aa r X i v : . [ m a t h . A C ] M a y A NOTE ON THE SECOND VANISHING THEOREM
RAJSEKHAR BHATTACHARYYA
Abstract.
Let R be a regular local ring of dimension d and H d − J ( R ) bethe ( d − J . The secondvanishing theorem for the local cohomology or the SVT is well known in regularlocal rings containing field and for complete unramified regular local ring ofmixed characteristic. It states that under certain conditions on J (see belowfor detail), H d − J ( R ) vanishes if and only if the punctured spectrum R/J isconnected. In this paper, we extend the result of SVT partially to completeramified regular local ring only for the extended ideals. When puncturedspectrum is not connected but it has t number of connected components, weshow that the Matlis duals of these local cohmology modules are free moduleswith t − introduction Recall that the cohomological dimension of an ideal J of a Noetherian ring R isthe maximum index i ≥ H iJ ( R ) is nonzero.The cohomological dimension of the maximal ideal of a local ring coincides withthe ring’s dimension [Har67].In this context we mention Hartshorne-Lichtenbaum vanishing theorem or “HLVT”[Har68]. It states that: For any complete local domain R of dimension d , H dJ ( R )vanishes if and only if dim( R/J ) >
0. One may regard the HLVT as a “firstvanishing theorem” for local cohomology.If the ring R contains a field, the “second vanishing theorem” or “SVT” oflocal cohomology states the following: Let R be a complete regular local ring ofdimension d with a separably closed residue field, which it contains. Let J ⊆ R bean ideal such that dim( R/J ) ≥
2. Then H d − J ( R ) = 0 if and only if the puncturedspectrum of R/J is connected [Har68, Ogu73, PS73, HL90].In [HNBPW18], the SVT has been extended to complete unramified regular localring of mixed characteristic: Let R be a d -dimensional complete unramified regularlocal ring of mixed characteristic, whose residue field is separably closed. Let J be an ideal of R for which dim( R/ p ) ≥ p of J . Then H d − J ( R ) = 0 if and only if the punctured spectrum of R/J is connected.In this paper, we extend the result of the SVT partially to complete ramifiedregular local rings only for the extended ideals. In proving the main result, wereduce the conditions given on ramified case to the unramified case and then weapply the result of [HNBPW18], Theorem 3.8. The main result is the following (seeTheorem 2.2):
Mathematics Subject Classification.
Key words and phrases.
Local Cohomology. Let ( S, n ) be a d -dimensional complete ramified regular local ring of mixed char-acteristic, with a separably closed residue field and I be an ideal of it. Assume I = JS is an extension of an ideal J of any unramified complete regular local ring R such that S is an Eisenstein extension of R . If dim( S/ q ) ≥ q of I , then H d − I ( S ) = 0 if and only if the punctured spectrum of S/I isconnected.When punctured spectrum is not connected but it has t number of connectedcomponents, we show that the Matlis duals of these above local cohmology modulesare free modules with t − the main result For local ring ( R, m ) of mixed characteristic. We say R is unramified if p / ∈ m and it is ramified if p ∈ m . For normal local ring ( R, m ), consider the extensionring defined by S = R [ X ] /f ( X ) where f ( X ) = X n + a X n − + . . . + a n with a i ∈ m for every i = 1 , . . . , n and a n / ∈ m . This ring ( S, n ) is local and it is defined asan Eisenstein extension of R and f ( X ) is known as an Eisenstein polynomial (seepage 228-229 of [Mat80]).Here we note down the following important results regarding Eisenstein exten-sions:(1) An Eisenstein extension of regular local ring is regular local (see Theorem29.8 (i) of [Mat80]) and in this context, we observe that an Eisenstein extensionsof an unramified regular local ring is a ramified regular local ring.(2) Every ramified regular local ring is an Eisenstein extension of some unramifiedregular local ring (see Theorem 29.8 (ii) of [Mat80].(3) The Eisenstein extension mentioned in (1) and (2) are faithfully flat (applyTheorem 23.1 of [Mat80]).Before presenting the main result, we need this lemma. Lemma 2.1.
