Stellar subdivisions and Stanley-Reisner rings of Gorenstein complexes
SSTELLAR SUBDIVISIONS AND STANLEY-REISNERRINGS OF GORENSTEIN COMPLEXES
JANKO B ¨OHM AND STAVROS ARGYRIOS PAPADAKIS
Abstract.
Unprojection theory analyzes and constructs complicatedcommutative rings in terms of simpler ones. Our main result is that,on the algebraic level of Stanley–Reisner rings, stellar subdivisions ofGorenstein* simplicial complexes correspond to unprojections of typeKustin–Miller. As an application of our methods we study the mini-mal resolution of Stanley–Reisner rings associated to stacked polytopes,recovering results of Terai, Hibi, Herzog and Li Marzi. Introduction
Stanley–Reisner rings of simplicial complexes form an important class ofcommutative rings whose theory has provided spectacular applications tocombinatorics; see [35] and [12, Chapter 5] [26]. The Stanley–Reisner ringof a simplicial complex ∆, defined as the quotient of a polynomial ring by acertain ideal, depends only on the combinatorics of ∆. Given a combinatorialoperation on ∆ which produces another simplicial complex, it is natural toask how the Stanley–Reisner ring of the new complex is related to that of∆. Stellar subdivision, which is one of the simplest ways to subdivide asimplicial complex, is such an operation. It has been used successfully, forinstance, to give a method for transforming the boundary of a polytopeinto that of any other polytope of the same dimension by operations whichpreserve interesting invariants [17], to construct polytopes whose f -vectors,or flag f -vectors, span a certain ‘Euler’ or ‘Dehn-Sommerville’ space [19,Chapter 9] [4] and to construct simplicial polytopes with prescribed facelattices [24, 34].On a different tone, unprojection theory aims to analyze and constructcommutative rings in terms of simpler ones. The first kind of unprojectionwhich appeared in the literature is that of type Kustin–Miller, studied orig-inally by Kustin and Miller [21] and later by Reid and the second author[30, 32]. Starting from a codimension one ideal I of a Gorenstein ring R such that the quotient R/I is Gorenstein, Kustin–Miller unprojection uses
Mathematics Subject Classification.
Primary 13D02; Secondary 13F55, 13H10,05E40.J. B. supported by DFG (German Research Foundation) through Grant BO3330/1-1.S. P. is a participant of the Project PTDC/MAT/099275/2008, a member of CAMGSD(IST/UTL), and was supported from the Portuguese Funda¸c˜ao para a Ciˆencia e a Tecno-logia (FCT) under research grant SFRH/BPD/22846/2005 of POCI2010/FEDER. a r X i v : . [ m a t h . A C ] J a n JANKO B ¨OHM AND STAVROS A. PAPADAKIS the information contained in Hom R ( I, R ) to construct a new Gorensteinring S which is ‘birational’ to R and corresponds to the ‘contraction’ of V ( I ) ⊂ Spec R . It has been used in the classification of Tor algebras inGorenstein codimension 4 [22]; in the birational geometry of Fano 3-folds[14, 15]; in the study of Mori flips [10]; in the study of algebraic surfacesof general type [27], [29]; in the construction of weighted K3 surfaces andFano 3-folds [1], [9]; and in the construction of Calabi–Yau 3-folds of highcodimension [5, 28]. A general discussion of unprojection theory and its ap-plications is contained in [33], while a precise general definition of unprojec-tion is proposed in [31]. The Kustin–Miller unprojection and the associatedcomplex construction has been implemented in the package KustinMiller [7] for the computer algebra system
Macaulay2 [18].The main objective of this paper is to show that the Stanley–Reisnerrings of stellar subdivisions of a Gorenstein* simplicial complex ∆ can beconstructed from the Stanley–Reisner ring of ∆ by unprojections of typeKustin–Miller. As an application, we inductively calculate the minimalgraded free resolution of the Stanley–Reisner rings of the boundary sim-plicial complexes of stacked polytopes, recovering results by Terai and Hibi[37] and Herzog and Li Marzi [20].To state our main result, we need to introduce some notation and ter-minology (see Section 2 for more details). We denote by k [∆] the Stanley–Reisner ring of a simplicial complex ∆ with coefficients in a fixed field k .Recall that ∆ is said to be Gorenstein* over k if k [∆] is Gorenstein andgiven a vertex i of ∆ there exists σ ∈ ∆ such that σ ∪ { i } is not a face of ∆.Given a face σ of ∆, we denote by ∆ σ the stellar subdivision of ∆ on σ , by x σ the square-free monomial in k [∆] with support σ and by J σ theannihilator of the principal ideal of k [∆] generated by x σ . Recall also from[32, Definition 1.2] that if I = ( f , . . . , f r ) ⊂ R is a homogeneous codimen-sion 1 ideal of a graded Gorenstein ring R such that the quotient R/I isGorenstein, then there exists φ ∈ Hom R ( I, R ) such that φ together with theinclusion I (cid:44) → R generate Hom R ( I, R ) as an R -module. The Kustin–Millerunprojection ring of the pair I ⊂ R is defined as the quotient of R [ y ] by theideal generated by the elements yf i − φ ( f i ), where y is a new variable. Theorem 1.1.
