Intersections and sums of Gorenstein ideals Joint work with Oana - - PowerPoint PPT Presentation

Intersections and sums

• f 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 ⊆ n2

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 ⊆ n2 What information about R is encoded in F : 0 − → Fc − → Fc−1 − → · · · − → F0 − → 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 ⊆ n2 What information about R is encoded in F : 0 − → Fc − → Fc−1 − → · · · − → F0 − → 0 the minimal free resolution of R over Q ? c = pdQ R = depth Q − depthQ 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 : 0 − → Q

f

− → Q − → 0 I = (f ) f ∈ n2 R is (abstract) hypersurface

Lars Winther Christensen Intersections and sums of Gorenstein ideals

Low codepth

Codepth 1 F : 0 − → Q

f

− → Q − → 0 I = (f ) f ∈ n2 R is (abstract) hypersurface Codepth 2 F : 0 − → Qn−1

Φ

− − → Qn − → Q − → 0 I = f · In−1(Φ) R either (abstract) complete intersection (c.i.) (e.g. k[[x, y]]/(x2, y2) ) Golod (e.g. k[[x, y]]/(x2, xy, y2) )

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 = (f1, f2) then F : 0 − → Q

−f2 f1

• −

− − − − → Q2

( f1 f2 )

− − − − − → Q − → 0 = KoszulQ(f1, f2) =

• Qe1, e2 , ∂(ei) = fi
• 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 = (f1, f2) then F : 0 − → Q

−f2 f1

• −

− − − − → Q2

( f1 f2 )

− − − − − → Q − → 0 = KoszulQ(f1, f2) =

• Qe1, e2 , ∂(ei) = fi
• 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 TorQ

∗ (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 TorQ

∗ (k, R) = H(k ⊗Q F)

Example (R = k[[x, y]]/(x2, y2)) TorQ

∗ (k, R) = H(k ⊗ (0 −

→ Q

• −y2

x2

• −

− − − − − → Q2

( x2 y2 )

− − − − − → Q − → 0)) =

• k2

Lars Winther Christensen Intersections and sums of Gorenstein ideals

Products in homology

The product on F yields product on TorQ

∗ (k, R) = H(k ⊗Q F)

Example (R = k[[x, y]]/(x2, y2)) TorQ

∗ (k, R) = H(k ⊗ (0 −

→ Q

• −y2

x2

• −

− − − − − → Q2

( x2 y2 )

− − − − − → Q − → 0)) =

• k2

Example (R = k[[x, y]]/(x2, xy, y2)) TorQ

∗ (k, R) = H(k ⊗ (0 → Q −y x −y x

• −

− − − − − − → Q2

( x2 xy y2 )

− − − − − − − → Q → 0)) = k ⋉ (Σ1k3 ⊕ Σ2k2)

Lars Winther Christensen Intersections and sums of Gorenstein ideals

Local rings by products in homology

Remark There is always a multiplicative structure on A = TorQ

∗ (k, R) ∼

= H(KoszulQ ⊗R) = H(KoszulR).

Lars Winther Christensen Intersections and sums of Gorenstein ideals

Local rings by products in homology

Remark There is always a multiplicative structure on A = TorQ

∗ (k, R) ∼

= H(KoszulQ ⊗R) = H(KoszulR). Theorem (Assmus, 1957) R is complete intesection if and only if A is the exterior algebra

• ver A1.

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 A = TorQ

∗ (k, R) ∼

= H(KoszulQ ⊗R) = H(KoszulR). Theorem (Assmus, 1957) R is complete intesection if and only if A is the exterior algebra

• ver A1.

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 Ai × Ac−i → Ac is non-degenerate and rankk Ac = 1.)

Lars Winther Christensen Intersections and sums of Gorenstein ideals

Local rings by character of singularity

regular

• hypersurface
• Golod

complete intersection

• Gorenstein

Lars Winther Christensen Intersections and sums of Gorenstein ideals

Local rings by character of singularity

regular

• hypersurface
• Golod

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. ab = (−1)|a||b|ba and a2 = 0 for |a| odd all such structures yield the same graded-commutative structure on A = TorQ

∗ (k, R) ∼

= H(KoszulR)

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. ab = (−1)|a||b|ba and a2 = 0 for |a| odd all such structures yield the same graded-commutative structure on A = TorQ

∗ (k, R) ∼

= H(KoszulR) Theorem (Avramov, Kustin, and Miller; Weyman) Let c = 3. For fixed m = µ(I) = rankk A1 and n = type R = rankk A3 there are only finitely many possible structures

Lars Winther Christensen Intersections and sums of Gorenstein ideals

Possible structures

There exist bases e1, . . . , em for A1 f1, . . . , fm′ for A2 (m′ = m + n − 1) g1, . . . , gn for A3 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 e1, . . . , em for A1 f1, . . . , fm′ for A2 (m′ = m + n − 1) g1, . . . , gn for A3 such that multiplicative structure on A completely given by one of C e1e2 = f3 e2e3 = f1 e3e1 = f2 eifi = g1 1 i 3 T e1e2 = f3 e2e3 = f1 e3e1 = f2 B e1e2 = f3 eifi = g1 1 i 2 G(r) [r 2] eifi = g1 1 i r H(p, q) eiep+1 = fi 1 i p ep+1fp+j = gj 1 j q

Lars Winther Christensen Intersections and sums of Gorenstein ideals

Possible structures

There exist bases e1, . . . , em for A1 f1, . . . , fm′ for A2 (m′ = m + n − 1) g1, . . . , gn for A3 such that multiplicative structure on A completely given by one of C e1e2 = f3 e2e3 = f1 e3e1 = f2 eifi = g1 1 i 3 T e1e2 = f3 e2e3 = f1 e3e1 = f2 B e1e2 = f3 eifi = g1 1 i 2 G(r) [r 2] eifi = g1 1 i r H(p, q) eiep+1 = fi 1 i p ep+1fp+j = gj 1 j q C ← → c.i.

