Connectedness of square-free Groebner Deformations
Lilia Alanís-López, Luis Núñez-Betancourt, Pedro Ramírez-Moreno
aa r X i v : . [ m a t h . A C ] M a y CONNECTEDNESS OF SQUARE-FREE GROEBNER DEFORMATIONS
LILIA ALANÍS-LÓPEZ, LUIS NÚÑEZ-BETANCOURT , AND PEDRO RAMÍREZ-MORENO Abstract.
Let I ⊆ S = K [ x , . . . , x n ] be a homogeneous ideal equipped with a mono-mial order < . We show that if in < ( I ) is a square-free monomial ideal, then S/I and S/ in < ( I ) have the same connectedness dimension. We also show that graphs related toconnectedness of these quotient rings have the same number of components. We also pro-vide consequences regarding Lyubeznik numbers. We obtain these results by furtheringthe study of connectedness modulo a parameter in a local ring. Contents
1. Graphs and Connectedness Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22. Connectedness Dimension and Γ t Graphs Modulo a Parameter . . . . . . . . . . . . . . . 53. Applications to Groebner deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104. Applications to Lyubeznik numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Let I ⊆ S = K [ x , . . . , x n ] be a homogeneous ideal in a standard graded polynomialring over a field. A theme of research is to obtain properties of I from the monomialideal in < ( I ) . For instance, it is well known that these two ideals have the same Hilbertfunction, and so, the same dimension.In this work we focus on the connectedness dimension, c( X ) , of an algebraic variety, X . This number measures how connected X is. Explicitly, c( X ) = min { dim( Z ) | Z is closed and X \ Z is diconnected } . This line of research was initiated by Varbaro [Var09], who showed that c ( S/ in < ( I )) ≥ c ( S/I ) (see also [KS95]). If X = Spec( R ) is an equidimensional affine variety, there is asequence of graphs, Γ t ( R ) , that detect c ( X ) (see Definition 1.1). Explicitly, t ( R ) = max { X \ Z ) | Z is a closed subvariety } , where denotes the number of connected components [NnBSW19]. We recall that Γ ( R ) is called the dual graph [Har62] or Hochster-Huneke graph [HH94] of R . Similarly, Γ t ( R ) played a role in the study of vanishing of local cohomology [Har68, HL90, Ogu73]. Mathematics Subject Classification.
Primary 13E05, 05E40, 13C15, 13H99, 13J10; Secondary13B25,13P10.
Key words and phrases.
Connectedness Dimension; Noetherian Equidimensional Complete LocalRings; Gröbner Bases; Monomial Orders; Non zero divisors. This author was partially supported by CONACyT Grant 284598 and Cátedras Marcos Moshinsky. This author was partially supported by CONACyT Grants 706865 and 284598.
Recently, Conca and Varbaro [CV20] showed that if in < ( I ) is a square-free monomialideal more properties of I are reflected in in < ( I ) . In particular, they showed that the a -invariants and extremal Betti numbers of in < ( I ) and I are equal. In this work, we showthat if in < ( I ) is a square-free monomial ideal, then c ( S/ in < ( I )) = c ( S/I ) . Theorem A (Theorem 3.2) . Let S = K [ x , . . . , x n ] be a polynomial ring over a field. Let I be an equidimensional ideal and < a monomial order such that dim S/I ≥ and in < ( I ) is square-free. Then, t ( S/I ) = t ( S/ in( I )) . As a consequence, c( S/I ) = c( S/ in( I )) . Theorem A extends previous work of Nadi and Varbaro [NV19] in which they showedthat Γ t ( S/I ) is connected if and only if Γ t ( S/ in( I )) is connected. We show that, under theconditions in Theorem A, t ( S/I ) does not change after field extensions (see Corollary3.3).The graphs Γ t ( R ) have played an important role in the study of Lyubeznik numbers[Lyu06, NnBSW19, Wal01, Zha07], which are numerical invariants obtained from localcohomology [Lyu93a]. We refer to a survey on Lyubeznik numbers for more informationabout this topic [NnBWZ16]. As a consequence of Theorem A, we are able show thatcertain Lyubeznik numbers of S/I and S/ in < ( I ) are equal if in < ( I ) is square-free (seeTheorem 4.4).The main technique to show Theorem A is the study of connectedness under a parameterideal. In fact, Theorem A is an application of the following result. Theorem B (Theorem 2.12) . Let ( R, m ) be an equidimensional complete local ring con-taining a field of dim( R ) = d ≥ , with separably closed residue field. Suppose there exists x ∈ m such that x is a non zero divisor of R and ( x ) is a radical ideal. Let t be an integersuch that ≤ t ≤ d − . Then, t ( R ) = t ( R/ ( x )) . As a consequence, c( R ) = c( R/ ( x )) + 1 . Theorem B is on the same line of research studied by Spiroff, Witt, and the second-named author [NnBSW19], where different conditions were imposed on the parameter toobtain the equality of connectedness dimension.
