Koszul homology and syzygies of Veronese subalgebras
aa r X i v : . [ m a t h . A C ] N ov KOSZUL HOMOLOGY AND SYZYGIES OF VERONESE SUBALGEBRAS
WINFRIED BRUNS, ALDO CONCA, AND TIM R ¨OMER
To J¨urgen Herzog, friend and teacher A BSTRACT . A graded K -algebra R has property N p if it is generated in degree 1, hasrelations in degree 2 and the syzygies of order ≤ p on the relations are linear. The Green-Lazarsfeld index of R is the largest p such that it satisfies the property N p . Our mainresults assert that (under a mild assumption on the base field) the c -th Veronese subringof a polynomial ring has Green-Lazarsfeld index ≥ c +
1. The same conclusion also holdsfor an arbitrary standard graded algebra, provided c ≫
1. I
NTRODUCTION
A classical problem in algebraic geometry and commutative algebra is the study ofthe equations defining projective varieties and of their syzygies. Green and Lazarsfeld[18, 19] introduced the property N p for a graded ring as an indication of the presence ofsimple syzygies. Let us recall the definition. A finitely generated N -graded K -algebra R = ⊕ i R i over a field K satisfies property N if R is generated (as a K -algebra) in degree1. Then R can be presented as a quotient of a standard graded polynomial ring S and onesays that R satisfies property N p for some p > b Si , j ( R ) = j > i + ≤ i ≤ p .Here b Si , j ( R ) denote the graded Betti numbers of R over S . For example, property N means that R is defined by quadrics, N means that R is defined by quadrics and that thefirst and second syzygies of the quadrics are linear. We define the Green-Lazarsfeld index of R , denoted by index ( R ) , to be the largest p such that R has N p , with index ( R ) = ¥ if R satisfies N p for every p . It is, in general, very difficult to determine the precise valueof the Green-Lazarsfeld index. Important conjectures, such as Green’s conjecture on thesyzygies of canonical curves [14, Chap.9], predict the value of the Green-Lazarsfeld indexfor certain families of varieties.The goal of this paper is to study the Green-Lazarsfeld index of the Veronese embed-dings v c : P n − → P N of degree c of projective spaces and, more generally, of the Veroneseembeddings of arbitrary varieties. Let S denote the polynomial ring in n variables over thefield K . The coordinate ring of the image of v c is the Veronese subring S ( c ) = L i ∈ N S ic of S . If n ≤ c ≤ S ( c ) is a determinantal ring whose resolution is well understoodand the Green-Lazarsfeld index can be easily deduced. If n = S ( c ) is resolved by theEagon-Northcott complex and so index ( S ( c ) ) = ¥ . The resolution of S ( ) in characteristic0 is described by Jozefiak, Pragacz and Weyman in [22]; it follows that index ( S ( ) ) = n > ( S ( ) ) = ¥ if n ≤
3. For n ≤ ( S ( ) ) isindependent on char K , but for n > K = ( S ( ) ) = For n > c > c ≤ index ( S ( c ) ) ≤ c − . The lower bound is due to Green [17] (and holds for any c and n ). Ottaviani and Paoletti[23] established the upper bound in characteristic 0. They also showed that index ( S ( c ) ) = c − n = ( S ( c ) ) = c − n ≥ n = c = ( S ( ) ) ≥ K =
0. Our main results are the following:(i) c + ≤ index ( S ( c ) ) if char K = > c +
1; see Corollary 4.2.(ii) If R is a quotient of S then index ( R ( c ) ) ≥ index ( S ( c ) ) for every c ≥ slope S ( R ) ; seeTheorem 5.2. In particular, if R is Koszul then index ( R ( c ) ) ≥ index ( S ( c ) ) for every c ≥ n =
3; see Theorem 4.7. Our approach is based on the study of the Koszul complexassociated to the c -th power of the maximal ideal. Let R be a standard graded K -algebrawith maximal homogeneous ideal m . Let K ( m c , R ) denote the Koszul complex associ-ated to m c , Z t ( m c , R ) the module of cycles of homological degree t and H t ( m c , R ) thecorresponding homology module. In Section 2 we study the homological invariants of Z t ( m c , R ) . Among other facts, we prove, in a surprisingly simple way, a generalizationof Green’s theorem [17, Thm. 2.2] to arbitrary standard graded algebras; see Corollary2.5. If R is a polynomial ring (or just a Koszul ring), then it follows that the regularity of Z t ( m c , R ) is ≤ t ( c + ) ; see Proposition 2.4.In Section 3 we investigate more closely the modules Z t ( m c , S ) in the case of a polyno-mial ring S . Lemma 3.4 describes certain cycles which then are used to prove a vanishingstatement in Theorem 3.6. In Section 4 we improve the lower bound (1) by one, seeCorollary 4.2. Proposition 4.4 states a duality of Avramov-Golod type, which is the al-gebraic counterpart of Serre duality. The duality is then used to establish Ottaviani andPaoletti’s upper bound index ( S ( c ) ) ≤ c − n = ( S ( c ) ) = c − R to be a quotient of a Koszul algebra D and prove that for every c ≥ slope D ( R ) we have index ( R ( c ) ) ≥ index ( D ( c ) ) ; see Theorem 5.2. Here slope D ( R ) = sup { t Di ( R ) / i : i ≥ } where t Di ( R ) is the largest degree of an i -th syzygy of R over D . Inparticular, slope D ( R ) = R is Koszul (Avramov, Conca and Iyengar [4]) and, when D = S is a polynomial ring, slope S ( R ) ≤ a if the defining ideal of R has a Gr¨obner basisof elements of degree ≤ a . Similar results have been obtained by Park [24] under somerestrictive assumptions. In the last section we discuss multigraded variants of the resultspresented. 2. G ENERAL BOUNDS
