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
a1, . . . , an with gcd(a1, . . . , an) = 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
a1, . . . , an with gcd(a1, . . . , an) = 1
Computing the Frobenius number of g(a1, . . . , an) of Z≥0a1 + · · · + Z≥0an
is hard. The only value of n for which there is a formula is n = 2 :
g(a1, a2) = a1a2 − a1 − a2
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
a1, . . . , an with gcd(a1, . . . , an) = 1
Computing the Frobenius number of g(a1, . . . , an) of Z≥0a1 + · · · + Z≥0an
is hard. The only value of n for which there is a formula is n = 2 :
g(a1, a2) = a1a2 − a1 − a2
Huge existing literature – Postage Stamp Problem, Coin Problem,
McNugget Problem (special case), Arnold Conjecture (on asymptotics
of g(a1, . . . , an) ), etc
Conductors and K -theory
• Joseph Gubeladze
7
Affine monoids, normalization, seminormalization
An affine monoid is a finitely generated submonoid M ⊂ Zd
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 ⊂ Zd
A positive affine monoid is an affine monoid with no non-zero ±x ∈ M
A positive affine monoid M ⊂ Zd defines a rational polyhedral cone
C(M ) := R≥0M ⊂ Rd
Conductors and K -theory
• Joseph Gubeladze
9
Affine monoids, normalization, seminormalization
An affine monoid is a finitely generated submonoid M ⊂ Zd
A positive affine monoid is an affine monoid with no non-zero ±x ∈ M
A positive affine monoid M ⊂ Zd defines a rational polyhedral cone
C(M ) := R≥0M ⊂ Rd
The subgroup of Zd , 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 ⊂ Zd
A positive affine monoid is an affine monoid with no non-zero ±x ∈ M
A positive affine monoid M ⊂ Zd defines a rational polyhedral cone
C(M ) := R≥0M ⊂ Rd
The subgroup of Zd , 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 ⊂ Zd
A positive affine monoid is an affine monoid with no non-zero ±x ∈ M
A positive affine monoid M ⊂ Zd defines a rational polyhedral cone
C(M ) := R≥0M ⊂ Rd
The subgroup of Zd , 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 ) & 2x, 3x ∈ 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
The normalization of M is the smallest normal submonoid M̄ ⊂ Zd
containing M , i.e., M̄ = 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
The normalization of M is the smallest normal submonoid M̄ ⊂ Zd
containing M , i.e., M̄ = C(M ) ∩ gp(M ) – ‘saturation’ of M
The seminormalization of M is the smallest seminormal submonoid sn(M ) ⊂
Zd containing M , i.e.,
M̄ = {x ∈ Zd | 2x, 3x ∈ 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
The normalization of M is the smallest normal submonoid M̄ ⊂ Zd
containing M , i.e., M̄ = C(M ) ∩ gp(M ) – ‘saturation’ of M
The seminormalization of M is the smallest seminormal submonoid sn(M ) ⊂
Zd containing M , i.e.,
M̄ = {x ∈ Zd | 2x, 3x ∈ 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
The normalization of M is the smallest normal submonoid M̄ ⊂ Zd
containing M , i.e., M̄ = C(M ) ∩ gp(M ) – ‘saturation’ of M
The seminormalization of M is the smallest seminormal submonoid sn(M ) ⊂
Zd containing M , i.e.,
M̄ = {x ∈ Zd | 2x, 3x ∈ M }
– ‘saturation’ of M along the rational rays inside the cone C(M )
REMARK.
FACT.
F ∩ sn(M ) = sn(F ∩ M ) for every face F ⊂ C(M )
M̄ ∩ 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
cM̄ /M := {x ∈ M̄ | x + M̄ ⊂ M } ⊂ M
It is an ideal of M because cM̄ /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
cM̄ /M := {x ∈ M̄ | x + M̄ ⊂ M } ⊂ M
It is an ideal of M because cM̄ /M +M ⊂ M
FACT.
cM̄ /M 6= ∅
Conductors and K -theory
• Joseph Gubeladze
19
Conductor ideals & gaps in affine monoids
The conductor ideal of an affine monoid M is
cM̄ /M := {x ∈ M̄ | x + M̄ ⊂ M } ⊂ M
It is an ideal of M because cM̄ /M +M ⊂ M
FACT.
cM̄ /M 6= ∅
Proof. Let M̄ is module finite over M . Let {x1 −y1, . . . , xn −yn} ⊂ gp(M )
be a generating set xi, yi ∈ M . Then y1 + · · · + yn ∈ cM̄ /M . 2
Conductors and K -theory
• Joseph Gubeladze
20
Conductor ideals & gaps in affine monoids
The conductor ideal of an affine monoid M is
cM̄ /M := {x ∈ M̄ | x + M̄ ⊂ M } ⊂ M
It is an ideal of M because cM̄ /M +M ⊂ M
FACT.
