Intersections and sums of Gorenstein ideals Joint work with Oana Veliche and Jerzy Weyman Lars Winther Christensen Texas Tech University 13 June 2015
Setup ( R , m , k ) a commutative noetherian local ring R ∼ � = Q / I with ( Q , n , k ) regular and I ⊆ n 2 Lars Winther Christensen Intersections and sums of Gorenstein ideals
Setup ( R , m , k ) a commutative noetherian local ring R ∼ � = Q / I with ( Q , n , k ) regular and I ⊆ n 2 What information about R is encoded in F : 0 − → F c − → F c − 1 − → · · · − → F 0 − → 0 the minimal free resolution of � R over Q ? Lars Winther Christensen Intersections and sums of Gorenstein ideals
Setup ( R , m , k ) a commutative noetherian local ring R ∼ � = Q / I with ( Q , n , k ) regular and I ⊆ n 2 What information about R is encoded in F : 0 − → F c − → F c − 1 − → · · · − → F 0 − → 0 the minimal free resolution of � R over Q ? c = pd Q � R = depth Q − depth Q � R = edim Q − depth � R = edim R − depth R = codepth R Lars Winther Christensen Intersections and sums of Gorenstein ideals
Low codepth Codepth 1 f F : 0 − → Q − → Q − → 0 f ∈ n 2 I = ( f ) R is (abstract) hypersurface Lars Winther Christensen Intersections and sums of Gorenstein ideals
Low codepth Codepth 1 f F : 0 − → Q − → Q − → 0 f ∈ n 2 I = ( f ) R is (abstract) hypersurface Codepth 2 → Q n − Φ → Q n − 1 F : 0 − − − → Q − → 0 I = f · I n − 1 (Φ) R either (abstract) complete intersection (c.i.) (e.g. k [[ x , y ]] / ( x 2 , y 2 ) ) Golod (e.g. k [[ x , y ]] / ( x 2 , xy , y 2 ) ) Lars Winther Christensen Intersections and sums of Gorenstein ideals
Multiplicative structures Theorem (Herzog) If c � 2 then F has unique structure of differential graded (DG) algebra, i.e. ∂ ( ab ) = ∂ ( a ) b + ( − 1) | a | a ∂ ( b ) Lars Winther Christensen Intersections and sums of Gorenstein ideals
Multiplicative structures Theorem (Herzog) If c � 2 then F has unique structure of differential graded (DG) algebra, i.e. ∂ ( ab ) = ∂ ( a ) b + ( − 1) | a | a ∂ ( b ) If R complete intersection I = ( f 1 , f 2 ) then � − f 2 � ( f 1 f 2 ) f 1 → Q 2 F : 0 − → Q − − − − − − − − − − → Q − → 0 = Koszul Q ( f 1 , f 2 ) �� � = Q � e 1 , e 2 � , ∂ ( e i ) = f i Lars Winther Christensen Intersections and sums of Gorenstein ideals
Multiplicative structures Theorem (Herzog) If c � 2 then F has unique structure of differential graded (DG) algebra, i.e. ∂ ( ab ) = ∂ ( a ) b + ( − 1) | a | a ∂ ( b ) If R complete intersection I = ( f 1 , f 2 ) then � − f 2 � ( f 1 f 2 ) f 1 → Q 2 F : 0 − → Q − − − − − − − − − − → Q − → 0 = Koszul Q ( f 1 , f 2 ) �� � = Q � e 1 , e 2 � , ∂ ( e i ) = f i If R Golod the algebra structure is different Lars Winther Christensen Intersections and sums of Gorenstein ideals
Products in homology The product on F yields product on Tor Q ∗ ( k , R ) = H( k ⊗ Q F ) Lars Winther Christensen Intersections and sums of Gorenstein ideals
