Cohomological dimension of ideals defining Veronese subrings
aa r X i v : . [ m a t h . A C ] M a y COHOMOLOGICAL DIMENSION OF IDEALSDEFINING VERONESE SUBRINGS
VAIBHAV PANDEYA
BSTRACT . Given a standard graded polynomial ring over a commutative Noetherianring A , we prove that the cohomological dimension and the height of the ideals definingany of its Veronese subrings are equal. This result is due to Ogus when A is a field ofcharacteristic zero, and follows from a result of Peskine and Szpiro when A is a field ofpositive characteristic; our result applies, for example, when A is the ring of integers.
1. I
NTRODUCTION
Throughout this paper, all rings are assumed to be commutative, Noetherian, and withan identity element.Let T = Z [ x , x , . . . , x k ] be the standard graded polynomial ring in n indeterminantsover the integers. Consider a minimal minimal presentation of its n -th Veronese subring T ( n ) = ⊕ i ≥ T in as T ( n ) ∼ = Z [ t , . . . , t d ] / I . We say that I is the ideal defining the n -th Veronesesubring of T . For A a ring, we set T A = T ⊗ Z A .Ogus in [Ogu73, Example 4.6] proved that when A is a field of characteristic zero, thecohomological dimension of I is the same as its height. The same result also follows when A is a field of positive characteristic by a result of Peskine and Szpiro [PS73, PropositionIII.4.1]. We prove that this continues to hold for any commutative Noetherian ring A .The critical step is the calculation of local cohomology of the polynomial ring Z [ t , . . . t d ] supported at the ideal I . More precisely, we prove: Theorem 1.1.
Let T = Z [ x , . . . , x k ] be a polynomial ring with the N -grading [ T ] = Z anddeg x i = for each i. Consider a minimal presentation of T ( n ) as R / I. ThenH iI ( R ) = for i = height I . Towards the above result, we establish a condition for the injectivity of multiplicationby a prime integer on local cohomology modules over the ring Z [ t , . . . t d ] in Lemma 2.3.This strengthens [LSW16, Corollary 2.18] and is a result of independent interest.It is worth mentioning that in the above context, the arithmetic rank may vary with thecharacteristic of the ring A : Example 1.2.
Let k [ x , . . . , x n ] be a standard graded polynomial ring over a field k . Let R be a polynomial ring over k in indeterminants that map entrywise to the distinct elementsof the matrix Date : May 8, 2020.2010
Mathematics Subject Classification.
Key words and phrases. cohomological dimension, local cohomology. x x x · · · x x n x x x · · · x x n ... ... . . . ... x x n x x n · · · x n .Thus, R is a polynomial ring in (cid:0) n + (cid:1) indeterminants. The relations between the generatorsof R under the above map are precisely those corresponding to the size two minors of thismatrix. These relations define an ideal I of R , with R / I being a minimal presentation.Barile proved that the arithmetic rank of I , i.e., the minimum number of equations definingthe affine variety V ( I ) set-theoretically, isara I = ((cid:0) n (cid:1) if char k = 2 , (cid:0) n + (cid:1) − t minors of a symmetric n × n matrix of indeterminants over a field in [Bar95,Theorems 3.1, 5.1] and remarked: This seems to be the first class of ideals defined over Z for which, after specialization to a field k, the arithmetical rank depends on k. Thisdependence of the arithmetic rank of I on the characteristic of the field makes it interestingto investigate the local cohomology of polynomial rings over the integers such as thoseexamined here.2. I NJECTIVITY OF MULTIPLICATION BY A PRIME INTEGERON LOCAL COHOMOLOGY MODULES
The following lemma gives a criterion for integer torsion in local cohomology modulesof a standard graded polynomial ring over the integers:
Lemma 2.1. [LSW16, Corollary 2.18]
Let R = Z [ x , . . . , x n ] be a polynomial ring with the N -grading [ R ] = Z and deg x i = for each i. Let I be a homogeneous ideal, p a primeinteger, and h a nonnegative integer. Suppose that the Frobenius action on [ H n − h ( x ,..., x n ) ( R / ( I + pR ))] is nilpotent, and that the multiplication by p mapH h + I ( R ) x i . p → H h + I ( R ) x i is injective for each i. Then the multiplication by p map on H h + I ( R ) is injective. The proof of this lemma largely relies on the following theorem. For an overview of D -modules and F -modules, we refer the reader to [LSW16]. Theorem 2.2. [LSW16, Theorem 2.16]
Let R be a standard graded polynomial ring, where [ R ] is a field of prime characteristic. Let m be the homogeneous maximal ideal of R,and I an arbitrary homogeneous ideal. For each nonnegative integer k, the following areequivalent:(1) Among the composition factors of the Eulerian D -module ξ ( H kI ( R )) , there is atleast one composition factor with support { m } .(2) Among the composition factors of the graded F -finite module H kI ( R ) , there is atleast one composition factor with support { m } .(3) H kI ( R ) has a graded F -module homomorphic image with support { m } .(4) The natural Frobenius action on [ H dimR − k m ( R / I )] is not nilpotent. We strengthen Lemma 2 . OHOMOLOGICAL DIMENSION OF IDEALS DEFINING VERONESE SUBRINGS 3
Lemma 2.3.
