The second vanishing theorem for Stanley-Reisner ring and its generalization
aa r X i v : . [ m a t h . A C ] F e b THE SECOND VANISHING THEOREM FOR STANLEY-REISNERRING AND ITS GENERALIZATION
RAJSEKHAR BHATTACHARYYA
Abstract.
For regular local ring, the “second vanishing theorem” or “SVT” oflocal cohomology has been proved in several cases. In this paper, we extend theresult of the SVT to Stanley-Reisner ring with an interpretation from combina-torial topology. Finally, in this context, we generalize the result of SVT. Introduction
For Noetherian ring S and for an ideal J ⊂ S , we have the local cohomologymodule H iJ ( S ) supported at J . It is a very mysterious object, even it is quite hardto know when it will vanish. Recently there have been several instances wheremore explicit information on local cohomology modules was obtained in specialcases using combinatorial approach and one of such place is Stanley-Reisner ring.Recall that the cohomological dimension of an ideal J of a Noetherian ring S isthe maximum index i ≥ H iJ ( S ) is nonzero.In this context we mention Hartshorne-Lichtenbaum vanishing theorem or “HLVT”[Har68]. It states that: For any complete local domain S of dimension d , H dJ ( S )vanishes if and only if dim( S/J ) >
0. One may regard the HLVT as the “firstvanishing theorem” for local cohomology. In [RWY], HLVT has been extended forStanley-Reisner ring with an interpretation from combinatorial topology.If the ring S contains a field, the “second vanishing theorem” or “SVT” of localcohomology states the following: Let S be a complete regular local ring of dimension d with a separably closed residue field, which it contains. Let J ⊆ S be an idealsuch that dim( S/J ) ≥
2. Then H d − J ( S ) = 0 if and only if the punctured spectrumof S/J is connected [Har68, HL, Ogu73, PS73]. In [HNBPW], the SVT has beenextended to complete unramified regular local ring of mixed characteristic. In [Bh],it has been realized in the ramified case only the for extended ideals.In this paper, we extend the result of the SVT to Stanley-Reisner ring with acombinatorial topological interpretetion, (see Theorem 3.2.):Let Σ ⊂ ∆ be simplicial complexes where dim k [∆] = d with J = I Σ . Then H d − J ( k [∆]) = 0 if and only if every ( d − d − Let Σ ⊂ ∆ be simplicial complexes where dim k [∆] = d with J = I Σ . For somefixed n ≤ d − H d − n +1 J ( k [∆]) = 0 if and only if for every 1 ≤ i ≤ n , every( d − i )-face of ∆ contains n − i + 1 vertices of Σ.2. Preliminaries
In this section, we recall some basic results from combinatorial topology. Forgeneral references we refer [Bj] and [Mu].For [ n ] = { , , . . . , n } , let ∆ ⊂ [ n ] be a simplicial complex i.e., if F ∈ ∆and G ⊂ F , then G ∈ ∆. For a ∈ Z n , we define the following support subsetsof [ n ] = { , , . . . , n } : supp + a = { i : a i > } , supp − a = { i : a i < } andsupp a = { i : a i = 0 } . Let S = k [ x , . . . , x n ] be a polynomial ring over a field k . If F ∈ ∆ ⊂ [ n ] , then we write square-free monomial x a . . . x a n n = x F , when F = supp a .For a simplicial complex ∆, the Stanley-Reisner ideal of ∆ is the square-freemonomial ideal, I ∆ = { x F : F / ∈ ∆ } . We define the Stanley-Reisner ring as S/I ∆ = k [∆]. Let Σ ⊂ ∆ ⊂ [ n ] be two simplicial complexes. Then we haveStanley-Reisner ideals I ∆ ⊂ I Σ . In k [∆], we denote the image of I Σ by J .Given a face F ∈ ∆, we can define three subcomplexes, called the star, deletion,and link of F inside ∆, as follows: star ∆ ( F ) = { G ∈ ∆ : G ∪ F ∈ ∆ } , del ∆ ( F ) = { G ∈ ∆ : G F } and lk ∆ ( F ) = { G ∈ ∆ : G ∩ F = φ, G ∪ F ∈ ∆ } Now to prove the result of this paper, we need the following result, Theorem 3.2of [RWY]:
Theorem A.
Let Σ ⊂ ∆ be simplicial complexes and let a ∈ Z n , F + = supp + a and F − = supp − a . Then H iJ ( k [∆]) a = e H i − ( || star ∆ ( F + ) || − || Σ || , || del star ∆ ( F + ) ( F − ) || −|| Σ || ; k ) where e H i ( − , − ; k ) denotes i − th singular relative reduced cohomology and || ∆ || denotes the geometric realization of a simplicial complex ∆.Let Sd(∆) denote the barycentric subdivision of the simplicial complexes ∆, seeSection 15 of [Mu] and also section 9 of [Bj] for abstract simplicial complexes. Givena subcomplex Σ ⊂ ∆, Sd(∆ − Σ) is the subcomplex of barycentric subdivision ofSd(∆) whose vertices are not the barycentre of any face of Σ.We also need the following result, Lemma 4.7.27 of [BLSWZ].
Theorem B.
Let ∆ be a simplicial complex and let ∆ ′ and Σ be two subcomplexes.Then the pair of spaces ( || ∆ || − || Σ || , || ∆ ′ || − || Σ || ) is relatively homotopy equivalentto the pair (Sd(∆ − Σ) , Sd(∆ ′ − Σ))3. main results
Before proving our main result, we need the following Lemma. In this context itshould be mentioned that this proof is greatly influenced by Theorem 3.5 of [RWY]and goes similarly.
