Toric rings and discrete convex geometryToric rings and discrete convex geometry Lectures for the School on Commutative Algebra and Interactions with Algebraic Geometry and Combinatorics

Toric rings and discrete convexgeometry

Lectures for the School on

Commutative Algebra and Interactions

with Algebraic Geometry and Combinatorics

Trieste, May/June 2004

Winfried Bruns

FB Mathematik/Informatik

Universitat Osnabruck

Toric rings and discrete convex geometry – p.1/118

Preface

This text contains the computer presentation of 4 lectures:

1. Affine monoids and their algebras

2. Homological properties and combinatorial applications

3. Unimodular covers and triangulations

4. From vector spaces to polytopal algebras

Lecture 1 introduces the affine monoids and relates them to the geometry

of rational convex cones. Lecture 2 contains the homological theory of

normal affine semigroup rings and their applications to enumerative

combinatorics developed by Hochster and Stanley.

Lectures 3 and 4 are devoted to lines of research that have been pursued in

joint work with Joseph Gubeladze (Tbilisi/San Francisco).


A rather complete expository treatment of Lectures 1 and 2 is contained in

W. BRUNS. Commutative algebra arising from the

Anand-Dumir-Gupta conjectures. Preprint.

Most of Lecture 3 and much more – in particular basic notions and results

of polyhedral convex geometry – is to be found in

W. BRUNS AND J. GUBELADZE. K-theory, rings, and polytopes.

Draft version of Part 1 of a book in progress.

For Lecture 4 there exists no coherent expository treatment so far, but a

brief overview is given in

W. BRUNS AND J. GUBELADZE. Polytopes and K-theory.

Preprint.


A previous exposition, covering various aspects of these lectures is to be

found in

W. BRUNS AND J. GUBELADZE. Semigroup algebras and

discrete geometry. In L. Bonavero and M. Brion (eds.), Toric

geometry. Seminaires et Congres 6 (2002), 43–127

All these texts can be downloaded from (or via)

http://www.math.uos.de/staff/phpages/brunsw/course.htm

They contain extensive lists of references.

Osnabruck, May 2004 Winfried Bruns


http://www.math.uos.de/staff/phpages/brunsw/course.htm

Lecture 1

Affine monoids and their algebras


Affine monoids and their algebras

An affine monoid M is (isomorphic to) a finitely generated

submonoid of Zd for some d ≥ 0, i. e.

M +M ⊂ M (M is a semigroup);

0 ∈ M (now M is a monoid);

there exist x1, . . . ,xn ∈ M such that M = Z+x1 + . . .Z+xn.

Often affine monoids are called affine semigroups.

gp(M) = ZM is the group generated by M.

gp(M) ∼= Zr for some r = rankM = rankgp(M).


Proposition 1.1. The Krull dimension of K[M] is given by

dimK[M] = rankM.

Proof. K[M] is an affine domain over K. Therefore

dimK[M] = trdegQF(K[M])

= trdegQF(K[gp(M)])

= trdegQF(K[Zr])

= r

where r = rankM.


Let K be a field (or a commutative ring). Then we can form the

monoid algebra

K[M] =⊕a∈M

KXa, XaXb = Xa+b

Xa = the basis element representing a ∈ M.

M ⊂ Zd ⇒ K[M] ⊂K[Zd] = K[X±11 , . . . ,X±1

d ] is a monomial

subalgebra.

Proposition 1.2. Let M be a monoid.

(a) M is finitely generated ⇐⇒ K[M] is a finitely generated

K-algebra.

(b) M is an affine monoid ⇐⇒ K[M] is an affine domain.


Sources for affine monoids (and their algebras) are

monoid theory,

ring theory,

initial algebras with respect to monomial orders,

invariant theory of torus actions,

enumerative theory of linear diophantine systems,

lattice polytopes and rational polyhedral cones,

coordinate rings of toric varieties.


Polytopal monoids

Definition 1.3. The convex hull conv(x1, . . . ,xm) of points xi ∈ Zn is

called a lattice polytope.

P


With a lattice polytope P ⊂ Rn we associate the polytopal monoid

MP ⊂ Zn+1 generated by (x,1),x ∈ P∩Zn.

P

Vertical cross-section of a polytopal monoid

Such monoids will play an important role in Lectures 3 and 4.


Presentation of an affine monoid algebra

Let R = K[x1, . . . ,xn]. Then we have a presentation

π : K[X ] = K[X1, . . . ,Xn] → K[x1, . . . ,xn], Xn → xn.

Let I = Kerπ and M = {π(Xa) : a ∈ Zn+}

Theorem 1.4. The following are equivalent:

(a) M is an affine monoid and R = K[M];

(b) I is prime, generated by binomials Xa −Xb, a,b ∈ Zn+;

(c) I = I ∩K[X±1], I is generated by binomials Xa −Xb, and

U = {a−b : Xa −Xb ∈ I} is a direct summand of Zn.


Proof. (a) ⇒ (b) Since R is a domain, I is prime. Let

f = c1Xa1 + · · ·+ cmXam ∈ I, ci ∈ K, ci = 0, a1 >lex · · · >lex am.

There exists j > 1 with π(Xa1) = π(Xa j), and so Xa1 −Xa j ∈ I.

