since ev ( I ) ∈ N by choice of e , if we need to check membership to a rational power,we need only check that a finite number of functions with values in N are uniformlybounded below by n . Critically, this allows for use of the techniques often foundin arguments involving the symbolic powers of squarefree monomial ideals (e.g. in[15, 19]), see Proposition 4.4.We now turn our attention to monomial ideals where the Rees valuations aregiven by hyperplane equations. Because of this added convex geometry, almostany filtration of ideals given by hyperplane equations can be described as rationalpowers. We note here that convex geometry yielding properties about a filtrationhas seen some recent advancements, for example in [3]. We now prove Theorem A. Theorem 3.2.
Let non-redundant hyperplanes h , . . . , h r with coefficients in Q d > define a family of monomial ideals { I σ } σ ∈ Q + in R = K [ x , . . . , x d ] so that x α ∈ I σ ifand only if h i ( α ) > σ for all i r Then (1)
There exists a monomial ideal J and g ∈ N so that for σ ∈ Q + we have I σ = J σg . JAMES LEWIS (2)
The family { I σ } σ ∈ Q + can be indexed by N so that it can be written as { I nf } n ∈ N for some f ∈ N . (3) The indexing of the rational powers of J with e = lcm { w ( J ) | w ∈ RV ( J ) } , { J ne } n ∈ N appears as a subsequence of { I nf } n ∈ N via J ne = I fge · nf where fge ∈ N .Proof. We begin with the proof of (1). We will show that the polyhedron formedby the hyperplanes defining the family { I σ } σ ∈ Q + can be scaled to be the NewtonPolyhedron of some monomial ideal. Without loss of generality we can write eachhyperplane h i = 1 in the reduced integral form as a i X + · · · + a id X d = f i where a ij , f i ∈ N for 1 i r and 1 j d and with gcd( a i , . . . , a id , f i ) = 1 for 1 i r . Let f = lcm( f , . . . , f d ).Consider the convex hull, C , of the hyperplanes h = f , . . . , h r = f r . Since thecoefficients of the hyperplanes are positive, C is the set of all the points β ∈ Q d + with h ( β ) > f , . . . , and h r ( β ) > f r . Let C t be the convex set given by all thepoints β ∈ Q d + with h ( β ) > f t, . . . , and h r ( β ) > f r t . Notice we have that thescaling of C by t is the same as C t by these definitions.Let P , . . . , P t ∈ Q d + be the vertices of C . Then, as the coefficients of thehyperplanes are rational, the vertices are rational as well. Let g be the least commonmultiple of all the denominators of all the components of the P j in reduced form.Scaling the convex hull by g we have that gC = C g has as lattice point vertices gP , . . . , gP t by the construction of g . Notice that C g is the minimal such (non-zero)scaling of C with lattice point vertices.Using these lattice point vertices to construct an ideal, let J be the monomialideal generated by the set { x gP , . . . , x gP t } which is well defined since gP j ∈ N d for1 j t . We can use the generators of J to construct the Newton polyhedron,i.e. NP( J ) = conv( G ( { x gP , . . . , x gP t } ) + Q d + where addition is Minkowski addition.Thus, by construction, the Newton Polyhedron of J is gC = C g . Hence we havethat J = I g .To prove the first claim, we need to find the Rees valuations of J . Following themethod of [12, Corollary 3.3], the Rees valuations of J are given by the reducedintegral form of the bounding faces of the Newton polyhedron. By constructionNP( J ) = C g and so these bounding faces come from the non-redundant hyperplanes h = f g, . . . , h r = f r g . To get the reduced integral form of these hyperplanes, let m i = gcd( a i , . . . , a id , f i g ) for 1 i r be the reducing factor. Notice that sincegcd( a i , . . . , a id , f i ) = 1 for 1 i r , then we have that m i | g for 1 i r . Thenthe bounding faces of the Newton polyhedron of J in the reduced integral