Suppose that ∆ is a Gorenstein* simplicial complex and that σ ∈ ∆ is a face of dimension d − for some d ≥ . Let z be a new variableof degree d − and set M = Hom k [∆][ z ] (( J σ , z ) , k [∆][ z ]) . (a) M is generated as a k [∆][ z ] -module by the elements i and φ σ , where i : ( J σ , z ) → k [∆][ z ] is the natural inclusion morphism, and φ σ isuniquely specified by φ σ ( z ) = x σ and φ σ ( u ) = 0 for u ∈ J σ . (b) Denote by S the Kustin–Miller unprojection ring of the pair ( J σ , z ) ⊂ k [∆][ z ] . Then z is a S -regular element and k [∆ σ ] is isomorphic to S/ ( z ) as a k -algebra. An example demonstrating Theorem 1.1 is the following. Assume ∆ isthe boundary simplicial complex of the 2-simplex and σ is a facet of ∆. In TELLAR SUBDIVISIONS AND STANLEY-REISNER RINGS 3 coordinates, k [∆] = k [ x , x , x ] / ( x x x ), σ = { , } and J σ = 0 : ( x x ) =( x ). Then S = k [ x , . . . , x , z ]( x z − x x , x x ) , where x denotes the new unprojection variable. Notice that when z = 0, S | z =0 is isomorphic to k [∆ σ ], while when a ∈ k ∗ , S | z = a is isomorphic (asungraded k -algebra) to k [∆]. A toric face ring interpretation of S is discussedin Example 1.The paper is organised as follows: Theorem 1.1 is proved in Section 3.Section 2 includes some definitions and background related to the conceptswhich appear in Theorem 1.1. Section 4 contains an interpretation of The-orem 1.1 using the theory of toric face rings. In Section 5, we apply Theo-rem 1.1 to inductively calculate the minimal graded free resolutions of theStanley–Reisner rings of the boundary simplicial complexes of stacked poly-topes, which were originally given in [20]. The graded Betti numbers of theserings were first calculated in [37]. When the parameter value d is not 3, ourmethods allow us to obtain these Betti numbers without using Hochster’sformula or Alexander duality. We conclude in Section 6 with some remarksand directions for future research.The applications of unprojection theory to Stanley–Reisner rings are notlimited to the case of stellar subdivisions, and in the paper [6] we use un-projection techniques for an inductive treatment of Stanley–Reisner ringsassociated to cyclic polytopes.2. Preliminaries
Let m be a positive integer and set E = { , , . . . , m } . An (abstract) simplicial complex on the vertex set E is a collection ∆ of subsets of E such that (i) all singletons { i } with i ∈ E belong to ∆ and (ii) σ ⊂ τ ∈ ∆implies σ ∈ ∆. The elements of ∆ are called faces and those maximal withrespect to inclusion are called facets . The dimension of a face σ is definedas one less than the cardinality of σ . The dimension of ∆ is the maximumdimension of a face. The complex ∆ is called pure if all facets of ∆ havethe same dimension. Any abstract simplicial complex ∆ has a geometricrealization, which is unique up to linear homeomorphism. When we referto a topological property of ∆, we mean the corresponding property of thegeometric realization of ∆.For any subset ρ of E , we denote by x ρ the square-free monomial in thepolynomial ring k [ x , . . . , x m ] with support ρ . The ideal I ∆ of k [ x , . . . , x m ]which is generated by the square-free monomials x ρ with ρ / ∈ ∆ is called the Stanley-Reisner ideal of ∆. The face ring , or