Convention.
All rings in this manuscript are commutative Noetherian with one. Graphs and Connectedness Dimension
We start by recalling the construction of certain graphs that detect the connectednessof a variety or a ring.
Definition 1.1 ([NnBSW19]) . Let R be a ring of dimension d and let t be an integersuch that ≤ t ≤ d . We define the graph Γ t ( R ) as a simple graph whose vertices are theminimal primes of R and there is an edge between p and q distinct minimal primes if andonly if ht( p + q ) ≤ t . ONNECTEDNESS OF SQUARE-FREE GROEBNER DEFORMATIONS 3
We note that Γ s ( R ) is a subgraph of Γ t ( R ) for every s and t such that ≤ s ≤ t ≤ d .As a consequence, if Γ s ( R ) is connected, then Γ t ( R ) is also connected. We observe that Γ ( R ) is connected if and only if R has only one minimal prime. In addition, Γ d ( R ) isconnected if R is local.Connectedness dimension is a ring invariant and one of our main objects of study. Wedefine it in the following way: Definition 1.2.
Let R be a ring. We define the connectedness dimension of R , anddenote it as c( R ) , as c( R ) = min { dim( R/I ) | Spec( R ) − V ( I ) is disconnected } We take the convention that the empty set is disconnected.
Theorem 1.3.
Let R be an equidimensional complete local ring. Let I be a proper idealof R . Then c( R/I ) ≥ min { c( R ) , dim( R ) − } − ara( I ) We state some well-known properties to have that are helpful in several results.
Remark 1.4 ([NnBSW19, Remark 2.6]) . Let R be a ring and let I , . . . , I n , J , . . . , J m beideals of R , then(1) qT ni =1 I i + T mj =1 J j = qT ni =1 T mj =1 ( I i + J j ) . (2) ht( T ni =1 I i + T mj =1 J j ) = min { ht( I i + J j ) | ≤ i ≤ n, ≤ j ≤ m } . Remark 1.5.
Let A be a complete local equidimensional ring and let I be an ideal of A .Then ht( I ) + dim( A/I ) = dim( A ) . A graph is connected if and only if no matter how we partition its set of indices in twonon empty sets, we can always find an edge between a vertex of one of these two disjointsets and a vertex of the other one.
Proposition 1.6.
Let ( R, m ) be a local ring of dimension d with more than one minimalprime. Let t be an integer such that ≤ t ≤ d − . Then Γ t ( R ) is connected if and only if ht( T p ∈ S p + T q ∈ T q ) ≤ t for every ( S, T ) partition of Min( R ) such that S and T are nonempty.Proof. By Remark 1.4, we know that given a ( S, T ) partition of Min( R ) , we have that ht \ p ∈ S p + \ q ∈ T q ! = min { ht( p + q ) | p ∈ S, q ∈ T } , so there must be p ′ ∈ S and q ′ ∈ T such that ht (cid:16)T p ∈ S p + T q ∈ T q (cid:17) = ht( p ′ + q ′ ) . Thismeans that for every ( S, T ) partition of Min( R ) such that S and T are non empty , wehave that ht \ p ∈ S p + \ q ∈ T q ! ≤ t ⇔ ∃ p ′ ∈ S, q ′ ∈ T : ht( p ′ + q ′ ) ≤ t. So, for any such partition ( S, T ) , you can find an edge between S and T . This happens ifand only if Γ t ( R ) is connected. (cid:3) L. ALANÍS-LÓPEZ, L. NÚÑEZ-BETANCOURT, AND P. RAMÍREZ-MORENO
Remark 1.7.
Let ( R, m ) be an equidimensional local ring with dim( R ) = d ≥ . Let x ∈ m such that x is not an element of any minimal prime of R . Then, dim R/ ( x ) = d − ,because x is a parameter. We know that the minimal primes of R/ ( x ) are of the form q / ( x ) ,with q a minimal prime of ( x ) . Then dim (cid:16) R/ ( x ) q / ( x ) (cid:17) = dim( R/ q ) , and ht( q ) = 1 by Krull’sPrincipal Ideal Theorem. We know that in R the equality ht( q )+dim( R/ q ) = dim( R ) = d holds from Remark 1.5, we conclude that dim( R/ q ) = d − . So dim (cid:16) R/ ( x ) q / ( x ) (cid:17) = d − .This means R/ ( x ) is equidimensional.Given a Γ graph we focus our attention in its subgraph corresponding to certain subsetof minimal primes. One way to study such subgraph is by doing specific quotients of thering. Proposition 1.8.