Let R = Z [ x , . . . , x n ] be a polynomial ring with the N -grading [ R ] = Z anddeg x i = for each i. Let I be a homogeneous ideal, p a prime integer, and h a nonnegativeinteger. Let t , . . . , t k be homogeneous elements in R such that p ( t , . . . , t k ) R / I = ( x , . . . , x n ) R / I . Further, suppose that the Frobenius action on [ H n − h ( t ,..., t k ) ( R / ( I + pR ))] is nilpotent and that the multiplication by p mapH h + I ( R ) t i . p → H h + I ( R ) t i is injective for each i. Then the multiplication by p map on H h + I ( R ) is injective.Proof. Since local cohomology modules depend only on the radical of the ideal definingthe support, H n − h ( t ,..., t k ) ( R / ( I + pR )) = H n − h ( x ,..., x n ) ( R / ( I + pR )) . Therefore, the natural Frobenius action on [ H n − h ( x ,..., x n ) ( R / ( I + pR ))] is nilpotent. The shortexact sequence 0 → R . p → R → R / pR → · · · → H iI ( R ) → H iI ( R / pR ) δ → H i + I ( R ) . p → H i + I ( R ) → · · · . Let K denote the kernel of the multiplication by p map in the above display, and m denotethe homogeneous maximal ideal of R / pR .By hypothesis, the localization K t i is zero for each i . Thus, any prime ideal in the supportof K must contain each t i . We may assume that I is a proper ideal of R . Thus, prime ideals p in the support of K are such that ( t , . . . , t k ) R ⊆ p and I ⊆ p . Therefore, p ( t , . . . , t k ) R + I = m is contained in p . Thus, Supp( K ) is contained in { m } .The kernel K is a D Z ( R ) -module; since it is annihilated by p , it is also a module over D Z ( R ) / p D Z ( R ) ∼ = D F p ( R / pR ) . This isomorphism follows, for example, from [BBL +
14, Lemma 2.1]. If K is nonzero, thenit is a homomorphic image of H iI ( R / pR ) in the category of Eulerian graded D F p ( R / pR ) -modules, supported precisely at the homogeneous maximal ideal m of R / pR . But thisis not possible, since the D F p ( R / pR ) -module H iI ( R / pR ) has no composition factor withsupport { m } by Theorem 2 . (cid:3) We illustrate Lemma 2 . Definition 2.4.
Let I be an ideal of a ring R . For each R -module M , setcd R ( I , M ) = sup { n ∈ N : H nI ( M ) = } . The cohomological dimension of I iscd ( I ) = sup { cd R ( I , M ) : M is an R -module } . By the right exactness of the functor H cd ( I ) I ( − ) , we get cd R ( I ) = cd R ( I , R ) . VAIBHAV PANDEY
Example 2.5.
Consider the ring T = Z [ x , x y , xy , y ] , which has a minimal presentation: T ∼ = Z [ t , t , t , t ] / ( t t − t t , t t − t , t t − t t , t t − t ) = R / I . We calculate the cohomological dimension of the ideal I . For any field k , we denote T ⊗ Z k by T k . Hartshorne in [Har79, Theorem] showed that for k , a field of positive characteristic,the arithmetic rank of IR k is two. Since the ideal I has height two, it follows that thecohomological dimension of IR k is also two.We denote by T ′ k the ring k [ x , x y , x y , xy , y ] , which is the normalization of T k . Theshort exact sequence of T k -modules0 → T k → T ′ k → T ′ k / T k → H ( x , x y , xy , y ) ( T k ) ∼ = H ( x , x y , xy , y ) ( T ′ k ) , since T ′ k / T k is a zero-dimensional T k -module. As T ′ k is a direct summand of the polynomialring k [ x , y ] , it follows that [ H ( t , t , t , t ) ( R / ( I + pR ))] = p ( t , t ) R / I = ( x , x y , xy , y ) R / I . Further, IR t = ( t − t / t , t − t / t ) and IR t = ( t − t / t , t − t / t ) are both two generated ideals. Thus, by Lemma 2.3, the map H I ( R ) . p → H I ( R ) is injec-tive for each nonzero prime integer p . The exact sequence of local cohomology modulesinduced by 0 → R . p → R → R / pR → H I ( R ) . p → H I ( R ) is surjective since H I ( R / pR ) =
0. Therefore, H I ( R ) is a Q -vector space. But the cohomological dimension of IR Q is known to be two. We concludethat the cohomological dimension of I is two. It is worth noting that T / pT ∼ = R / ( I + pR ) is not F -pure, since, ( x y ) / ∈ ( x ) T / pT but ( x y ) p ∈ ( x p ) T / pT .
