Local cohomology with support in determinantal ideals Claudiu Raicu - PowerPoint PPT Presentation

Local cohomology with support in determinantal ideals Claudiu Raicu ∗ and Jerzy Weyman Fort Collins, August 2013

Resolutions Example I 2 = 2 × 2 minors of a 3 × 3 matrix . S = Sym ( C 3 ⊗ C 3 ) . 1 . . . . β ( S / I 2 ) : . 9 16 9 . . . . . 1

Resolutions Example I 2 = 2 × 2 minors of a 3 × 3 matrix . S = Sym ( C 3 ⊗ C 3 ) . 1 . . . . β ( S / I 2 ) : . 9 16 9 . . . . . 1 More generally, I p = p × p minors of m × n matrix, S = Sym ( C m ⊗ C n ) . β ( S / I p ) : Lascoux, J´ ozefiak, Pragacz, Weyman ’80 . Feature: I p is a GL m × GL n –representation. Assume m ≥ n .

Resolutions Example I 2 = 2 × 2 minors of a 3 × 3 matrix . S = Sym ( C 3 ⊗ C 3 ) . 1 . . . . β ( S / I 2 ) : . 9 16 9 . . . . . 1 More generally, I p = p × p minors of m × n matrix, S = Sym ( C m ⊗ C n ) . β ( S / I p ) : Lascoux, J´ ozefiak, Pragacz, Weyman ’80 . Feature: I p is a GL m × GL n –representation. Assume m ≥ n . � S x C m ⊗ S x C n . S = Cauchy’s formula: x =( x 1 ≥ x 2 ≥···≥ x n ) � p � p � � C m ⊗ � S ( 1 p ) C m ⊗ S ( 1 p ) C n � C n I p = = .

The ideals I x � I x = ( S x C m ⊗ S x C n ) = S y C m ⊗ S y C n . x ⊂ y [De Concini–Eisenbud–Procesi ’80]

The ideals I x � I x = ( S x C m ⊗ S x C n ) = S y C m ⊗ S y C n . x ⊂ y [De Concini–Eisenbud–Procesi ’80] Problem Compute the resolution of all I x .

The ideals I x � I x = ( S x C m ⊗ S x C n ) = S y C m ⊗ S y C n . x ⊂ y [De Concini–Eisenbud–Procesi ’80] Problem Compute the resolution of all I x . Example m = n = 3, I x = I ( 2 , 2 ) . 1 . . . . . . . . . . . . . . . . . . . . β ( S / I 2 , 2 ) : . 36 90 84 36 9 1 . . . . . . . . . . . 1 . .

Regularity of the ideals I x Unfortunately, we don’t know how to compute β ( S / I x ) for arbitrary x ! Question What about the regularity? Effective bounds?

Regularity of the ideals I x Unfortunately, we don’t know how to compute β ( S / I x ) for arbitrary x ! Question What about the regularity? Effective bounds? Theorem (–, Weyman ’13) ( n · ( x p − p ) + p 2 + 2 · ( p − 1 ) · ( n − p )) . reg ( I x ) = max p = 1 , ··· , n x p > x p + 1 In particular, the only ideals I x which have a linear resolution are those for which x 1 = · · · = x n or x 1 − 1 = x 2 = · · · = x n .

Regularity of the ideals I x Unfortunately, we don’t know how to compute β ( S / I x ) for arbitrary x ! Question What about the regularity? Effective bounds? Theorem (–, Weyman ’13) ( n · ( x p − p ) + p 2 + 2 · ( p − 1 ) · ( n − p )) . reg ( I x ) = max p = 1 , ··· , n x p > x p + 1 In particular, the only ideals I x which have a linear resolution are those for which x 1 = · · · = x n or x 1 − 1 = x 2 = · · · = x n . Example For m = n = 3, reg ( I ( 1 , 1 ) ) = 3 , reg ( I ( 2 , 2 ) ) = 6 .