In this section we consider a standard graded K -algebra R with maximal homogeneousideal m , which is a quotient of a polynomial ring S , say R = S / I where I is homogeneous OSZUL HOMOLOGY AND SYZYGIES OF VERONESE SUBALGEBRAS 3 (and may contain elements of degree 1). For a finitely generated graded R -module M let b Ri , j ( M ) = dim K Tor Ri ( M , K ) j be the graded Betti numbers of M over R . We define thenumber t Ri ( M ) = max { j ∈ Z : b Ri , j ( M ) = } , if Tor Ri ( M , K ) = t Ri ( M ) = − ¥ otherwise. The (relative) regularity of M over R isgiven by reg R ( M ) = sup { t Ri ( M ) − i : i ∈ N } and the Castelnuovo-Mumford regularity of M isreg ( M ) = reg S ( M ) = sup { t Si ( M ) − i : i ∈ N } ;it has also the cohomological interpretationreg ( M ) = max { j : H i m ( M ) j − i = i ∈ N } where H i m ( M ) denotes the i -th local cohomology module of M . One defines the slope of M over R by slope R ( M ) = sup { t Ri ( M ) − t R ( M ) i : i ∈ N , i > } , and the Backelin rate of R byRate ( R ) = slope R ( m ) = sup { t Ri ( K ) − i − i ∈ N , i > } . The Backelin rate measures the deviation from being Koszul: in general, Rate ( R ) ≥ R is Koszul if and only if Rate ( R ) =
1. Finally, the
Green-Lazarsfeld index of R isgiven by index ( R ) = sup { p ∈ N : t Si ( R ) ≤ i + i ≤ p } . It is the largest non-negative integer p such that R satisfies the property N p . Note thatwe have index ( R ) = ¥ if and only if reg ( R ) ≤
1, that is, the defining ideal of R hasa 2-linear resolution. On the other hand, index ( R ) ≥ R is defined byquadrics. In general, reg ( M ) and slope R ( M ) are finite (see [4]) while reg R ( M ) can beinfinite. However, reg R ( M ) is finite if R is Koszul, see Avramov and Eisenbud [5]. Remark 2.1.
The invariants reg ( M ) and index ( R ) are defined in terms of a presenta-tion of R as a quotient of a polynomial ring but do not depend on it. The assertion is aconsequence of the following formula which is obtained, for example, from the gradedanalogue of [3, Theorem 2.2.3]: if x ∈ R and xM =
0, then b Ri , j ( M ) = b R / ( x ) i , j ( M ) + b R / ( x ) i − , j − ( M ) . We record basic properties of these invariants. The modules are graded and finitelygenerated and the homomorphisms are of degree 0.
Lemma 2.2.
Let R be a standard graded K-algebra, N and M j , j ∈ N , be R-modules andi ∈ N . WINFRIED BRUNS, ALDO CONCA, AND TIM R ¨OMER (a)
Let → M → M → M → be an exact sequence. Thent Ri ( M ) ≤ max { t Ri ( M ) , t Ri + ( M ) } , t Ri ( M ) ≤ max { t Ri ( M ) , t Ri ( M ) } , t Ri ( M ) ≤ max { t Ri ( M ) , t Ri − ( M ) } . (b) Let · · · → M k + → M k → M k − → · · · → M → M → N → be an exact complex. Thent Ri ( N ) ≤ max { t Ri − j ( M j ) : j = , . . . , i } and reg R ( N ) ≤ sup { reg R ( M j ) − j : j ≥ } . (c) If N vanishes in degree > a then t Ri ( N ) ≤ t Ri ( K ) + a. (d) Let J be a homogeneous ideal of R. If reg R ( R / J ) = , then index ( R / J ) ≥ index ( R ) . Proof.
To prove (a) one just considers the long exact homology sequence for Tor R ( · , K ) .For (b) one uses induction on i and applies (a). Part (c) is proved by induction on a − min { j : N j = } . For (d) one applies (c) to the minimal free resolution of R / J asan R -module. For every i one gets t Si ( R / J ) ≤ max { t Si − j ( R ( − j )) : j = , . . . i } . But wehave t Si − j ( R ( − j )) = t Si − j ( R ) + j . If i ≤ index ( R ) then t Si − j ( R ) ≤ i − j +
1. It follows that t Si ( R / J ) ≤ i + i ≤ index ( R ) . Hence index ( R / J ) ≥ index ( R ) . (cid:3) Let M be an R -module and let K ( m c , M ) be the graded Koszul complex associated tothe c -th power of the maximal ideal of R . Let Z i ( m c , M ) , B i ( m c , M ) , H i ( m c , M ) denote the i -th cycles, boundaries and homology of K ( m c , M ) , respectively. We have: Lemma 2.3.
Set Z i = Z i ( m c , M ) . For every a ≥ and i ≥ we have:t Ra ( Z i + ) ≤ max (cid:8) t Ra ( M ) + ( i + ) c , t Ra + ( Z i ) , t R ( Z i ) + c + ( a + ) Rate ( R ) (cid:9) . Proof.