Let ( R, m ) be a d -dimensional complete unramified regular local ringof mixed characteristic. Let ( S, n ) be a ramified complete regular local ring obtainedvia Eisenstein extension of R where f ( X ) ∈ R [ X ] is the Eisenstein polynomial.(1) If the residue field of S is separably closed then so is the residue field of R (2) For minimal prime q ⊃ p S in S , where p is a prime in R , if dim( S/ q ) ≥ then dim( R/ p ) ≥ .(3) If the image Eisenstein polynomial f ( X ) ∈ R [ X ] is prime in the ring ( R/ p )[ X ] ,then p S is also prime in S .Proof. (1) In S , n = m S + XS . This gives S/ n = ( R [ X ] /f ) / ( m + X )( R [ X ] /f ) =( R [ X ] / m R [ X ]) / ( X )( R [ X ] / m R [ X ]) since f − X n ∈ m R [ X ]. So the result follows.(2) Set R [ X ] / p R [ X ] = R ′ . It is given that 3 ≤ dim( S/ q ) ≤ dim( S/ p S ) =dim( R ′ /f R ′ ). Now dim( R ′ /f R ′ ) ≤ dim( R ′ ) − ht f R ′ . Since p R [ X ] is a prime idealin R [ X ], f is R ′ -regular. So ht f R ′ = 1. Thus dim( R ′ ) ≥ dim( R ′ /f R ′ )+ht f R ′ ≥ R ′ = ( R/ p )[ X ], we get dim( R/ p ) ≥ S/ q = S/ p S = ( R [ X ] /f R [ X ]) / p R [ X ]( R [ X ] /f R [ X ]) = ( R/ p )[ X ] /f ( R/ p )[ X ].From the assertion, ( R/ p )[ X ] /f ( R/ p )[ X ] is a domain, hence q ⊂ S is a prime. (cid:3) Now we prove the SVT over complete ramified regular local rings of mixed char-acteristic, only for the extended ideals.
Theorem 2.2.
Let ( S, n ) be a d -dimensional complete ramified regular local ringof mixed characteristic, with a separably closed residue field and I be an ideal of it.Assume I = JS is an extension of an ideal J of any unramified complete regularlocal ring R such that S is an Eisenstein extension of R . If dim( S/ q ) ≥ for everyminimal prime q of I , then H d − I ( S ) = 0 if and only if the punctured spectrum of S/I is connected.Proof.
Let f ( X ) = X n + a X n − + . . . + a n be the Eisenstein polynomial and S is obtained via Eisenstein extension of R , i.e. S = R [ X ] /f ( X ). Set f ( X ) = f .Moreover from (1) of above Lemma 2.1, we have that R also has separably closedresidue field.We first assume H d − I ( S ) = 0. Assume that the punctured spectrum of S/I isdisconnected. Let n denote the maximal ideal of S , so that there exist ideals I and I of S that are not n -primary for which rad ( I ∩ I ) = rad I and rad ( I + I ) = n .Consider the Mayer-Vietoris sequence, · · · → H d − I ( S ) → H dI + I ( S ) → H dI ( S ) ⊕ H dI ( S ) → H dI ( S ) → . Now H d − I ( S ) = 0, and H dI ( S ) = H nI ( S ) = H dI ( S ) = 0 by the HLVT. Then H d n ( S ) = H dI + I ( S ) = 0, which contradics the HLVT.It should be mentioned that the part of the proof in the above paragraph, issame to that of [HNBPW18], Theorem 3.8, but for the sake of completeness wekeep it here.To prove the other direction, we can proceed as follows: Let q ⊂ S be theminimal prime of I = JS . Now p = q ∩ R is a minimal prime ideal of J , otherwise if J ⊂ p ′ ⊂ p , then by going down theorem there exists q ′ ⊂ S such that JS ⊂ q ′ ⊂ q .Conversely, for any minimal prime p of J , we claim that any q that lying over p is a minimal prime of JS as well as of p S . Since S is integral over R , such q always exists. To see this, suppose there exists some JS ⊂ q ′ ⊂ q . This gives J ⊂ q ′ ∩ R ⊂ p . But, p is minimal over R hence by lying over theorem, q = q ′ .Thus there is a one to one correspondence between the minimal primes over J andthose over JS . So, if p , . . . , p t be the minimal primes of J and q , . . . , q t be theminimal primes of JS then q i ∩ R = p i and q i is minimal prime over p i S as wellfor every i = 1 , . . . , t . Moreover, from (2) of Lemma 2.1 we get that dim( R/ p ) ≥ J .Next, we assert that the punctured spectrum of R/J is connected or equivalentlythe graph Θ