Local cohomology The ˇ Cech complex C • ( f 1 , · · · , f t ) is defined by � � 0 − → S − → S f i − → S f i f j − → · · · − → S f 1 ··· f t − → 0 . 1 ≤ i ≤ t 1 ≤ i < j ≤ t

Local cohomology The ˇ Cech complex C • ( f 1 , · · · , f t ) is defined by � � 0 − → S − → S f i − → S f i f j − → · · · − → S f 1 ··· f t − → 0 . 1 ≤ i ≤ t 1 ≤ i < j ≤ t For I = ( f 1 , · · · , f t ) , i ≥ 0, the local cohomology modules H i I ( S ) are defined by H i I ( S ) = H i ( C • ( f 1 , · · · , f t )) .

Local cohomology The ˇ Cech complex C • ( f 1 , · · · , f t ) is defined by � � 0 − → S − → S f i − → S f i f j − → · · · − → S f 1 ··· f t − → 0 . 1 ≤ i ≤ t 1 ≤ i < j ≤ t For I = ( f 1 , · · · , f t ) , i ≥ 0, the local cohomology modules H i I ( S ) are defined by H i I ( S ) = H i ( C • ( f 1 , · · · , f t )) . Problem Compute H • I ( S ) for all I

Local cohomology The ˇ Cech complex C • ( f 1 , · · · , f t ) is defined by � � 0 − → S − → S f i − → S f i f j − → · · · − → S f 1 ··· f t − → 0 . 1 ≤ i ≤ t 1 ≤ i < j ≤ t For I = ( f 1 , · · · , f t ) , i ≥ 0, the local cohomology modules H i I ( S ) are defined by H i I ( S ) = H i ( C • ( f 1 , · · · , f t )) . Problem Compute H • I ( S ) for all I = I x .

Local cohomology The ˇ Cech complex C • ( f 1 , · · · , f t ) is defined by � � 0 − → S − → S f i − → S f i f j − → · · · − → S f 1 ··· f t − → 0 . 1 ≤ i ≤ t 1 ≤ i < j ≤ t For I = ( f 1 , · · · , f t ) , i ≥ 0, the local cohomology modules H i I ( S ) are defined by H i I ( S ) = H i ( C • ( f 1 , · · · , f t )) . Problem Compute H • I ( S ) for all I = I x . Note that H • I ( S ) = H • √ I ( S ) , and � I x = I p , where p is the number of non-zero parts of x .

Characters Problem For each p = 1 , · · · , n, determine H • I p ( S ) .

Characters Problem For each p = 1 , · · · , n, determine H • I p ( S ) . I p ( S ) is an example of a doubly-graded module M j H • i , equivariant with respect to the action of GL m × GL n . i − → internal degree , j − → cohomological degree .

Characters Problem For each p = 1 , · · · , n, determine H • I p ( S ) . I p ( S ) is an example of a doubly-graded module M j H • i , equivariant with respect to the action of GL m × GL n . i − → internal degree , j − → cohomological degree . For such M , we define the character χ M by � i ] · z i · w j , [ M j χ M ( z , w ) = i , j where [ M j i ] is the class of M j i in the representation ring of GL m × GL n .

Dominant weights We define the set of dominant weights in Z r (for r = m or n ) dom = { λ ∈ Z r : λ 1 ≥ · · · ≥ λ r } . Z r

Dominant weights We define the set of dominant weights in Z r (for r = m or n ) dom = { λ ∈ Z r : λ 1 ≥ · · · ≥ λ r } . Z r For λ ∈ Z n dom , s = 0 , 1 , · · · , n − 1 , let λ ( s ) = ( λ 1 , · · · , λ s , s − n , · · · , s − n , λ s + 1 + ( m − n ) , · · · , λ n + ( m − n )) . � �� � m − n For m > n , λ ( s ) is dominant if and only if λ s ≥ s − n and λ s + 1 ≤ s − m .