Set B i = B i ( m c , M ) and H i = H i ( m c , M ) . Recall that m c H i = H i van-ishes in degrees > t ( Z i ) + c −
1. It follows from Lemma 2.2(c) that t Ra ( H i ) ≤ t R ( Z i ) + c − + t Ra ( K ) . The short exact sequences 0 → B i → Z i → H i → → Z i + → K i + → B i → OSZUL HOMOLOGY AND SYZYGIES OF VERONESE SUBALGEBRAS 5 now imply that t Ra ( Z i + ) ≤ max { t Ra ( M ) + ( i + ) c , t Ra + ( B i ) }≤ max { t Ra ( M ) + ( i + ) c , t Ra + ( Z i ) , t Ra + ( H i ) }≤ max { t Ra ( M ) + ( i + ) c , t Ra + ( Z i ) , t R ( Z i ) + c − + t Ra + ( K ) } . Since, by the very definition, t Ra + ( K ) ≤ + ( a + ) Rate ( R ) the desired result follows. (cid:3) Lemma 2.3 allows us to bound t Ra ( Z i ) inductively in terms of the various t Rj ( M ) and ofRate ( R ) : Proposition 2.4.
Set Z i = Z i ( m c , M ) . (a) Assume c ≥ slope R ( M ) . Then for all a , i ∈ N we havet Ra ( Z i ) ≤ t R ( M ) + ic + max { a slope R ( M ) , ( a + i ) Rate ( R ) } . In particular, slope R ( Z i ) ≤ max { slope R M , ( + i ) Rate ( R ) } . (b) Assume R is Koszul, i.e.,
Rate ( R ) = . Then for all a , i ∈ N we havet Ra ( Z i ) ≤ a + i ( c + ) + reg R ( M ) . In particular, reg R ( Z i ) ≤ i ( c + ) + reg R ( M ) .Proof. To show (a) one uses that t Ra ( M ) ≤ t R ( M ) + a slope R ( M ) in combination withLemma 2.3 and induction on i . For (b) one observes that t Ra ( M ) ≤ a + reg R ( M ) in combi-nation with Lemma 2.3 and induction on i . (cid:3) In particular, we have the following corollary that generalizes Green’s theorem [17,Theorem 2.2] to arbitrary standard graded K -algebras. Corollary 2.5.
Set Z i = Z i ( m c , R ) . Then: (a) t R ( Z i ) ≤ ic + min { i Rate ( R ) , i + reg ( R ) } . (b) H i ( m c , R ) ic + j = for every j ≥ min { i Rate ( R ) , i + reg ( R ) } + c.Proof. To prove (a) one notes that setting M = R and a = t R ( Z i ) ≤ ic + i Rate ( R ) . Then one considers R as an S -module and sets M = R and a = t S ( Z i ) ≤ i ( c + ) + reg ( R ) . Since t S ( Z i ) = t R ( Z i ) we aredone. To prove (b) one uses (a) and the fact that m c H i ( m c , R ) = (cid:3)
3. K
OSZUL CYCLES
In this section we concentrate our attention on the Koszul complex K ( m c ) = K ( m c , S ) where S = K [ X , . . . , X n ] is a standard graded polynomial ring over a field K and m =( X , . . . , X n ) is its maximal homogeneous ideal. The Koszul complex K ( m c ) is indeed an S -algebra, namely the exterior algebra S ⊗ K V . S c ∼ = V . F where F is the free S -moduleof rank equal to dim K S c = (cid:0) n − + cn − (cid:1) . The differential of K ( m c ) is denoted by ¶ ; it is anantiderivation of degree −
1. We consider the cycles Z t ( m c , S ) , simply denoted by Z t ( m c ) ,of the Koszul complex K ( m c ) , and the S -subalgebra Z ( m c ) = L t Z t ( m c ) of K ( m c ) .For f , . . . , f t ∈ S c and g ∈ S we set g [ f , . . . , f t ] = g ⊗ f ∧ · · · ∧ f t ∈ K t ( m c ) . WINFRIED BRUNS, ALDO CONCA, AND TIM R ¨OMER
The elements [ u , . . . , u t ] for distinct monomials u , u , . . . , u t of degree c (ordered in someway) form a basis of K t ( m c ) as an S -module. We call them monomial free generators of K t ( m c ) . The elements v [ u , . . . , u t ] , where u , u , . . . , u t are distinct monomials of degree c and v is a monomial of arbitrary degree, form a basis of the K -vector space K t ( m c ) .They are called monomials of K t ( m c ) . Evidently K ( m c ) is a Z -graded complex, but itis also Z n -graded with the following assignment of degrees: deg v [ u , . . . , u t ] = a where vu · · · u t = X a .Every element z ∈ K t ( m c ) can be written uniquely as a linear combination z = (cid:229) f i [ u i , . . . , u it ] of monomial free generators [ u i , . . . , u it ] with coefficients f i ∈ S . We call f i the coef-ficient of [ u i , . . . , u it ] in z . Note that z is Z -homogenous of degree ct + j if every f i ishomogeneous of degree j . In this case z has coefficients of degree j . Note also that z ishomogeneous of degree a ∈ Z n in the Z n -grading if for every i one has f i = l i v i suchthat l i ∈ K and v i is a monomial with v i u i · · · u it = X a . Given z ∈ K ( m c ) and a mono-mial v [ u , . . . , u t ] we say that v [ u , . . . , u t ] appears in z if it has a non-zero coefficient inthe representation of z as K -linear combination of monomials of K ( m c ) . An immediateconsequence of Proposition 2.4 is: Lemma 3.1.
We have reg ( Z t ( m c )) ≤ t ( c + ) . In particular, Z t ( m c ) is generated by ele-ments of degree ≤ t ( c + ) . Remark 3.2.
It is easy to see and well known that Z ( m c ) is generated by the elements X i [ X j b ] − X j [ X i b ] where b is a monomial of degree c − Z ( m c ) t for V t Z ( m c ) ⊂ Z t ( m c ) , and similarly for other products. Theorem 3.3.