Stanley-Reisner ring , k [∆] of∆ over k , is defined as the quotient ring of k [ x , . . . , x m ] by the ideal I ∆ .For a face σ of ∆ denote by lk ∆ ( σ ) = { τ : τ ∪ σ ∈ ∆, τ ∩ σ = ∅} the link ,and by star ∆ ( σ ) = { τ : τ ∪ σ ∈ ∆ } the star of σ in ∆. Given a face σ of∆ of dimension at least 1, the stellar subdivision of ∆ on σ is the simplicial JANKO B ¨OHM AND STAVROS A. PAPADAKIS complex ∆ σ on the vertex set E ∪ { m + 1 } obtained from ∆ by removing allfaces containing σ and adding all sets of the form τ ∪ { m + 1 } , where τ ∈ ∆does not contain σ and τ ∪ σ ∈ ∆. The complex ∆ σ is homeomorphic to ∆.We denote by J σ the ideal (0 : ( x σ )) of k [∆], in other words J σ = { y ∈ k [∆] : yx σ = 0 } . The complex ∆ is said to be Gorenstein* (over k ) if k [∆] is a Gorensteinring and given a vertex i of ∆ there exists σ ∈ ∆ such that σ ∪ { i } is not aface of ∆.It is known [35, Section II.5] that ∆ is Gorenstein* if and only if for any σ ∈ ∆ (including the empty face) we have(1) (cid:101) H i (lk ∆ ( σ ) , k ) ∼ = (cid:40) k, if i = dim(lk ∆ ( σ ))0 , otherwise,where (cid:101) H ∗ (lk ∆ ( σ ) , k ) denotes simplicial homology of lk ∆ ( σ ) with coefficientsin the field k . By [12, Corollary 5.1.5], any Gorenstein* complex ∆ is pure.It follows from (1) that the Gorenstein* property is inherited by links. Inparticular, any codimension 1 face of ∆ is contained in exactly 2 facets of ∆.The class of Gorenstein* complexes includes all triangulations of spheres.Assume R is a polynomial ring over a field k with the degrees of allvariables positive, and M is a finitely generated graded R -module. Let0 → F g → F g − → . . . → F → F → M → M as R -module. Write F i = ⊕ j R ( − j ) b ij , then b ij is called the ij -th graded Betti number of M , and we also denote itby b ij ( M ). For more details about free resolutions and Betti numbers see,for example, [16, Sections 19, 20].Assume R is a ring. An element r ∈ R will be called R -regular if themultiplication by r map R → R, u (cid:55)→ ru is injective. A sequence r , . . . , r n of elements of R will be called a regular R -sequence if r is R -regular, and,for 2 ≤ i ≤ n , we have that r i is R/ ( r , . . . , r i − )-regular.3. Proof of Theorem 1.1
In this section, ∆ denotes an ( n − { , , . . . , m } . Remark 1.
We will use the fact that k [∆] has no nonzero nilpotent elementsand that if I , I are monomial ideals of k [∆], then so is the ideal quotient( I : I ) = { y ∈ k [∆] : yI ⊂ I } . Remark 2.
Assume that ∆ is Gorenstein*. If e is a vertex of ∆ and σ ∈ ∆is a face that does not contain e , then there exists a facet of ∆ that contains σ but not e . Indeed, let τ be a facet of ∆ containing σ . If τ contains e , TELLAR SUBDIVISIONS AND STANLEY-REISNER RINGS 5 then there exists a facet τ distinct from τ containing τ \ { e } . This facetcontains σ and does not contain e . Proposition 3.1.
Let ∆ be a Gorenstein* simplicial complex on the vertexset { , , . . . , m } and let σ be a face of ∆ of dimension at least . The ideal J σ is a codimension ideal of k [∆] and the quotient k [∆] /J σ is Gorenstein.Moreover, (0 : J σ ) = ( x σ ) . Proof.
The first claim is well-known, cf. [16, Theorem 21.23], and the secondfollows from the observation that k [∆] /J σ = k [ x , . . . , x m ] /I , where I = I star ∆ ( σ ) + ( x i : i is not a vertex of star ∆ ( σ )),and the fact that lk ∆ ( σ ) is also Gorenstein*.We now prove that (0 : J σ ) = ( x σ ). It is clear that ( x σ ) ⊂ (0 : J σ ). Since(0 : J σ ) is a monomial ideal (Remark 1), it suffices to show that for anynonzero monomial u ∈ (0 : J σ ) we have u ∈ ( x σ ). Let ρ ∈ ∆ be the supportof u . By the way of contradiction, suppose that u is not in ( x σ ), so we maychoose i ∈ ( σ \ ρ ). By Remark 2, there exists a facet τ of ∆ which does notcontain i and contains ρ . Since i is not in τ and τ is a facet, we have x i x τ = 0in k [∆] and hence x τ ∈ J σ . This fact and the assumption u ∈ (0 : J σ ) implythat x τ u = 0 in k [∆]. Since each variable which appears in u also appearsin x τ , we conclude that x τ is a nonzero nilpotent element of k [∆]. Thiscontradicts Remark 1 and completes the proof of the proposition. (cid:3) Remark 3.
