Conductor ideals of affine monoids and K -theory Joseph Gubeladze San Francisco State University AMS Special Session: Combinatorial Ideals and Applications Fargo, 2016
Outline • Frobenius number of a numerical semigroup • Affine monoid, normalization, seminormalization • Conductor ideals & gaps in affine monoids • Crash course in K -theory • Affine monoid rings and their K -theory • Nilpotence of higher K -theory of toric varieties • Conjecture Conductors and K -theory • Joseph Gubeladze 2
Frobenius number of a numerical semigroup Numerical semigroup is a sub-semigroup S ⊂ Z ≥ 0 such that the set of gaps Z \ S is finite Conductors and K -theory • Joseph Gubeladze 3
Frobenius number of a numerical semigroup Numerical semigroup is a sub-semigroup S ⊂ Z ≥ 0 such that the set of gaps Z \ S is finite The Frobenius number of a numerical semigroup S is the largest gap of S Conductors and K -theory • Joseph Gubeladze 4
Frobenius number of a numerical semigroup Numerical semigroup is a sub-semigroup S ⊂ Z ≥ 0 such that the set of gaps Z \ S is finite The Frobenius number of a numerical semigroup S is the largest gap of S Every numerical semigroup S is generated by finitely many integers a 1 , . . . , a n with gcd( a 1 , . . . , a n ) = 1 Conductors and K -theory • Joseph Gubeladze 5
Frobenius number of a numerical semigroup Numerical semigroup is a sub-semigroup S ⊂ Z ≥ 0 such that the set of gaps Z \ S is finite The Frobenius number of a numerical semigroup S is the largest gap of S Every numerical semigroup S is generated by finitely many integers a 1 , . . . , a n with gcd( a 1 , . . . , a n ) = 1 Computing the Frobenius number of g ( a 1 , . . . , a n ) of Z ≥ 0 a 1 + · · · + Z ≥ 0 a n is hard . The only value of n for which there is a formula is n = 2 : g ( a 1 , a 2 ) = a 1 a 2 − a 1 − a 2 Conductors and K -theory • Joseph Gubeladze 6
Frobenius number of a numerical semigroup Numerical semigroup is a sub-semigroup S ⊂ Z ≥ 0 such that the set of gaps Z \ S is finite The Frobenius number of a numerical semigroup S is the largest gap of S Every numerical semigroup S is generated by finitely many integers a 1 , . . . , a n with gcd( a 1 , . . . , a n ) = 1 Computing the Frobenius number of g ( a 1 , . . . , a n ) of Z ≥ 0 a 1 + · · · + Z ≥ 0 a n is hard . The only value of n for which there is a formula is n = 2 : g ( a 1 , a 2 ) = a 1 a 2 − a 1 − a 2 Huge existing literature – Postage Stamp Problem , Coin Problem , McNugget Problem (special case) , Arnold Conjecture (on asymptotics of g ( a 1 , . . . , a n ) ), etc Conductors and K -theory • Joseph Gubeladze 7
Affine monoids, normalization, seminormalization An affine monoid is a finitely generated submonoid M ⊂ Z d A positive affine monoid is an affine monoid with no non-zero ± x ∈ M Conductors and K -theory • Joseph Gubeladze 8
Affine monoids, normalization, seminormalization An affine monoid is a finitely generated submonoid M ⊂ Z d A positive affine monoid is an affine monoid with no non-zero ± x ∈ M A positive affine monoid M ⊂ Z d defines a rational polyhedral cone C ( M ) := R ≥ 0 M ⊂ R d Conductors and K -theory • Joseph Gubeladze 9
Affine monoids, normalization, seminormalization An affine monoid is a finitely generated submonoid M ⊂ Z d A positive affine monoid is an affine monoid with no non-zero ± x ∈ M A positive affine monoid M ⊂ Z d defines a rational polyhedral cone C ( M ) := R ≥ 0 M ⊂ R d The subgroup of Z d , generated by M , is the group of differences of M and denoted gp( M ) Conductors and K -theory • Joseph Gubeladze 10
Affine monoids, normalization, seminormalization An affine monoid is a finitely generated submonoid M ⊂ Z d A positive affine monoid is an affine monoid with no non-zero ± x ∈ M A positive affine monoid M ⊂ Z d defines a rational polyhedral cone C ( M ) := R ≥ 0 M ⊂ R d The subgroup of Z d , generated by M , is the group of differences of M and denoted gp( M ) A (positive) affine monoid M is normal if x ∈ gp( M ) & nx ∈ C ( M ) for some n = ⇒ x ∈ M Conductors and K -theory • Joseph Gubeladze 11
Affine monoids, normalization, seminormalization An affine monoid is a finitely generated submonoid M ⊂ Z d A positive affine monoid is an affine monoid with no non-zero ± x ∈ M A positive affine monoid M ⊂ Z d defines a rational polyhedral cone C ( M ) := R ≥ 0 M ⊂ R d The subgroup of Z d , generated by M , is the group of differences of M and denoted gp( M ) A (positive) affine monoid M is normal if x ∈ gp( M ) & nx ∈ C ( M ) for some n = ⇒ x ∈ M A (positive) affine monoid M is seminormal if x ∈ gp( M ) & 2 x, 3 x ∈ M for some n = ⇒ x ∈ M Conductors and K -theory • Joseph Gubeladze 12
