VIETNAM NATIONAL UNIVERSITY UNIVERSITY OF SCIENCE FACULTY OF MATHEMATICS, MECHANICS AND INFORMATICS Nguyen Duy Khanh THE ASYMPTOTIC LINEARITY OF CASTELNUOVO-MUMFORD REGULARITY Undergraduate Thesis. Advanced Undergraduate Program in Mathematics. Hanoi - 2012 VIETNAM NATIONAL UNIVERSITY UNIVERSITY OF SCIENCE FACULTY OF MATHEMATICS, MECHANICS AND INFORMATICS Nguyen Duy Khanh THE ASYMPTOTIC LINEARITY OF CASTELNUOVO-MUMFORD REGULARITY Undergraduate Thesis. Advanced Undergraduate Program in Mathematics. Thesis Advisor: Prof.Dr.Sc. Ngo Viet Trung. Hanoi - 2012 Acknowledgements I would like to thank my advisor, Prof. Ngo Viet Trung, for his limitless patience in making abstract mathematics so easy to be perceived, and for his supporting my work on this thesis. I also would like to thank Prof. Ha Huy Tai for his very helpful conversation on Castelnuovo-Mumford regularity. Finally, I would like to thank my family for always believing in me, supporting me, and encouraging me spiritually and mentally. 1 Contents 1. Introduction 3 2. Castelnuovo-Mumford regularity 3 2.1. Graded rings and graded modules 3 2.2. Castelnuovo-Mumford regularity 4 3. Bigraded module and regularity 7 4. Asymptotic Linearity 9 5. Open problem 10 6. Conclusion 13 References 14 2 1. Introduction Castelnuovo-Mumford regularity is an important invariant in commutative algebra. It was first defined by D.Mumford-who attributes the idea to G.Castelnuovo-for coherent sheaves on projectives space. In 1982, Ooshi[O] and in 1984, D.Eisenbud, and S.Goto[EG] both turned Castelnuovo-Mumford regularity into algebraic side. Ooishi characterized this regularity by local cohomology and Eisenbud and Goto made explicite the link between algebraic regularity of a graded module over a polynomial ring and its minimal free reso- lution. Denote by reg(M) the Castelnuvo-Mumford regularity of M. The behaviour of reg(I n ) where I is a homogeneous ideal in a polynomial ring over a field interests many mathemati- cians. Bertram, Ein, and Lazarsfeld[BEL] discovered firstly that if I is the defining ideal of a smooth complex variety, reg(I n ) is bounded by a linear function. Later, Chandler[C] has conjectured that: reg(I n ) ≤ nreg(I). Swanson[S] prove that for any homogenous ideal I and integer n ≥ 1 there exists a number D such that : reg(I n ) ≤ nD. This result sup- ported Chandler’s conjecture, however the method of the proof there makes it difficult to find such a constant. Finally, Cutkosky et al [CHT] and Kodiyalam[Kod] found out that reg(I n ) is asymptotically a linear function, in particular, Kodiyalam pointed out that the slope of that funtion is the least maximal degreee of reductions of I. However, the free coefficient is now still a mysterious, there are only some special results in some special situations of I. The main task of this writing is proving the following result, some other related results are also derived. Theorem: Let R be a standard graded algebra over a commutative Noetherian ring with unity and I is a graded ideal of R. Define: d(I) := max{degf|f belongs to a minimal generating set of I} ρ M (I) = min{d(J)|J is an M reduction of I} Let M be a finitely generated graded R-module, (M) denote the smallest degree of the homogeneous element of M. Then there exists an integer e ≥ (M) such that for n  0, reg(I n M) = ρ M (I)n + e 2. Castelnuovo-Mumford regularity 2.1. Graded rings and graded modules. In this section we study the bigraded struc- ture, which will play the important role in the proof for the main theorem. We start with a definition. 