The conclusion of Proposition 3.1 is not true under the weakerhypothesis that k [∆] is Gorenstein. For a counterexample consider∆ = {{ , } , { , } , { } , { } , { } , ∅} and σ = { , } . We have k [∆] = k [ x , x , x ] / ( x x ), J σ = (0 : x x ) = ( x ),but (0 : J σ ) = ( x ). We believe that this is also a counterexample to thesecond claim of Part a) of [16, Theorem 21.23], this is the reason we did notuse this claim in the proof of Proposition 3.1.Let σ ∈ ∆ be a face of dimension d − d ≥
2. We recall thatthe stellar subdivision ∆ σ of ∆ on σ is a simplicial complex on the vertexset { , , . . . , m + 1 } . We will use the (easy) fact that(2) k [∆ σ ] ∼ = k [ x , . . . , x m +1 ]( I ∆ , x σ , x m +1 u , . . . , x m +1 u r ) , where { u , . . . , u r } is a generating set of monomials for the ideal J σ of k [∆]. Proof of Theorem 1.1.
Clearly there exists a unique element φ σ of M sat-isfying φ σ ( z ) = x σ and φ σ ( u ) = 0 for u ∈ J σ . Given f ∈ M , we write f ( z ) = w z + w with w ∈ k [∆][ z ] and w ∈ k [∆] and set g = f − w i ∈ M ,so that g ( z ) = w . For u ∈ J σ we have zg ( u ) = g ( zu ) = ug ( z ) = uw ∈ k [∆] . JANKO B ¨OHM AND STAVROS A. PAPADAKIS
Hence g ( u ) = 0 for all u ∈ J σ , which implies w ∈ (0 : J σ ). By Proposi-tion 3.1 we have (0 : J σ ) = ( x σ ). As a consequence, there exist w ∈ k [∆] suchthat w = wx σ and hence g = wφ σ . This proves part (a) of the theorem.By Proposition 3.1, the ring k [∆] /J σ is Gorenstein of the same dimensionas k [∆]. Therefore ( J σ , z ) is a codimension 1 homogeneous ideal of thegraded Gorenstein ring k [∆][ z ], so the general theory of [32] applies. Usingpart (a) we get S ∼ = k [ x , . . . , x m +1 , z ]( I ∆ , x m +1 z − x σ , x m +1 u , . . . , x m +1 u r ) , where the new variable x m +1 has degree equal to 1. It follows from (2)that S/ ( z ) ∼ = k [∆ σ ]. By [32, Theorem 1.5], S is Gorenstein of dimensionequal to the dimension of k [∆][ z ]. As a consequence dim S/ ( z ) = dim S − z is an S -regular element. This completes the proof of thetheorem. (cid:3) Toric face ring interpretation
Is it clear that Theorem 1.1 is equivalent to the following theorem.
Theorem 4.1.