For every t the module Z t ( m c ) / Z ( m c ) t is generated in degree < t ( c + ) .Proof. The assertion is proved by induction on t . For t = Z t ( m c ) / Z ( m c ) Z t − ( m c ) is generated in degree < t ( c + ) . Since Z t ( m c ) is Z n -graded and generated in degree ≤ t ( c + ) , it suffices to showthat every Z n -graded element f ∈ Z t ( m c ) of total degree t ( c + ) can be written modulo Z ( m c ) Z t − ( m c ) as a multiple of an element in Z t ( m c ) of total degree < t ( c + ) . Let a ∈ Z n be the Z n -degree of f . Permuting the coordinates if necessary, we may assume a n > u ∈ S be a monomial of degree c with X n | u . We write f = a + b [ u ] with a ∈ K t ( m c ) and b ∈ K t − ( m c ) such that a , b involve only free generators [ u , . . . , u s ] ( s = t , t −
1) with u i = u for all i . Since 0 = ¶ ( f ) = ¶ ( a ) + ¶ ( b )[ u ] ± bu it follows that ¶ ( b ) =
0, i.e., b ∈ Z t − ( m c ) . Note that b has coefficients of degree t . Since Z t − ( m c ) is generated by elements with coefficients of degree ≤ t − b = s (cid:229) j = l j v j z j where l j ∈ K , z j ∈ Z t − ( m c ) and the v j are monomials of positive degree. OSZUL HOMOLOGY AND SYZYGIES OF VERONESE SUBALGEBRAS 7
Let l j v j z j be a summand in (2). If X n does not divide v j , then choose i < n such that X i | v j . We set z ′ = X i [ u ] − X n [ u ′ ] ∈ Z ( m c ) where u ′ = uX i / X n , and subtract from f theelement l j v j X i z j z ′ ∈ Z t − ( m c ) Z ( m c ) . Repeating this procedure for each l j v j z j in (2) such that X n does not divide v j we obtaina cycle f ∈ Z t ( m c ) of degree a such that(i) f = f mod Z ( m c ) Z t − ( m c ) ;(ii) if a monomial v [ u , . . . , u t ] appears in f and u ∈ { u , . . . , u t } , then X n | v .We repeat the described procedure for each monomial u of degree c with X n | u . We end upwith an element f ∈ Z t ( m c ) of degree a such that(iii) f = f mod Z ( m c ) Z t − ( m c ) ;(iv) if a monomial v [ u , . . . , u t ] appears in f and X n | u · · · u t , then X n | v .Note that if v [ u , . . . , u t ] appears in f and X n ∤ u · · · u t , then X n | v by degree reasons.Hence for every monomial v [ u , . . . , u t ] appearing in f we have X n | v . Therefore f = X n g ,and g ∈ Z t ( m c ) has degree < t ( c + ) . This completes the proof. (cid:3) Next we describe some cycles which are needed in the following. For t ∈ N , t ≥ S t be the group of permutations of { , . . . , t } . Lemma 3.4.
Let s , t be integers such that ≤ s ≤ c and t > . Let a , a . . . , a t + ∈ S bemonomials of degree s and b , b . . . , b t ∈ S monomials of degree c − s. Then (3) (cid:229) s ∈ S t + ( − ) s a s ( t + ) [ b a s ( ) , b a s ( ) , . . . , b t a s ( t ) ] belongs to Z t ( m c ) .Proof. We apply the differential of K ( m c ) to (3) and observe that for distinct integers j , j , . . . , j i − , j i + , . . . , j t in the range of 1 to t + [ b a j , b a j , . . . , b i − a j i − , b i + a j i + . . . , b t a i t ] appears twice in the image. The coefficients differ just by − (cid:3) Remark 3.5. (a) Of course, it may happen that a cycle described in Lemma 3.4 is identically 0. Butfor t = s = X i [ bX j ] − X j [ bX i ] , and, as said already in Remark 3.2, they generate Z ( m c ) . For s = c the cycles inLemma 3.4 are the boundaries of K t ( m c ) (multiplied by t !). Hence for c = Z ( m ) . So there is some evidence that the cyclesin Lemma 3.4 might generate Z ( m c ) in general. WINFRIED BRUNS, ALDO CONCA, AND TIM R ¨OMER (b) For n = , c = , t = s = a i = X i for i = , , b i = X i for i = , + X [ X , X ] − X [ X , X X ] − z }| { X [ X X , X X ]+ X [ X X , X X ] + X [ X X , X X ] − X [ X X , X ] , a non-zero element in Z ( m ) .Let B i ( m c ) ⊂ Z i ( m c ) denote the S -module of boundaries in K i ( m c ) . Theorem 3.6.
We have ( c + ) ! m c − Z ( m c ) c ⊂ B c ( m c ) . Proof.
