[PPT] - Conductor ideals of affine monoids and K -theory Joseph Gubeladze PowerPoint Presentation

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Conductor ideals of affine monoids and K-theory

Joseph Gubeladze San Francisco State University AMS Special Session: Combinatorial Ideals and Applications Fargo, 2016

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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

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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

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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

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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

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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

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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

f g(a1, . . . , an) ), etc

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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

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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

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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)

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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

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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

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Affine monoids, normalization, seminormalization

All affine affine monoids from this point on are positive

Conductors and K-theory

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Affine monoids, normalization, seminormalization

All affine affine monoids from this point on are positive The normalization of M is the smallest normal submonoid ¯ M ⊂ Zd containing M , i.e., ¯ M = C(M) ∩ gp(M) – ‘saturation’ of M

Conductors and K-theory

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Affine monoids, normalization, seminormalization

All affine affine monoids from this point on are positive The normalization of M is the smallest normal submonoid ¯ M ⊂ Zd containing M , i.e., ¯ M = C(M) ∩ gp(M) – ‘saturation’ of M The seminormalization of M is the smallest seminormal submonoid sn(M) ⊂ Zd containing M , i.e., ¯ M = {x ∈ Zd | 2x, 3x ∈ M} – ‘saturation’ of M along the rational rays inside the cone C(M)

Conductors and K-theory

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Affine monoids, normalization, seminormalization

All affine affine monoids from this point on are positive The normalization of M is the smallest normal submonoid ¯ M ⊂ Zd containing M , i.e., ¯ M = C(M) ∩ gp(M) – ‘saturation’ of M The seminormalization of M is the smallest seminormal submonoid sn(M) ⊂ Zd containing M , i.e., ¯ M = {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

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Affine monoids, normalization, seminormalization

All affine affine monoids from this point on are positive The normalization of M is the smallest normal submonoid ¯ M ⊂ Zd containing M , i.e., ¯ M = C(M) ∩ gp(M) – ‘saturation’ of M The seminormalization of M is the smallest seminormal submonoid sn(M) ⊂ Zd containing M , i.e., ¯ M = {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) FACT. ¯ M ∩ int C(M) = sn(M) ∩ int C(M)

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Conductor ideals & gaps in affine monoids

The conductor ideal of an affine monoid M is c ¯

M/M := {x ∈ ¯

M | x + ¯ M ⊂ M} ⊂ M It is an ideal of M because c ¯

M/M +M ⊂ M

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Conductor ideals & gaps in affine monoids

The conductor ideal of an affine monoid M is c ¯

M/M := {x ∈ ¯

M | x + ¯ M ⊂ M} ⊂ M It is an ideal of M because c ¯

M/M +M ⊂ M

FACT. c ¯

M/M = ∅

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Conductor ideals & gaps in affine monoids

The conductor ideal of an affine monoid M is c ¯

M/M := {x ∈ ¯

M | x + ¯ 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 {x1−y1, . . . , xn−yn} ⊂ gp(M) be a generating set xi, yi ∈ M . Then y1 + · · · + yn ∈ c ¯

M/M . ✷

Conductors and K-theory

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Conductor ideals & gaps in affine monoids

The conductor ideal of an affine monoid M is c ¯

M/M := {x ∈ ¯

M | x + ¯ 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 {x1−y1, . . . , xn−yn} ⊂ gp(M) be a generating set xi, yi ∈ M . Then y1 + · · · + yn ∈ c ¯

M/M . ✷

(Katth¨ an, 2015) ¯ M \ M =

l

j=1

(qj + gp(M ∩ F)) ∩ C(M), where the Fj are faces of the cone C(M) and qj ∈ ¯ M

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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

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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

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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

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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

(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 .

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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

(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 c ¯

M/M = g +

int C(M) ∩ gp(M)
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 ¯ 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

(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 c ¯

M/M = g +

int C(M) ∩ gp(M)
where

g = d+1

i=1

divi

/2 −

d+1

i=1

vi di being the order of Zd modulo v1, . . . , vi−1, vi+1, . . . , vd+1

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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’’)

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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’’) 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

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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’’) 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

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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’’) 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

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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’’) 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

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K-theory of monoid rings

Let R be a (commutative) regular ring and M ⊂ Zd an affine monoid (Grothendieck) K0(R) = K0(R[X1, . . . , Xd]) (= K0(R[Zd

≥0]))

Conductors and K-theory

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K-theory of monoid rings

Let R be a (commutative) regular ring and M ⊂ Zd an affine monoid (Grothendieck) K0(R) = K0(R[X1, . . . , Xd]) (= K0(R[Zd

≥0]))

(Quillen) K∗(R) = K∗(R[X1, . . . , Xd]) (= K∗(R[Zd

≥0]))

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 (Grothendieck) K0(R) = K0(R[X1, . . . , Xd]) (= K0(R[Zd

≥0]))

(Quillen) K∗(R) = K∗(R[X1, . . . , Xd]) (= K∗(R[Zd

≥0]))

(G., 1988) K0(R) = K0(R[M]) iff M = sn(M)

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 (Grothendieck) K0(R) = K0(R[X1, . . . , Xd]) (= K0(R[Zd

≥0]))

(Quillen) K∗(R) = K∗(R[X1, . . . , Xd]) (= K∗(R[Zd

≥0]))

(G., 1988) K0(R) = K0(R[M]) iff M = sn(M) Corollary: K0(R[M])/K0(R) ∼ = R(sn(M) \ M) when Q ⊂ R

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 (Grothendieck) K0(R) = K0(R[X1, . . . , Xd]) (= K0(R[Zd

≥0]))

(Quillen) K∗(R) = K∗(R[X1, . . . , Xd]) (= K∗(R[Zd

≥0]))

(G., 1988) 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

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K-theory of monoid rings

Let R be a (commutative) regular ring and M ⊂ Zd an affine monoid (Grothendieck) K0(R) = K0(R[X1, . . . , Xd]) (= K0(R[Zd

≥0]))

(Quillen) K∗(R) = K∗(R[X1, . . . , Xd]) (= K∗(R[Zd

≥0]))

(G., 1988) 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 → cm , kill K∗(R[M])/K∗(R)

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 (Grothendieck) K0(R) = K0(R[X1, . . . , Xd]) (= K0(R[Zd

≥0]))

(Quillen) K∗(R) = K∗(R[X1, . . . , Xd]) (= K∗(R[Zd

≥0]))

(G., 1988) 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 → cm , kill K∗(R[M])/K∗(R) (Corti˜ nas, Haesemayer, Walker, Weibel, announced in 2016) The condition Q ⊂ R in the statement above can be dropped

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Conjecture

Conjecture. R a regular ring, containing Q : for every finitely generated

monomial algebra R[M] without nontrivial units we have the equality

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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 → cm , acts by dilating the M

degrees by factor c .

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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 → cm , acts by dilating the M

degrees by factor c .

Informally, the mentioned thinness means that every element

f

Ki(R[M])/Ki(R) is pushed by sufficiently high iterations of the map M → M , m → cm , to the M -graded zero zone. In particular, this conjecture implies the aforementioned nilpotence of Ki(R[M])/Ki(R) .

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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 → cm , acts by dilating the M

degrees by factor c .

Informally, the mentioned thinness means that every element

f

Ki(R[M])/Ki(R) is pushed by sufficiently high iterations of the map M → M , m → 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.

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Conjecture

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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. Katth¨

an, 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

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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. Katth¨

an, 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

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