Lars Winther Christensen Intersections and sums of Gorenstein ideals

Possible structures

There exist bases e1, . . . , em for A1 f1, . . . , fm′ for A2 (m′ = m + n − 1) g1, . . . , gn for A3 such that multiplicative structure on A completely given by one of C e1e2 = f3 e2e3 = f1 e3e1 = f2 eifi = g1 1 i 3 T e1e2 = f3 e2e3 = f1 e3e1 = f2 B e1e2 = f3 eifi = g1 1 i 2 G(r) [r 2] eifi = g1 1 i r H(p, q) eiep+1 = fi 1 i p ep+1fp+j = gj 1 j q C ← → c.i. G(r = m) ← → Gorenstein

Lars Winther Christensen Intersections and sums of Gorenstein ideals

Possible structures

There exist bases e1, . . . , em for A1 f1, . . . , fm′ for A2 (m′ = m + n − 1) g1, . . . , gn for A3 such that multiplicative structure on A completely given by one of C e1e2 = f3 e2e3 = f1 e3e1 = f2 eifi = g1 1 i 3 T e1e2 = f3 e2e3 = f1 e3e1 = f2 B e1e2 = f3 eifi = g1 1 i 2 G(r) [r 2] eifi = g1 1 i r H(p, q) eiep+1 = fi 1 i p ep+1fp+j = gj 1 j q C ← → c.i. G(r = m) ← → Gorenstein H(0, 0) ← → Golod

Lars Winther Christensen Intersections and sums of Gorenstein ideals

Realizable structures

Conjecture (Avramov, 2011) Every ring of type G(r) is Gorenstein.

Lars Winther Christensen Intersections and sums of Gorenstein ideals

Realizable structures

Conjecture (Avramov, 2011) Every ring of type G(r) is Gorenstein. Example ( and Veliche) Let k be a field R = k[[x, y, z]] (xy, yz, x3, x2z, xz2 − y3, z3) has rankk A1 = 6 and rankk A3 = 2 and it belongs to G(3). Among G rings that are not Gorenstein it is minimal w.r.t Dimension Type Number of generators of I Number of non-monomial generators of I?

Lars Winther Christensen Intersections and sums of Gorenstein ideals

Observation: G rings are ubiquitous

Look among artinian quotients R of Q = k[x, y, z]

Lars Winther Christensen Intersections and sums of Gorenstein ideals

Observation: G rings are ubiquitous

Look among artinian quotients R of Q = k[x, y, z] Type of R = Q/I must be n 2

Lars Winther Christensen Intersections and sums of Gorenstein ideals

Observation: G rings are ubiquitous

Look among artinian quotients R of Q = k[x, y, z] Type of R = Q/I must be n 2 Irreducible decomposition I = I1 ∩ · · · ∩ In with each Q/Ij artinian Gorenstein

Lars Winther Christensen Intersections and sums of Gorenstein ideals

Observation: G rings are ubiquitous

Look among artinian quotients R of Q = k[x, y, z] Type of R = Q/I must be n 2 Irreducible decomposition I = I1 ∩ · · · ∩ In with each Q/Ij artinian Gorenstein For generic Gorenstein ideals I1 and I2 look at I = I1 ∩ I2

Lars Winther Christensen Intersections and sums of Gorenstein ideals

Observation: G rings are ubiquitous

Look among artinian quotients R of Q = k[x, y, z] Type of R = Q/I must be n 2 Irreducible decomposition I = I1 ∩ · · · ∩ In with each Q/Ij artinian Gorenstein For generic Gorenstein ideals I1 and I2 look at I = I1 ∩ I2 The class of R = Q/I is determined by the socle degrees s1 and s2 of Q/I1 and Q/I2:

Lars Winther Christensen Intersections and sums of Gorenstein ideals

Observation: G rings are ubiquitous

Look among artinian quotients R of Q = k[x, y, z] Type of R = Q/I must be n 2 Irreducible decomposition I = I1 ∩ · · · ∩ In with each Q/Ij artinian Gorenstein For generic Gorenstein ideals I1 and I2 look at I = I1 ∩ I2 The class of R = Q/I is determined by the socle degrees s1 and s2 of Q/I1 and Q/I2:

For s1 ≈ s2: Golod H(0, 0) For s1 ≪ s2: Gorenstein G(r = m)

Lars Winther Christensen Intersections and sums of Gorenstein ideals

Observation: G rings are ubiquitous

Look among artinian quotients R of Q = k[x, y, z] Type of R = Q/I must be n 2 Irreducible decomposition I = I1 ∩ · · · ∩ In with each Q/Ij artinian Gorenstein For generic Gorenstein ideals I1 and I2 look at I = I1 ∩ I2 The class of R = Q/I is determined by the socle degrees s1 and s2 of Q/I1 and Q/I2:

For s1 ≈ s2: Golod H(0, 0) For s1 ≪ s2: Gorenstein G(r = m) Inbetween: G(r) not Gorenstein G(r < m)

Lars Winther Christensen Intersections and sums of Gorenstein ideals