Products in homology The product on F yields product on Tor Q ∗ ( k , R ) = H( k ⊗ Q F ) Example ( R = k [[ x , y ]] / ( x 2 , y 2 ) ) � � − y 2 ( x 2 y 2 ) x 2 → Q 2 Tor Q ∗ ( k , R ) = H( k ⊗ (0 − → Q − − − − − − − − − − − → Q − → 0)) � k 2 = Lars Winther Christensen Intersections and sums of Gorenstein ideals
Products in homology The product on F yields product on Tor Q ∗ ( k , R ) = H( k ⊗ Q F ) Example ( R = k [[ x , y ]] / ( x 2 , y 2 ) ) � � − y 2 ( x 2 y 2 ) x 2 → Q 2 Tor Q ∗ ( k , R ) = H( k ⊗ (0 − → Q − − − − − − − − − − − → Q − → 0)) � k 2 = Example ( R = k [[ x , y ]] / ( x 2 , xy , y 2 ) ) � − y � 0 x − y ( x 2 xy y 2 ) 0 x Tor Q → Q 2 ∗ ( k , R ) = H( k ⊗ (0 → Q − − − − − − − − − − − − − − → Q → 0)) = k ⋉ (Σ 1 k 3 ⊕ Σ 2 k 2 ) Lars Winther Christensen Intersections and sums of Gorenstein ideals
Local rings by products in homology Remark There is always a multiplicative structure on = H(Koszul Q ⊗ R ) = H(Koszul R ) . ∗ ( k , R ) ∼ A = Tor Q Lars Winther Christensen Intersections and sums of Gorenstein ideals
Local rings by products in homology Remark There is always a multiplicative structure on = H(Koszul Q ⊗ R ) = H(Koszul R ) . ∗ ( k , R ) ∼ A = Tor Q Theorem (Assmus, 1957) R is complete intesection if and only if A is the exterior algebra over A 1 . Theorem (Golod, 1962) R is Golod if and only if A admits a trivial Massey operation. (Golod means that the minimal free resolution of k has extremal growth.) Lars Winther Christensen Intersections and sums of Gorenstein ideals
Local rings by products in homology Remark There is always a multiplicative structure on = H(Koszul Q ⊗ R ) = H(Koszul R ) . ∗ ( k , R ) ∼ A = Tor Q Theorem (Assmus, 1957) R is complete intesection if and only if A is the exterior algebra over A 1 . Theorem (Golod, 1962) R is Golod if and only if A admits a trivial Massey operation. (Golod means that the minimal free resolution of k has extremal growth.) Theorem (Avramov and Golod, 1971) R is Gorenstein if and only if A is a Poincar´ e duality algebra. (The pairing A i × A c − i → A c is non-degenerate and rank k A c = 1.) Lars Winther Christensen Intersections and sums of Gorenstein ideals
� � � Local rings by character of singularity regular � Golod hypersurface complete intersection Gorenstein Lars Winther Christensen Intersections and sums of Gorenstein ideals
� � � � Local rings by character of singularity regular � Golod hypersurface complete intersection Gorenstein Cohen–Macaulay (CM) Lars Winther Christensen Intersections and sums of Gorenstein ideals
Codepth 3 Theorem (Buchsbaum and Eisenbud) If c � 3 then F has a structure of an associative graded-commutative DG algebra, i.e. a 2 = 0 for | a | odd ab = ( − 1) | a || b | ba and all such structures yield the same graded-commutative structure on ∗ ( k , R ) ∼ A = Tor Q = H(Koszul R ) Lars Winther Christensen Intersections and sums of Gorenstein ideals