Suppose that ∆ is a Gorenstein* simplicial complex and that σ ∈ ∆ is a face of dimension d − for some d ≥ . Let z , . . . , z d − be d − new variables of degree and set M = Hom k [∆][ z ,...,z d − ] (( J σ , z z · · · z d − ) ,k [∆][ z , . . . , z d − ]) . (a) M is generated as a k [∆][ z , . . . , z d − ] -module by the elements i and φ σ , where i : ( J σ , z z · · · z d − ) → k [∆][ z , . . . , z d − ] is the natural in-clusion morphism, and φ σ is uniquely specified by φ σ ( z z · · · z d − ) = x σ and φ σ ( u ) = 0 for u ∈ J σ . (b) Denote by S the Kustin–Miller unprojection ring of the pair ( J σ , z z · · · z d − ) ⊂ k [∆][ z , . . . , z d − ] . Then z , z , . . . , z d − is an S -regular sequence, and k [∆ σ ] is isomorphic to S / ( z , z , . . . , z d − ) as a k -algebra. We remark that, unlike in Theorem 1.1, in Theorem 4.1 all variableshave degree 1 which is the usual grading in the theory of Stanley–Reisnerrings. Compare also [6, Section 4], where a similar product z z appears ina natural way when relating unprojection and cyclic polytopes.Consider the Kustin–Miller unprojection ring S = k [ x , . . . , x m +1 , z , . . . , z d − ]( I ∆ , x m +1 z · · · z d − − x σ , x m +1 u , . . . , x m +1 u r )appearing in Theorem 4.1, where as in Section 3 { u , . . . , u r } denotes agenerating set of monomials for the ideal J σ = (0 : x σ ) of k [∆]. We willnow give a combinatorial interpretation of S using the notion of toric facerings as defined by Stanley in [36, p. 202], compare also [11, Section 4] and[13]. Let M be a free Z -module of rank m + d −
1, and consider the R -vectorspace M R = M ⊗ Z R . We will define a (finite, pointed) rational polyhedral TELLAR SUBDIVISIONS AND STANLEY-REISNER RINGS 7 fan F in M R , such that S is isomorphic to the toric face ring k [ F ]. Forsimplicity of notation we assume in the following that σ = { , , . . . , d } .Denote by e x, , . . . , e x,m , e z, , . . . , e z,d − a fixed Z -basis of M , and set e a = ( e x, + · · · + e x,d ) − ( e z, + · · · + e z,d − ) ∈ M. Assume τ = { a , . . . , a p } is a face of ∆. If σ is not a face of τ we set c τ tobe the cone in M R spanned by the basis vectors e x,a , . . . , e x,a p , e z, , . . . , e z,d − , while if σ is a face of τ we set c τ to be the cone in M R spanned by the(non-affinely independent) vectors e x,a , . . . , e x,a p , e z, , . . . , e z,d − , e a . It is easy to see that the collection of cones { c τ (cid:12)(cid:12) τ face of ∆ } together withtheir faces form a fan F in M R and that the toric face ring k [ F ] is isomorphicas a k -algebra to S . Example 1.
Consider the example given after the statement of Theo-rem 1.1. That is, let ∆ be the boundary of a triangle with vertices cor-responding to the variables x , x , x , and denote by ∆ σ the stellar subdi-vision of ∆ with respect to the face x x . We embed the fan F into R by assigning to the variables x , x , x , x the rays generated by (1 , , , , − , − , − , , ∈ Z , i.e., those of the standard fan of P as a toric variety. Then the ray associated to z is generated by (1 , , − S ∼ = k [ F ] via representing each cone of the embedded fan F by a poly-tope spanning it. There are 3 polytopes of maximal dimension, spanned by { x , x , z } , { x , x , z } and { x , x , x , z } . Notice that subdividing the conecorresponding to x , x , x , z into x , x , z and x , x , z amounts to passingfrom S to the polynomial ring in the variable z over k [∆ σ ].5. Application to stacked polytopes
The Kustin–Miller complex construction.
The following construc-tion, which is due to Kustin and Miller [21], will be important in the appli-cations to stacked polytopes contained in Subsection 5.2.Assume R is a polynomial ring over a field with the degrees of all variablespositive, and I ⊂ J ⊂ R are two homogeneous ideals of R such that bothquotient rings R/I and
R/J are Gorenstein and dim
R/J = dim
R/I − k , k ∈ Z such that ω R/I = R/I ( k ) and ω R/J = R/J ( k ),compare [12, Proposition 3.6.11], and assume that k > k . Moreover, let0 → A g → A g − → · · · → A → A → R/J → → B g − → · · · → B → B → R/I → R/J and
R/I respectively as R -modules. Denote by S = R [ T ] /Q the Kustin–Miller unprojection ring of JANKO B ¨OHM AND STAVROS A. PAPADAKIS
Figure 1.
Unprojection via toric face ringsthe pair J ⊂ R/I , where T is a new variable of degree k − k . Kustin andMiller constructed in [21] a graded free resolution of S as R [ T ]-module ofthe form 0 → F g → F g − → · · · → F → F → S → , where, when g ≥ F = B (cid:48) , F = B (cid:48) ⊕ A (cid:48) ( k − k ) ,F i = B (cid:48) i ⊕ A (cid:48) i ( k − k ) ⊕ B (cid:48) i − ( k − k ) , for 2 ≤ i ≤ g − ,F g − = A (cid:48) g − ( k − k ) ⊕ B (cid:48) g − ( k − k ) , F g = B (cid:48) g − ( k − k ) , cf. [21, p. 307, Equation (3)]. When g = 2 we have F = B (cid:48) , F = A (cid:48) ( k − k ) , F = B (cid:48) ( k − k ) . In the above expressions, for an R -module M we denoted by M (cid:48) the R [ T ]-module M ⊗ R R [ T ]. This resolution is, in general, not minimal, see Ex-ample 3 below. However, in the case of stacked and cyclic polytopes it isminimal, see Theorem 5.1 and [6]. We call the complex consisting of the F i the Kustin–Miller complex construction . For more details and an imple-mentation of this construction see [8].
