Local Cohomology with Support in Ideals of Maximal Minors and sub–Maximal Pfaffians Claudiu Raicu ∗ , Jerzy Weyman, and Emily E. Witt Alba Iulia, June 2013
Overview Cohen–Macaulayness of modules of covariants 1 Local cohomology 2 Ext modules via the geometric technique and duality 3
Modules of covariants Theorem (Hochster–Roberts ’74) Consider a reductive group H in characteristic zero, and a finite dimensional H–representation W. Write S = Sym ( W ) , and let S H be the ring of invariants with respect to the natural action of H on S. S H is a Cohen–Macaulay ring.
Modules of covariants Theorem (Hochster–Roberts ’74) Consider a reductive group H in characteristic zero, and a finite dimensional H–representation W. Write S = Sym ( W ) , and let S H be the ring of invariants with respect to the natural action of H on S. S H is a Cohen–Macaulay ring. More generally, to any H –representation U we can associate the module of covariants ( S ⊗ U ) H . Question Which modules of covariants are Cohen–Macaulay? [Stanley ’82, Brion ’93, Van den Bergh ’90s.]
Modules of covariants Theorem (Hochster–Roberts ’74) Consider a reductive group H in characteristic zero, and a finite dimensional H–representation W. Write S = Sym ( W ) , and let S H be the ring of invariants with respect to the natural action of H on S. S H is a Cohen–Macaulay ring. More generally, to any H –representation U we can associate the module of covariants ( S ⊗ U ) H . Question Which modules of covariants are Cohen–Macaulay? [Stanley ’82, Brion ’93, Van den Bergh ’90s.] For us: G finite dimensional vector space, dim ( G ) = n . H = SL ( G ) . W = G ⊕ m .
Theorem on covariants of the special linear group S = Sym ( W ) = C [ x ij ] , where x ij are the entries of the generic matrix · · · x 11 x 21 x m 1 . . . ... . . . X = . . . . x 1 n x 2 n · · · x mn C , m < n ; S H = C [ n × n minors of X ] = C [ det ( X )] , m = n ; more interesting , m > n .
Theorem on covariants of the special linear group S = Sym ( W ) = C [ x ij ] , where x ij are the entries of the generic matrix · · · x 11 x 21 x m 1 . . . ... . . . X = . . . . x 1 n x 2 n · · · x mn C , m < n ; S H = C [ n × n minors of X ] = C [ det ( X )] , m = n ; more interesting , m > n . Theorem (–WW ’13) If µ = ( µ 1 ≥ µ 2 ≥ · · · ≥ µ n = 0 ) is a partition and U = S µ G, then ( S ⊗ U ) H is Cohen–Macaulay if and only if µ s − µ s + 1 < m − n for all s = 1 , · · · , n − 1 .
Theorem on covariants of the special linear group S = Sym ( W ) = C [ x ij ] , where x ij are the entries of the generic matrix · · · x 11 x 21 x m 1 . . . ... . . . X = . . . . x 1 n x 2 n · · · x mn C , m < n ; S H = C [ n × n minors of X ] = C [ det ( X )] , m = n ; more interesting , m > n . Theorem (–WW ’13) If µ = ( µ 1 ≥ µ 2 ≥ · · · ≥ µ n = 0 ) is a partition and U = S µ G, then ( S ⊗ U ) H is Cohen–Macaulay if and only if µ s − µ s + 1 < m − n for all s = 1 , · · · , n − 1 . [B’93: m = n + 1; VdB’94: n = 2, arbitrary W ; VdB’99: n = 3.]
Local cohomology If R is a ring, J = ( f 1 , · · · , f t ) an ideal, and M an R –module, we define the ˇ Cech complex C • ( M ; f 1 , · · · , f t ) by � � 0 − → M − → M f i − → M f i f j − → · · · − → M f 1 ··· f t − → 0 . 1 ≤ i ≤ t 1 ≤ i < j ≤ t
Local cohomology If R is a ring, J = ( f 1 , · · · , f t ) an ideal, and M an R –module, we define the ˇ Cech complex C • ( M ; f 1 , · · · , f t ) by � � 0 − → M − → M f i − → M f i f j − → · · · − → M f 1 ··· f t − → 0 . 1 ≤ i ≤ t 1 ≤ i < j ≤ t For j ≥ 0, the local cohomology modules H j J ( M ) are defined by H j J ( M ) = H j ( C • ( M ; f 1 , · · · , f t )) .
Local cohomology If R is a ring, J = ( f 1 , · · · , f t ) an ideal, and M an R –module, we define the ˇ Cech complex C • ( M ; f 1 , · · · , f t ) by � � 0 − → M − → M f i − → M f i f j − → · · · − → M f 1 ··· f t − → 0 . 1 ≤ i ≤ t 1 ≤ i < j ≤ t For j ≥ 0, the local cohomology modules H j J ( M ) are defined by H j J ( M ) = H j ( C • ( M ; f 1 , · · · , f t )) . If R is local or graded, with maximal ideal m , then M is said to be Cohen–Macaulay if H j m ( M ) = 0 for j < dim ( M ) .
