Monomial Resolutions Dave Bayer, Irena Peeva, Bernd Sturmfels April - - PowerPoint PPT Presentation

Monomial Resolutions

Dave Bayer, Irena Peeva, Bernd Sturmfels April 25, 2018

Dave Bayer, Irena Peeva, Bernd Sturmfels Monomial Resolutions April 25, 2018 1 / 14

Let S = k[x1, . . . , xn], and M be a monomial ideal of S.

Dave Bayer, Irena Peeva, Bernd Sturmfels Monomial Resolutions April 25, 2018 2 / 14

Let S = k[x1, . . . , xn], and M be a monomial ideal of S.

Central Question

How can we find a minimal free resolution of S/M over S?

Dave Bayer, Irena Peeva, Bernd Sturmfels Monomial Resolutions April 25, 2018 2 / 14

Let S = k[x1, . . . , xn], and M be a monomial ideal of S.

Central Question

How can we find a minimal free resolution of S/M over S? Why is this interesting?

Dave Bayer, Irena Peeva, Bernd Sturmfels Monomial Resolutions April 25, 2018 2 / 14

Let S = k[x1, . . . , xn], and M be a monomial ideal of S.

Central Question

How can we find a minimal free resolution of S/M over S? Why is this interesting? Uniqueness (i.e., useful as an invariant)

Dave Bayer, Irena Peeva, Bernd Sturmfels Monomial Resolutions April 25, 2018 2 / 14

Let S = k[x1, . . . , xn], and M be a monomial ideal of S.

Central Question

How can we find a minimal free resolution of S/M over S? Why is this interesting? Uniqueness (i.e., useful as an invariant) Encodes structural information

Dave Bayer, Irena Peeva, Bernd Sturmfels Monomial Resolutions April 25, 2018 2 / 14

Existing general constructions: Taylor’s resolution Lyubeznik’s subcomplex Both are far from minimal for a large number of generators.

Dave Bayer, Irena Peeva, Bernd Sturmfels Monomial Resolutions April 25, 2018 3 / 14

Existing general constructions: Taylor’s resolution Lyubeznik’s subcomplex Both are far from minimal for a large number of generators. Additional construction (for generic monomial ideals): Scarf complex

Dave Bayer, Irena Peeva, Bernd Sturmfels Monomial Resolutions April 25, 2018 3 / 14

Existing general constructions: Taylor’s resolution Lyubeznik’s subcomplex Both are far from minimal for a large number of generators. Additional construction (for generic monomial ideals): Scarf complex

Definition

M ⊂ k[x1, . . . , xn] is a generic monomial ideal if no variable xi appears in two distinct minimal generators of M with the same non-zero exponent

Dave Bayer, Irena Peeva, Bernd Sturmfels Monomial Resolutions April 25, 2018 3 / 14

Theorem 3.2

When M is a generic monomial ideal, the minimal free resolution of S/M

• ver S is defined by the Scarf complex ∆M.

Dave Bayer, Irena Peeva, Bernd Sturmfels Monomial Resolutions April 25, 2018 4 / 14

Theorem 3.2

When M is a generic monomial ideal, the minimal free resolution of S/M

• ver S is defined by the Scarf complex ∆M.

Consider M = m1, . . . , mr ⊂ k[x1, . . . , xn]. For I ⊂ {1, . . . , r}, set mI := lcm(mi : i ∈ I).

Dave Bayer, Irena Peeva, Bernd Sturmfels Monomial Resolutions April 25, 2018 4 / 14

Theorem 3.2

When M is a generic monomial ideal, the minimal free resolution of S/M

• ver S is defined by the Scarf complex ∆M.

Consider M = m1, . . . , mr ⊂ k[x1, . . . , xn]. For I ⊂ {1, . . . , r}, set mI := lcm(mi : i ∈ I).

Definition (Scarf Complex)

∆M := {I ⊂ {1, . . . , r} : mI = mJ ⇐ ⇒ I = J}

Dave Bayer, Irena Peeva, Bernd Sturmfels Monomial Resolutions April 25, 2018 4 / 14

Example: I = a2, ab, b3 ⊂ k[a, b]

Dave Bayer, Irena Peeva, Bernd Sturmfels Monomial Resolutions April 25, 2018 5 / 14

Example: I = a2, ab, b3 ⊂ k[a, b] I mI {a2, ab} a2b {ab, b3} ab3 {a2, b3} a2b3 {a2, ab, b3} a2b3

a2 b3 ab a2b ab3 a2b3 a2b3

SI : 0 → S[a2, ab] ⊕ S[ab, b3]

  

−b a −b2 a

  

− − − − − − − − − − → S[a2] ⊕ S[ab] ⊕ S[b3]

• a2

ab b3 − − − − − − − − − − − → S[∅] → S/I → 0 Dave Bayer, Irena Peeva, Bernd Sturmfels Monomial Resolutions April 25, 2018 5 / 14