Example 2.
Assume k − k = −
1, and that the 2 complexes are0 → A → A → A → A → A → → B → B → B → B → → B (cid:48) ( − → B (cid:48) ( − ⊕ A (cid:48) ( − → B (cid:48) ( − ⊕ A (cid:48) ( − ⊕ B (cid:48) → A (cid:48) ( − ⊕ B (cid:48) → B (cid:48) → TELLAR SUBDIVISIONS AND STANLEY-REISNER RINGS 9
Example 3.
Let ∆ be the simplicial complex with Stanley–Reisner ideal( x x x , x x ), ∆ is just the stellar subdivision of a facet of the boundarycomplex of the 3-simplex. Then σ = { , } is a face of ∆. Since the Stanley-Reisner ideal of ∆ σ is minimally generated by 3 monomials and not by 5,the Kustin–Miller complex construction gives a graded resolution of k [∆ σ ]which is not minimal.5.2. The minimal resolution for stacked polytopes.
Assume d ≥ d -simplex one canadd new vertices by building shallow pyramids over facets to obtain a sim-plicial convex d -polytope with m vertices, called a stacked polytope P d ( m ).We denote by ∆ P d ( m ) the boundary simplicial complex of the simplicialpolytope P d ( m ). By definition, ∆ P d ( m ) has as elements the empty set andthe sets of vertices of the proper faces of P d ( m ), cf. [12, Corollary 5.2.7].There is a slight abuse of notation here, since the combinatorial type of∆ P d ( m ) does not depend only on d and m but also on the specific choicesof the sequence of facets we used when building the shallow pyramids. Thegraded Betti numbers b ij of the Stanley-Reisner ring k [∆ P d ( m )] have beencalculated by Terai and Hibi in [37, Theorem 1.1], and it turns out thatthey only depend on d and m . Later Herzog and Li Marzi [20] constructedthe minimal graded free resolution of k [∆ P d ( m )]. In Theorem 5.1 we give adifferent proof of their result based on Theorem 1.1.It is clear that, for d < m , the simplicial complex ∆ P d ( m + 1) can beconsidered as the stellar subdivision of the boundary simplicial complex∆ P d ( m ) of a stacked polytope P d ( m ) with respect to a facet σ of ∆ P d ( m ).Since σ is a facet, the ideal ( J σ , z ) is generated by the regular sequence x ρ , z , where ρ takes values in the set of vertices of ∆ P d ( m ) which are notvertices of σ . Hence, the minimal graded free resolution of ( J σ , z ) is aKoszul complex. Combining Theorem 1.1 with the Kustin–Miller complexconstruction described in Subsection 5.1 we can get, starting with the Koszulcomplex and the minimal graded free resolution of k [∆ P d ( m )], a graded freeresolution of k [∆ P d ( m + 1)]. The following theorem states that we indeedget the minimal graded free resolution of k [∆ P d ( m + 1)]. In this way werecover the result from [20] using different ideas. We remark that, when d = 2 or d ≥
4, we do not use in the proof of the theorem the calculationof the graded Betti numbers of k [∆ P d ( m )] given in [37], and, moreover, weobtain these numbers in Proposition 5.5. The proof of the theorem will begiven in Subsection 5.3. Theorem 5.1.
Assume d ≥ and d + 1 < m . The resolution of k [∆ P d ( m +1)] , obtained using the Kustin–Miller complex construction starting fromthe minimal graded free resolution of k [∆ P d ( m )] and the Koszul complexresolving ( J σ , z ) is minimal. Proof of Theorem 5.1.
We need the following combinatorial defini-tion. Assume d ≥ d < m . For 1 ≤ i ≤ m − d − θ ( d, m, i ) = i (cid:18) m − di + 1 (cid:19) , compare [37, p. 448]. Moreover we set θ ( d, m,
0) = θ ( d, m, m − d ) = 0. Lemma 5.2. (Compare [37, p. 451] ). Assume ≤ i ≤ m − d . Then θ ( d, m + 1 , i ) = θ ( d, m, i ) + (cid:18) m − di (cid:19) + θ ( d, m, i − . (By our conventions, for i = 1 the equality becomes θ ( d, m +1 ,
1) = θ ( d, m, m − d ) , while for i = m − d it becomes θ ( d, m + 1 , m − d ) = θ ( d, m, m − d −
1) + 1 ).Proof.