Let ( R, m ) be an equidimensional complete local ring and let I be aproper ideal of R such that Min(
R/I ) ⊆ Min( R ) . Then R/I is also an equidimensionalcomplete local ring and dim(
R/I ) = dim( R ) . Furthermore, if J is an ideal of R such that I ⊆ J then ht( J ) = ht( J/I ) . In addition, if Σ is the subgraph of Γ t ( R ) whose vertices are Min( I ) , then Σ ∼ = Γ t ( R/I ) . Proof.
We know that
R/I is a complete local ring. We also know that the minimal primesof
R/I are the ideals of the form p /I with p minimal prime of R .Observe that dim (cid:16) R/I p /I (cid:17) = dim( R/ p ) = dim( R ) for every minimal prime p of R since R is equidimensional. This means R/I is also equidimensional and dim(
R/I ) = dim( R ) .Let J be an ideal of R such that I ⊆ J . By Remark 1.5 we have that ht( J/I ) +dim (cid:0) RI / JI (cid:1) = dim( R/I ) , so ht( J/I ) = dim( R ) − dim( R/J ) . Remark 1.5 implies that ht( J ) = dim( R ) − dim( R/J ) . Thus ht( J ) = ht( J/I ) .The correspondence between vertices of Σ and vertices of Γ t ( R/I ) is given by assigningeach minimal prime p of I to the minimal prime p /I of R/I . Thus the vertices arepreserved. Notice that edges are also preserved since if there is a edge between p and q minimal primes of I , then ht( p + q ) ≤ t . This is the same as saying that ht( p /I + q /I ) =ht(( p + q ) /I ) ≤ t since ht( p + q ) = ht(( p + q ) /I ) by the previous paragraph. (cid:3) In particular we choose I to be exactly the intersection of the minimal primes corre-sponding to the part of the graph we want to focus our attention on.The next proposition gives us more information about how the graphs work when westudy the quotient ring with different ideals but with the same radical. Observation 1.9.
Let ( R, m ) be an equidimensional complete local ring and let I , J beideals of R such that √ I = √ J . Then both R/I and
R/J are complete local rings ofthe same dimension and if
R/I is equidimensional, then
R/J is also equidimensional and Γ t ( R/I ) ∼ = Γ t ( R/J ) . Now we are ready to begin exploring the relations between connectedness dimensionand the Γ graphs. Proposition 1.10 ([NnBSW19, Proposition 2.5]) . Let ( R, m ) be an equidimensional com-plete local ring with dim( R ) = d ≥ . let t be an integer such that ≤ t ≤ d − . Then Γ t ( R ) is connected ⇔ c( R ) ≥ d − t. ONNECTEDNESS OF SQUARE-FREE GROEBNER DEFORMATIONS 5
As a consequence, the connectedness dimension is given by c( R ) = max { i | Γ d − i ( R ) is connected } . We can compute connectedness dimension by counting how many of the Γ t ( R ) graphswith t ∈ [0 , d − are connected. Even if a graph Γ t ( R ) is not connected we can alsoobtain information regarding its connected components. Notation 1.11.
Let G be a graph and let X be a topological space. We denote G tothe amount of connected components of G and denote X to the amount of connectedcomponents of the space X . Corollary 1.12 ([NnBSW19, Corollary 2.7]) . Let ( R, m ) be an equidimensional completelocal ring of dimension d ≥ . Let t be an integer such that ≤ t ≤ d − . Then: t ( R ) = max { R ) − V ( I )) | dim( R/I ) < d − t } . Connectedness Dimension and Γ t Graphs Modulo a Parameter
In this section we study our connectedness graphs modulo a parameter. In order to dothis, we need to develop need additional tools. The following lemma gives us informationabout the behavior between minimal primes of a ring and the minimal primes of an idealgenerated by a parameter.
Lemma 2.1.
Let ( R, m ) be an equidimensional complete local ring with dim( R ) = d ≥ .Let x ∈ m such that x is not an element of any minimal prime of R . Then(1) For every minimal prime q of ( x ) , there is a minimal prime p of R such that p ⊆ q .(2) For every minimal prime p of R , there is a minimal prime q of ( x ) such that p ⊆ q .Proof. For the first part, we observe that ht( q ) = 1 . Then, there must be a minimal prime p of A such that p ⊆ q by prime avoidance.We now show the second part. Let p be a minimal prime of R . Notice that R/ p that x is not contained in the unique minimal prime of R/ p , so by Remark 1.7, we get that R/ p ( x ) ∼ = R/ ( p + ( x )) is equidimensional of dimension d − and that ht( x ) = 1 .Since ht( x ) + dim (cid:18) R/ p ( x ) (cid:19) = dim ( R/ p ) , we conclude that dim ( R/ ( p + ( x ))) = d − . We also know that ht( p + ( x )) + dim ( R/ ( p + ( x ))) = dim( R ) , and so, ht( p + ( x )) = 1 . Now take q ∈ Min( p + ( x )) such that ht( q ) = ht( p + ( x )) . Since ( x ) ⊆ p + ( x ) and ht( x ) =ht( p + ( x )) by Remark 1.7, then q ∈ Min( x ) . Finally p ⊆ q because q ∈ Min( p + ( x )) . (cid:3) From Lemma 2.1, we are able to characterize minimal primes of R/ ( x ) . L. ALANÍS-LÓPEZ, L. NÚÑEZ-BETANCOURT, AND P. RAMÍREZ-MORENO
Proposition 2.2.