Dominant weights We define the set of dominant weights in Z r (for r = m or n ) dom = { λ ∈ Z r : λ 1 ≥ · · · ≥ λ r } . Z r For λ ∈ Z n dom , s = 0 , 1 , · · · , n − 1 , let λ ( s ) = ( λ 1 , · · · , λ s , s − n , · · · , s − n , λ s + 1 + ( m − n ) , · · · , λ n + ( m − n )) . � �� � m − n For m > n , λ ( s ) is dominant if and only if λ s ≥ s − n and λ s + 1 ≤ s − m . The following Laurent power series are the key players in the description of H • I p ( S ) . � [ S λ ( s ) C m ⊗ S λ C n ] · z | λ | . h s ( z ) = λ ∈ Z n dom λ s ≥ s − n λ s + 1 ≤ s − m

An example Take m = 11, n = 9, s = 5, λ = ( 4 , 2 , 1 , − 2 , − 3 , − 6 , − 8 , − 8 , − 10 ) . We have m − n = 2 and λ ( s ) = ( λ 1 , · · · , λ s , s − n , · · · , s − n , λ s + 1 + ( m − n ) , · · · , λ n + ( m − n )) � �� � m − n = ( 4 , 2 , 1 , − 2 , − 3 , − 4 , − 4 , − 4 , − 6 , − 6 , − 8 ) .

An example Take m = 11, n = 9, s = 5, λ = ( 4 , 2 , 1 , − 2 , − 3 , − 6 , − 8 , − 8 , − 10 ) . We have m − n = 2 and λ ( s ) = ( λ 1 , · · · , λ s , s − n , · · · , s − n , λ s + 1 + ( m − n ) , · · · , λ n + ( m − n )) � �� � m − n = ( 4 , 2 , 1 , − 2 , − 3 , − 4 , − 4 , − 4 , − 6 , − 6 , − 8 ) . The coefficient of z − 30 in h s ( z ) involves (among other terms)                     C 11 C 9   ⊗                    

Local cohomology with support in determinantal ideals Theorem (–, Weyman, Witt ’13) n − 1 � h s ( z ) · w 1 +( n − s ) · ( m − n ) . χ H • In ( S ) ( z , w ) = s = 0

Local cohomology with support in determinantal ideals Theorem (–, Weyman, Witt ’13) n − 1 � h s ( z ) · w 1 +( n − s ) · ( m − n ) . χ H • In ( S ) ( z , w ) = s = 0 � a � We define the Gauss polynomial to be the generating function b � a � � � w t 1 + ··· + t a − b = p ( a − b , b ; c ) · w c , ( w ) = b b ≥ t 1 ≥ t 2 ≥···≥ t a − b ≥ 0 c ≥ 0 where p ( a − b , b ; c ) = # { t ⊢ c : t ⊂ ( b a − b ) } .

Local cohomology with support in determinantal ideals Theorem (–, Weyman, Witt ’13) n − 1 � h s ( z ) · w 1 +( n − s ) · ( m − n ) . χ H • In ( S ) ( z , w ) = s = 0 � a � We define the Gauss polynomial to be the generating function b � a � � � w t 1 + ··· + t a − b = p ( a − b , b ; c ) · w c , ( w ) = b b ≥ t 1 ≥ t 2 ≥···≥ t a − b ≥ 0 c ≥ 0 where p ( a − b , b ; c ) = # { t ⊢ c : t ⊂ ( b a − b ) } . Theorem (–, Weyman ’13) p − 1 � n − s − 1 � � h s ( z ) · w ( n − p + 1 ) 2 +( n − s ) · ( m − n ) · ( w 2 ) . χ H • Ip ( S ) ( z , w ) = p − s − 1 s = 0

Ext modules reg ( M ) = max {− r − j : Ext j S ( M , S ) r � = 0 } .

Ext modules reg ( S / I x ) = max {− r − j : Ext j S ( S / I x , S ) r � = 0 } .