For a monomial b ∈ S of degree c − X i , X j we set z b ( X i , X j ) = X i [ bX j ] − X j [ bX i ] . As observed in Remark 3.2, the elements z b ( X i , X j ) generate Z ( m c ) . Let a , b ∈ S bemonomials of degree c −
1. We note that az b ( X i , X j ) + bz a ( X i , X j ) = ¶ (cid:0) [ aX i , bX j ] + [ bX i , aX j ] (cid:1) ∈ B ( m c ) , that is,(4) az b ( X i , X j ) = − bz a ( X i , X j ) mod B ( m c ) . Let b , . . . , b c + ∈ S be monomials of degree c −
1, and let X i j ∈ { X , . . . , X n } for i = , . . . , c and j = , f = b c + c (cid:213) i = z b i ( X i , X i ) ∈ Z c ( m c ) generate m c − Z ( m c ) c . We have to show that ( c + ) ! f ∈ B c ( m c ) .Let s ∈ S c + be an arbitrary permutation. From Equation (4) and from the inclusion B ( m c ) Z c − ( m c ) ⊂ B c ( m c ) it follows that f = ( − ) s b s ( c + ) c (cid:213) i = z b s ( i ) ( X i , X i ) mod B c ( m c ) . Hence(5) ( c + ) ! f = (cid:229) s ∈ S c + ( − ) s b s ( c + ) c (cid:213) i = z b s ( i ) ( X i , X i ) mod B c ( m c ) . In the right-hand side of (5) we replace z b s ( i ) ( X i , X i ) with X i [ b s ( i ) X i ] − X i [ b s ( i ) X i ] ,then expand the product and collect the multiples of X j · · · X c j c for j = ( j , . . . , j c ) ∈{ , } c . We can rewrite Equation (5) as(6) ( c + ) ! f = (cid:229) j ∈{ , } c ( − ) j + ··· + j c X j · · · X c j c W j mod B c ( m c ) , where W j = (cid:229) s ∈ S c + ( − ) s b s ( c + ) [ X i b s ( ) , . . . , X ci c b s ( c ) ] OSZUL HOMOLOGY AND SYZYGIES OF VERONESE SUBALGEBRAS 9 with i k = − j k for k = , . . . , c . Lemma 3.4 yields W j ∈ Z c ( m c ) . Since m c Z c ( m c ) ⊂ B c ( m c ) we get X j · · · X c j c W j = B c ( m c ) . Thus Equation (6) implies ( c + ) ! f ∈ B c ( m c ) as desired. (cid:3) As a consequence we obtain:
Corollary 3.7.
The homology H t ( m c ) tc + j vanishes if j ≥ t + c. If t ≥ c and the charac-teristic of K is either or > c + , then H t ( m c ) tc + j = for j ≥ t + c − .Proof. The first statement is a special case of Corollary 2.5. For the second, set j = t + c −
1. We have to prove that H t ( m c ) tc + j =
0. Theorem 3.3 implies that Z t ( m c ) isgenerated by some elements z i of degree < t ( c + ) and by some elements w i of Z ( m c ) t of degree t ( c + ) . Hence an element f ∈ Z t ( m c ) tc + j can be written as f = (cid:229) a i z i + (cid:229) b i w i with a i ∈ m c and b i ∈ m c − by degree reasons. Now (cid:229) a i z i ∈ m c Z t ( m c ) ⊂ B t ( m c ) . In viewof Theorem 3.6 we furthermore have (cid:229) b i w i ∈ m c − Z ( m c ) t = m c − Z ( m c ) c Z ( m c ) t − c ⊂ B c ( m c ) Z ( m c ) t − c ⊂ B t ( m c ) . Summing up, f ∈ B t ( m c ) and hence H t ( m c ) tc + j = (cid:3) Remark 3.8.
The coefficient ( c + ) ! in Theorem 3.6 and the assumption on the character-istic in Corollary 3.7 are necessary. For n = , c = K = m Z ( m ) B ( m ) and that dim H ( m ) =
1. More precisely, H ( m ) has dimension 1in the multidegree ( , , , , , , ) if char K = Corollary 3.9.
Assume char
K is or > c + . Then reg Z t + ( m c ) ≤ ( t + )( c + ) − forevery t ≥ c. In particular, Z ( m c ) c + ⊂ m Z c + ( m c ) .Proof. To prove the first assertion, let us denote by Z t the module Z t ( m c ) and similarlyfor B t , H t and K t . The short exact sequences0 → B t → Z t → H t → → Z t + → K t + → B t → ( Z t + ) ≤ max { reg ( Z t ) + , reg ( H t ) + } . Using Lemma 3.1 and Corollary3.7 one obtains reg ( Z t + ) ≤ ( t + )( c + ) − t ≥ c . The second assertionfollows immediately from the first. (cid:3)
4. T HE G REEN -L AZARSFELD INDEX OF V ERONESE SUBRINGS OF POLYNOMIALRINGS
Again we consider a standard graded K -algebra R of the form R = S / I where K is afield, S = K [ X , . . . , X n ] is a polynomial ring over K graded by deg ( X i ) = I ⊂ S is agraded ideal.Given c ∈ N , c ≥ ≤ k < c , we set V R ( c , k ) = M i ∈ N R k + ic . Observe that R ( c ) = V R ( c , ) is the usual c -th Veronese subring of R , and that the V R ( c , k ) are R ( c ) -modules known as the Veronese modules of R . For a finitely generated graded R -module M we similarly define M ( c ) = M i ∈ Z M ic . We consider R ( c ) as a standard graded K -algebra with homogeneous component of degree i equal to R ic , and M ( c ) as a graded R ( c ) -module with homogeneous components M ic . Thegrading of the R ( c ) -module V R ( c , k ) is given by V R ( c , k ) i = R k + ic . In particular, the lattermodules are all generated in degree 0 with respect to this grading.Let T = Sym ( R c ) be the symmetric algebra on vector space R c , that is, T = K [ Y u : u ∈ B c ] where B c is any K -basis of R c . When R = S the basis B c can be taken as the set ofmonomials of degree c . The canonical map T → R ( c ) is surjective, and the modules V R ( c , k ) are also finitely generated graded T -modules (generated in degree 0).With the notation of the preceding sections we have: Lemma 4.1.
For i ∈ N , j ∈ Z and ≤ k < c we have b Ti , j ( V R ( c , k )) = dim K H i ( m c , R ) jc + k . Proof.