formsare m i h i = gm i f i for 1 i r . Thus, the (normalized) Rees valuations of J are w i = m i h i and w i ( J ) = gm i f i for 1 i r .Then we have that for any σ ∈ Q + , x α ∈ I σ if and only if h i ( α ) > σf i for all1 i r . We can rewrite h i ( α ) > σf i with the valuations as w i ( x α ) > σ f i m i for each1 i r . Then notice that we can rewrite this further as w i ( x α ) > σ f i gm i g = σg w i ( J ) for all 1 i r. Hence, x α ∈ I σ if and only if w i ( x α ) > σg w i ( J ) for all 1 i r . Thus, by the rationalvaluative criterion of Theorem 2.2, I σ = J σg for any σ ∈ Q + , finishing (1).For (2), notice that x α ∈ I σ if and only if f i h i ( α ) > σ for all 1 i r . Bythe choice of e we have ff i h i ( α ) ∈ N so that f i h i ( α ) ∈ f N for all 1 i r . Thus, IMIT BEHAVIOR OF THE RATIONAL POWERS OF MONOMIAL IDEALS 71 f i h i ( α ) > σ implies f i h i ( α ) is greater than or equal to the nearest element of f N ,i.e. f ⌈ f σ ⌉ . Hence, f i h i ( α ) > σ for each 1 i r if and only if f i h i ( α ) > f ⌈ f σ ⌉ foreach 1 i r . Thus(3.1) I σ = I ⌈ fσ ⌉ f . Hence, we can index { I σ } σ ∈ Q + by N via { I nf } n ∈ N . Equivalently, we can describethe filtration via the lattice points inside the convex sets nf C for n ∈ N .For (3), we form the canonical indexing of the rational powers of J by lookingat the least common multiple of the Rees valuations of J evaluated at J , i.e. e = lcm( w ( J ) , . . . , w r ( J )) = lcm( gm f , . . . , gm r f r ) . Notice that f i gm i | f g for each 1 i r so that e | f g . From the rational valuativecriterion of Theorem 2.2 we can describe the rational powers of J via the latticepoints inside the convex sets ne NP( J ) for n ∈ N . Hence, for any n ∈ N , ne NP( J ) = ne C g = nge C and using Equation (3.1) we have: J ne = I nge = I f · ⌈ fgne ⌉ = I fge · nf , where the last equality follows from e dividing f g and thus completing the proof. (cid:3) Remark 3.3.
If gcd( a i , . . . , a id ) = 1 (or m i = 1) for all 1 i r , then we canimprove our understanding of the relationship between { J σ } σ ∈ Q + and { I σ } σ ∈ Q + .Specifically, if m i = 1 we have e = lcm( w ( J ) , . . . , w r ( J )) = lcm( f g, . . . , f r g ) = lcm( f , . . . , f r ) g = f g. Then, using part (1) and Equation (3.1), for any σ ∈ Q + we have J σ = J ⌈ fgσ ⌉ fg = J g · ⌈ fgσ ⌉ f = I ⌈ fgσ ⌉ f = I gσ An important corollary of this theorem establishes a strong connection betweenintegral closure theory and rational powers to symbolic powers.
Corollary 3.4.
Let I be a squarefree monomial ideal. Then there exists g ∈ N sothat the rational powers of I ( g ) are the symbolic powers of I , that is I ( n ) = ( I ( g ) ) ng for every n ∈ N .Proof. Since I is squarefree, the minimal primes, Min( I ), are complete intersectionsand hence we can use their powers to compute the symbolic powers. That is, wecan write I ( n ) = T Min( I ) p n . Then we have for a monomial x α ∈ R , x α ∈ I ( n ) if andonly if x α ∈ p n for all p ∈ Min( I ). As the p are monomial primes, we have that x α ∈ p n if and only if the degree of x α supported in the variables of p is at least n .That is, if we write p = ( x i , . . . , x i l ) for 1 i < . . . < i l d , then x α ∈ p n if and onlyif α i + · · · + α i l > n . Hence, we can associate a hyperplane equation to each p via h p ( α ) = α i + · · · + α i l . Thus we have that x α ∈ I ( n ) if and only if x α ∈ p n for all p ∈ Min( I ) if and only if h p ( α ) > n for all p ∈ Min( I ). Hence, { I ( n ) } n ∈ N satisfies thehypotheses of Theorem 3.2, finishing the proof. (cid:3) The polyhedron formed by the minimal primes here can be made more generalfor non-squarefree monomial ideals and is called the symbolic polyhedron . For aconstruction and exposition of the symbolic polyhedron, we refer the reader to [3].
