The weak Lefschetz property for monomial complete intersections in - PowerPoint PPT Presentation

The weak Lefschetz property for monomial complete intersections in positive characteristic Adela Vraciu University of South Carolina joint work with Andy Kustin

Definitions Let A = ⊕ A i be a standard graded algebra over an algebraically closed field k . We say that A has the weak Lefschetz property (WLP) if there exists a linear form L ∈ A 1 such that the map × L : A i → A i +1 has maximal rank (i.e. it is injective or surjective) for all i . Such a linear form L is called a Lefschetz element. The set of Lefschetz elements forms a (possibly empty) Zarisky subset of A 1 .

A key fact Migliore, Miro-Roig, and Nagel (2011) show that if A = k [ x 1 , . . . , x n ] /I is a standard graded Artinian Gorenstein algebra, then A has WLP if and only if L = x 1 + . . . + x n is a Lefschetz element, and this is if and only if the map × L : A ⌊ e − 1 2 ⌋ → A ⌊ e +1 2 ⌋ is injective, where e is the degree of the socle generator of A . In the case when I = ( x d 1 1 , . . . , x d n n ) , we translate this into a condition on the degrees of the non-Koszul relations on n − 1 , ( x 1 + . . . + x n − 1 ) d n ∈ k [ x 1 , . . . , x n − 1 ] : x d 1 1 , . . . , x d n − 1

Proposition A = k [ x 1 , . . . , x n ] has WLP if and only if ( x d 1 1 , . . . , x d n n ) the smallest total degree of a non-Koszul relation on n − 1 , ( x 1 + . . . + x n − 1 ) d n is ⌊ Σ n i =1 d i − n + 3 1 , . . . , x d n − 1 x d 1 ⌋ . 2 We use the convention that the total degree of a relation 1 + · · · + a n ( x 1 + . . . x n − 1 ) d n is deg( a 1 ) + d 1 . a 1 x d 1

The role of the characteristic Theorem [Stanley - J. Watanabe]: If char( k ) = 0 , then A = k [ x 1 , . . . , x n ] has WLP for every ( x d 1 1 , . . . , x d n n ) d 1 , . . . , d n ≥ 1 . THIS IS NO LONGER TRUE IN POSITIVE CHARACTERISTIC! Li-Zanello (2010) found a surprising connection between the monomial complete intersections in three variables that have WLP (as a function of the characteristic) and enumerations of plane partitions. Brenner-Kaid (2011) gave an explicit description of the values of d k [ x, y, z ] (in terms of p ) such that ( x d , y d , z d ) has WLP.

WLP in three variables and finite projective dimension Theorem [Kustin - Rahmati -V.] k [ x, y, z ] Let R = ( x n + y n + z n ) where n ≥ 2 , and let N ≥ n be an integer not divisible by n . Then R/ ( x N , y N , z N ) R has finite projective dimension as an R -module if and only if k [ x, y, z ] A = ( x a , y a , z a ) DOES NOT have WLP for at least one of the values a = ⌊ N n ⌋ or a = ⌈ N n ⌉ .

Our main results Theorem 1 [Kustin - V.] Assume char( k ) = p ≥ 3 , and let d = lp e + d ′ , where l ≤ p − 1 and d ′ < p e . ( x d , y d , z d , w d ) has WLP if and only if l ≤ p − 1 k [ x, y, z, w ] Then and 2 d ′ ∈ { p e − 1 , p e + 1 } . 2 2

Main results - continued Theorem 2 [Kustin - V.] Let n ≥ 5 and char( k ) = p > 0 . Then A = k [ x 1 , . . . , x n ] ( x d 1 , . . . , x d n ) has WLP if and only if ⌊ n ( d − 1) + 3 ⌋ ≤ p 2 In particular, A does not have WLP for any d ≥ p .

Necessary conditions for WLP We use the Frobenius endomorphism to create relations of small degree. The following is the key observation: Write d = lp e + d ′ and fix 1 ≤ i ≤ n . If n − 1 + b n ( x 1 + . . . + x n − 1 ) l +1 = 0 a 1 x l 1 + . . . + a i x l i + b i +1 x l +1 + . . . + b n − 1 x l +1 i i , x l +1 i +1 , . . . , x l +1 n − 1 ( x 1 + . . . + x n − 1 ) l +1 , then is a relation on x l 1 , . . . , x l ( x 1 · · · x i ) d ′ � n − 1 x ( l +1) p e n ( x 1 + . . . + x n − 1 ) ( l +1) p e � a p e 1 x lp e + . . . + b p e + b p e = 1 n − 1 is a relation on x d 1 , . . . , x d n − 1 , ( x 1 + . . . , x n − 1 ) d .

Necessary conditions - continued If l = ( l 1 , . . . , l n ) , we use N ( l ) to denote the smallest total degree of a non-Koszul relation on x l 1 1 , . . . , x l n − 1 n − 1 , ( x 1 + . . . + x n − 1 ) l n . We have shown that N ( d ) ≤ p e N ( l ) + id ′ where d = ( d, . . . , d ) , l = ( l, . . . , l, l + 1 , . . . , l + 1) ( i l ’s and n − i l + 1 ’s), and d = lp e + d ′ .