Let K ( T ) be the Koszul complex (of T -modules) associated to the elements Y u with u ∈ B c . We observe that b Ti , j ( V R ( c , k )) = Tor Ti ( K , V R ( c , k )) j = dim K H i ( K ( T )) ⊗ T V R ( c , k )) j . But the last homology is H i ( m c ) jc + k , the i -th homology of the complex K ( m c ) jc + k . (cid:3) Lemma 4.1 and Corollary 3.7 imply:
Corollary 4.2.
For all integers i ≥ and k = , . . . , c − we havet Ti ( V S ( c , k )) < + i + i − kc . If K has characteristic or > c + and i ≥ c, thent Ti ( V S ( c , k )) < + i + i − k − c . Remark 4.3.
Let S = K [ X , . . . , X n ] . Andersen [1] proved that the graded Betti numbers b Ti j ( S ( ) ) do not depend on the characteristic of K if i ≤ i = n ≤
6. She alsoproved that, for n ≥
7, one has b T , ( S ( ) ) = b T , ( S ( ) ) = n ≥ ( S ( ) ) = ( , char K = , , char K = . Also note that already b , ( V S ( , )) depends on the characteristic if n ≥
7, as followsfrom the data in Remark 3.8.
OSZUL HOMOLOGY AND SYZYGIES OF VERONESE SUBALGEBRAS 11
We now record a duality on H ( m c ) . It can be seen as an Avramov-Golod type duality(see [8, Theorem 3.4.5]) or as an algebraic version of Serre duality. Proposition 4.4.
Let N = (cid:0) n + c − c (cid:1) . Then dim K H i ( m c ) j = dim K H N − n − i ( m c ) Nc − n − j , i , j ∈ Z , i , j ≥ . Proof.
For this proof (and only here) we consider the grading on the polynomial ring T = K [ Y u : u ∈ S monomial, deg u = c ] in which Y u has degree c . The polynomial ring S in its standard grading is a finitely generated graded T -module as usual.Note that the canonical module of S is w S = S ( − n ) , and that the canonical module of T is w T = T ( − Nc ) . Recall thatExt jT ( S , T ( − Nc )) = ( j < N − n , S ( − n ) if j = N − n . (See, e.g., [8, Theorem 3.3.7 and Theorem 3.3.10].) Let F be a minimal graded free T -resolution of S . Computing Ext iT ( S , T ( − Nc )) via Hom T ( F , T ( − Nc )) , the minimal gradedfree T -resolution of S ( − n ) , we see immediately that b Ti , j ( S ) = b TN − n − i , Nc − j ( S ( − n )) . Thendim K H i ( m c ) j = b Ti , j ( S ) = b TN − n − i , Nc − n − j ( S ) = dim K H N − n − i ( m c ) Nc − n − j . (cid:3) Example 4.5.
Let char K =
0. Computer algebra systems as CoCoA [10], Macaulay 2[16] or Singular [20] can easily compute the following diagram for dim K H ( m ) in thecase n =
3: 0 1 2 3 4 5 6 70 1 - - - - - - - ←
27 105 189 189 105 27 - ←
21 105 147 105 49 65 - - ← The ( i , j ) -entry of the table is dim K H i ( m c ) ic + j and - indicates that entry is 0. By selectingthe rows whose indices are multiples of c = S ( ) . Green’s theorem [17, Thm.2.2] implies the vanishingin the positions of the boldface zeros and below. Our result implies the vanishing in thepositions of the non-bold zeros and below. (Also see Weyman [29, Example 7.2.7] for thecase n = c = ( S ( c ) ) due to Ottaviani and Paoletti[23] (in arbitrary characteristic). To this end we need a variation of [25, Corollary 2.10]. Proposition 4.6.
Let ( e i : i = , . . . , m ) be a basis of the vector space V t S c (thus m = (cid:0) Nt (cid:1) with N = (cid:0) n − + cn − (cid:1) ). Let z = m (cid:229) i = f i e i be a non-zero element in Z t ( m c ) . Then the K-vector space generated by the coefficients f i of z has dimension ≥ t + . Proof.
Since the K -vector space dimension of the space of coefficients does not dependon the basis, it is enough to prove the assertion for the monomial basis ( e i ) . We useinduction on t .The case t = t >
0. Fix a term order, for example the lexico-graphic term order, on S . Let C ( z ) denote the vector space generated by the coefficientsof z . As already discussed in the proof of Theorem 3.3, for every monomial u of degree c we may write z = a + b [ u ] with b ∈ Z t − ( m c ) . Choose u to be the largest monomial(with respect to the term order) such that the corresponding b is non-zero. By inductiondim K C ( b ) ≥ t and C ( b ) ⊂ C ( z ) .If C ( b ) = C ( z ) then clearly dim K C ( z ) ≥ t +
1. If instead C ( b ) = C ( z ) , then C ( a ) ⊂ C ( b ) . Let v be the largest monomial appearing in the elements of C ( b ) . The inclusion C ( a ) ⊂ C ( b ) implies that every monomial appearing in the elements of C ( a ) is ≤ v . But ¶ ( a ) ± bu = C ( ¶ ( a )) = C ( bu ) = uC ( b ) . The monomial vu appears in C ( bu ) .Every monomial in C ( ¶ ( a )) is of the form wu where w is a monomial appearing in C ( a ) and u is a monomial of degree c which is an “exterior” factor of some free generatorappearing in z . By construction w ≤ v and u < u . It follows that wu = vu , a contradictionwith C ( ¶ ( a )) = uC ( b ) . (cid:3) Theorem 4.7.