3 Definition 2.1. Let (G, +) be an abelian group. A ring R is called G graded if there exists a family of Z-modules R g , g ∈ G such that R =  g∈G R g as a Z module with R g R h ∈ R g+h for all g, h ∈ R. Let R be a graded ring. A R-module M is called G− graded if there exists a family of Z-modules M g , g ∈ G such that M =  g∈G M g as a Z-module with R g M h ∈ M g+h for any g, h ∈ G. The element x ∈ M is called homogeneous of degree g if x ∈ M g for some g ∈ G and we set d(x) = g. M g is called the homogeneous component of M, an ideal I of R is G-graded if I =  g∈G I g where I g = I ∩ R g . From this definition, it is easy to check that if I is graded and finitely generated then the degree of generators of I is uniquely determined by I. Definition 2.2. Let R be a G-graded ring, M be a G-graded R-module, and g ∈ G. We define the M(g) to be the G-graded R-module M by shifting its grading g steps, i.e : M(g) h = M g+h for all h ∈ G. This module M(g) is isomorphic to M as a module and called the g th twist of M. For a N-graded module M, the maximal non-vanishing degree of M is defined to be the maximal number g such that M g = 0. If G is Z or Z 2 we say that R is a graded or bigraded ring and M is a graded or bigraded R-module. In particular, the following examples will be considered in this writing: i- Let R = k[x 1 , , x n ] be a polynomial ring of n-variable over a field k. Then R has a graded structure by setting d(x i ) = 1 for every i = 1, , n. ii- Let S = k[x 1 , , x n , y 1 , , y m ] be the polynomial ring in n + m variables. Then S has bigraded structure by setting d(x i ) = (1, 0) and d(y j ) = (0, 1) for any 1 ≤ i ≤ n, 1 ≤ j ≤ m. The above polynomial rings are usually called the standard graded/bigraded polynomial ring. Note that there are other grading structures that can be defined on them. iii- Let R = k[x 1 , , x n ], I is a graded ideal of R. Define R[It] = { n  i=0 a i t i |n ∈ N, a i ∈ I i } =  n≥0 I n t n This is called the Rees algebra of I. There is a bigraded structure on this algebra defined by two degree, one by elements of I, one by the variable t. 2.2. Castelnuovo-Mumford regularity. In this section, we will define the Castelnuvo- Mumford regularity in two way: one in term of local cohomology and one in term of filter regular sequence. We begin with a definition 4 Definition 2.3. Let A be a commutative Noetherian ring with unity, R be a standard graded algebra over A, R + be the ideal generated by the element of positive degree. For any finitely generated graded R-module M, define: Γ R + (M) = {m ∈ M|mR t + = 0 for some t ∈ N} This is a functor from the category of R-module to R-module and it is left exact. The right derived funtor of Γ R + is called the local cohomology functor with respect to R + and denoted by H R + . Definition 2.4. Denote by a(M) the maximal non-vanishing degree of M, for i ≥ 0 we denote by H i R + (M) to be the i th local cohomology module of M with respect to R + . The Castelnuvo-Mumford regularity of M is the invariant defined to be : reg(M) := max{a(H i R + (M)) + i|i ≥ 0} Let z 1 , , z s be linear form in R. This set of element is called M-filter-regular sequence if z i /∈ p for any associated prime p  R + of (z 1 , , z i−1 )M for i = 1, , s. Then the Castelnuovo-Mumford regularity can be characterized as follows: Proposition 2.1. Let z 1 , , z s be an M-filter regular sequence of linear forms which gen- erate an M-reduction of R + . Then : reg(M) = max{a((z 1 , , z i )M : R + /(z 1 , , z i )M)|i = 1, , s} In order to prove this proposition, we need a lemma: Lemma 2.2. Let z ∈ R 1 be an M-filter regular element. Define a i (M) = max{r|H i R + (M) r = 0}. Then, for all k ≥ 0, we have a k+1 (M) + 1 ≤ a k (M/zM) ≤ max{a k (M), a k+1 (M) + 1}. Proof. Consider the exact sequence: 0 → (M/0 : z)(−1) z −→ M → M/zM → 0. This exact sequence induces the following exact sequence : H k R + (M) i → H k R + (M/zM) i → H k+1 R + (M) i−1 → H k+1 R + (M) i → H k+1 R + (M/zM) i → From this exact sequence we get : a k (M/zM) ≤ max{a k (M), a k+1 (M) + 1} Note that for all i > a k (M/zM) we have : H k+1 R + (M) i−1 → H k+1 R + (M) i → H k+1 R + (M) i+1 → → 0 Then, we get a k+1 (M) < a k (M/zM). This completes the proof for lemma.  