cM̄ /M 6= ∅
Proof. Let M̄ is module finite over M . Let {x1 −y1, . . . , xn −yn} ⊂ gp(M )
be a generating set xi, yi ∈ M . Then y1 + · · · + yn ∈ cM̄ /M . 2
(Katthän, 2015)
M̄ \ M =
l
[
(qj + gp(M ∩ F )) ∩ C(M ),
j=1
where the Fj are faces of the cone C(M ) and qj ∈ M̄
Conductors and K -theory
• Joseph Gubeladze
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Conductor ideals & gaps in affine monoids
The elements of sn(M ) \ M are gaps of M . Different from the set M̄ \ 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 M̄ \ M
For a numerical semigroup S , this is the same as S̄ \ 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 M̄ \ M
For a numerical semigroup S , this is the same as S̄ \ S
Moreover, cS̄/S = g(S) + Z>0 , where g(S) is the Frobenius number of S
Conductors and K -theory
• Joseph Gubeladze
24
Conductor ideals & gaps in affine monoids
The elements of sn(M ) \ M are gaps of M . Different from the set M̄ \ M
For a numerical semigroup S , this is the same as S̄ \ S
Moreover, cS̄/S = g(S) + Z>0 , where g(S) is the Frobenius number of S
(Reid-Roberts, 2001) Let {v1, . . . , vd, vd+1} ⊂ Zd≥0 be a circuit (no d
elements are linearly dependent) and M = Z≥0v1 + · · · + Z≥0v1 .
Conductors and K -theory
• Joseph Gubeladze
25
Conductor ideals & gaps in affine monoids
The elements of sn(M ) \ M are gaps of M . Different from the set M̄ \ M
For a numerical semigroup S , this is the same as S̄ \ S
Moreover, cS̄/S = g(S) + Z>0 , where g(S) is the Frobenius number of S
(Reid-Roberts, 2001) Let {v1, . . . , vd, vd+1} ⊂ Zd≥0 be a circuit (no d
elements are linearly dependent) and M = Z≥0v1 + · · · + Z≥0v1 . Then
cM̄ /M
Conductors and K -theory
= g + int C(M ) ∩ gp(M )
• Joseph Gubeladze
26
Conductor ideals & gaps in affine monoids
The elements of sn(M ) \ M are gaps of M . Different from the set M̄ \ M
For a numerical semigroup S , this is the same as S̄ \ S
Moreover, cS̄/S = g(S) + Z>0 , where g(S) is the Frobenius number of S
(Reid-Roberts, 2001) Let {v1, . . . , vd, vd+1} ⊂ Zd≥0 be a circuit (no d
elements are linearly dependent) and M = Z≥0v1 + · · · + Z≥0v1 . Then
cM̄ /M
= g + int C(M ) ∩ gp(M )
where
g=
d+1
X
i=1
divi /2 −
d+1
X
vi
i=1
di being the order of Zd modulo v1, . . . , vi−1, vi+1, . . . , vd+1
Conductors and K -theory
• Joseph Gubeladze
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Crash course in K-theory
Grothendieck’s group K0(R) of a ring R measures how far the projective R
-modules overall are from being free (actually, stably free, which is a certain
functorial weakening of ‘free’’)
Conductors and K -theory
• Joseph Gubeladze
28
Crash course in K-theory
Grothendieck’s group K0(R) of a ring R measures how far the projective R
-modules overall are from being free (actually, stably free, which is a certain
functorial weakening of ‘free’’)
Bass-Whitead group K1(R) measures how far the invertible matrices over
R overall are from being diagonalizable via elementary row (or column)
transformations. (Again, in the stable sense)
Conductors and K -theory
• Joseph Gubeladze
29
Crash course in K-theory
Grothendieck’s group K0(R) of a ring R measures how far the projective R
-modules overall are from being free (actually, stably free, which is a certain
functorial weakening of ‘free’’)
Bass-Whitead group K1(R) measures how far the invertible matrices over
R overall are from being diagonalizable via elementary row (or column)
transformations. (Again, in the stable sense)
Milnor’s group K2(R) measures how many essentially different
diagonalizations overall there exist for all possible diagonalizable invertible
R -matrices
Conductors and K -theory
• Joseph Gubeladze
29
Crash course in K-theory
Grothendieck’s group K0(R) of a ring R measures how far the projective R