JAMES LEWIS
Example 3.5. In R = K [ x, y, z ] let I = ( xy, yz, zx ) = ( x, y ) ∩ ( x, z ) ∩ ( y, z ). Followingthe argument of Theorem 3.2, we have that the symbolic polyhedron is given bythe valuations v ( x a y b z c ) = a + b , v ( x a y b z c ) = a + c , and v ( x a y b z c ) = b + c . Thenthe vertices are ( , , ), (1 , , , , , , g = 2 so that therational powers of I (2) = ( x y , xyz, x z , y z ) coincide with the symbolic powersof I . Specifically, for all n ∈ N we have I ( n ) = ( I (2) ) n .Using Theorem 3.2 as a characterization of rational powers, we can use the meth-ods of symbolic powers of squarefree monomial ideals to investigate invariants ofrational powers. One of the recent methods in this direction comes from [15].4. Convergence of Depth and Regularity via Splittings
While this section is used to prove facts about rational powers, we present thisgeneralization of [14, 15] as it can hold for more general filtrations. Throughoutthis section, let R = K [ x , . . . , x d ] for some field K , and I ⊂ R be a monomial ideal.Using the set-up from [15], let m ∈ N and denote R m = K [ x m , . . . , x m d ], which isa m N -graded ring. Notice that R m is isomorphic to R as rings. Let i : R → R m be the natural inclusion given by i ( x α ) = ( x mα ) /m . Following the method of [15]we define the splitting R -homomorphism induced by the map φ Rm : R m → R givenby φ Rm (( x α ) /m ) = ( x α/m α ≡ m ) , , with 0 = (0 , . . . , ∈ N d . Notice that φ Rm restricted to R is the identity, hence with i forms a split map. Definition 4.1.
We call a filtration of monomial ideals { I n } n ∈ N asymptoticallystable if it satisfies the following conditions:(1) The associated Rees algebra L n ∈ N I n t n ⊂ R [ t ] is Noetherian.(2) For an unbounded sequence { m f } f ∈ N ⊆ N , i and φ Rm f above induce a splitinjection ι : R/I n +1 → R mf / ( I nm f + j ) mf for all n, f ∈ N and 1 j m f .We can make the second requirement easier to check via the following remark. Remark 4.2.
For the second condition, to check that ι is a split injection, since φ Rm is a splitting we need only check that i ( I n +1 ) ⊆ ( I nm + j ) m and that φ Rm (( I nm + j ) m ) = I n +1 for infinitely many m ∈ N , all n ∈ N , and 1 j m . Notice for the former that i ( I n ) ⊆ ( I mn ) m ⊆ ( I nm ) m by the definition of i and the second condition. Hence i ( I n +1 ) ⊆ ( I nm + m ) m ⊆ ( I nm + j ) m , and thus it suffices to show the latter conditionthat φ Rm (( I nm + j ) m ) = I n +1 . The containment i ( I n +1 ) ⊆ ( I nm + j ) m also yields that I n +1 ⊆ φ Rm (( I nm + j ) m ) since φ Rm is split and R -linear. Hence we need only showthe other containment, that φ Rm (( I nm + j ) m ) ⊆ I n +1 . Example 4.3.
From [14], we note that the symbolic powers of squarefree monomialideals form an asymptotically stable filtration.The following also shows that the rational powers of a monomial ideal forman asymptotically stable family, extending the ideas of [15] and the connection ofCorollary 3.4.
IMIT BEHAVIOR OF THE RATIONAL POWERS OF MONOMIAL IDEALS 9
Proposition 4.4.