Lemma 3.1.
For Σ ⊂ ∆ and dim ∆ = d − , if every ( d − -face of ∆ containstwo vertices of Σ and if every ( d − -face of ∆ contains one vertex of Σ , then wecan remove every ( d − -face and every ( d − -face of Sd(∆ − Σ) . HE SECOND VANISHING THEOREM FOR STANLEY-REISNER RING AND ITS GENERALIZATION3
Proof.
Denote Sd(∆ − Σ) = X and set D = d − , d −
2. To remove a D -face of X ,we adopt the way of elementary collapse [Bj], which states that, if we find a D -facewhich contains some D − D -face, we removeboth of them. We claim that this process removes all D -faces of Sd(∆ − Σ).If not, assume that after removing few D -faces, when we reach the simplicialcomplex X ′ and there exists one D -face in X ′ which can not be removed, i.e. thereis a D -face in X ′ such that all of its ( D − D -face, sothat no further collapse is possible. Let σ be one of such D -faces in X ′ , and let G be a D -face of X in which it lies, so that the barycenter b G of G is one of itsvertex. . Then lk X ′ ( b G ) is a subcomplex of lk Sd(∆) ( b G ) which is the boundary ofSd G (since being a barycentre only for F ⊂ Sd G , { b G } ∪ F is a face of Sd(∆)).Since σ is in X ′ , dim(lk X ′ ( b G )) = dim(lk Sd(∆) ( b G )), and moreover any ( D − τ , ( D − τ ∪ { b G } is in two D -faces (otherwise, we could remove the face σ ).Thus we find lk X ′ ( b G ) = lk Sd(∆) ( b G ). But this is a contradiction, since lk Sd(∆) ( b G )should contain atleast one vertex of Σ, while lk X ′ ( b G ) should not contain any vertexof Σ. (cid:3) Now we state the main result of this paper.
Theorem 3.2.
Let Σ ⊂ ∆ be simplicial complexes where dim k [∆] = d with J = I Σ .Then H d − J ( k [∆]) = 0 if and only if every ( d − -face contains two vertices of Σ and every ( d − -face contains one vertex of Σ .Proof. For the forward direction, we assume that some ( d − F contains novertex of Σ and we want to prove that for some multidegree a , H d − J ( k [∆]) a =0. We consider a multidegree a having F + = supp + a = φ , F − = supp − a = F . From Theorem A given in Section 2, we get H d − J ( k [∆]) a = e H d − ( || ∆ || −|| Σ || , || del ∆ ( F ) || − || Σ || ; k ) where we use the fact that star ∆ ( φ ) = ∆. Now if weremove || ∆ || − || Σ || − || F || , using ”exision” we get e H d − ( || ∆ || − || Σ || , || del ∆ ( F ) || −|| Σ || ; k ) = e H d − ( || F || , || ∂F || ; k ). Finally, going to the quotient space || F || / || ∂F || ,we get e H d − ( || F || , || ∂F || ; k ) = e H d − ( || F || / || ∂F || ; k ) = e H d − ( S d − ; k ) = k . Thus H d − J ( k [∆]) a = 0.For the other direction, we assume that every ( d − d − H d − J ( k [∆]) a = e H d − ( || star ∆ ( F + ) || − || Σ || , || del star ∆ ( F + ) ( F − ) || −|| Σ || ; k ) = 0 with F + ∪ F − ∈ ∆. Since star ∆ ( F + ) can be of dimension atmostthat of ∆, we can assume F + = φ and above reduces that we only need to show e H d − ( || ∆ || − || Σ || , || del ∆ ( F ) || − || Σ || ; k ) = 0.Now instead of taking simplicial homology for the pair of spaces ( || ∆ ||−|| Σ || , || del ∆ ( F ) ||−|| Σ || ), we can take the pair of spaces (Sd(∆ − Σ) , Sd(del ∆ ( F ) − Σ)), where Sd(∆ − Σ)is the subcomplex of barycentric subdivision of Sd(∆) whose vertices are not thebarycentre of any face of Σ. Using Theorem B of Section 2, we get that e H d − ( || ∆ ||−|| Σ || , || del ∆ ( F ) || − || Σ || ; k ) = e H d − (Sd(∆ − Σ) , Sd(del ∆ ( F ) − Σ)).Now from above Lemma we have shown that we can remove every ( d − d − − Σ) and this leads to the desired result. (cid:3)
THE SECOND VANISHING THEOREM FOR STANLEY-REISNER RING AND ITS GENERALIZATION generalization of the SVT The result of the last section can be generalized in the following
Lemma 4.1.
Let Σ ⊂ ∆ and dim ∆ = d − . For some fixed n ≤ d − and forevery ≤ i ≤ n , if every ( d − i ) -face of ∆ contains n − i + 1 vertices of Σ , then forevery i ≤ n , we can remove every ( d − i ) -face of Sd(∆ − Σ) .Proof. Similar to that of Lemma 3.1. (cid:3)
Using above lemma in a similar way we can prove the generalization of Theorem3.2.
Theorem 4.2.
Let Σ ⊂ ∆ be simplicial complexes where dim k [∆] = d with J = I Σ .For some fixed n ≤ d − , H d − n +1 J ( k [∆]) = 0 if and only if for every ≤ i ≤ n ,every ( d − i ) -face of ∆ contains n − i + 1 vertices of Σ .Proof. Similar to that of Theorem 3.2. (cid:3)
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