Let ( R, m ) be an equidimensional complete local ring with dim( R ) = d ≥ . Let x ∈ m such that x is not an element of any minimal prime of R . For everyminimal prime p of R , we have that Min( p + ( x )) = { q ∈ Min( x ) | p ⊆ q } . Proof.
We proceed by double containment. Let A = { q ∈ Min( x ) | p ⊆ q } .Take a minimal prime q of ( x ) that contains p . Since p +( x ) ⊆ q and ht( p +( x )) = ht( q ) ,then q must be a minimal prime of p + ( x ) . Then, A ⊆ Min( p + ( x )) .Now, let p ∈ Min( R ) and let q ∈ Min( p + ( x )) . We show that ht( q ) = 1 . From Remark1.7, we know that R p +( x ) is an equidimensional ring of dimension d − . We know fromRemark 1.5 that ht ( q / ( p + ( x ))) + dim (cid:18) R/ ( p + ( x )) q / ( p + ( x )) (cid:19) = dim ( R/ ( p + ( x ))) . Since q / ( p + ( x )) ∈ Min ( R/ ( p + ( x ))) , we get that ht ( q / ( p + ( x ))) = 0 . We concludethat dim (cid:18) R/ ( p + ( x )) q / ( p + ( x )) (cid:19) = dim ( R/ q ) = d − . We also know that ht( q ) + dim( R/ q ) = dim( R ) , and so ht( q ) = 1 . This means that q isa minimal prime of ( x ) and contains p . Hence, Min( p + ( x )) ⊆ A . (cid:3) We know D ( p ) = Min( p + ( x )) by Proposition 2.2. From Lemma 2.1, we deduce that S p ∈ Min( R ) D ( p ) = Min( x ) . Corollary 2.3.
Let ( R, m ) be an equidimensional complete local ring of dim( R ) = d ≥ .Let x ∈ m such that x is a not an element of any minimal prime of R . Let S be a nonempty subset of Min( R ) and let I = T p ∈ S p . Then Min( I + ( x )) = [ p ∈ S Min( p + ( x )) . Proof.
We proceed by double containment.First we prove that
Min( I + ( x )) ⊆ S p ∈ S Min( p + ( x )) . Take a minimal prime q of I + ( x ) . By prime avoidance q contains a prime p ∈ S . But I + ( x ) ⊆ p + ( x ) ⊆ q . Thisimplies that q is also a minimal prime of p + ( x ) . This also means that ht( I + ( x )) = 1 ,because all the minimal primes of p + ( x ) are of height by Proposition 2.2.Now, we prove that S p ∈ S Min( p + ( x )) ⊆ Min( I + ( x )) . Let p ∈ S and q be a minimalprime of p + ( x ) . Since I + ( x ) ⊆ p + ( x ) and ht( I + ( x )) = ht( p + ( x )) = ht( q ) , we deducethat q must also be a minimal prime of I + ( x ) . (cid:3) The following definition plays a key role in the rest of the section, in particular, inTheorem B.
Definition 2.4.
Let ( R, m ) be an equidimensional complete local ring with dim( R ) = d ≥ . Let x ∈ m be such that x is not an element of any minimal prime of R . Given aminimal prime p of R , we define the dust of p modulo x by D x ( p ) = { q ∈ Min( x ) | p ⊆ q } . ONNECTEDNESS OF SQUARE-FREE GROEBNER DEFORMATIONS 7
Furthermore if Σ is a subgraph of Γ t ( R ) , then D x (Σ) = [ p ∈ Σ D x ( p ) . If x is clear from the context, we omit the subscript. Definition 2.5.
Let ( R, m ) be an equidimensional complete local ring with dim( R ) = d ≥ . Let x ∈ m be such that x is not an element of any minimal prime of R . Let Σ bea subgraph of Γ t ( R ) . Let Σ ′ be the subgraph of Γ t ( R/ ( x )) such that its vertices are givenby q / ( x ) such that q ∈ D (Σ) . We call Σ ′ the associated graph to Σ .In the previous setting let Σ be the subgraph of Γ t ( R ) whose vertices are the elementsof S . We observe that D (Σ) = Min( I + ( x )) , because D ( p ) = Min( p + ( x )) .Now we are ready for our study of connectedness dimension modulo a parameter. Itturns out that if Γ t ( R ) is connected, then Γ t ( R/ ( x )) is also connected. The only casewhen this is not necessarily true is when t = 0 , as the following example shows. Example 2.6.