Let I be a monomial ideal and e = lcm { v ( I ) | v ∈ RV ( I ) } . Thenthe rational powers of I , { I ne } n ∈ N are asymptotically stable.Proof. From Theorem 2.2, the rational powers form a filtration. Hence we mustnow show that the associated Rees algebra is Noetherian. From [11, Lemma 1.1(a)],using our polynomial ring R , the algebra L n ∈ Z I ne u n ⊆ R [ u, u − ] is a finitelygenerated R -algebra. Hence, its positively graded piece must also be a finitelygenerated R -algebra.Now, to show condition (2), we use Remark 4.2. Let RV ( I ) = { v , . . . , v r } , thenusing Remark 3.1, let w i = ev i ( I ) v i for each 1 i r . Then let w i ≡ a i X + · · · + a id X d be the hyperplane equation corresponding to w i with a ik ∈ N for 1 k d and 1 i r as guaranteed in Remark 3.1 and Remark 2.3. Let m, n ∈ N and 1 j m .Let ( x α ) /m ∈ ( I nm + je ) m such that φ Rm (( x α ) /m ) = 0. Then x α ∈ I nm + je so that w i ( x α ) > nm + j for each 1 i r . Let β = α/m ∈ N d as α ≡ m ). In otherwords φ Rm (( x α ) /m ) = x β . So, since x β = x α/m , applying φ Rm yields that for each1 i rw i ( x α/m ) = a i α m + · · · + a id α d m = 1 m ( a i α + · · · + a id α d ) = w i ( x α ) m > n + jm . But since w i has coefficients in N we have w i ( β ) > n + 1 for each 1 i r . Hence x β ∈ I n +1 e , finishing the proof. (cid:3) The following lemma and theorem are needed to obtain our desired convergences.
Lemma 4.5. If { I n } n ∈ N is an asymptotically stable family, then for all n ∈ N andan infinite sequence { m f } f ∈ N ⊆ N as in Definition 4.1: (1) depth( R/I n ) depth( R/I l nmf m ) , (2) a i ( R/I n ) > m f a i ( R/I l nmf m ) for i dim R/ √ I .Proof. Notice that without loss of generality we may assume that the sequence { m f } f ∈ N ⊆ N is increasing. Now, we have the splitting map ι : R/I n +1 → R mf /I mf nm f + j for all n ∈ N and 1 j m f . Hence, for 0 i dim R/ √ I , themodule H i m ( R/I n +1 ) is a direct summand of H i m ( R mf /I mf nm f + j ). Now, by the iso-morphism of rings R ∼ = R mf we have an equivalence of categories (of R -modulesand R mf -modules) and that if we set m = ( x , . . . , x d ) to be the homogeneousmaximal ideal then m R mf = m mf . Thus we have the following equality:(4.1) (H i m ( R/I nm f + j )) mf = H i m mf ( R mf /I mf nm f + j ) = H i m ( R mf /I mf nm f + j ) . Hence H i m ( R/I nm f + j ) = 0 implies H i m ( R mf /I mf nm f + j ) = 0 and hence, as it is a directsummand, H i m ( R/I n +1 ) = 0. Thusdepth( R/I n +1 ) depth( R/I nm f + j )proving the first part. Again from the splitting of ι and hence the direct summand of local cohomology,and Equation (4.1) above, we have a i ( R/I n +1 ) a i ( R mf /I mf nm f + j ) = 1 m f a i ( R/I nm f + j )showing the second part. (cid:3) Theorem 4.6. If { I n } n ∈ N is an asymptotically stable family, then (1) lim n →∞ reg( I n ) n exists, (2) lim n →∞ depth( R/I n ) exists.Proof. Let { m f } e ∈ N be the sequence of natural numbers that satisfy the infinitelymany splittings of Definition 4.1. Similarly to before, we may assume without lossof generality that the sequence is increasing.For (1), note thatlim n →∞ reg( I n ) n = lim n →∞ reg( R/I n ) + 1 n = lim n →∞ max { a i ( R/I n ) + i + 1 | i dim R/ √ I } n = lim n →∞ max { a i ( R/I n ) | i dim R/ √ I } n = max { lim n →∞ a i ( R/I n ) n | i dim R/ p I } . (4.2)Thus we set α n = max i { a i ( R/I n ) } . Then, as the associated Rees algebra is Noe-therian, using the lemma inside the proof from [4, Proof of Theorem 4.3], weknow that reg( I n ) must be quasi-linear for large enough n . Hence, there are c , . . . , c r , b , . . . , b r ∈ N such that reg( R/I n ) = c j n + b j for n ≡ j (mod r ) for n ≫
0. By Equation (4.2) and the quasi-linearity we have lim k →∞ α rk + j rk + j = c j for each j . From this, fix 1 i, j r and m f > r . Let ǫ > t ∈ N such that | c j − α rk + j rk + j | < ǫ for all k > t . Thus, c j − α rk + j rk + j < ǫ . From Lemma 4.5 part (2) we have that α rk + j α m fs ( rk + j − b m fs for every s ∈ N and 1 b m fs . Thus,(4.3) c j − ǫ α rk + j rk + j α m fs ( rk + j − b m fs ( rk + j ) α m fs ( rk + j − b m fs ( rk + j −
1) + b .