Assume first 2 ≤ i ≤ m − d −
1. Using the Pascal triangle identity (cid:0) md (cid:1) = (cid:0) m − d (cid:1) + (cid:0) m − d − (cid:1) we have θ ( d, m + 1 , i ) = i (cid:18) m + 1 − di + 1 (cid:19) = i ( (cid:18) m − di + 1 (cid:19) + (cid:18) m − di (cid:19) )= i (cid:18) m − di + 1 (cid:19) + (cid:18) m − di (cid:19) + ( i − (cid:18) m − di (cid:19) = θ ( d, m, i ) + (cid:18) m − di (cid:19) + θ ( d, m, i − . The special cases i = 1 and i = m − d are proven by the same argument. (cid:3) The following proposition is well-known.
Proposition 5.3. ( [12, Proposition 1.1.5] ). Assume R = k [ x , . . . , x n ] is apolynomial ring over a field k with the degrees of all variables positive, and I ⊂ R a homogeneous ideal. Moreover, assume that x n is R/I -regular. De-note by cF the minimal graded free resolution of R/I as R -module. We thenhave that cF ⊗ R R/ ( x n ) is the minimal graded free resolution of R/ ( I, x n ) as k [ x , . . . , x n − ] -module, where we used the natural isomorphisms R ⊗ R R/ ( x n ) ∼ = R/ ( x n ) ∼ = k [ x , . . . , x n − ] . The proof of the following proposition is an immediate corollary of theconstruction of the Koszul complex in [12, Section 1.6].
Proposition 5.4.
Assume R = k [ x , . . . , x n ] is a polynomial ring over afield k with the degrees of all variables positive, p ≤ n a fixed integer, and g , . . . , g p , an R -regular sequence consisting of homogeneous elements of R ,with deg g i = 1 , for ≤ i ≤ p − , and deg g p = q ≥ . Then, the minimalresolution of R/ ( g , . . . , g p ) is of the form → F p → F p − . . . → F → F , with F ∼ = R , F p ∼ = R ( − p − q + 1) , and F i ∼ = R ( − i ) b i ⊕ R ( − q − i + 1) b p − i for ≤ i ≤ p − , where b i = (cid:0) p − i (cid:1) . TELLAR SUBDIVISIONS AND STANLEY-REISNER RINGS 11
Theorem 5.1 is an immediate consequence of the following more preciseproposition. Notice that as we have already mentioned the statements aboutthe graded Betti numbers of k [∆ P d ( m )] have been proven before in [37]. For d (cid:54) = 3 we do not use in our proof the results of [37]. Proposition 5.5.
Assume d ≥ and d + 1 < m . Set b ij = b ij ( k [∆ P d ( m )]) .Then the statement of Theorem 5.1 is true for ( d, m ) . Moreover, we havethat if d = 2 then b ij = 1 for ( i, j ) ∈ { (0 , , ( m − d, m ) } , b i,i +1 = θ ( d, m, i ) + θ ( d, m, m − d − i ) , for ≤ i ≤ m − d − , and b ij = 0 otherwise. If d ≥ , we have b ij = 1 for ( i, j ) ∈ { (0 , , ( m − d, m ) } , b i,i +1 = θ ( d, m, i ) , b i,d + i − = θ ( d, m, m − d − i ) , for ≤ i ≤ m − d − , and b ij = 0 otherwise.Proof. We fix d ≥ m . If d ≥ m = d + 2then k [∆ P d ( m )] is a type (2 , d ) codimension 2 complete intersection andeverything is clear.Assume d (cid:54) = 3, and that Proposition 5.5 is true for ( d, m ). By Theo-rem 1.1, the extension ring S of k [∆ P d ( m +1)] is the Kustin–Miller unprojec-tion ring of the pair ( J σ , z ) ⊂ k [∆ P d ( m )][ z ]. As we noticed above, the ideal( J σ , z ) is generated by a regular sequence, so the Koszul complex describedin Proposition 5.4 is the minimal resolution of k [∆ P d ( m )][ z ] / ( J σ , z ). Com-bining Proposition 5.3 and the discussion of Subsection 5.1, starting fromthe Koszul complex and the minimal graded free resolution of k [∆ P d ( m )],the Kustin–Miller complex construction gives a graded free resolution of k [∆ P d ( m + 1)]. Using Lemma 5.2 this complex has the conjectured gradedBetti numbers, and since there are no degree 0 morphisms it is necessarilyminimal.When d = 3 the above arguments work except for the minimality argu-ment, since there are degree 0 morphisms. But comparing the graded Bettinumber of the Kustin–Miller complex construction with the graded Bettinumbers of k [∆ P d ( m + 1)] calculated in [37] we again obtain the minimalityof the Kustin–Miller complex construction. (cid:3) Remarks and open questions
In the following we use the notation introduced in Section 1 and Theorem1.1.