Example: I = a2, ab, b3 ⊂ k[a, b] I mI {a2, ab} a2b {ab, b3} ab3 {a2, b3} a2b3 {a2, ab, b3} a2b3

a2 b3 ab a2b ab3

Dave Bayer, Irena Peeva, Bernd Sturmfels Monomial Resolutions April 25, 2018 6 / 14

Example: I = a2, ab, b3 ⊂ k[a, b] I mI {a2, ab} a2b {ab, b3} ab3 {a2, b3} a2b3 {a2, ab, b3} a2b3

a2 b3 ab a2b ab3

SI : 0 → S[a2, ab] ⊕ S[ab, b3]

  

−b a −b2 a

  

− − − − − − − − − − → S[a2] ⊕ S[ab] ⊕ S[b3]

• a2

ab b3 − − − − − − − − − − − → S[∅] → S/I → 0 Dave Bayer, Irena Peeva, Bernd Sturmfels Monomial Resolutions April 25, 2018 6 / 14

What if M isn’t a generic monomial ideal?

Dave Bayer, Irena Peeva, Bernd Sturmfels Monomial Resolutions April 25, 2018 7 / 14

What if M isn’t a generic monomial ideal? Then we proceed via “deformation of exponents”.

Dave Bayer, Irena Peeva, Bernd Sturmfels Monomial Resolutions April 25, 2018 7 / 14

What if M isn’t a generic monomial ideal? Then we proceed via “deformation of exponents”. For non-generic M = m1, . . . , mr: let {ai = (ai1, . . . , ain) : 1 ≤ i ≤ r} be the exponent vectors of the minimal generators of M.

Dave Bayer, Irena Peeva, Bernd Sturmfels Monomial Resolutions April 25, 2018 7 / 14

What if M isn’t a generic monomial ideal? Then we proceed via “deformation of exponents”. For non-generic M = m1, . . . , mr: let {ai = (ai1, . . . , ain) : 1 ≤ i ≤ r} be the exponent vectors of the minimal generators of M. Choose ǫi = (ǫi1, . . . , ǫin) ∈ Rn for 1 ≤ i ≤ r such that ais + ǫis and ait + ǫit are distinct for all i and s = t, and ais + ǫis < ait + ǫit implies ais ≤ ait.

Dave Bayer, Irena Peeva, Bernd Sturmfels Monomial Resolutions April 25, 2018 7 / 14

What if M isn’t a generic monomial ideal? Then we proceed via “deformation of exponents”. For non-generic M = m1, . . . , mr: let {ai = (ai1, . . . , ain) : 1 ≤ i ≤ r} be the exponent vectors of the minimal generators of M. Choose ǫi = (ǫi1, . . . , ǫin) ∈ Rn for 1 ≤ i ≤ r such that ais + ǫis and ait + ǫit are distinct for all i and s = t, and ais + ǫis < ait + ǫit implies ais ≤ ait.

Definition

The generic deformation of M is Mǫ := m1 · xǫ1, m2 · xǫ2, . . . , mr · xǫr

Dave Bayer, Irena Peeva, Bernd Sturmfels Monomial Resolutions April 25, 2018 7 / 14

Definition

The generic deformation of M is Mǫ := m1 · xǫ1, m2 · xǫ2, . . . , mr · xǫr

Dave Bayer, Irena Peeva, Bernd Sturmfels Monomial Resolutions April 25, 2018 8 / 14

Definition

The generic deformation of M is Mǫ := m1 · xǫ1, m2 · xǫ2, . . . , mr · xǫr Let ∆Mǫ := the Scarf complex of Mǫ. Label the vertex of ∆Mǫ corresponding to mi · xǫi with the original monomial mi. Let Fǫ denote the complex defined by this labeling.

Dave Bayer, Irena Peeva, Bernd Sturmfels Monomial Resolutions April 25, 2018 8 / 14

Definition

The generic deformation of M is Mǫ := m1 · xǫ1, m2 · xǫ2, . . . , mr · xǫr Let ∆Mǫ := the Scarf complex of Mǫ. Label the vertex of ∆Mǫ corresponding to mi · xǫi with the original monomial mi. Let Fǫ denote the complex defined by this labeling.

Theorem

The complex Fǫ is a free resolution of S/M over S. Note: Fǫ is not necessarily minimal.

Dave Bayer, Irena Peeva, Bernd Sturmfels Monomial Resolutions April 25, 2018 8 / 14

M = x2, xy2z, y2z2, yz2w, w2

Dave Bayer, Irena Peeva, Bernd Sturmfels Monomial Resolutions April 25, 2018 9 / 14

M = x2, xy2z, y2z2, yz2w, w2 Mǫ = x2, xy2z, y3z3, yz2w, w2

Dave Bayer, Irena Peeva, Bernd Sturmfels Monomial Resolutions April 25, 2018 9 / 14

M = x2, xy2z, y2z2, yz2w, w2 Mǫ = x2, xy2z, y3z3, yz2w, w2 So ǫ3 = (0, 1, 0, 1) and ǫ1 = ǫ2 = ǫ4 = ǫ5 = (0, 0, 0, 0).