R/J of p , . . . , p t is connected [HL90], Theorem 2.9. From the hypoth-esis, applying [HL90], Theorem 2.9 once again we get that, the puntured spectrumof S/I is connected or equivalently the graph Θ
S/I of q , . . . , q t is connected, i.e forany pair of q i , q j , q i + q j is not n -primary. So we would like to show for any pair of p i , p j , p i + p j is not m -primary. To see this, assume for some pair rad ( p i + p j ) = m . Since S is an Eisenstein extension of R , we have S/ m S = ( R/ m )[ X ] / ( X n ).This gives dim( S/ m S ) = dim(( R/ m )[ X ] / ( X n )) = 0. Thus rad ( m S ) = n . Nowrad ( q i + q j ) ⊃ rad ( p i S + p j S ) ⊃ rad (( p i + p j ) S ) ⊃ (rad ( p i + p j )) S ⊃ m S . Thisgives rad ( q i + q j ) ⊃ rad ( m S ) = n . This gives that the graph Θ S/I of q , . . . , q t isnot connected.So, all conditions on ramified ring S and on the ideal JS reduces to those onunramified ring R and on the ideal J . Now, by [HNBPW18], Theorem 3.8, we get H d − J ( R ) = 0. Since S is flat over R , H d − JS ( S ) = H d − J ( R ) ⊗ S = 0. This finishes the proof. (cid:3) when punctured spectrum is not connected For Noetherian local ring ( R, m ), let E R be the R -injective hull of the residuefield. Then for any R -module M , we set D R ( M ) = Hom R ( M, E R ) as the Matlisdual of M. In this section, we show that, for some restricted class of ideals, whenpunctured spectrum is not connected but it has t number of connected components,the Matlis duals of those local cohmology modules of Theorem 2.2 are free moduleswith t − Theorem 3.1.
Let ( S, n ) be a d -dimensional complete ramified regular local ringof mixed characteristic, with a separably closed residue field and I be an ideal of it.Assume I = JS is an extension of an ideal J of any unramified complete regularlocal ring R such that S is an Eisenstein extension of R and for every minimalprime p of J , R/ p is normal. If dim( S/ q ) ≥ for every minimal prime q of I andthe punctured spectrum of S/I has t connected components, then(1) D R ( H d − I ( S )) = S ⊕ t − ,(2) D R ( H d − J ( R )) = R ⊕ t − ,(3) D R ( H d − J ( R/f R )) = D R/fR ( H d − J ( R/fR ) ( R/f R )) = (
R/f R ) ⊕ t − = D R ( H d − J ( S/XS )) = D R/fR ( H d − J ( S/XS ) ( S/XS )) = (
S/XS ) ⊕ t − .Proof. For ideal J ⊂ R , let p , . . . , p t be the minimal primes of J . We choosean Eisenstein polynomial f ( X ) = X n + a X n − + . . . + a n and S is obtainedvia Eisenstein extension of R , i.e. S = R [ X ] /f ( X ). From hypothesis and usingLemma 1 in page 228 of [Mat80], the image of f ( x ) is prime in every ( R/ p i )[ X ] forall i = 1 , . . . , t . Moreover, from (3) of Lemma 2.1, we get that p i S is a prime in S for all i = 1 , . . . , t .Now we claim that q , . . . , q t be the minimal primes of JS where q i = p i S forevery i = 1 , . . . , t . To see this, let q ⊂ S be the minimal prime of I = JS . Now p = q ∩ R is a minimal prime ideal of J , otherwise if J ⊂ p ′ ⊂ p , then by going downtheorem there exists q ′ ⊂ S such that JS ⊂ q ′ ⊂ q and this is a contradiction. Thus p is minimal over J and JS ⊂ p S ⊂ q . Since q is minimal, we get that q = p S .Conversely if p be a minimal prime of J , then p S is the minimal prime of JS ,otherwise, for JS ⊂ q ′ ⊂ p S , we get J ⊂ q ′ ∩ R ⊂ p . Since p is minimal over J , p = q ′ ∩ R , but then q ′ can not be inside p S by lying over