Let K be a field and consider the power series ring R = K [[ x, y, z ]] . Γ t ( R ) is connected for every t since R is a domain, but Γ ( R/ ( xyz )) is not connected since ithas more than one minimal prime.Additionally, we restrict t to be less or equal than d − . We do so because Γ d − ( R/ ( x )) is connected regardless the connectedness of Γ d − ( R ) . We now recall a result on thisregarding connectedness modulo a parameter. We poit out that the original statementrequires that the residue field is separably closed [NnBSW19, Proposition 3.2]; however,this hypothesis is not necessary in the proofs. Theorem 2.7 ([NnBSW19, Proposition 3.2]) . Let ( R, m ) be an equidimensional completelocal ring containing a field,of dim( R ) = d ≥ . Let x ∈ m such that x is a not an elementof any minimal prime of R . Let t be an integer such that ≤ t ≤ d − . Then, Γ t ( R ) is connected ⇒ Γ t ( R/ ( x )) is connected . We now give a slightly more general version of Theorem 2.7. This is needed in ourproof of Theorem B.
Corollary 2.8.
Let ( R, m ) be an equidimensional complete local ring containing a field,of dim( R ) = d ≥ . Let x ∈ m such that x is a not an element of any minimal prime of A . Let t be an integer such that ≤ t ≤ d − . Let Σ be a subgraph of Γ t ( R ) and let Σ ′ be the subgraph of Γ t ( R/ ( x )) associated to D (Σ) . Then, Σ is connected ⇒ Σ ′ is connected . Proof.
Suppose Σ is connected. Let I be the intersection of all the vertices of Σ . FromProposition 1.8 we know that Σ ∼ = Γ t ( R/I ) , so Γ t ( R/I ) is also connected and Theorem2.7 implies that Γ t ( R/ ( I + ( x ))) is also connected.From Corollary 2.3 we know that p I + ( x ) = T q ∈ Min( I +( x )) q = T q ∈ D (Σ) q . Observation1.9 implies that Γ t ( R/ ( I + ( x ))) ∼ = Γ t (cid:16) R/ T q ∈ D (Σ) q (cid:17) . Then, Γ t R T q ∈ D (Σ) q ! ∼ = Γ t R/ ( x ) T q ∈ D (Σ) q / ( x ) ! ∼ = Σ ′ by Proposition 1.8. We conclude that Σ ′ is also connected. (cid:3) L. ALANÍS-LÓPEZ, L. NÚÑEZ-BETANCOURT, AND P. RAMÍREZ-MORENO
Lemma 2.9.
Let ( R, m ) be an equidimensional complete local ring of dim( R ) = d ≥ .Let x ∈ m such that x is not in any minimal prime of R . Let t be an integer such that ≤ t ≤ d − . Let Σ i and Σ j be subgraphs of Γ t ( R ) . If the graphs Σ i and Σ j do not shareany vertices and there are no edges between them, then D (Σ i ) and D (Σ j ) are disjoint. Inparticular the subgraphs Σ ′ i and Σ ′ j of Γ t ( R/ ( x )) associated to Σ i and Σ j respectively donot share vertices.Proof. Let q ∈ D (Σ i ) ∩ D (Σ j ) , then there are p i ∈ Σ i and p j ∈ Σ j such that q ∈ D ( p i ) ∩ D ( p j ) . So p i + p j ⊆ q , this means that ht( p i + p j ) ≤ ht( q ) = 1 ≤ t . So there is anedge between Σ i and Σ j , a contradiction. (cid:3) The following lemma plays the main role in the proof Theorem B.
Lemma 2.10.