Affine monoids, normalization, seminormalization All affine affine monoids from this point on are positive Conductors and K -theory • Joseph Gubeladze 13
Affine monoids, normalization, seminormalization All affine affine monoids from this point on are positive ¯ M ⊂ Z d The normalization of M is the smallest normal submonoid containing M , i.e., ¯ M = C ( M ) ∩ gp( M ) – ‘saturation’ of M Conductors and K -theory • Joseph Gubeladze 14
Affine monoids, normalization, seminormalization All affine affine monoids from this point on are positive ¯ M ⊂ Z d The normalization of M is the smallest normal submonoid containing M , i.e., ¯ M = C ( M ) ∩ gp( M ) – ‘saturation’ of M The seminormalization of M is the smallest seminormal submonoid sn( M ) ⊂ Z d containing M , i.e., M = { x ∈ Z d | 2 x, 3 x ∈ M } ¯ – ‘saturation’ of M along the rational rays inside the cone C ( M ) Conductors and K -theory • Joseph Gubeladze 15
Affine monoids, normalization, seminormalization All affine affine monoids from this point on are positive ¯ M ⊂ Z d The normalization of M is the smallest normal submonoid containing M , i.e., ¯ M = C ( M ) ∩ gp( M ) – ‘saturation’ of M The seminormalization of M is the smallest seminormal submonoid sn( M ) ⊂ Z d containing M , i.e., M = { x ∈ Z d | 2 x, 3 x ∈ M } ¯ – ‘saturation’ of M along the rational rays inside the cone C ( M ) REMARK. F ∩ sn( M ) = sn( F ∩ M ) for every face F ⊂ C ( M ) Conductors and K -theory • Joseph Gubeladze 16
Affine monoids, normalization, seminormalization All affine affine monoids from this point on are positive ¯ M ⊂ Z d The normalization of M is the smallest normal submonoid containing M , i.e., ¯ M = C ( M ) ∩ gp( M ) – ‘saturation’ of M The seminormalization of M is the smallest seminormal submonoid sn( M ) ⊂ Z d containing M , i.e., M = { x ∈ Z d | 2 x, 3 x ∈ M } ¯ – ‘saturation’ of M along the rational rays inside the cone C ( M ) REMARK. F ∩ sn( M ) = sn( F ∩ M ) for every face F ⊂ C ( M ) ¯ FACT. M ∩ int C ( M ) = sn( M ) ∩ int C ( M ) Conductors and K -theory • Joseph Gubeladze 17
Conductor ideals & gaps in affine monoids The conductor ideal of an affine monoid M is M/M := { x ∈ ¯ M | x + ¯ c ¯ M ⊂ M } ⊂ M It is an ideal of M because c ¯ M/M + M ⊂ M Conductors and K -theory • Joseph Gubeladze 18
Conductor ideals & gaps in affine monoids The conductor ideal of an affine monoid M is M/M := { x ∈ ¯ M | x + ¯ c ¯ M ⊂ M } ⊂ M It is an ideal of M because c ¯ M/M + M ⊂ M FACT. c ¯ M/M � = ∅ Conductors and K -theory • Joseph Gubeladze 19
Conductor ideals & gaps in affine monoids The conductor ideal of an affine monoid M is M/M := { x ∈ ¯ M | x + ¯ c ¯ M ⊂ M } ⊂ M It is an ideal of M because c ¯ M/M + M ⊂ M FACT. c ¯ M/M � = ∅ Proof. Let ¯ M is module finite over M . Let { x 1 − y 1 , . . . , x n − y n } ⊂ gp( M ) be a generating set x i , y i ∈ M . Then y 1 + · · · + y n ∈ c ¯ M/M . ✷ Conductors and K -theory • Joseph Gubeladze 20
Conductor ideals & gaps in affine monoids The conductor ideal of an affine monoid M is M/M := { x ∈ ¯ M | x + ¯ c ¯ M ⊂ M } ⊂ M It is an ideal of M because c ¯ M/M + M ⊂ M FACT. c ¯ M/M � = ∅ Proof. Let ¯ M is module finite over M . Let { x 1 − y 1 , . . . , x n − y n } ⊂ gp( M ) be a generating set x i , y i ∈ M . Then y 1 + · · · + y n ∈ c ¯ M/M . ✷ (Katth¨ an, 2015) l ¯ � M \ M = ( q j + gp( M ∩ F )) ∩ C ( M ) , j =1 where the F j are faces of the cone C ( M ) and q j ∈ ¯ M Conductors and K -theory • Joseph Gubeladze 21
Conductor ideals & gaps in affine monoids The elements of sn( M ) \ M are gaps of M . Different from the set ¯ M \ M Conductors and K -theory • Joseph Gubeladze 22
Conductor ideals & gaps in affine monoids The elements of sn( M ) \ M are gaps of M . Different from the set ¯ M \ M For a numerical semigroup S , this is the same as ¯ S \ S Conductors and K -theory • Joseph Gubeladze 23
Conductor ideals & gaps in affine monoids The elements of sn( M ) \ M are gaps of M . Different from the set ¯ M \ M For a numerical semigroup S , this is the same as ¯ S \ S Moreover, c ¯ S/S = g ( S ) + Z > 0 , where g ( S ) is the Frobenius number of S Conductors and K -theory • Joseph Gubeladze 24
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