Because Equation (4.3) holds for all 1 b m fs and that r < m f m fs , we can findinfinitely many s, k , and b such that m fs ( rk + j −
1) + b ≡ i (mod r ). We use thepairs in Equation (4.3) to yield in the limit that c j − ǫ c i for all ǫ . Hence c j c i and as i and j were arbitrary, we have c = . . . = c r , showing that the regularitylimit exists.For (2), again as the associated Rees algebra is Noetherian, there is a k ∈ N suchthat I n + k = I n I k for every n > k (see [17, Remark 2.4.3]). Thus, for 0 j k − d j , h j ∈ N withdepth( I ( n +1) k + j ) = depth(( I k ) n I k + j ) = d j so that depth( R/I ( n +1) k + j ) = d j − n > h j from [9, Theorem 1.1]. IMIT BEHAVIOR OF THE RATIONAL POWERS OF MONOMIAL IDEALS 11
Let d = min n { depth( R/I n ) } and fix l ∈ N with d = depth( R/I l ). Let f ∈ N suchthat m f > k and m f ( l − > ( h j + 1) k for 0 j k −
1. Now, Lemma 4.5 part (2)implies that depth(
R/I m f ( l − i ) depth( R/I l )for 1 i m f . By choice of m f , for each 0 j k − t > h j + 1 and1 i m f with m f ( l −
1) + i = tk + j . Then d = depth( R/I l ) > depth( R/I m f ( l − i ) = depth( R/I tk + j ) = d j − . Hence, d = d j − j k −
1, showing the convergence. (cid:3) Asymptotics of Rational Powers
This section focuses on using the connection between rational powers and sym-bolic powers to find the convergence of Stanley depth, and the connection withintegral closure powers to find the convergence of length for some local cohomologymodules. First, as we can gleam from the previous section, we present the proof ofTheorem B.
Theorem 5.1. If I is any monomial ideal and e = lcm { v ( I ) | v ∈ RV ( I ) } , then (1) lim n →∞ reg( I ne ) n exists and is equal to e lim n →∞ reg( I n ) n . (2) lim n →∞ depth( R/I ne ) exists and is equal to d − ℓ ( I ) , where ℓ ( I ) is the analyticspread of I .Proof. From Proposition 4.4 and Theorem 4.6, both of these limits exist. For theregularity, notice that for n ∈ N thatreg( I n ) n = reg( I nee ) ene = e reg( I nee ) ne so that if lim n →∞ reg( I ne ) n = p e and lim n →∞ reg( I n ) n = p then ep e = p . From this equality, p e = e p , finishing (1).For depth, notice that from the correspondence I n = I nee , { depth( R/I n ) } n ∈ N appears as subsequences of the I ne depth sequence. Hence both sequences have thesame limit and so by [10, Lemma 1.5], we have finished (2). (cid:3) Notice due to the connection with symbolic powers of squarefree monomial ideals,we can conclude a convex-geometric computation of the symbolic analytic spread.