For n ≥ and c ≥ one has index ( S ( c ) ) ≤ c − , and equality holds forn = .Proof. We first consider the case n =
3. By an inspection of the Hilbert function of (theCohen-Macaulay ring) S ( c ) one sees immediately that reg S ( c ) ≤
2, that is, t Ti ( S ( c ) ) ≤ i + i ≥
0. From Theorem 4.1 and Proposition 4.4 we have b Ti , j ( S ( c ) ) = dim K H i ( m c ) jc = dim K H N − − i ( m c ) Nc − − jc . Therefore t Ti ( S ( c ) ) ≤ i + H N − − i ( m c ) ( N − − i ) c + c − = , and, since the boundaries have coefficients of degree ≥ c , this is equivalent to Z N − − i ( m c ) ( N − − i ) c + c − = . So we have to analyze the cycles in Z N − − i ( m c ) with coefficients of degree c − N − − i + ≤ dim K S c − = (cid:18) c − (cid:19) if there exists a non-zero cycle z ∈ Z N − − i ( m c ) with coefficients of degree c −
3. Thusthere are no cycles in that degree if N − − i ≥ (cid:0) c − (cid:1) . Hence t Ti ( S ( c ) ) ≤ i + ≤ i ≤ c − , that is, index ( S ( c ) ) ≥ c −
3. It remains to show that S ( c ) does not satisfy the property N c − . We have to find a non-zero cycle in Z j ( m c ) with coefficients of degree c − j = N − − i . Note that j = (cid:0) c − (cid:1) − j + = dim S c − . Take the monomials u ′ , . . . , u ′ j + of degree c − u k = u ′ k X X X for k = , . . . , j +
1. Then w = ¶ ([ u , . . . , u j + ]) ∈ Z j ( m c ) OSZUL HOMOLOGY AND SYZYGIES OF VERONESE SUBALGEBRAS 13 is non-zero boundary with coefficients of degree c . But we can divide each coefficient of w by X X X to obtain a non-zero cycle z ∈ Z j ( m c ) with coefficients of degree c −
3. Itfollows that S ( c ) does not satisfy the property N c − . This concludes the proof for n = n >
3. Recall that H i ( m c ) is multigraded. For a vector a = ( a , . . . , a n ) ∈ N n with a i = i > b = ( a , a , a ) . We may identify H i ( m c ) a = H i ( m c ) b where H i ( m c ) is the corresponding Koszul homology in 3 variables. Since for n = c -th Veronese does not satisfy N c − it follows that the same is true for all n ≥
3, provingthat index ( S ( c ) ) ≤ c − (cid:3) Remark 4.8.
It is well-known that reg S ( c ) ≤ n − t i ( S ( c ) ) ≤ i + n −
1. Analogously to the proof of Theorem 4.7 one can determine the largest i such that t i ( S ( c ) ) < i + n −
1. Again this is determined by elements in Z i ( m c ) with coefficients ofdegree c − n . It remains to count the monomials of S in that degree. For example, for c ≥ n = t i ( S ( c ) ) < i + i ≤ c − HE G REEN -L AZARSFELD INDEX OF V ERONESE SUBRINGS OF STANDARDGRADED RINGS
Let D be a Koszul K -algebra and I be a homogeneous ideal of D . Set R = D / I . Wewant to relate the Green-Lazarsfeld index of R ( c ) to that of D ( c ) . For a polynomial ring S Aramova, Bˇarcˇanescu and Herzog proved in [2, Theorem 2.1] that the Veronese modules V S ( c , k ) have a linear resolution over the Veronese ring S ( c ) . We show that this propertyholds for Koszul algebras in general. Lemma 5.1.
Assume D is a Koszul algebra and, for a given c, let T = Sym ( D c ) be thesymmetric algebra of D c . (a) The Veronese module V D ( c , k ) has a linear resolution as a D ( c ) -module. (b) For every k = , . . . , c − we havet Ti ( V D ( c , k )) ≤ t Ti ( D ( c ) ) . Proof. (a) Let m denote the homogeneous maximal ideal of D , and set A = D ( c ) and V k = V D ( c , k ) . We prove by induction on i that t Ai ( V k ) ≤ i for all i and k . For i = k = i ≥
0, too. Assume that i >
0. The ideal m k is generated in degree k and, since D is Koszul, it has a linear resolution over D . Shiftingthat resolution by k , we obtain a complex · · · → F i → F i − → · · · → F → F → m k ( k ) and such that F i = D ( − i ) b i . Applying the exact functor ( ) ( c ) to it we getan exact complex of A -modules · · · → F ( c ) i → F ( c ) i − → · · · → F ( c ) → A b → V k → . Note that D ( − j ) ( c ) = V e ( −⌈ j / c ⌉ ) where e = c ⌈ j / c ⌉ − j . Therefore F ( c ) j = V e j ( −⌈ j / c ⌉ ) b j where e j = c ⌈ j / c ⌉ − j . Applying Lemma 2.2 (b) to the complex above we have t Ai ( V k ) ≤ max { t Ai − j ( V e j ) + ⌈ j / c ⌉ : j = , . . . , i } . Obviously t Ai ( V e ) = t Ai ( A ) = − ¥ and, by induction, t Ai − j ( V e j ) ≤ i − j for j = , . . . , i .Therefore t Ai ( V k ) ≤ max { i − j + ⌈ j / c ⌉ : j = , . . . , i } = i and this concludes the proof of (a). For (b) we may apply Lemma 2.2(b) to the minimal A -free resolution of V k and to get the desired inequality. (cid:3) Now we prove the main result of this section.
Theorem 5.2.