theorem. Thus there is aone to one correspondence between the minimal primes over J and those over JS ,and if p , . . . , p t be the minimal primes of J and q , . . . , q t be the minimal primesof JS then q i = p i S for every i = 1 , . . . , t . Moreover, from (2) of Lemma 2.1 weget that dim( R/ p ) ≥ J .From (2) of Lemma 2.1 we get that dim( R/ p ) ≥ J .Moreover from (1) of above Lemma 2.1, we have that S also has separably closedresidue field.From the proof of Theorem 2.2, we know that if the punctured spectrum of S/JS is connected then the punctured spectrum of
R/J is also connected. Now we assertthat the converse, i.e. if the punctured spectrum of
R/J is connected then so isthe puntured spectrum of
S/JS or equivalently the graph Θ
S/JS of q , . . . , q t isconnected, i.e for any pair of q i , q j , q i + q j is not n -primary [HL90], Theorem 2.9. To see this, assume otherwise that for some pair q i and q j , rad ( q i + q j ) = n . Since q i = p i S and q j = p j S , we get m = n ∩ R = rad ( p i S + p j S ) ∩ R = (rad ( p i + p j ) S ) ∩ R = rad (( p i + p j ) S ∩ R ) = rad ( p i + p j ). This is a contradiction. Thus, we concludethat S/JS is connected. Thus we find the number of connected components in thepunctured spectrum of
R/J and that of
S/JS are same.Set f ( X ) = f . Now, ( X, f ) as well as ( f, X ) are two R [ X ]-regular sequences (forpolynomials g, h ∈ R [ X ], if f h = Xg then X | h and f | g , since R[X] is a UFD). Thisgives the following commutative diagram of short exact sequences whose rows andcolumns are exact.0 0 0 y y y −−−−→ R [ X ] f −−−−→ R [ X ] −−−−→ R [ X ] /f R [ X ] −−−−→ X y y X y X −−−−→ R [ X ] f −−−−→ R [ X ] −−−−→ R [ X ] /f R [ X ] −−−−→ y y y −−−−→ R [ X ] /XR [ X ] f −−−−→ R [ X ] /XR [ X ] −−−−→ R [ X ] / ( X, f ) R [ X ] −−−−→ y y y D R ( − ), gives the fol-lowing diagram of long exact sequences where all the rows and columns are exact. y y y y D R ( H i − JR [ X ] ( R [ X ])) f −−−−→ D R ( H i − JR [ X ] ( R [ X ])) −−−−→ D R ( H i − JS ( S )) −−−−→ D R ( H iJR [ X ] ( R [ X ])) X y X y X y X y D R ( H i − JR [ X ] ( R [ X ])) f −−−−→ D R ( H i − JR [ X ] ( R [ X ])) −−−−→ D R ( H i − JS ( S )) −−−−→ D R ( H iJR [ X ] ( R [ X ])) y y y y D R ( H i − J ( R )) f −−−−→ D R ( H i − J ( R )) −−−−→ D R ( H i − J ( R [ X ]( X,f ) R [ X ] )) −−−−→ D R ( H iJ ( R )) y y y y D R ( H iJR [ X ] ( R [ X ])) f −−−−→ D R ( H iJR [ X ] ( R [ X ])) −−−−→ D R ( H iJS ( S )) −−−−→ D R ( H i +1 JR [ X ] ( R [ X ])) y y y y By [HNBPW18], Proposition 3.12, we get H d − J ( R ) = E ⊕ ( t − R , where E R isthe R -injective hull of its residue field. Further, by HLVT, H dJ ( R ) = 0. Now H d − JR [ X ] ( R [ X ]) = ( H d − J ( R ))[ X ] = E R [ X ] ⊕ ( t − . Similarly, H dJR [ X ] ( R [ X ]) = 0.So, from above diagram, for i = d , we get D R ( H d − J ( R )) = R ⊕ ( t − , and hence D R ( H d − JR [ X ] ( R [ X ])) = R [[ X ]] ⊕ ( t − , and since f is a nonzero divisor of R [ X ] as wellas of R [[ X ]] and S is complete, D R ( H d − JS ( S )) = S ⊕ ( t − .This proves (1) and (2).For (3), we can observe that R [ X ]( X,f ) R [ X ] = R/f R = S/XS . Since f and X arenonzero divisor in R and S respectively, from above diagram we get D R ( H d − J ( R/f R )) =(
R/f R ) ⊕ t − = D R ( H d − J ( S/XS )) = (
S/XS ) ⊕ t − . For any φ ∈ D R ( H d − J ( R/f R ), f φ = 0 and also φ maps into (0 : E R f ) ⊂ E R and it is well known that (0 : E R f ) = E R/fR . Thus D R ( H d − J ( R/f R )) = D R/fR ( H d − J ( R/fR ) ( R/f R )) and similar is truefor the ring
S/XS . This finishes the proof. (cid:3)
Remark 1.