Remark 5.2.
Let I be a squarefree monomial ideal. Then there exists a g ∈ N such that the symbolic analytic spread ℓ s ( I ) = ℓ ( I ( g ) ). Notice that this allows usto have ℓ s ( I ) computed by using the Symbolic Polyhedron. To see this, notice thatfrom Corollary 3.4 we have that such a g exists. Then the rational powers of I ( g ) are the symbolic powers of I . By Theorem 5.1 we have thatlim n →∞ depth( R/I ( n ) ) = lim n →∞ depth( R/ ( I ( g ) ) ng ) = d − ℓ ( I ( g ) ) . On the other hand, from [22, Theorem 3.6]lim n →∞ depth( R/I ( n ) ) = d − ℓ s ( I ) . and thus ℓ s ( I ) = ℓ ( I ( g ) ).Then, from [1, Theorem 2.3] we can compute ℓ ( I ( g ) ) by the maximal dimensionof a compact face of NP( I ( g ) ) plus one. However, as we have seen, NP( I ( g ) ) is a multiple of the symbolic polyhedron, SP( I ), and thus, since taking multiples of aconvex hull does not change the dimension of the faces, we can look at the dimensionof the faces of SP( I ) to compute ℓ ( I ( g ) ) = ℓ s ( I ).Notice also that Theorem 5.1 allows us to use the connection between the rationaland integral powers. We can now use this connection for Stanley Depth. Most ofthe study of Stanley depth of integral closure powers has been limited (e.g. [20]),but this connection allows us to conclude new facts about integral closure powersof monomial ideals. We now prove Theorem C. Theorem 5.3. If I is a monomial ideal and e = lcm { v ( I ) | v ∈ RV ( I ) } . Then thelimits lim n →∞ sdepth( R/I ne ) and lim n →∞ sdepth( I ne ) exist. In particular, these limitsmust exist for { I n } n ∈ N . Furthermore: lim n →∞ sdepth( R/I ne ) = min n sdepth( R/I n ), andlim n →∞ sdepth( I ne ) = min n sdepth( I n ) . Proof.
Using Remark 3.1, suppose that the Rees valuations of I are v , . . . , v r , let w i = ev i ( I ) v i for each 1 i r .For ease of notation, let f denote a monomial of R . Using [22, Lemma 4.1,Theorem 4.2] let m ∈ N and k m , then for j with m − k j m we claim that f ∈ I me if and only if f k +1 ∈ I km + je . Indeed, from Theorem 2.2, f ∈ I me implies f k +1 ∈ ( I me ) k +1 ⊆ I m ( k +1) e ⊆ I km + je .Conversely, assume f k +1 ∈ I km + je and f / ∈ I me . Then there is an 1 i r suchthat w i ( f ) < m . Since, by Remark 2.3 and Remark 3.1, we can consider w i as ahyperplane equation with coefficients in N we must have w i ( f ) m −
1. Hence w i ( f k +1 ) ( k + 1)( m −
1) = km + m − k − < km + ( m − k ) km + j which implies that f k +1 / ∈ I km + je a contradiction; proving the equivalence.Thus, by [22, Theorem 4.2], we have(5.1) sdepth( I me ) > sdepth( I km + je ) and sdepth( R/I me ) > sdepth( R/I km + je ) . Similarly, for any s, k ∈ N we have f ∈ I se if and only if f s ∈ I kse by the monomialvaluation inequalities. Indeed, w i ( f ) > se for each 1 i r if and only if w i ( f k ) > ske for each 1 i r since valuations are a multiplicative group homomorphism. Thus,from [19, Theorem 3.1] we must have(5.2) sdepth( R/I kse ) sdepth( R/I se ) and sdepth( I kse ) sdepth( I se )Set m = min { sdepth( R/I ke ) | k ∈ N } and t = min { s | sdepth( R/I se ) = m } . If t = 1then from Equation (5.2) we havesdepth( R/I e ) sdepth( R/I ke ) = sdepth( R/I k · e ) sdepth( R/I e )proving the limit. So, suppose that t >
1. Then sdepth(
R/I t − te ) sdepth( R/I te ) = m again from Equation (5.2). For every k > t − t write k = lt + j with 1 j t .Then notice that l > t − lt + j > t − t . Then by Equation (5.1) we have thatsdepth( R/I ke ) = sdepth( R/I lt + je ) sdepth( R/I te ) = m and thus sdepth( R/I ke ) = m for all k > t − t , proving the convergence. The sameargument then holds for sdepth( I ne ) as well. IMIT BEHAVIOR OF THE RATIONAL POWERS OF MONOMIAL IDEALS 13
For { I n } n ∈ N and the limit equalities, notice that the correspondence I n = I nee yields { sdepth( R/I n ) } n ∈ N as a subsequence of the sequence { sdepth( R/I ne ) } n ∈ N .Hence both sequences have the same limit, and thuslim n →∞ sdepth( R/I ne ) = min n sdepth( R/I ne ) min n sdepth( R/I n ) lim n →∞ sdepth( R/I n )completes the proof. (cid:3) We can use the fact that rational powers are integrally closed to conclude otherdesirable properties of a filtration, for example, the finiteness of associated primes.We will present this argument in full generality.