Remark 4.
It follows from Theorem 1.1 that S is a 1-parameter deformationring of k [∆ σ ], compare [16, Exerc. 18.18]. The fact that such a deformationring of k [∆ σ ] exists is a special case of more general results due to Altmannand Christophersen [2, 3]. Remark 5.
Using the Kustin–Miller complex construction described in Sub-section 5.1, we can construct a graded free resolution of S , therefore using Proposition 5.3 also of k [∆ σ ], starting from graded free resolutions of k [∆]and k [∆] /J σ . In particular, it follows that F ( k [∆ σ ] , t ) = F ( k [∆] , t ) + ( t + t + · · · + t d − ) F ( k [∆] /J σ , t ) , where F ( R, t ) stands for the Hilbert series of R and d − σ . This equality can be rewritten as(3) h (∆ σ , t ) = h (∆ , t ) + ( t + t + · · · + t d − ) h ( lk ∆ ( σ ) , t ) , where h (Γ , t ) stands for the h -polynomial [35, Section II.2] of the simplicialcomplex Γ. It is not hard to see that (3) holds for any pure simplicialcomplex ∆. Indeed, one can check directly that (3) is equivalent to theformula f j (∆ σ ) = f j (∆) − f j − d ( lk ∆ ( σ )) + (cid:88) i ≥ (cid:18) dj − (cid:19) f i − ( lk ∆ ( σ )) , where f j (Γ) denotes the number of j -dimensional faces of a complex Γ. Thatformula follows from the definition of ∆ σ . Remark 6.
In [27], Neves and the second author introduced the (cid:0) n (cid:1) Pfaf-fians format, starting from a certain hypersurface ideal. We give a mono-mial interpretation of the construction. Start with the boundary simplicialcomplex ∆ of the ( n − the simplicial complexobtained by the stellar subdivisions of all facets of ∆. It is easy to checkthat the Stanley–Reisner ideal of ∆ is equal to (cid:101) I n , where (cid:101) I n denotes theideal obtained by substituting z i = 0, for 1 ≤ i ≤ n , and r d ,...,d n = 1, for( d , . . . , d n ) ∈ { , } n , to the ideal I n defined in [27, Definition 2.2].Similarly, in [28, Section 4.3], Neves and the second author constructeda codimension 11 Gorenstein ideal starting from a certain codimension 2complete intersection ideal. The monomial interpretation of the constructionis as follows. Denote by ∆ the simplicial complex which is the join [12, p. 221]of 2 copies of the boundary simplicial complex of the 2-simplex. ∆ hasStanley–Reisner ideal equal to ( x x x , x x x ) and exactly 9 facets.Denote by ∆ the simplicial complex obtained by the stellar subdivisions of∆ on these 9 facets. Using the notations of [28, Section 2], denote by I L the kernel of the surjection R [ y u (cid:12)(cid:12) u ∈ L ] → R L . It is easy to check thatthe Stanley–Reisner ideal of ∆ is equal to (cid:102) I L , where (cid:102) I L denotes the idealobtained by substituting x i = 0, for 1 ≤ i ≤
3, to I L . Remark 7.
It is plausible that our ideas also generalize to non-Gorensteinsimplicial complexes. To do this a more detailed study of non-Gorensteinunprojections would be necessary.
Remark 8.
Combining our results with those of [23] we get a link betweenstellar subdivisions of Gorenstein* simplicial complexes and linkage theory[25]. Is it possible to use this connection to define new combinatorial invari-ants of simplicial complexes?
TELLAR SUBDIVISIONS AND STANLEY-REISNER RINGS 13
Acknowledgements . The authors are grateful to Christos Athanasiadis forimportant discussions and suggestions. They also thank Jo˜ao Martins foruseful discussions, Tim R¨omer for useful comments on an earlier version,and an anonymous referee for informing us about [20].
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E-mail address : [email protected] Stavros Argyrios Papadakis, Centro de An´alise Matem´atica, Geometria eSistemas Dinˆamicos, Departamento de Matem´atica, Instituto Superior T´ecnico,Av. Rovisco Pais, 1049-001 Lisboa, Portugal
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