Necessary conditions - continued Lemma: Let l = ( l, . . . , l, l + 1 , . . . , l + 1) be as above. Then we have N ( l ) ≤ ⌊ nl − i + 3 ⌋ . 2 Sketch of Proof: k [ x 1 , . . . , x n ] Let A l = . The socle generator of A l i , x l +1 i +1 , . . . , x l +1 ( x l 1 , . . . , x l ) n has degree e = nl − i and the Hilbert function of A l becomes strictly decreasing after step ⌊ e + 1 ⌋ , which implies that the map 2 × L : [ A l ] ⌊ e +1 2 ⌋ → [ A l ] ⌊ e +3 2 ⌋ is not injective. This gives rise to a non-Koszul relation of degree ⌊ e + 3 ⌋ . 2

Necessary conditions - conclusion Corollary: Assume that d = lp e + d ′ (where p = char( k ) ) and 1 ≤ i ≤ n . If A = k [ x 1 , . . . , x n ] ( x d 1 , . . . , x d n ) has WLP, then ⌊ n ( d − 1) + 3 ⌋ ≤ N ( d ) ≤ p e ⌊ nl − i + 3 ⌋ + id ′ . 2 2

Sufficient conditions for WLP Lemma: Let c ≤ d . The ideal ( x d , y d ) : ( x + y ) 2 c in k [ x, y ] is generated in degrees ≥ d − c if and only if ∆ c ( d ) � = 0 in k , where � d � d � d � � � � � � � . . . � � 1 2 c � � d � d � � � � � � � d � � . . . � � 2 3 c + 1 ∆ c ( d ) = � � � . . . . � . . . . � � . . . . � � d � � � � � � � � d d � � . . . � � c c + 1 2 c − 1 � �

Proof of Lemma The statement is equivalent to ( x d , y d ) ∩ ( x + y ) 2 c has no non-zero elements of degree ≤ d − c − 1 if and only if ∆ c ( d ) � = 0 . Write a general element of ( x d , y d ) of degree d − c − 1 as a polynomial in the variables x and x + y : a 1 x c − 1 + a 2 x c − 2 ( x + y ) + . . . + a c ( x + y ) c − 1 � x d + � H = b 1 x c − 1 + b 2 x c − 2 ( x + y ) + . . . + b c ( x + y ) c − 1 � � y d Write y = ( x + y ) − x and use the binomial expansion for y d ; the condition that H ∈ ( x + y ) 2 c amounts to saying that the coefficients for ( x + y ) 2 c − 1 , ( x + y ) 2 c − 2 , . . . , ( x + y ) , 1 that are obtained when the expression for H is expanded are equal to zero. This gives rise to a homogeneous system of 2 c equations in the unknowns a 1 , . . . , a c , b 1 , . . . , b c . The first c equations tell us that the a i ’s can be expressed as linear combinations of the b i ’s. The last c equations (in the unknowns b 1 , . . . , b c ) have determinant ∆ c ( d ) , so ∆ c ( d ) � = 0 ⇔ there is no non-trivial solution.

Sufficient conditions for WLP in four variables Theorem: Assume that ∆ c ( d ) � = 0 in k for every c ∈ { 1 , . . . , d } . Then k [ x, y, z, w ] A = ( x d , y d , z d , w d ) has WLP. Sketch of Proof: WLP is equivalent to the statement that ( x d , y d , z d ) : ( x + y + z ) d has no non-zero elements of degree d − 2 . Let u = u d − 2 + u d − 3 z + . . . + u 0 z d − 2 ∈ ( x d , y d , z d ) : ( x + y + z ) d , with u i ∈ k [ x, y ] homogeneous of degree i . We want to show � d � u = 0 . Expand ( x + y + z ) d = Σ d ( x + y ) i z d − i and i =0 i multiply u ( x + y + z ) d ; the condition is that the coefficients of 1 , z, . . . , z d − 1 in the resulting expression are in ( x d , y d ) .

The coefficient of z d − 1 is � d � ( x + y ) 2 u d − 3 + . . . + d ( x + y ) d − 1 u 0 . d ( x + y ) u d − 2 + 2 Since it has degree d − 1 , the only way it can be in ( x d , y d ) is if it is zero; this implies u d − 2 ∈ ( x + y ) . Now the coefficient of z d − 2 � d � d � � ( x + y ) 2 u d − 2 + ( x + y ) 3 u d − 3 + . . . + u 0 ( x + y ) d 2 3 must be in ( x d , y d ) ∩ ( x + y ) 2 . According to the lemma, it has insufficient degree, so it must be zero; this implies u d − 2 ∈ ( x + y ) 2 , u d − 3 ∈ ( x + y ) , etc.

The determinants ∆ c ( d ) It is known that � d � � d + 1 � d + c − 1 � � · · · c c c ∆ c ( d ) = � c � � c + 1 � 2 c − 1 � � · · · c c c and d ′ ∈ { p e − 1 , p e + 1 If d = lp e + d ′ , with l ≤ p − 1 } , we show 2 2 2 that ∆ c ( d ) � = 0 in k (where char( k ) = p ) for all c = 1 , . . . , d by counting the powers of p in the numerator and in the denominator.