Let ( R, m ) be an equidimensional complete local ring. Suppose there is an x ∈ m such that x is a non zero divisor of R and that ( x ) is a radical ideal. Let ( S, T ) bea partition of Min( R ) such that S and T are non empty, and I = T p ∈ S p and J = T q ∈ T q .Then, x is a non zero divisor of RI + J . In particular x is not in any minimal prime of I + J .Proof. Consider the exact sequence → RI ∩ J → RI ⊕ RJ → RI + J → . This sequence induces a long exact sequence of Tor of the form · · ·
Tor (cid:16) RI ∩ J , R ( x ) (cid:17) Tor (cid:16) RI ⊕ RJ , R ( x ) (cid:17) Tor (cid:16) RI + J , R ( x ) (cid:17) RI ∩ J ⊗ R ( x ) (cid:0) RI ⊕ RJ (cid:1) ⊗ R ( x ) RI + J ⊗ R ( x ) Since x is a non zero divisor of R , we have that Tor ( R/I, R/ ( x )) = Ann R/I ( x ) , and Ann
R/I ( x ) = 0 . Similarly Tor ( R/J, R/ ( x )) = Ann R/J ( x ) = 0 . Then, Tor (cid:18) RI ⊕ RJ , R ( x ) (cid:19) = Tor (cid:18) RI , R ( x ) (cid:19) ⊕ Tor (cid:18) RJ , R ( x ) (cid:19) = 0 . We have that
Tor (cid:16) RI + J , R ( x ) (cid:17) = Ann RI + J ( x ) , and R a ⊗ R ( x ) = R a +( x ) for every ideal a ⊆ R. Then, → Ann RI + J ( x ) → RI ∩ J + ( x ) → RI + ( x ) ⊕ RJ + ( x ) → RI + J + ( x ) → . ONNECTEDNESS OF SQUARE-FREE GROEBNER DEFORMATIONS 9
Observe that ( x ) ⊆ √ x )= I ∩ J + ( x ) ⊆ ( I + ( x )) ∩ ( J + ( x )) ⊆ p ( I + ( x )) ∩ ( J + ( x ))= p I ∩ J + ( x )= q √ p ( x )= p x )= p ( x )= ( x ) . This implies that I ∩ J + ( x ) = ( I + ( x )) ∩ ( J + ( x )) . Since the sequence → R ( I + ( x )) ∩ ( J + ( x )) → RI + ( x ) ⊕ RJ + ( x ) → R ( I + ( x )) + ( J + ( x )) → is exact, we conclude that Ann RI + J ( x ) = 0 . This means that x is a non zero divisor of RI + J ,and so, x is not in any minimal prime of I + J . (cid:3) Theorem 2.11.
Let ( R, m ) be a equidimensional complete local ring with dim( R ) = d ≥ .Suppose there exists an x ∈ m such that x is a non zero divisor of R and that ( x ) is aradical ideal. Let t be an integer such that ≤ t ≤ d − . Then Γ t ( R/ ( x )) is connected ⇒ Γ t ( R ) is connected.As a consequence c( R ) = c( R/ ( x )) + 1 . Proof.
Let ( S, T ) be a partition of Min( R ) . We know c( R ) = dim( R/ ( I + J )) where I and J are the intersection of all the elements of S and T respectively.From Lemma 2.10, we know that x is not an element of any minimal prime of I + J .Then, c( R ) = dim( R/ ( I + J ))= dim( R/ ( I + J + ( x ))) + 1 ≥ dim R T p ∈ S T q ∈ D ( p ) q + T p ∈ T T q ∈ D ( p ) q ! + 1= dim R/ ( x ) T p ∈ S T q ∈ D ( p ) q / ( x ) + T p ∈ T T q ∈ D ( p ) q / ( x ) ! + 1 ≥ c( R/ ( x )) + 1 . From Proposition 1.10 we have the inequality c( R/ ( x )) ≥ ( d − − t , so c( R/x )+1 ≥ d − t .From our previous chain of inequalities, we get that c( R ) ≥ d − t . We conclude that Γ t ( R ) is connected. (cid:3) Theorem 2.12.
Let ( R, m ) be an equidimensional complete local ring containing a field,of dim( R ) = d ≥ . Suppose there exists x ∈ m such that x is a non zero divisor of R and that ( x ) is a radical ideal. Let t be an integer such that ≤ t ≤ d − . Then t ( R ) = t ( R/ ( x )) . Proof.
Suppose t ( R ) = s . Let Σ , . . . , Σ s be the s connected components of Γ t ( R ) . Let Σ ′ , . . . , Σ ′ s be the subgraphs of Γ t ( R/ ( x ) associated to the sets D (Σ ) , . . . , D (Σ s ) respec-tively. We show that the associated graphs are the connected components of Γ t ( R/ ( x )) .Let a i = T p ∈ Σ i p . From Corollary 2.8 and its proof we know that Σ ′ i ∼ = Γ t ( R/ ( a i + ( x ))) is also connected for each i .From Lemma 2.9 we know that for distinct i and j , the graphs Σ ′ i and Σ ′ j do not sharevertices. Thus they are distinct connected subgraphs of Γ t ( R/ ( x )) .It remains to show that for every pair of distinct Σ ′ i and Σ ′ j there are no edges betweenthem, so they are indeed the connected components of Γ t ( R/ ( x )) .For i = j , suppose there is an edge between q / ( x ) ∈ Σ ′ i and q / ( x ) ∈ Σ ′ j . Let S be theset of vertices of Γ t ( R ) in Σ i and let T be the set of vertices of Γ t ( R ) which are not in Σ i . Note that ( S, T ) is a partition of Min( R ) . Let I and J be the intersection of all theelements of S and T respectively. Take p and p such that q ∈ D ( p ) and q ∈ D ( p ) .Since I + J ⊆ p + p + ( x ) , we have that ht( I + J ) ≤ ht( p + p + ( x )) . Suppose that theequality holds. Take a minimal prime q of p + p + ( x ) such that ht( q ) = ht( p + p + ( x )) .Since I + J and p + p + ( x ) have the same height, q must also be a minimal prime of I + J . This is not possible by Lemma 2.10, because x ∈ q . We have that ht( I + J ) + 1 ≤ ht( p + p + ( x )) ≤ ht( q + q )= ht( q / ( x ) + q / ( x )) + 1 . Thus ht( I + J ) ≤ ht( q / ( x ) + q / ( x )) ≤ t . From the proof of Proposition 1.6 we knowthis means there is an edge between some prime in S and some prime in T . Then, thereis an edge between a vertex of Σ i and a vertex of another connected component of Γ t ( R ) ,which is a contradiction. We conclude that Σ ′ , Σ ′ , . . . , Σ ′ s are the connected componentsof Γ t ( R/ ( x )) . (cid:3) Applications to Groebner deformations
In this section we apply the results regarding a parameter to initial ideals and squarefree Groebner deformations.