Proposition 5.4. If { I n } n ∈ N is a filtration whose associated Rees algebra is Noe-therian and such that I n is integrally closed for all n ≫ , then there exists an m ∈ N with Ass(
R/I n ) ⊆ Ass(
R/I m ) for all n > m . Hence ∪ n ∈ N Ass(
R/I n ) is finite.Proof. By [17, 2.4.4], for all c ≫ n > I cn = I nc . Now let m ≫ I k is integrally closed for all k > m .Then for n m ≫ R/I mn ) = Ass( R/I nm ) = Ass( R/I nm ) = Ass( R/I n m m ) = Ass( R/I mn m )for all n > n m by the stabilizing of associated primes for integral closure powersfrom [13, 6.8.8]. Now let r > mn m , then as I r is integrally closed we haveAss( R/I r ) ⊆ Ass(
R/I n r r ) = Ass( R/I rn r ) = Ass( R/I rn r mn m ) = Ass( R/I mn m )finishing the proof. (cid:3) Clearly, we can apply Proposition 4.4 to rational powers as follows.
Corollary 5.5.
Let I be a monomial ideal. Then ∪ n ∈ N Ass(
R/I ne ) is finite with e = lcm { v ( I ) | v ∈ RV ( I ) } . We next show the existence of the limit of the lengths of local cohomologymodules. In order to do this, we will be using the method of [5]. First we mustestablish some lemmas that will allow us to use their results.
Lemma 5.6.
For a commutative ring A , multiplicatively closed set W ⊆ A , ideal I ⊆ A , and α ∈ Q + , we have W − I α = ( W − I ) α .Proof. Write α = pq for coprime p, q ∈ N . We will use [13, Proposition 1.1.4] andtheir argument for persistence. We will first show that taking rational powers ispersistent, that is, if φ : A → S is a ring homomorphism to some commutative ring S , then φ ( I α ) ⊆ ( φ ( I ) S ) α . Indeed if s ∈ φ ( I α ), then s = φ ( r ) for some r ∈ I α . Thenthere is an integral dependence equation of r q over I p since r q ∈ I p . Then applying φ to the equation gives an integral dependence equation of φ ( r q ) = s q over ( φ ( I ) S ) p so that s ∈ ( φ ( I ) S ) α .Now for the localization case, persistence yields one containment, for the otherlet x ∈ ( W − I ) α . Since x q ∈ ( W − I ) p = W − I p there exists a w ∈ W with wx q ∈ I p and with wx ∈ A . Thus, w q − wx q = ( wx ) q ∈ I p so that wx ∈ I pq = I α and hence x ∈ W − I α , finishing the proof. (cid:3) Notice here that we do not need a polynomial ring for the localization, makingthe lemma a general result on rational powers. We now must appeal to the convexgeometry of monomial ideals as used in [5].
Lemma 5.7.