Codepth 3 Theorem (Buchsbaum and Eisenbud) If c � 3 then F has a structure of an associative graded-commutative DG algebra, i.e. a 2 = 0 for | a | odd ab = ( − 1) | a || b | ba and all such structures yield the same graded-commutative structure on ∗ ( k , R ) ∼ A = Tor Q = H(Koszul R ) Theorem (Avramov, Kustin, and Miller; Weyman) Let c = 3 . For fixed m = µ ( I ) = rank k A 1 and n = type R = rank k A 3 there are only finitely many possible structures Lars Winther Christensen Intersections and sums of Gorenstein ideals
Possible structures There exist bases e 1 , . . . , e m for A 1 ( m ′ = m + n − 1) f 1 , . . . , f m ′ for A 2 g 1 , . . . , g n for A 3 such that multiplicative structure on A completely given by one of Lars Winther Christensen Intersections and sums of Gorenstein ideals
Possible structures There exist bases e 1 , . . . , e m for A 1 ( m ′ = m + n − 1) f 1 , . . . , f m ′ for A 2 g 1 , . . . , g n for A 3 such that multiplicative structure on A completely given by one of C e 1 e 2 = f 3 e 2 e 3 = f 1 e 3 e 1 = f 2 e i f i = g 1 1 � i � 3 T e 1 e 2 = f 3 e 2 e 3 = f 1 e 3 e 1 = f 2 B e 1 e 2 = f 3 e i f i = g 1 1 � i � 2 G( r ) [ r � 2] e i f i = g 1 1 � i � r H( p , q ) e i e p +1 = f i 1 � i � p e p +1 f p + j = g j 1 � j � q Lars Winther Christensen Intersections and sums of Gorenstein ideals
Possible structures There exist bases e 1 , . . . , e m for A 1 ( m ′ = m + n − 1) f 1 , . . . , f m ′ for A 2 g 1 , . . . , g n for A 3 such that multiplicative structure on A completely given by one of C e 1 e 2 = f 3 e 2 e 3 = f 1 e 3 e 1 = f 2 e i f i = g 1 1 � i � 3 T e 1 e 2 = f 3 e 2 e 3 = f 1 e 3 e 1 = f 2 B e 1 e 2 = f 3 e i f i = g 1 1 � i � 2 G( r ) [ r � 2] e i f i = g 1 1 � i � r H( p , q ) e i e p +1 = f i 1 � i � p e p +1 f p + j = g j 1 � j � q C ← → c.i. Lars Winther Christensen Intersections and sums of Gorenstein ideals
Possible structures There exist bases e 1 , . . . , e m for A 1 ( m ′ = m + n − 1) f 1 , . . . , f m ′ for A 2 g 1 , . . . , g n for A 3 such that multiplicative structure on A completely given by one of C e 1 e 2 = f 3 e 2 e 3 = f 1 e 3 e 1 = f 2 e i f i = g 1 1 � i � 3 T e 1 e 2 = f 3 e 2 e 3 = f 1 e 3 e 1 = f 2 B e 1 e 2 = f 3 e i f i = g 1 1 � i � 2 G( r ) [ r � 2] e i f i = g 1 1 � i � r H( p , q ) e i e p +1 = f i 1 � i � p e p +1 f p + j = g j 1 � j � q C ← → c.i. G( r = m ) ← → Gorenstein Lars Winther Christensen Intersections and sums of Gorenstein ideals
Possible structures There exist bases e 1 , . . . , e m for A 1 ( m ′ = m + n − 1) f 1 , . . . , f m ′ for A 2 g 1 , . . . , g n for A 3 such that multiplicative structure on A completely given by one of C e 1 e 2 = f 3 e 2 e 3 = f 1 e 3 e 1 = f 2 e i f i = g 1 1 � i � 3 T e 1 e 2 = f 3 e 2 e 3 = f 1 e 3 e 1 = f 2 B e 1 e 2 = f 3 e i f i = g 1 1 � i � 2 G( r ) [ r � 2] e i f i = g 1 1 � i � r H( p , q ) e i e p +1 = f i 1 � i � p e p +1 f p + j = g j 1 � j � q C ← → c.i. G( r = m ) ← → Gorenstein H(0 , 0) ← → Golod Lars Winther Christensen Intersections and sums of Gorenstein ideals
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