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

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

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 )


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Intersections and sums

  • f Gorenstein ideals

Joint work with Oana Veliche and Jerzy Weyman

Lars Winther Christensen

Texas Tech University

13 June 2015

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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Local rings by character of singularity

regular

  • hypersurface
  • Golod

complete intersection

  • Gorenstein

Lars Winther Christensen Intersections and sums of Gorenstein ideals

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Local rings by character of singularity

regular

  • hypersurface
  • Golod

complete intersection

  • Gorenstein
  • Cohen–Macaulay (CM)

Lars Winther Christensen Intersections and sums of Gorenstein ideals

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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

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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

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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

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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

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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

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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

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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

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Realizable structures

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

Lars Winther Christensen Intersections and sums of Gorenstein ideals

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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

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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

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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

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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

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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

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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

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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

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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