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 [ x 1 , . . . , x n ], and M be a monomial ideal of S . Dave Bayer, Irena Peeva, Bernd Sturmfels Monomial Resolutions April 25, 2018 2 / 14

Let S = k [ x 1 , . . . , x n ], 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 [ x 1 , . . . , x n ], 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 [ x 1 , . . . , x n ], 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 [ x 1 , . . . , x n ], 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 [ x 1 , . . . , x n ] is a generic monomial ideal if no variable x i 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 over 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 over S is defined by the Scarf complex ∆ M . Consider M = � m 1 , . . . , m r � ⊂ k [ x 1 , . . . , x n ]. For I ⊂ { 1 , . . . , r } , set m I := lcm ( m i : 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 over S is defined by the Scarf complex ∆ M . Consider M = � m 1 , . . . , m r � ⊂ k [ x 1 , . . . , x n ]. For I ⊂ { 1 , . . . , r } , set m I := lcm ( m i : i ∈ I ). Definition (Scarf Complex) ∆ M := { I ⊂ { 1 , . . . , r } : m I = m J ⇐ ⇒ I = J } Dave Bayer, Irena Peeva, Bernd Sturmfels Monomial Resolutions April 25, 2018 4 / 14

Example: I = � a 2 , ab , b 3 � ⊂ k [ a , b ] Dave Bayer, Irena Peeva, Bernd Sturmfels Monomial Resolutions April 25, 2018 5 / 14

Example: I = � a 2 , ab , b 3 � ⊂ k [ a , b ] ab I m I { a 2 , ab } a 2 b { ab , b 3 } ab 3 a 2 b ab 3 a 2 b 3 { a 2 , b 3 } a 2 b 3 { a 2 , ab , b 3 } a 2 b 3 a 2 a 2 b 3 b 3 − b  0  S [ a 2 ] − b 2  a  S [ a 2 , ab ]   ⊕ � a 2 b 3 � 0 ab a S I : 0 → ⊕ − − − − − − − − − − → S [ ab ] − − − − − − − − − − − → S [ ∅ ] → S / I → 0 S [ ab , b 3 ] ⊕ S [ b 3 ] Dave Bayer, Irena Peeva, Bernd Sturmfels Monomial Resolutions April 25, 2018 5 / 14

Example: I = � a 2 , ab , b 3 � ⊂ k [ a , b ] ab I m I { a 2 , ab } a 2 b { ab , b 3 } ab 3 a 2 b ab 3 { a 2 , b 3 } a 2 b 3 { a 2 , ab , b 3 } a 2 b 3 a 2 b 3 Dave Bayer, Irena Peeva, Bernd Sturmfels Monomial Resolutions April 25, 2018 6 / 14

Example: I = � a 2 , ab , b 3 � ⊂ k [ a , b ] ab I m I { a 2 , ab } a 2 b { ab , b 3 } ab 3 a 2 b ab 3 { a 2 , b 3 } a 2 b 3 { a 2 , ab , b 3 } a 2 b 3 a 2 b 3 − b  0  S [ a 2 ] − b 2  a  S [ a 2 , ab ]   ⊕ � a 2 b 3 � 0 ab a S I : 0 → ⊕ − − − − − − − − − − → S [ ab ] − − − − − − − − − − − → S [ ∅ ] → S / I → 0 S [ ab , b 3 ] ⊕ S [ b 3 ] 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 = � m 1 , . . . , m r � : let { a i = ( a i 1 , . . . , a in ) : 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 = � m 1 , . . . , m r � : let { a i = ( a i 1 , . . . , a in ) : 1 ≤ i ≤ r } be the exponent vectors of the minimal generators of M . Choose ǫ i = ( ǫ i 1 , . . . , ǫ in ) ∈ R n for 1 ≤ i ≤ r such that a is + ǫ is and a it + ǫ it are distinct for all i and s � = t , and a is + ǫ is < a it + ǫ it implies a is ≤ a it . 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 = � m 1 , . . . , m r � : let { a i = ( a i 1 , . . . , a in ) : 1 ≤ i ≤ r } be the exponent vectors of the minimal generators of M . Choose ǫ i = ( ǫ i 1 , . . . , ǫ in ) ∈ R n for 1 ≤ i ≤ r such that a is + ǫ is and a it + ǫ it are distinct for all i and s � = t , and a is + ǫ is < a it + ǫ it implies a is ≤ a it . Definition The generic deformation of M is M ǫ := � m 1 · x ǫ 1 , m 2 · x ǫ 2 , . . . , m r · x ǫ r � Dave Bayer, Irena Peeva, Bernd Sturmfels Monomial Resolutions April 25, 2018 7 / 14

