The dual module of Gorenstein k -algebras Maria Evelina Rossi Department of Mathematics, University of Genoa, Italy The dual module of Gorenstein k -algebras 38th Symposium on Commutative Algebra in Japan 9th Japan-Vietnam Joint Seminar on Commutative Algebra Maria Evelina Rossi, University of Genoa Gorenstein k -algebras, Japan November 18-22, 2016
MAIN GOAL We will survey some recent results on: an extension of Macaulay’s Inverse System theorem to Gorenstein d -dimensional k -algebras The results are obtained jointly with J. Elias Maria Evelina Rossi, University of Genoa Gorenstein k -algebras, Japan November 18-22, 2016
GORENSTEIN RINGS P r e l i m i n a r i e s Let S be a regular ring ( k [ x 1 , . . . , x n ] or k [[ x 1 , . . . , x n ]]). Consider R = S / I The codimension of R (of I ) is defined by codim ( R ) = dimS − dimR = n − dimR R is said a complete intersection (c.i.) if I can be generated by codim ( R ) elements. c . i . = ⇒ Gorenstein Maria Evelina Rossi, University of Genoa Gorenstein k -algebras, Japan November 18-22, 2016
GORENSTEIN RINGS P r e l i m i n a r i e s Based on the famous paper by H. Bass (’63) (On the ubiquity of Gorenstein rings), there are many equivalent definitions of Gorenstein rings: Definition. R is Gorenstein if R is Cohen-Macaulay and its dualizing module (or canonical module) Ext n − d ( R , S ) is free (of rank 1) where S d = dimR . In terms of free resolutions Proposition. Let 0 → F c → F c − 1 → · · · → F 0 → R → 0 a minimal free S -resolution of R . Then R is Gorenstein ⇐ ⇒ c = codim ( R ) and F c ≃ S 0 → S → S 2 → S → R → 0 Codim ( R ) = 2 Gorenstein ⇐ ⇒ c.i. Maria Evelina Rossi, University of Genoa Gorenstein k -algebras, Japan November 18-22, 2016
GORENSTEIN RINGS P r e l i m i n a r i e s Theorem. [Buchsbaum-Eisenbud] codim ( R ) = 3 R is Gorenstein ⇐ ⇒ I is generated by 2 m -order Pfaffians of a skew-symmetric (2 m + 1) alternating matrix A . In this case a minimal free resolution of R over S has the form 0 → S → S 2 m +1 → S 2 m +1 → S → R → 0 A A. Kustin, M. Reid studied the projective resolution of Gorenstein ideals of codimension 4, aiming to extend the previous famous theorem by Buchsbaum and Eisenbud. To date a geometric or an algebraic description of Gorenstein rings of any dimension and codimension is not understood. Maria Evelina Rossi, University of Genoa Gorenstein k -algebras, Japan November 18-22, 2016
MATLIS DUALITY I n v e r s e s y s t e m Let k = k of arbitrary characteristic. Let S = k [[ x 1 , . . . , x n ]] (or k [ x 1 , . . . , x n ]) and let E S ( k ) the injective hull of k as R -module. Gabriel (58) observed that an injective hull of k = S / ( x 1 , . . . , x n ) E S ( k ) ≃ D k · ( S 1 ) ≃ k [ X 1 , . . . , X n ] := D a divided power ring. D is a S -module by a contraction action: x i ◦ X [ a ] = x i ◦ ( X a 1 n ) = X a 1 1 · · · X a i − 1 1 · · · X a n · · · X a n n i if a i > 0 . If a i = 0 , then is 0 . Maria Evelina Rossi, University of Genoa Gorenstein k -algebras, Japan November 18-22, 2016
INVERSE SYSTEM Example: X 2 x 1 ◦ 1 X 2 = X 1 X 2 X 2 x 1 ◦ 2 = 0 If we assume char ( k ) = 0 , then ( D , ◦ ) ≃ ( k [ X 1 , . . . , X n ] , ∂ ) X [ a ] X [ a ] ��� a ! where a ! = � ( a i !) and ∂ is the usual partial derivative (with coefficients). Maria Evelina Rossi, University of Genoa Gorenstein k -algebras, Japan November 18-22, 2016
INVERSE SYSTEM We denote · ∨ = Hom S ( · , D ) the exact functor in the category of the S -modules. Matlis (’58) showed that the functor ∨ defines an equivalence between { Artinian S -modules } − → { Noetherian S -modules } ( S / I ) ∨ := I ⊥ = < { g ( X ) ∈ D | I ◦ g ( X ) = 0 } > S / I ��� Maria Evelina Rossi, University of Genoa Gorenstein k -algebras, Japan November 18-22, 2016
INVERSE SYSTEM Let I = ( x 2 , y 3 ) ⊂ S = k [[ x , y ]] . Then I ⊥ is a Example : S -submodule of D = k [ X , Y ] and I ⊥ = < { g ∈ D | x 2 ◦ g = 0 and y 3 ◦ g = 0 } > = < XY 2 > If I ⊂ S is an ideal (not necessarily 0-dimensional), then ( S / I ) ∨ = Hom S ( R / I , D ) ≃ I ⊥ = < { g ( X ) ∈ D | I ◦ g ( X ) = 0 } >, a S -submodule of D and called Macaulay’s inverse system of I . I ⊥ is finitely generated ⇐ ⇒ S / I is 0-dimensional I ⊥ is cyclic. S / I is 0-dimensional Gorenstein ⇐ ⇒ Maria Evelina Rossi, University of Genoa Gorenstein k -algebras, Japan November 18-22, 2016