Let I , . . . , I r , J , . . . , J s be monomial ideals in R and e ∈ N . Forevery n > consider the set S n = { α ∈ N d | x α ∈ I ne i for every 1 i r , and x α / ∈ J ne i for every 1 i s } . Assume | S n | < ∞ for n ≫ . Then lim n →∞ | S n | n d exists and is equal to a rational number.Proof. Let Γ := ∩ ri =1 1 e NP( I i ) and Γ i := Γ ∩ e NP( J i ) for 1 i s . Furthermore, let nC := n Γ − ∪ si =1 n Γ i . Then notice that α ∈ S n if and only if α ∈ ne NP( I i ) for each1 i r and α / ∈ ne NP( J i ) for each 1 i s . Using the new notation, α ∈ S n if andonly if α ∈ nC . Hence, by assumption C is bounded and so by [5, Lemma 4.1] thelimit exists and is rational. (cid:3) Before proceeding to apply this lemma, we must establish some notation from[5]. Let [ d ] = { , . . . , d } and F ⊆ [ d ]. Then we let π F : R → R be the map definedby π F ( x i ) = 1 if i ∈ F and π F ( x i ) = x i otherwise. For an ideal I we say I F := π F ( I ).For α ∈ Z d we let G α = { i | α i < } and α + = ( α +1 , . . . , α + d ) where α + i = α i if i / ∈ G α and α + i = 0 otherwise. For a monomial ideal I we let ∆ α ( I ) be thesimplicial complex of all subsets F of [ d ] − G α such that x α + / ∈ I F ∪ G α . Finally,for a monomial ideal I , let ∆( I ) be the simplicial complex of all F ⊆ [ d ] withΠ i ∈ F x i / ∈ √ I . We are now ready to prove Theorem D. Theorem 5.8.
Let I be a monomial ideal and e = lcm { v ( I ) | v ∈ RV ( I ) } . Assumethat λ (H i m ( R/I ne )) < ∞ for n ≫ . Then the limit lim n →∞ λ (H i m ( R/I ne )) n d exists and is rational. In particular, this limit must exist for { I ( n ) } n ∈ N when I issquarefree. Furthermore lim n →∞ λ (H i m ( R/I ne )) n d = e d lim n →∞ λ (H i m ( R/I n )) n d .Proof. From Proposition 4.4 the associated Rees algebra of the rational powers isNoetherian, and so by the application of Takayama’s formula from [5, Theorem 3.8]we have that λ (H i m ( R/I ne )) = Σ ∆ ′ ⊆ ∆( I ) dim k ˜ H i − (∆ ′ , k ) f ∆ ′ ( n )where f ∆ ′ ( n ) = |{ α ∈ N d | ∆ α ( I ne ) = ∆ ′ }| . We claim that for each ∆ ′ ⊂ ∆( I )such that ˜ H i − (∆ ′ , k ) = 0 the limit lim n →∞ f ∆ ′ ( n ) n d exists and is rational, which willfinish the proof. By assumption we must have f ∆ ′ ( n ) < ∞ for n ≫ H i − (∆ ′ , k ) = 0. For α ∈ N d we have ∆ α ( I ne ) = ∆ ′ if and only if x α / ∈ ( I ne ) F forevery facet F of ∆ ′ , and x α ∈ ( I ne ) G for every minimal non-face G of ∆ ′ . Then byLemma 5.6 we have ( I ne ) F = ( I F ) ne and ( I ne ) G = ( I G ) ne for each F and G . Thus,we may use Lemma 5.7 to show the existence of the limit.By Corollary 3.4, the limit must exist under the same assumptions for { I ( n ) } n ∈ N when I is squarefree. Notice that the correspondence I n = I nee yields the limitequality via the argument of Theorem 5.1 part (1). (cid:3) Acknowledgements
The author would like to thank his advisor, Jonathan Monta˜no, for the manyinvaluable suggestions, comments, and discussions he provided. The author wouldalso like to thank Louiza Fouli for her helpful comments.
IMIT BEHAVIOR OF THE RATIONAL POWERS OF MONOMIAL IDEALS 15
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