As an example of primes p in R such that R/ p is normal as mentionedin the hypothesis of Theorem 3.1, consider p generated by part of regular system ofparameters of R . Clearly R/ p is regular local ring, hence normal. In the following corollary we show that Conjecture 1 of [LY18] is true in unrami-fied and ramified situation for certain local cohomology modules. The proof followsimmediately from Theorem 3.1.
Corollary 1.
Let R be a d -dimensional complete unramified regular local ring ofmixed characteristic and J be an ideal of it. Assume R and J satisfy hypothesis ofTheorem 3.1. Let f ( X ) ∈ R [ X ] be the Eisenstein Polynomial and S be the corre-sponding ramified regular local ring. Then for R -module, M = H d − JS ( S ) , H d − J ( R ) , ∈ Ass ( D R ( M )) and also for R/f R -module N = H d − J ( R/fR ) ( R/f R ) = H d − J ( S/XS ) ( S/XS ) , ∈ Ass ( D R/fR ( N )) . Thus M and N satisfy Conjecture 1 of [LY18] . The following corollary extends the result of Corollary 1.2 of [LY18] in unramifiedand ramified situation for certain local cohomology modules. The proof followsimmediately from Theorem 3.1.
Corollary 2.
Let R be a d -dimensional complete unramified regular local ring ofmixed characteristic and J be an ideal of it. Assume R and J satisfy hypothesis ofTheorem 3.1. Let f ( X ) ∈ R [ X ] be the Eisenstein Polynomial and S be the corre-sponding ramified regular local ring. Then for R -module, M = H d − J ( R ) , H d − JS ( S ) , Supp R ( D R ( M )) = Spec R . Also for R/f R -module N = H d − J ( R/fR ) ( R/f R ) = H d − J ( S/XS ) ( S/XS ) , Supp
R/fR ( D R/fR ( N )) = Spec R/f R . Similar result is true for
S/XS . References [Har67] Robin Hartshorne.
Local cohomology , volume 1961 of
A seminar given by A.Grothendieck, Harvard University, Fall . Springer-Verlag, Berlin, 1967.[Har68] Robin Hartshorne. Cohomological dimension of algebraic varieties.
Ann. of Math. (2) ,88:403–450, 1968.[HL90] Craig Huneke and Gennady Lyubeznik. On the vanishing of local cohomology modules.
Invent. Math. , 102(1):73–93, 1990.[HNBPW18] Daniel J. Hern´andez, Luis N´u˜nez-Betancourt, Juan F. P´erez, and Emily E. Witt.Cohomological dimension , Lyubeznik numbers, and connectedness in mixed characteristic
J.Algebra , 514 (2018), 442-467[LY18] Gennady Lyubeznik and Tu˘gba Yıldırım On the Matlis duals of local cohomology modulesProc. Amer. Math. Soc. 146 (2018), 3715-3720.[Mat80] Hideyuki Matsumura.
Commutative algebra , volume 56 of
Mathematics Lecture NoteSeries . Benjamin/Cummings Publishing Co., Inc., Reading, Mass., second edition, 1980.[Ogu73] Arthur Ogus. Local cohomological dimension of algebraic varieties.
Ann. of Math. (2) ,98:327–365, 1973. [PS73] C. Peskine and L. Szpiro. Dimension projective finie et cohomologie locale. Applications`a la d´emonstration de conjectures de M. Auslander, H. Bass et A. Grothendieck.
Inst. Hautes´Etudes Sci. Publ. Math. , (42):47–119, 1973.
Dinabandhu Andrews College, Garia, Kolkata 700084, India
E-mail address ::