ARTINIAN Gorenstein k -algebras M a c a u l a y’ s I n v e r s e S y s t e m Macaulay proved that there is the following 1-1 correspondence I ⊆ S ideal such that M = S ◦ F S / I is Artinian Gorenstein ↔ S -cyclic submodule of D with socledegree( S / I ) = s . with degree F = s I ⊥ I − → Ann S ( F ) ← − M = � F � S Given a S -submodule M of D then Ann S ( M ) = { f ( x ) ∈ S | f ( x ) ◦ M = 0 } . is an ideal of S . Maria Evelina Rossi, University of Genoa Gorenstein k -algebras, Japan November 18-22, 2016
EXAMPLES M a c a u l a y I n v e r s e S y s t e m Example 1. Let F = X 2 + Y 3 ∈ D = k [ X , Y ] and let S = k [[ x , y ]] . Then I = Ann S ( F ) = ( xy , x 2 − y 3 ) and R = S / I is Gorenstein c.i. e = ℓ ( S / I ) = dim k < F > = dim k < F , Y 2 , X , Y , 1 > = 5 . HF S / I ( j ) = dim k ( I ⊥ ) j : h = (1 , 2 , 1 , 1) Let F = X 2 + Y 2 + Z 2 ∈ D = k [ X , Y , Z ] and let Example 2. S = k [[ x , y , z ]] . Then I = Ann S ( F ) = ( x 2 − y 2 , y 2 − z 2 , xy , xz , yz ) and R = S / I is Gorenstein (not c.i.). Maria Evelina Rossi, University of Genoa Gorenstein k -algebras, Japan November 18-22, 2016
Isomorphism classes of 0-dimensional Gorenstein rings M a c a u l a y I n v e r s e S y s t e m We may translate in terms of F many properties of the corresponding Gorenstein ideal For instance we translate in an effective framework the analytic isomorphisms of Gorenstein 0-dimensional k -algebras in terms of the dual module < F > This topic plays an important role in studying the Hilbert scheme Hilb d ( P n ) parametrizing Gorenstein 0-dimensional subschemes of P n and the rationality of the Poincar´ e series of k as R -module, see [Emsalem], [Iarrobino], [Poonen], [Erman], [Cartwright-Erman-Velasco-Viray], [–, S ¸ega], ... Maria Evelina Rossi, University of Genoa Gorenstein k -algebras, Japan November 18-22, 2016
Isomorphism classes of 0-dimensional Gorenstein rings M a c a u l a y I n v e r s e S y s t e m Proposition. Let A = S / I and B = S / J be two local Artinian Gorenstein algebras so that I = Ann S ( F ) and J = Ann S ( G ) with F , G ∈ D . Then TFAE: 1. A ≃ B 2. ∃ φ ∈ Aut ( S ) such that φ ( I ) = J 3. ∃ φ ∈ Aut ( S ) such that φ ∨ ( G ) = u ◦ F with u ∈ S ∗ 4. F and G lie in the same G -orbit of D where G = Aut ( S ) ⋉ S ∗ . Maria Evelina Rossi, University of Genoa Gorenstein k -algebras, Japan November 18-22, 2016
Isomorphism classes of 0-dimensional Gorenstein rings E X A M P L E Example Consider the Gorenstein local rings R with h -vector: h = (1 , 2 , 2 , 1) ( x 2 , y 3 ) → xy 2 We have only two models: I = ( xy , x 3 − y 3 ) → x 3 + y 3 Remark that both the models are homogeneous!!! Hence A is canonically graded ( R ≃ gr m ( R )). Maria Evelina Rossi, University of Genoa Gorenstein k -algebras, Japan November 18-22, 2016
d -dimensional Gorenstein rings G O A L Characterize the S -submodules M of D ( not finitely generated! ) such that S / Ann S ( M ) is a d -dimensional Gorenstein ring (codimension n , multiplicity e , regularity r , .....) Maria Evelina Rossi, University of Genoa Gorenstein k -algebras, Japan November 18-22, 2016
d -dimensional Gorenstein rings G d -a d m i s s i b l e In the Artinian case M is cyclic, in positive dimension further conditions will be required. Notation: L = ( l 1 , . . . , l d ) ∈ N d + γ i = (0 , . . . , 0 , 1 i , 0 , . . . , 0) L i = ( l 1 , . . . , l i − 1 , 1 i , l i +1 , . . . , l d ) Definition. Let d > 0 and let M � = (0) be a S -submodule of the D = E S ( k ) . We say that M is G d -admissible, 1 ≤ d < n , if it admits a system of generators { H L } L ∈ N d + in D = k [ Z 1 , . . . , Z n ] satisfying for every L ∈ N d + and i = 1 , . . . , d the following conditions: H L − γ i if L − γ i > 0 1. z i ◦ H L = 0 otherwise. 2. < H L > ∩ k [ Z 1 , . . . , ∧ i , . . . , Z n ] ⊆ < H L i > Maria Evelina Rossi, University of Genoa Gorenstein k -algebras, Japan November 18-22, 2016
d -dimensional Gorenstein rings M a i n T h e o r e m (J. E l i a s, R.) With the above notation: There is a one-to-one correspondence between the following sets: M = � H L , L ∈ N d S / I Gorenstein + � ⊆ D d -dimensional rings G d -admissible ( graded ) ← → (homogeneous) multiplicity e dim k < H 1 d > = e regularity r deg H 1 d = r � ( I + ( z L )) ⊥ , L ∈ N d I − → + � z regular linear sequence mod I � H L , L ∈ N d � L Ann S ( � H L � ) ← − + � Maria Evelina Rossi, University of Genoa Gorenstein k -algebras, Japan November 18-22, 2016
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