Now we are ready to prove the proposition. 5 Proof. Note that from the definition of a i (M) we have reg(M) = max{a i (M) + i|i ≥ 0}. By using lemma successively we get a i (M) + i ≤ a 0 (M/(z 1 , . . . , z i )M) ≤ max{a j (M) + j| j = 0, . . . , i}. This implies that max{a i (M) + i| i = 0, . . . , t} = max{a 0 (M/(z 1 , . . . , z i )M)| i = 0, . . . , t} for all t ≤ s. Now we identify H 0 R + (M/(z 1 , . . . , z i )M)) with  n≥0 (z 1 , , z i )M : R n + /(z 1 , , z i )M. Set a = a 0 (M/(z 1 , . . . , z i )M). Then we have H 0 R + (M/(z 1 , . . . , z i )M)) a ⊆ (z 1 , , z i )M : R + /(z 1 , , z i )M ⊆ H 0 R + (M/(z 1 , . . . , z i )M)) and therefore a((z 1 , , z i )M : R + /(z 1 , , z i )M) = a.  The following lemma show that any M-reduction of R + can be generated by an M-filter regular sequence in a flat extension of A. Lemma 2.3. Let q be an M-reduction generated by the linear form x 1 , , x s . For i = 1, , s let z i =  s j=1 u ij x j , where U = {u ij |i, j = 1, , s} is a matrix of indeterminates.Put : A  = A[U, det(U) −1 ], R  = R ⊗ A A  , M  = M ⊗ A  . Then if we consider R  as a standard graded algebra over A  and M  as a graded R  module, then z 1 , , z s is an M  filter regular sequence. Proof. From the independence of indeterminates, it suffices to show that z 1 /∈ p for any associated prime p  R + of M  . From the definition of R  , we see that this prime ideal p must have have the form qR  for some associated prime q  R + of M. If Q ⊆ q then since M/qM is a quotient module of M/QM we have (M/qM) n = 0 for all n  0 . It follows that there is a number t such that R t + M ⊆ qM. But ann(M) ⊆ q we get R + ⊆ q which is a contradiction. Therefore Q  q. Since Q = (x 1 , , x s ), this implies that : z 1 = u 11 x 1 + + u 1s x s /∈ qR  = p  The following corollary is crucial in proving the main result : Corollary 2.4. The maximal degree of the generator of M does not exceed the regularity of M. Proof. Applying Lemma 2.2 with notice that reg(M) = max{a k (M)+k|k ≥ 0} we have: a k (M/zM) + k ≤ max{a k (M) + k, a k+1 (M) + k + 1} ↔ reg(M/zM) ≤ reg(M) 6 Denote by dim(M) the Krull dimension of M. If d = 0 then dim(R/ann R (M)) = 0 thus M is annihilated by some power of R + therefore M = Γ R + (M). Hence H i R + (M) = 0 for every i ∈ N and so reg(M) = a(M). Assume that a(Q) ≤ reg(Q) for every module Q of dimension less than dim(M), denote by r the regularity of M. Let N be the submodule of M generated by all the elements of degree less than r. Note that : dim(M/zM) ≤ dim(M) then by the induction assumption we have : a(M/zM) ≤ reg(M/zM) ≤ reg(M) Hence : M = N + zM. Applying the graded version of Nakayama lemma, we get N = M. Therefore the maximal degree of the generator of M does not exceed the regularity of M.  3. Bigraded module and regularity In this section we will apply the bigraded structure to study the behaviour of the regularity. Let S = A[X 1 , . . . , X s , Y 1 . . . , Y v ] be a polynomial ring over a commutative Noetherian ring A with unity. Then we can view S as a bigraded ring by define degX i = (1, 0), i = 1, . . . , s, and deg(Y j ) = (d j , 1), j = 1, . . . , v for a given sequence d 1 , . . . , d v of non-negative integers. Without loss of generality, assume that d v = max{d i |i = 1, . . . , v}. Let M be a finitely generated bigraded module over A[X 1 , . . . , X s , Y 1 . . . , Y v ]. For a fixed number n define M n :=  a≥0 M (a,n) . Then M n is a finitely generated graded module over the naturally graded polynomial ring A[X 1 . . . , X s ]. We will show that reg(M n ) is asymptotically a linear function. If s = 0, then reg(M n ) = max{a|M (a,n) = 0} := a(M n ). In [CHT-Theorem 3.4] they has already consider this case for the case of bigraded module over polynomial ring over a field. However the proof there can not be used to prove the general case. Proposition 3.1. Let M be a finitely generated bigraded module over the bigraded polynomial ring A[Y 1 . . . , Y v ]. Then a(M n ) is a linear function with slope ≤ d v for n  0. Proof. Since the case v = 0 is trivial then we only need to consider the case v ≥ 1. Consider the exact sequence of bigraded A[Y 1 . . . , Y v ]-modules: 0 −→ [0 M : Y v ] (a,n) −→ M (a,n) Y v −→ M (a+d v ,n+1) −→ [M/Y v M] (a+d v ,n+1) −→ 0. Assume that a([0 M : Y v ] n ) and a([M/Y v M] n ) are asymptotically linear functions with slopes ≤ d v , then we will prove the proposition inductively on n. We have a([0 M : Y v ] n ) + d v ≥ a([0 M : Y v ] n+1 ) 7 a([M/Y v M] n ) + d v ≥ a([M/Y v M] n+1 ) for all large n. The proof is completed if we show that a(M n ) = a([0 M : Y v ] n ) for all large n. Since a(M n ) ≥ a([0 M : Y v ] n ) for all n, it suffices to consider the case that there exists an infinite sequence of integers m such that a(M m ) ≥ a([0 M : Y v ] m ). Applying this condition into the exact sequence above we have : a(M m+1 ) = max{a(M m ) + d v , a([M/Y v M] m+1 )}. On the other hand, we have : a(M m ) + d v ≥ a([M/Y v M] m ) + d v ≥ a([M/Y v M] m+1 ). Then for all n ≥ m, a(M n ) > a([0 M : Y v ] n ), hence a(M n+1 = a(M n )) + d v . Therefore a(M n ) is asymptotically a linear function with slope d v in this case. The proof completed.  Using the above result and the technique of flat extension, we can prove the following lemma: Theorem 3.2 Let M be a finitely generated bigraded module over the bigraded poly- nomial A[X 1 , . . . , X s , Y 1 . . . , Y v ]. Then reg(M n ) is a linear function with slope ≤ d v for n  0. Proof. The case s = 0 is the Proposition 3.1. Now assume that s ≥ 1. Let U = {u ij |i, j = 1, . . . , s} is a matrix of indeterminates, let z i =  s j=1 u ij X j , i = 1, . . . , s. Define A  := A[U, det(U) −1 ] and M  := M n ⊗ A A  . Then M  is a finitely generated bi- graded module over A  [X 1 , . . . , X s , Y 1 , . . . , Y v ]. Since M  n = M n ⊗ A A  , then reg(M n ) = reg(M  n ) for all n ≥ 0. According to Lemma 2.3 , we have z 1 , . . . , z s is an M  n -filter- regular sequence. Let R = A  [X 1 , . . . , X s ]. Since (z 1 , . . . , z s ) = (X 1 , . . . , X s ) = R + , applying Proposition 3.1 we get reg(M  n ) = max{a((z 1 , . . . , z n )M  n : R + /(z 1 , . . . , z i )M  n )|i = 0, . . . , s}. We have (z 1 , . . . , z n )M  n : R + /(z 1 , . . . , z i )M  n ) = [(z 1 , . . . , z n )M  n : R + /(z 1 , . . . , z i )M  n )]. On the other hand note that (z 1 , . . . , z i )M  : R + /(z 1 , . . . , z i )M  can be viewed as a graded module over the bigraded polynomial ring A  [Y 1 , . . . , Y v ] then by Proposition 3.1, a([(z 1 , . . . , z i )M  : R + /(z 1 , . . . , z i )M  ] n ) is asymptotically a linear function with slope ≤ d v for i = 0, . . . , s. Hence reg(M n ) is asymptotically a linear function with slope ≤ d v .  8 [...]... 5.2 The free coefficent in case of regularity The problem of finding the constant e in case of regularity is very hard and in fact, there are only several results in some special situation.In [EH], D.Eisenbud and J.Harris has characterized Castelnuovo-Mumford regularity in more geometrical way, they consider the ideal of a projective scheme X in Pn , the information of regularity is received under the... Eisenbud, C Huneke and B Ulrich, The regularity of Tor and graded Betti numbers Amer J Math 128 (2006) [EH]-D Eisenbud and J Harris, Power of ideals and fibers of morphisms, Math Res Lett 17, 267-273, (2010) [EU]-D Eisenbud and N.Ulrich, Stabilization of the regularity of power of and ideal arXiv: 1012.0951v1 [Kod]-V Kodiyalam, Asymptotic behaviour of Castelnuovo-Mumford regularity Proceeedings of the AMS... Proceeedings of the AMS 128, 407411, (1999) [O]-A.Ooishi, Castelnuovo’s regularity of graded ring and modules, Hiroshima Math J.12, 627-644, (1982) [S]-I Swanson, Powers of ideals, primary decompositions, ArtinRees lemma and regularity Math Ann 307, 299313, (1997) [TW]-N V Trung and H.