Definition The generic deformation of M is M ǫ := � m 1 · x ǫ 1 , m 2 · x ǫ 2 , . . . , m r · x ǫ r � Dave Bayer, Irena Peeva, Bernd Sturmfels Monomial Resolutions April 25, 2018 8 / 14

Definition The generic deformation of M is M ǫ := � m 1 · x ǫ 1 , m 2 · x ǫ 2 , . . . , m r · x ǫ r � Let ∆ M ǫ := the Scarf complex of M ǫ . Label the vertex of ∆ M ǫ corresponding to m i · x ǫ i with the original monomial m i . Let F ǫ 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 M ǫ := � m 1 · x ǫ 1 , m 2 · x ǫ 2 , . . . , m r · x ǫ r � Let ∆ M ǫ := the Scarf complex of M ǫ . Label the vertex of ∆ M ǫ corresponding to m i · x ǫ i with the original monomial m i . Let F ǫ denote the complex defined by this labeling. Theorem The complex F ǫ is a free resolution of S / M over S . Note: F ǫ is not necessarily minimal . Dave Bayer, Irena Peeva, Bernd Sturmfels Monomial Resolutions April 25, 2018 8 / 14

M = � x 2 , xy 2 z , y 2 z 2 , yz 2 w , w 2 � Dave Bayer, Irena Peeva, Bernd Sturmfels Monomial Resolutions April 25, 2018 9 / 14

M = � x 2 , xy 2 z , y 2 z 2 , yz 2 w , w 2 � M ǫ = � x 2 , xy 2 z , y 3 z 3 , yz 2 w , w 2 � Dave Bayer, Irena Peeva, Bernd Sturmfels Monomial Resolutions April 25, 2018 9 / 14

M = � x 2 , xy 2 z , y 2 z 2 , yz 2 w , w 2 � M ǫ = � x 2 , xy 2 z , y 3 z 3 , yz 2 w , w 2 � So ǫ 3 = (0 , 1 , 0 , 1) and ǫ 1 = ǫ 2 = ǫ 4 = ǫ 5 = (0 , 0 , 0 , 0). Dave Bayer, Irena Peeva, Bernd Sturmfels Monomial Resolutions April 25, 2018 9 / 14

M = � x 2 , xy 2 z , y 2 z 2 , yz 2 w , w 2 � M ǫ = � x 2 , xy 2 z , y 3 z 3 , yz 2 w , w 2 � So ǫ 3 = (0 , 1 , 0 , 1) and ǫ 1 = ǫ 2 = ǫ 4 = ǫ 5 = (0 , 0 , 0 , 0). I m I I m I I m I x 2 y 2 z yz 2 w 2 y 3 z 3 w 2 { 1 , 2 } { 4 , 5 } { 3 , 4 , 5 } x 2 y 3 z 3 x 2 y 3 z 3 x 2 y 3 z 3 w { 1 , 3 } { 1 , 2 , 3 } { 1 , 2 , 3 , 4 } x 2 yz 2 w x 2 y 2 zw x 3 y 3 z 3 w 2 { 1 , 4 } { 1 , 2 , 4 } { 1 , 2 , 3 , 5 } x 2 w 2 x 2 y 2 zw 2 x 2 y 2 z 2 w 2 { 1 , 5 } { 1 , 2 , 5 } { 1 , 2 , 4 , 5 } xy 3 z 3 x 2 y 3 z 3 w x 2 y 3 z 3 w 2 { 2 , 3 } { 1 , 3 , 4 } { 1 , 3 , 4 , 5 } xy 3 z 2 w x 2 y 3 z 3 w 2 xy 3 z 3 w 2 { 2 , 4 } { 1 , 3 , 5 } { 2 , 3 , 4 , 5 } xy 2 zw x 2 yz 2 w 2 x 2 y 3 z 3 w 2 { 2 , 5 } { 1 , 4 , 5 } { 1 , 2 , 3 , 4 , 5 } y 3 z 3 w xy 3 z 3 w { 3 , 4 } { 2 , 3 , 4 } y 3 z 3 w 2 xy 3 z 3 w 2 { 3 , 5 } { 2 , 3 , 5 } Dave Bayer, Irena Peeva, Bernd Sturmfels Monomial Resolutions April 25, 2018 9 / 14