3. C
ALCULATION OF COHOMOLOGICAL DIMENSION
Definition 3.1.
Let R = ⊕ i ≥ R i be a graded ring, and n be a positive integer. We denoteby R ( n ) , the Veronese subring of R spanned by elements which have degree a multiple of n ,i.e., R ( n ) = ⊕ i ≥ R in .We now present the key result which helps us calculate the cohomological dimensionof ideals defining Veronese subrings. Proposition 3.2.
Let A be a domain. Let T = A [ x , . . . , x k ] be a polynomial ring with the N -grading [ T ] = A and deg x i = for each i. Consider the lexicographic ordering ofmonomials in T induced by x > x > · · · > x k .Write a minimal presentation of T ( n ) as R / I where R = A [ t , . . . , t d ] with t i mapping tothe i-th degree n monomial under the above monomial ordering. Then, for each i such thatt i x nj for some j, the ideal IR t i is generated by a regular sequence of length height I.Proof.
By symmetry, it is enough to consider t x n . We claim that the ideal IR t isgenerated by the regular sequence t k + − t / t , t k + − t t / t , t k + − t t / t , . . . , t ( k + ) − t k / t , t ( k + ) + − t / t , . . .. . . , t ( k + ) − t k / t , . . . , t d − − t k − t n − / t n − , t d − t nk / t n − . OHOMOLOGICAL DIMENSION OF IDEALS DEFINING VERONESE SUBRINGS 5
Note that the length of this regular sequence is equal to height I . Let J be the ideal ( t k + − α k + , t k + − α k + , . . . , t d − α d ) R t , where α k + , α k + , . . . α d are as above, i.e., α k + = t / t , α k + = t t / t , . . . , and α d = t nk / t n − . We claim that J = IR t . It is clear that the ideal J is contained in IR t . Since ( R / I ) t is a subring of the fraction field of R / I , it follows that the ideal IR t is prime ofheight d − k .Define a ring homomorphism φ : R t → A [ t , . . . , t k ][ t ] such that t i t i for 1 ≤ i ≤ k and t j α j for k + ≤ j ≤ d . Then the map φ is a surjective ring homomorphism withkernel J . Hence, J is a prime ideal of R t of height d − k . Thus, J ⊆ IR t are prime idealsof the same height in the ring R t . We conclude that the ideals J and IR t are equal. (cid:3) Remark 3.3.
In the notation of Proposition 3.2, assume that the ring A is regular. Then foreach t i with t i x nj , the ring ( R / I ) t i ∼ = T ( n ) x nj = A [ x nj , / x nj , x / x j , . . . , x j − / x j , x j + / x j , . . . x k / x j ] is regular.One of the most well-known vanishing results for local cohomology modules in positivecharacteristic was given by Peskine and Szpiro: Theorem 3.4. [PS73, Proposition III.4.1]
Let R be a regular domain of positive character-istic p and I be an ideal of R such that R / I is a Cohen-Macaulay ring. ThenH iI ( R ) = for i = height I . The proof uses the flatness of the Frobenius action on R which characterizes regularrings in positive characteristic.Before we proceed to our main result, we would like to remark that the cohomologicaldimension of ideals may depend on the coefficient ring: Remark 3.5.
Let k be a field. Let R = Z [ u , v , w , x , y , z ] and R k = R ⊗ Z k . Let I be the ideal ( ∆ , ∆ , ∆ ) R where ∆ = vz − wy , ∆ = wx − uz , and ∆ = uy − vx . It is easily checked thatheight I =
2. Then cd R / pR ( I , R / pR ) = H I ( R Q ) is nonzero, i.e., cd R Q ( I , R Q ) =
3. Since local cohomology commutes withlocalization, we also have H I ( R ) is nonzero, i.e., cd R ( I , R ) =
3. We point the reader to[ILL +
07, Example 21.31] for further details.In Theorem 1.1, we obtain a vanishing result for local cohomology modules over theintegers similar to Theorem 3.4.