Dave Bayer, Irena Peeva, Bernd Sturmfels Monomial Resolutions April 25, 2018 9 / 14

M = x2, xy2z, y2z2, yz2w, w2 Mǫ = x2, xy2z, y3z3, yz2w, w2 So ǫ3 = (0, 1, 0, 1) and ǫ1 = ǫ2 = ǫ4 = ǫ5 = (0, 0, 0, 0). I mI {1, 2} x2y2z {1, 3} x2y3z3 {1, 4} x2yz2w {1, 5} x2w2 {2, 3} xy3z3 {2, 4} xy3z2w {2, 5} xy2zw {3, 4} y3z3w {3, 5} y3z3w2 I mI {4, 5} yz2w2 {1, 2, 3} x2y3z3 {1, 2, 4} x2y2zw {1, 2, 5} x2y2zw2 {1, 3, 4} x2y3z3w {1, 3, 5} x2y3z3w2 {1, 4, 5} x2yz2w2 {2, 3, 4} xy3z3w {2, 3, 5} xy3z3w2 I mI {3, 4, 5} y3z3w2 {1, 2, 3, 4} x2y3z3w {1, 2, 3, 5} x3y3z3w2 {1, 2, 4, 5} x2y2z2w2 {1, 3, 4, 5} x2y3z3w2 {2, 3, 4, 5} xy3z3w2 {1, 2, 3, 4, 5} x2y3z3w2

Dave Bayer, Irena Peeva, Bernd Sturmfels Monomial Resolutions April 25, 2018 9 / 14

M = x2, xy2z, y2z2, yz2w, w2 Mǫ = x2, xy2z, y3z3, yz2w, w2 So ǫ3 = (0, 1, 0, 1) and ǫ1 = ǫ2 = ǫ4 = ǫ5 = (0, 0, 0, 0). I mI {1, 2} x2y2z {1, 3} x2y3z3 {1, 4} x2yz2w {1, 5} x2w2 {2, 3} xy3z3 {2, 4} xy3z2w {2, 5} xy2zw {3, 4} y3z3w {3, 5} y3z3w2 I mI {4, 5} yz2w2 {1, 2, 3} x2y3z3 {1, 2, 4} x2y2zw {1, 2, 5} x2y2zw2 {1, 3, 4} x2y3z3w {1, 3, 5} x2y3z3w2 {1, 4, 5} x2yz2w2 {2, 3, 4} xy3z3w {2, 3, 5} xy3z3w2 I mI {3, 4, 5} y3z3w2 {1, 2, 3, 4} x2y3z3w {1, 2, 3, 5} x3y3z3w2 {1, 2, 4, 5} x2y2z2w2 {1, 3, 4, 5} x2y3z3w2 {2, 3, 4, 5} xy3z3w2 {1, 2, 3, 4, 5} x2y3z3w2

Dave Bayer, Irena Peeva, Bernd Sturmfels Monomial Resolutions April 25, 2018 10 / 14

Scarf complex for Mǫ, labeled by original generators of M:

xy2z x2 y2z2 yz2w w2

Dave Bayer, Irena Peeva, Bernd Sturmfels Monomial Resolutions April 25, 2018 11 / 14

From which we get the free resolution:

0 → S[xy2z, x2, w2, yz2w] → S[xy2z, x2, w2] ⊕ S[xy2z, yz2w, x2] ⊕ S[xy2z, yz2w, y2z2] ⊕ S[yz2w, x2, w2] → S[x2, xy2z] ⊕ S[x2, w2] ⊕ S[x2, yz2w] ⊕ S[xyz , w2] ⊕ S[xy2z, yz2w] ⊕ S[xy2z, y2z2] ⊕ S[w2, yz2w] ⊕ S[yz2w, w2] → S[x2] ⊕ S[xy2z] ⊕ S[y2z2] ⊕ S[yz2w] ⊕ S[w2] → S/M → 0 Dave Bayer, Irena Peeva, Bernd Sturmfels Monomial Resolutions April 25, 2018 12 / 14

Corollary 3.3(a)

For a generic monomial ideal M, the number of j-faces of the Scarf complex ∆M equals the total Betti number βj+1.

Corollary 4.4

Suppose M is not generic. Then the Betti numbers of M are less than or equal to those of any deformation Mǫ - i.e., less than or equal to the face numbers of the Scarf complex ∆Mǫ.

Dave Bayer, Irena Peeva, Bernd Sturmfels Monomial Resolutions April 25, 2018 13 / 14

Sources:

• D. Bayer, I. Peeva, and B. Sturmfels, Monomial Resolutions. Math

Res Lett. 5 (1998), no 1-2, 31-46. Jeff Mermin, Three simplicial resolutions. arXiv:1102.5062 (2011).

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