Remark 3.1.
Let I ⊆ S = K [ x , . . . , x n ] /I , where I homogeneous under a grading givenby a vector w ∈ Z > . Let m = ( x , . . . , x n ) be the maximal homogeneous ideal. Let R = S/I . Let J k = S ≥ t . Let b R denote the localization of R with respect to { J k } . Wenote that b R is equal to the | m -adic completion of R and R m . Let p be a homogeneousprime ideal of R . We note that p b R is a prime ideal in b R , because M k ∈ N J k ( b R/ p b R ) J k +1 ( b R/ p b R ) = R/ p ONNECTEDNESS OF SQUARE-FREE GROEBNER DEFORMATIONS 11 is a domain. As a consequence, there is a one-to-one correspondence between the minimalprimes of R and the minimal primes of b R , and their sums have the same height. Then, Γ k ( R ) = Γ k ( b R ) . Theorem 3.2.
Let S = K [ x , . . . , x n ] be a polynomial ring over a field. Let I be anequidimensional ideal and < a monomial order such that in < ( I ) is square-free. Then, t ( S/I ) = t ( S/ in( I )) . As a consequence, c( S/I ) = c( S/ in( I )) .Proof. Let η = ( x , . . . , x n ) . There exists a vector w ∈ N n such that in < ( I ) = in w ( I ) [Stu96, Proposition 1.11]. Let A = K [ t ] be a polynomial ring, L = Frac( A ) , and T = A ⊗ K S . We consider T as a graded ring with deg( t ) = 1 , and deg( x i ) = w i for every i ∈ N . Given f = P α c α x α · · · x α n n ∈ S , we take f w = t deg w f P α c α (cid:0) x t (cid:1) α · · · (cid:0) x n t (cid:1) α n ∈ T .We set I w = ( f w | f ∈ I ) ⊆ T the homogenization of I , and R = T /I w . Then, R/tR ∼ = S/ in < ( I ) and R/ ( t − R ∼ = S/I . Let p , . . . , p ℓ be the minimal primes of I .We note that p w , . . . , p wℓ be the minimal primes of I w [Var09, Lemma 2.3]. We note that dim T / p wi = dim S/ p i +1 = dim S/I +1 [Var09, Lemma 2.3], and so, I w is equidimensional.Since I w is homogeneous, we have that p wi are also homogeneous. Thus, p wi + p wj and itsminimal primes are homogeneous for every i, j . We conclude that all these ideals arecontained in n = ( t, x , . . . , x n ) . Hence, k ( S/ in( I )) = k ( R/tR ) because S/ in( I ) ∼ = R/tR = k ( c R n /t c R n ) by Remark 3.1 = k ( c R n ) by Theorem 2.12 = k ( R ) by Remark 3.1We now show that k ( R ) = k ( S/I ) . We have that ht( p wi + p wj ) ≤ ht(( p i + p j ) w ) because p wi + p wj ⊆ ( p i + p j ) w = ht( p i + p j ) [Var09, Proof of Lemma 2.3(6)] . Let R = R/ ( t − R . Then, ht( p i + p j ) = ht( p wi R + p wj R ) because S/I ∼ = R = ht( p wi + p wj + ( t − − because T is a polynomial ring ≤ ht( p wi + p wj ) by Krull’s principal ideal Theorem.We conclude that ht( p wi + p wj )) = ht( p i + p j ) . Hence, Γ k ( S/I ) ∼ = Γ k ( T /I w ) . We conclude k ( S/I ) = k ( S/ in < ( I )) . (cid:3) It is known that the graphs Γ t ( S/I ) may vary after a field extension. Furthermore,the number of connected components might change. For instance, if S = R [ x, y ] and I = ( x + y ) , Γ ( S/I ) has only one vertex and it is connected. In contrast, Γ ( S/I ⊗ R C ) has two vertices and it is disconnected. As a consequence of Theorem 3.2, we obtain that T ( S/I ) does not change when the field is extended if in < ( I ) is square-free. Corollary 3.3.