-J Wang, On the asymptotic linearity of Castelnuovo-Mumford regularity, J.Pure Appl Algebra 201, no 1-3, 42-48, (2005) [Ro]-T.Romer,... a theorem of Severi, and the equations defining projective varieties J Amer Math Soc 4, 587602, (1991) [C]-K A Chandler, Regularity of the powers of an ideal Commun Algebra 25, 37733776, (1997) [CHT]-D Cutkosky, J Herzog and N.V Trung, Asymptotic behaviour of the Castelnuovo-Mumford regularity Compositio Math 118, 243261, (1999) [E]-Commutative Algebra with a view toward algebraic geometry, Springer,... Once again the role of the Rees algebra is very important The problem of finding out the number e and t0 in general is now still open 12 6 Conclusion In this thesis, I presented the asymptotic linearity of Castelnuovo-Mumford regularity, based almost on the paper [TW] I proved the Proposition 1.3 without using the idea in that paper(the authors did not prove it, they left the idea of the proof only), and... unity and I a graded ideal of R Then there exists an integer e ≥ 0 such that for all large n, reg(I n ) = ρR (I)n + e In particular, we can use Theorem 3.2 to prove that regI n+1 the regularity of integral closure of I n is asymptotically linear This result was also derived in [CHT] for the case R is a polynomial ring over a field however in there the slope was not determined Corollary 4.4 Let R be a standard... of the projective plane has regularity reg(I) = 3 however reg(I 2 ) = 7 The minimal number t0 such that the function reg(I t ) is linear for all t ≥ t0 also be interested, but we also know a little of it Eisenbud and Ulrich[EU] prove that as in the case I is equigenerated in degree d and m-primary, then we have a lower bound for t0 : 1 + reg(M ) , N }, in which N is the regularity of the Rees module... result 5 Open problem 5.1 Asymptotic linearity of maximal degree of power of ideal Recall that if I is an ideal in a finitely generated standard graded algebra A over a Noetherian ring A, then dt + ε(M ) ≤ d(I t ) ≤ reg(I t ) ≤ dt + e for every t ∈ N and some positive integer e Therefore it is reasonable to ask the question about the behaviour of d(I t ) for t large enough : Is it asymptotically a linear... function ? This question is still open now, but we can prove it in a particular situation : Theorem 5.1 Let I be a homogeneous ideal in the polynomial ring over a field R = k[x1 , , xs ] Then d(I n ) is asymptotically a linear function in slope d, i.e : d(I n ) = dn + e for n 0 Proof The linearity part was implicitly appeared in [CHT], we will recall and make it preciser here Define regi (I) := max{a|Tori...4 Asymptotic Linearity Let A be a commutative Noetherian ring with unity Let R be a standard graded algebra over A and I a graded ideal of R, M be a finitely generated graded R-module, let ε(M ) be the smallest . regularity of power of and ideal. arXiv: 1 012. 0951v1 [Kod]-V. Kodiyalam, Asymptotic behaviour of Castelnuovo-Mumford regularity. Proceeedings of the AMS 128 , 407411, (1999) [O]-A.Ooishi, Castelnuovo’s. Thesis. Advanced Undergraduate Program in Mathematics. Thesis Advisor: Prof.Dr.Sc. Ngo Viet Trung. Hanoi - 2 012 Acknowledgements I would like to thank my advisor, Prof. Ngo Viet Trung, for his limitless patience. is very important. The problem of finding out the number e and t 0 in general is now still open. 12 6. Conclusion In this thesis, I presented the asymptotic linearity of Castelnuovo-Mumford regularity, based