-modules overall are from being free (actually, stably free, which is a certain
functorial weakening of ‘free’’)
Bass-Whitead group K1(R) measures how far the invertible matrices over
R overall are from being diagonalizable via elementary row (or column)
transformations. (Again, in the stable sense)
Milnor’s group K2(R) measures how many essentially different
diagonalizations overall there exist for all possible diagonalizable invertible
R -matrices
Higher groups Ki(R) do not admit transparent definitions in terms of
classical algebraic objects, they are higher homotopy variants of K0, K1, K2
Conductors and K -theory
• Joseph Gubeladze
29
Crash course in K-theory
Grothendieck’s group K0(R) of a ring R measures how far the projective R
-modules overall are from being free (actually, stably free, which is a certain
functorial weakening of ‘free’’)
Bass-Whitead group K1(R) measures how far the invertible matrices over
R overall are from being diagonalizable via elementary row (or column)
transformations. (Again, in the stable sense)
Milnor’s group K2(R) measures how many essentially different
diagonalizations overall there exist for all possible diagonalizable invertible
R -matrices
Higher groups Ki(R) do not admit transparent definitions in terms of
classical algebraic objects, they are higher homotopy variants of K0, K1, K2
Informally, these groups are syzygies between elementary transformation of
invertibe matricces over R . Formally, they are higher homotopy groups of
a certain K -theoretical space, associated to R (Quillen, the 1970s)
Conductors and K -theory
• Joseph Gubeladze
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K-theory of monoid rings
Let R be a (commutative) regular ring and M ⊂ Zd an affine monoid
Conductors and K -theory
• Joseph Gubeladze
29
K-theory of monoid rings
Let R be a (commutative) regular ring and M ⊂ Zd an affine monoid
(Grothendieck)
Conductors and K -theory
K0(R) = K0(R[X1, . . . , Xd])
• Joseph Gubeladze
(= K0(R[Zd≥0]))
29
K-theory of monoid rings
Let R be a (commutative) regular ring and M ⊂ Zd an affine monoid
(Grothendieck)
(Quillen)
K0(R) = K0(R[X1, . . . , Xd])
K∗(R) = K∗(R[X1, . . . , Xd])
Conductors and K -theory
• Joseph Gubeladze
(= K0(R[Zd≥0]))
(= K∗(R[Zd≥0]))
29
K-theory of monoid rings
Let R be a (commutative) regular ring and M ⊂ Zd an affine monoid
(Grothendieck)
(Quillen)
(G., 1988)
K0(R) = K0(R[X1, . . . , Xd])
K∗(R) = K∗(R[X1, . . . , Xd])
(= K0(R[Zd≥0]))
(= K∗(R[Zd≥0]))
K0(R) = K0(R[M ]) iff M = sn(M )
Conductors and K -theory
• Joseph Gubeladze
29
K-theory of monoid rings
Let R be a (commutative) regular ring and M ⊂ Zd an affine monoid
(Grothendieck)
(Quillen)
(G., 1988)
Corollary:
K0(R) = K0(R[X1, . . . , Xd])
K∗(R) = K∗(R[X1, . . . , Xd])
(= K0(R[Zd≥0]))
(= K∗(R[Zd≥0]))
K0(R) = K0(R[M ]) iff M = sn(M )
K0(R[M ])/K0(R) ∼
= R(sn(M ) \ M ) when Q ⊂ R
Conductors and K -theory
• Joseph Gubeladze
29
K-theory of monoid rings
Let R be a (commutative) regular ring and M ⊂ Zd an affine monoid
(Grothendieck)
(Quillen)
(G., 1988)
K0(R) = K0(R[X1, . . . , Xd])
K∗(R) = K∗(R[X1, . . . , Xd])
(= K0(R[Zd≥0]))
(= K∗(R[Zd≥0]))
K0(R) = K0(R[M ]) iff M = sn(M )
Corollary:
K0(R[M ])/K0(R) ∼
= R(sn(M ) \ M ) when Q ⊂ R
(G., 1992)
K∗(R) = K∗(R[M ]) iff M ∼
= Zr≥0 for some r ≥ 0
Conductors and K -theory
• Joseph Gubeladze
29
K-theory of monoid rings
Let R be a (commutative) regular ring and M ⊂ Zd an affine monoid
(Grothendieck)
(Quillen)
(G., 1988)
K0(R) = K0(R[X1, . . . , Xd])
K∗(R) = K∗(R[X1, . . . , Xd])
(= K0(R[Zd≥0]))
(= K∗(R[Zd≥0]))
K0(R) = K0(R[M ]) iff M = sn(M )
Corollary:
K0(R[M ])/K0(R) ∼
= R(sn(M ) \ M ) when Q ⊂ R
(G., 1992)
K∗(R) = K∗(R[M ]) iff M ∼
= Zr≥0 for some r ≥ 0
(G., 2005)
Assume Q ⊂ R and c ≥ 2 . Then high iterations of the