Proof of Theorem 1.1.
Let h denote the height of the ideal I . Since R is regular, the gradeof I equals h so that H iI ( R ) = i < h . Further, by Grothendieck’s nonvanishing theorem, H hI ( R ) = p be a nonzero prime integer. The short exact sequence0 → R . p → R → R / pR → · · · → H iI ( R ) → H iI ( R / pR ) δ → H i + I ( R ) . p → H i + I ( R ) → · · · . Note that the height of the ideal IR / pR is also h . Hence, by Theorem 3.4 H iI ( R / pR ) = H iIR / pR ( R / pR ) = i = h . VAIBHAV PANDEY
It follows that the map H iI ( R ) . p → H iI ( R ) is an isomorphism for each i > h + H h + I ( R ) . p → H h + I ( R ) is surjective. The crucial part that remains to show is thatthe map H h + I ( R ) . p → H h + I ( R ) is also injective. For this, we appeal to Lemma 2.3. Afterreordering of indices, let t , . . . , t k denote the preimages of x n , . . . , x nk respectively.The ring R / ( I + pR ) is a direct summand of the polynomial ring T / pT . Therefore, [ H n − h ( t ,..., t k ) ( R / ( I + pR ))] is zero.By symmetry, it is enough to show that the multiplication by p map H h + I ( R ) t . p → H h + I ( R ) t is injective. Note that the R -module H h + I ( R ) t is isomorphic to H h + IR t ( R t ) . ApplyingProposition 3.2 with A = Z , we get that the ideal IR t is generated by a regular sequence oflength h . Therefore, H h + IR t ( R t ) = H h + I ( R ) . p → H h + I ( R ) is injective.For i > h , by [Ogu73, Example 4.6], the module H iI ( R ) ⊗ Z Q vanishes so that H iI ( R ) is equal to its Z -torsion submodule. But the Z -torsion submodule of H iI ( R ) is zero sincemultiplication by each nonzero prime integer is injective. We therefore conclude that thelocal cohomology modules H iI ( R ) vanish for i > h . (cid:3) Remark 3.6.
Following the notation of Theorem 1.1, all but finitely many prime integersare known to be nonzerodivisors on H iI ( R ) for any i by [BBL +
14, Theorem 3.1 (2)]. Notethat in Theorem 1.1, we proved that each nonzero prime integer is a nonzerodivisor on H iI ( R ) for every i . Consequently, any associated prime of the R -module H hI ( R ) contracts tothe zero ideal in the integers.In [Sin00, Section 4], Singh constructs an example of a local cohomology module overa six dimensional hypersurface, which has p -torsion elements for each prime integer p ,and consequently has infinitely many associated prime ideals.In [Rai17, Theorem 4.1], Raicu recovers the result due to Ogus in [Ogu73, Example4.6] which we used in proving Theorem 1.1; and also determines the D -module structureof the only nonvanishing local cohomology module.Finally, we extend Theorem 1.1 to standard graded polynomial rings with coefficientsfrom any commutative Noetherian ring. For this, we use the following proposition whichis proved in [BV88] more generally when R = Z [ t , . . . , t d ] / J is a faithfully flat Z -algebra. Proposition 3.7. [BV88, Proposition 3.14]
Let I be an ideal of the polynomial ring R = Z [ t , . . . , t d ] and A be a ring. If there exists an integer h such that grade I ( R ⊗ Z k ) = h forevery field k, then grade I ( R ⊗ Z A ) = h. Analogous statements hold for height. Theorem 3.8.
Let A be a commutative Noetherian ring and T = A [ x , . . . , x k ] be a poly-nomial ring with the N -grading [ T ] = A and deg x i = for each i. Consider a minimalpresentation of T ( n ) as R / I. ThenH iI ( R ) = for i = height I . Proof.
Theorem 1.1 and Proposition 3.7 together ensure that height I and grade I areequal. Therefore, H iI ( R ) = i < height I . Further, the map Z −→ A induces the map Z [ x , . . . , x n ] −→ R which makes R into a Z [ x , . . . , x n ] -module. By the right exactness ofthe top local cohomology, the cohomological dimension of I in R is at most the cohomo-logical dimension of I in Z [ x , . . . , x n ] , which, by Theorem 1.1, equals height I . (cid:3) OHOMOLOGICAL DIMENSION OF IDEALS DEFINING VERONESE SUBRINGS 7 A CKNOWLEDGEMENT
The author would like to thank Anurag Singh for many valuable discussions and for hisconstant encouragement and support. R
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