Let S = K [ x , . . . , x n ] be a polynomial ring over a field. Let I be anequidimensional ideal and < a monomial order such that in < ( I ) is square-free. Then, t ( S/I ) = t ( S/I ⊗ K L ) for every field extension K ⊆ L .Proof. We observe that the initial ideal of I ⊗ K L is in < ( I ) ⊗ K L , which is also square-free. Since the minimal primes of a square-free monomial ideal, and their sums, are idealsgenerated by variables, their heights are independent of the field. Then, t ( S/I ) = t ( S/ in( I )) by Theorem 3.2 = t (( S/ in( I )) ⊗ K L )= t (( S ⊗ K L/ in( I ) ⊗ K L )= t (( S ⊗ K L/I ⊗ K L ))) by Theorem 3.2 . (cid:3) Applications to Lyubeznik numbers
Lyubeznik [Lyu93b] defined numerical inviariants for local rings in equal characteristic.We now recall their definition.
Definition 4.1 ([Lyu93b]) . Let ( R, m , K ) be a local ring containing a field. Then, b R admits a surjection, π : S → R , from a regular local ring ( S, η, K ) containing a field. Let n = dim( S ) and I = ker( π ) . The i, j -Lyubeznik number of R is defined by λ i,j ( R ) := dim K Ext iS (cid:0) K, H n − jI ( S ) (cid:1) , We recall that the previous numbers depends only on R , i , and j . In particular, thisnumber is independent of the choice of S and of π [Lyu93b, Lemma 4.3].Let X be an equidimensional projective variety of dimension d over a field K . Bychoosing an embedding X ֒ → P nK , we can write X = Proj( R/I ) , where I is a homogeneousideal of the polynomial ring S = K [ x , . . . , x n ] . Let m = ( x , . . . , x n ) be the homogeneousmaximal ideal of S , and R = ( S/I ) m the local ring at the vertex of the affine cone over X .Lyubeznik asked whether the Lyubeznik numbers λ i,j ( R ) are independent of n , and thechoice of embedding of X into P nK , and so, we can write it as λ i,j ( X ) . The question hasbeen answered affirmatively for all Lyubeznik numbers by Zhang when K has prime char-acteristic [Zha11, Theorem 1.1]. In contrast, these numbers may vary with the embeddingin characteristric zero [RSW18, Theorem 1].The highest Lyubeznik number λ d +1 ,d +1 ( R ) is independent of the choice of embedding[Zha07, Theorem 2.7], and it is well known that λ , ( R ) is as well [Wal01, Proposition3.1]. It is also know that λ , ( R ) is also an invariant in all characteristics. These numbers λ , ( R ) , λ , ( R ) , and λ d +1 ,d +1 ( R ) are defined in terms of a geometric version of Γ i ( R ) . Definition 4.2.
For X an equidimensional projective variety of dimension d , given aninteger ≤ t ≤ d , define the graph Γ t ( X ) as follows:(1) The vertices of Γ t ( X ) are indexed by the irreducible components of X , and(2) There is an edge between distinct vertices Z and W if and only if dim( Z ∩ W ) ≥ d − t. Theorem 4.3.
Let X be an equidimensional projective variety of dimension d over a field K . Then, • λ , ( X ) = d ( X ⊗ K K ) − [Wal01, Proposition 3.1] ; ONNECTEDNESS OF SQUARE-FREE GROEBNER DEFORMATIONS 13 • λ , ( X ) = d − ( X ⊗ K K ) − d ( X ⊗ K K ) [NnBSW19, Theorem 7.4] ; • λ d +1 ,d +1 ( X ) = ( X ⊗ K K ) [Zha07, Theorem 2.7] ; As a consequence of Theorem B, we provide a way to compute certain Lyubezniknumbers from quare-free initial ideals. This type of questions was previously studied byNadi and Varbaro [NV19]. In particular, our next theorem extends one of their results[NV19, Proposition 2.11].
Theorem 4.4.
Let S = K [ x , . . . , x n ] be a polynomial ring over a field. Let I be anequidimensional ideal and < a monomial order such that in < ( I ) is square-free. Let X =Proj( S/I ) and Y = Proj( S/ in < ( I )) . Then, • λ , ( X ) = λ , ( Y ) ; • λ , ( X ) = λ , ( Y ) ; • λ d +1 ,d +1 ( X ) = λ d +1 ,d +1 ( Y ) .Proof. We have that Γ t ( X ) = Γ(( R/I ) m ) and Γ t ( Y ) = Γ(( S/ in < ( I )) m ) [NnBSW19,Lemma 7.3]. Then, the result follows from Theorems 3.2 and 4.3. (cid:3) Acknowledgments
We thank Raul Gómez Múñoz for valuable comments.
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