homothety M → M , defined by m 7→ cm , kill K∗(R[M ])/K∗(R)
Conductors and K -theory
• Joseph Gubeladze
29
K-theory of monoid rings
Let R be a (commutative) regular ring and M ⊂ Zd an affine monoid
(Grothendieck)
(Quillen)
(G., 1988)
K0(R) = K0(R[X1, . . . , Xd])
K∗(R) = K∗(R[X1, . . . , Xd])
(= K0(R[Zd≥0]))
(= K∗(R[Zd≥0]))
K0(R) = K0(R[M ]) iff M = sn(M )
Corollary:
K0(R[M ])/K0(R) ∼
= R(sn(M ) \ M ) when Q ⊂ R
(G., 1992)
K∗(R) = K∗(R[M ]) iff M ∼
= Zr≥0 for some r ≥ 0
(G., 2005)
Assume Q ⊂ R and c ≥ 2 . Then high iterations of the
homothety M → M , defined by m 7→ cm , kill K∗(R[M ])/K∗(R)
(Cortiñas, Haesemayer, Walker, Weibel, announced in 2016)
condition Q ⊂ R in the statement above can be dropped
Conductors and K -theory
• Joseph Gubeladze
The
29
Conjecture
Conjecture. R a regular ring, containing Q : for every finitely generated
monomial algebra R[M ] without nontrivial units we have the equality
Conductors and K -theory
• Joseph Gubeladze
29
Conjecture
Conjecture. R a regular ring, containing Q : for every finitely generated
monomial algebra R[M ] without nontrivial units we have the equality
Ki(R[M ])/Ki(R) ∼
= (a finitely generated M -graded thin R[M ] -module)
and on this module the map M → M , m 7→ cm , acts by dilating the M
-degrees by factor c .
Conductors and K -theory
• Joseph Gubeladze
29
Conjecture
Conjecture. R a regular ring, containing Q : for every finitely generated
monomial algebra R[M ] without nontrivial units we have the equality
Ki(R[M ])/Ki(R) ∼
= (a finitely generated M -graded thin R[M ] -module)
and on this module the map M → M , m 7→ cm , acts by dilating the M
-degrees by factor c .
Informally, the mentioned thinness means that every element of
Ki(R[M ])/Ki(R) is pushed by sufficiently high iterations of the map
M → M , m 7→ cm , to the M -graded zero zone. In particular, this
conjecture implies the aforementioned nilpotence of Ki(R[M ])/Ki(R) .
Conductors and K -theory
• Joseph Gubeladze
29
Conjecture
Conjecture. R a regular ring, containing Q : for every finitely generated
monomial algebra R[M ] without nontrivial units we have the equality
Ki(R[M ])/Ki(R) ∼
= (a finitely generated M -graded thin R[M ] -module)
and on this module the map M → M , m 7→ cm , acts by dilating the M
-degrees by factor c .
Informally, the mentioned thinness means that every element of
Ki(R[M ])/Ki(R) is pushed by sufficiently high iterations of the map
M → M , m 7→ cm , to the M -graded zero zone. In particular, this
conjecture implies the aforementioned nilpotence of Ki(R[M ])/Ki(R) .
It is known that Ki(R[M ])/Ki(R) is an R -module; this follows from the
Bloch-Stienstra action of the big Witt vectors.
Conductors and K -theory
• Joseph Gubeladze
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Conjecture
Conductors and K -theory
• Joseph Gubeladze
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REFERENCES
J. Gubeladze, K-theory of toric varieties revisited (survey), J. Homotopy
Relat. Struct. 9 (2014), 9–23, and many references therein
L. Katthän, Non-normal affine monoid algebras, manuscripta math. 146,
(2015) 223–233
L. Reid and L. Roberts, Monomial Subrings in Arbitrary Dimension,
Journal of Algebra 236, (2001)707–730
Conductors and K -theory
• Joseph Gubeladze
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REFERENCES
J. Gubeladze, K-theory of toric varieties revisited (survey), J. Homotopy
Relat. Struct. 9 (2014), 9–23, and many references therein
L. Katthän, Non-normal affine monoid algebras, manuscripta math. 146,
(2015) 223–233
L. Reid and L. Roberts, Monomial Subrings in Arbitrary Dimension,
Journal of Algebra 236, (2001)707–730
Thank you
Conductors and K -theory
• Joseph Gubeladze
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