Local cohomology If R is a ring, J = ( f 1 , · · · , f t ) an ideal, and M an R –module, we define the ˇ Cech complex C • ( M ; f 1 , · · · , f t ) by � � 0 − → M − → M f i − → M f i f j − → · · · − → M f 1 ··· f t − → 0 . 1 ≤ i ≤ t 1 ≤ i < j ≤ t For j ≥ 0, the local cohomology modules H j J ( M ) are defined by H j J ( M ) = H j ( C • ( M ; f 1 , · · · , f t )) . If R is local or graded, with maximal ideal m , then M is said to be Cohen–Macaulay if H j m ( M ) = 0 for j < dim ( M ) . � m � , f 1 , · · · , f t are the maximal minors of X , R = S H , For us t = n m = ( f 1 , · · · , f t ) ⊂ R is the homogeneous maximal ideal. We have � ( S ⊗ U ) H � � � H H j H j = m S ( S ) ⊗ U . m
Local cohomology and covariants Recall that H = SL ( G ) , W = G ⊕ m , S = Sym ( W ) , and X is the generic m × n matrix. We have S H = C [ maximal minors of X ] = C [ Grass ( n , m )] , so for every H –representation U , dim ( S H ) = dim ( S ⊗ U ) H = n · ( m − n ) + 1 .
Local cohomology and covariants Recall that H = SL ( G ) , W = G ⊕ m , S = Sym ( W ) , and X is the generic m × n matrix. We have S H = C [ maximal minors of X ] = C [ Grass ( n , m )] , so for every H –representation U , dim ( S H ) = dim ( S ⊗ U ) H = n · ( m − n ) + 1 . Let I ⊂ S be the ideal generated by the maximal minors of X . It follows that ( S ⊗ U ) H is Cohen–Macaulay iff � � H H j I ( S ) ⊗ U = 0 , for 0 ≤ j ≤ n · ( m − n ) .
Local cohomology and covariants Recall that H = SL ( G ) , W = G ⊕ m , S = Sym ( W ) , and X is the generic m × n matrix. We have S H = C [ maximal minors of X ] = C [ Grass ( n , m )] , so for every H –representation U , dim ( S H ) = dim ( S ⊗ U ) H = n · ( m − n ) + 1 . Let I ⊂ S be the ideal generated by the maximal minors of X . It follows that ( S ⊗ U ) H is Cohen–Macaulay iff � � H H j I ( S ) ⊗ U = 0 , for 0 ≤ j ≤ n · ( m − n ) . When U = S µ G is an irreducible H –representation, this is equivalent to saying that U ∗ = S ( µ 1 ,µ 1 − µ n − 1 , ··· ,µ 1 − µ 2 ) G doesn’t occur in the decomposition of H j I ( S ) into a sum of irreducible H –representations.
Theorem on Maximal Minors Write G ⊕ m = F ⊗ G for an m –dimensional vector space F , so that S = Sym ( F ⊗ G ) . I is generated by � n F ⊗ � n G ⊂ Sym n ( F ⊗ G ) .
Theorem on Maximal Minors Write G ⊕ m = F ⊗ G for an m –dimensional vector space F , so that S = Sym ( F ⊗ G ) . I is generated by � n F ⊗ � n G ⊂ Sym n ( F ⊗ G ) . Theorem (–WW ’13) For 1 ≤ s ≤ n and λ = ( λ 1 , · · · , λ n ) ∈ Z n a dominant weight, let λ ( s ) = ( λ 1 , · · · , λ n − s , − s , · · · , − s , λ n − s + 1 + ( m − n ) , · · · , λ n + ( m − n )) . � �� � m − n We let W ( r ; s ) denote the set of dominant weights λ ∈ Z n with | λ | = r and λ ( s ) ∈ Z m also dominant. We have the decomposition into a sum of GL ( F ) × GL ( G ) –representations �� λ ∈ W ( r ; s ) S λ ( s ) F ⊗ S λ G , if j = s · ( m − n ) + 1 , 1 ≤ s ≤ n ; H j I ( S ) r = 0 , otherwise .
Weights of local cohomology for maximal minors Take m = 11, n = 9, s = 4, λ = ( 4 , 2 , 1 , − 2 , − 3 , − 6 , − 8 , − 8 , − 10 ) . We have m − n = 2 and λ ( s ) = ( λ 1 , · · · , λ n − s , − s , · · · , − s , λ n − s + 1 + ( m − n ) , · · · , λ n + ( m − n )) � �� � m − n = ( 4 , 2 , 1 , − 2 , − 3 , − 4 , − 4 , − 4 , − 6 , − 6 , − 8 ) .
Weights of local cohomology for maximal minors Take m = 11, n = 9, s = 4, λ = ( 4 , 2 , 1 , − 2 , − 3 , − 6 , − 8 , − 8 , − 10 ) . We have m − n = 2 and λ ( s ) = ( λ 1 , · · · , λ n − s , − s , · · · , − s , λ n − s + 1 + ( m − n ) , · · · , λ n + ( m − n )) � �� � m − n = ( 4 , 2 , 1 , − 2 , − 3 , − 4 , − 4 , − 4 , − 6 , − 6 , − 8 ) . The local cohomology module H 9 I ( S ) contains in degree r = | λ | = − 30 the irreducible representation F ⊗ G
Theorem on sub–Maximal Pfaffians dim ( F ) = 2 n + 1, W = � 2 F , and S = Sym ( W ) . Let I be the ideal generated by � 2 n F ⊂ Sym n �� 2 F � (the 2 n × 2 n –Pfaffians of the generic ( 2 n + 1 ) × ( 2 n + 1 ) skew–symmetric matrix).
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