Assume D is a Koszul algebra and R = D / I. Let c ≥ slope D ( R ) . Then index ( R ( c ) ) ≥ index ( D ( c ) ) .Proof. To prove the statement we set A = D ( c ) and B = R ( c ) . By virtue of Lemma 2.2 (d)it is enough to show that reg A ( B ) =
0. Let · · · → F p → · · · → F → F → R → R over D . Since ( ) ( c ) is an exact functor, weobtain an exact complex of finitely generated graded A -modules(7) · · · → F ( c ) p → · · · → F ( c ) → F ( c ) → B → . Hence by virtue of Lemma 2.2 (b) we havereg A ( B ) ≤ max { reg A ( F ( c ) i ) − i : i ≥ } . Note that D ( − k ) ( c ) = V D ( c , e )( −⌈ k / c ⌉ ) where e = c ⌈ k / c ⌉ − k . Hence, by virtue of Lemma5.1, reg A ( D ( − k ) ( c ) ) = ⌈ k / c ⌉ . Therefore, since F i = L k ∈ Z S ( − k ) b Dik ( R ) we get reg A ( F ( c ) i ) = ⌈ t Di ( R ) / c ⌉ . Summing up,reg A ( B ) ≤ max {⌈ t Di ( R ) / c ⌉ − i : i ≥ } . If c ≥ slope D ( R ) , then t Di ( R ) ≤ ci and hence reg A ( B ) =
0. This concludes the proof. (cid:3)
As a corollary we have:
Corollary 5.3.
Let S be a polynomial ring and R = S / I a standard graded algebra quo-tient of it and let c ≥ slope S ( R ) . Then index ( R ( c ) ) ≥ index ( S ( c ) ) . In particular, (a) index ( R ( c ) ) ≥ c. Furthermore, if K has characteristic or > c + , then we have index ( R ( c ) ) ≥ c + . (b) If dim R = , then index ( R ( c ) ) ≥ c − . Note that slope S ( R ) = R is Koszul; see [4]. Furthermore slope S ( R ) ≤ a if R isdefined by either a complete intersection of elements of degree ≤ a or by a Gr¨obner basisof elements of degree ≤ a . Remark 5.4.
Sometimes the bound in Theorem 5.2 can be improved by a more carefulargumentation. Let R = S / I and let T be the symmetric algebra of S c . For instance, usingthe argument of the proof of Theorem 5.2 one shows that t Ti ( R ( c ) ) ≤ max { t Ti − j ( S ( c ) ) + ⌈ t Sj ( R ) / c ⌉} : j = , , . . . , i } . OSZUL HOMOLOGY AND SYZYGIES OF VERONESE SUBALGEBRAS 15
It follows that index ( R ( c ) ) ≥ p if c ≥ p and index ( R ) ≥ p , a result proved by Rubei in[26]. It is very easy to show that R ( c ) is defined by quadrics, i.e. index ( R ( c ) ) ≥ c ≥ t S ( R ) /
2. Similarly, one can prove that index ( R ( c ) ) ≥ p if c ≥ max { p , max { t Sj ( R ) / j : j = , . . . , p − } , t Sp ( R ) / ( p + ) } . Remark 5.5. (a) Let us say that a positively graded K -algebra is almost standard if R is Noetherianand a finitely generated module over K [ R ] . If K is infinite, then this propertyis equivalent to the existence of a Noether normalization generated by elementsof degree 1. Galliego and Purnaprajna [15, Theorem 1.3] proved a general resulton the property N p of Veronese subalgebras of almost standard K -algebras R ofdepth ≥ R ( c ) has N p for all c ≥ max { reg ( R ) + p − , reg ( R ) , p } . If reg ( R ) ≥ p ≥
1, this amounts to the property N c − reg ( R )+ of R ( c ) for all c ≥ reg ( R ) . Thus Theorem 5.2 gives a stronger result for standardgraded algebras.(b) Eisenbud, Reeves and Totaro [12] proved that the Veronese subalgebras R ( c ) ofstandard graded K -algebra R are defined by an ideals with Gr¨obner bases of degree2 for all c ≥ ( reg ( R ) + ) /
2. It follows that these algebras are Koszul.(c) If R is almost standard and Cohen-Macaulay, then R ( c ) is defined by an ideal with aGr¨obner bases of degree 2 for every c ≥ reg ( R ) . See Bruns, Gubeladze and Trung[7, Theorem 1.4.1] or Bruns and Gubeladze [6, Theorem 7.41].6. T HE MULTIGRADED CASE
The results presented in this paper have natural extensions to the multigraded case.Here we just formulate the main statements. Detailed proofs will be given in the forthcom-ing article [9]. Suppose S = K [ X ( ) , . . . , X ( m ) ] is a Z m -graded polynomial ring in whicheach X ( i ) is the set of variables of degree e i ∈ Z m . For a vector c ∈ ( c , . . . , c m ) ∈ N m + con-sider the c -th diagonal subring S ( c ) = ⊕ i ∈ N S ic , the coordinate ring of the correspondingSegre-Veronese embedding. The following result improves the bound of Hering, Schenckand Smith [21] by one: Theorem 6.1.
With the notation above one has: min ( c ) ≤ index ( S ( c ) ) . Moreover, we have min ( c ) + ≤ index ( S ( c ) ) if char K = or char K > + min ( c ) . Similarly one has the multigraded analog of Theorem 5.2. Here one uses the fact,proved in [11], given any Z m -graded standard graded algebra quotient of S then if the c j ’sare big enough (in terms of the multigraded Betti numbers of R over S ) then reg S ( c ) ( R ( c ) ) = Proposition 6.2.
Assume that for all j = . . . , m one has c j ≥ max { a j / i : i > , a ∈ Z m and b Si , a ( R ) = , } then index ( R ( c ) ) ≥ index ( S ( c ) ) . R EFERENCES [1] J. L. Andersen,
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