Apply lexicographic induction to f − c1(Xa1 −Xa j).

(b) ⇒ (c) Since I is an ideal, U is a subgroup. Since I is prime and

Xi /∈ I for all i, I = I ∩K[X±11 , . . . ,X±1

n ]. Let u ∈ Zn, m > 0 such that

mu ∈U , u = v−w with v,w ∈ Zn+. Clearly Xum −Xvm ∈ I. We can

assume charK � m. Then

Xum −Xvm = (Xu −Xv)(Xu(m−1) +Xu(m−2)v + · · ·+X (m−1)),

and the second term is not in (X1 −1, . . . ,Xn −1) ⊃ I.

(c) ⇒ (a) Consider K[X ] →K[X±1] = K[Zn] → K[Zn/U ].


Cones

An affine monoid M generates the cone

R+M ={

∑aixi : xi ∈ M, ai ∈ R+

}Since M = ∑n

i=1 Z+xi is finitely generated, R+M is finitely generated:

R+M ={ n

∑i=1

aixi : a1, . . . ,an ∈ R+

}.

The structures of M and R+M are connected in many ways. It is

necessary to understand the geometric structure of R+M.


Finite generation ⇐⇒ cut out by finitely many halfspaces:

Theorem 1.5. Let C = /0 be a subset of Rm. Then the following are

equivalent:

there exist finitely many elements y1, . . . ,yn ∈ Rm such that

C = R+y1 + · · ·+R+yn;

there exist finitely many linear forms λ1, . . . ,λs such that C is the

intersection of the half-spaces H+i = {x : λi(x) ≥ 0}.

For full-dimensional cones the (essential) support hyperplanes

Hi = {x : λi(x) = 0} are unique:


Proposition 1.6. If C generates Rm as a vector space and the

representation C = H+1 ∩·· ·∩H+

s is irredundant, then the

hyperplanes Hi are uniquely determined (up to enumeration).

Equivalently, the linear forms λi are unique up to positive scalar

factors.

rational generators ⇐⇒ rationality of the support hyperplanes:

Proposition 1.7. The generating elements y1, . . . ,yn can be chosen

in Qm (or Zm) if and only if the λi can be chosen as linear forms with

rational (or integral) coefficients.

Such cones are called rational.

Proposition 1.8. If Y = {y1, . . . ,yn} ⊂ Qm, then Qm ∩R+Y = Q+Y .


Gordan’s lemma and normality

As seen above, affine monoids define rational cones. The converse

is also true.

Lemma 1.9 (Gordan’s lemma). Let L ⊂ Zd be a subgroup and

C ⊂ Rd a rational cone. Then U ∩C is an affine monoid.

Proof. Let V = RL ⊂ Rd . Then:

V ∩Qd = QL;

C∩V is a rational cone in V

⇒ We may assume that L = Zd .

C is generated by elements y1, . . . ,yn ∈ M = C∩Zd.

x ∈C ⇒ x = a1y1 + · · ·+anyn ai ∈ R+.


x = x′ + x′′, x′ = a1�y1 + · · ·+ an�yn.

Clearly x′ ∈ M. But

x ∈ M ⇒ x′′ ∈ gp(M)∩C ⇒ x′′ ∈ M.

x′′ lies in a bounded set B ⇒

M generated by y1, . . . ,yn and the finite set M∩B.

The monoid M = U ∩C has a special property:

Definition 1.10. A monoid M is normal ⇐⇒

x ∈ gp(M), kx ∈ M for some k ∈ Z,k > 0 ⇒ x ∈ M.


Proposition 1.11.

M ⊂ Zd normal affine monoid ⇐⇒ there exists a rational cone

C such that M = gp(M)∩C;

M ⊂ Zd affine monoid ⇒ the normalization M = gp(M)∩R+M

is affine.

Briefly: Normal affine monoids are discrete cones.

0Toric rings and discrete convex geometry – p.19/118

Positivity, gradings and purity

Definition 1.12. A monoid M is positive if x,−x ∈ M ⇒ x = 0.

Definition 1.13. A grading on M is a homomorphism deg : M → Z. It

is positive if degx > 0 for x = 0.

Proposition 1.14. For M affine the following are equivalent:

(a) M is positive;

(b) R+M is pointed (i. e. contains no full line);

(c) M is isomorphic to a submonoid of Zs for some s;

(d) M has a positive grading.

Proof. (c) ⇒ (d) ⇒ (a) trivial.


(a) ⇒ (b) Set C = R+M. One shows:

{x ∈C : −x ∈C} = R{x ∈ M : −x ∈ M}.

Therefore: M positive ⇒C pointed.

(b) ⇒ (c) Let C be positive. For each facet F of C there exists a

unique linear form σF : Rd → R with the following properties:

F = {x ∈C : σF(x) = 0}, σF(x) ≥ 0 for all x ∈C;

σ has integral coefficients, σ(Zd) = Z.

These linear forms are called the support forms of C. Let

s = # facets(C) and define

σ : Rd → Rs, σ(x) =(σF(x) : F facet

).

Then σ(M) ⊂ σ(M) ⊂ Zs+. Since C is positive, σ is injective!

We call σ the standard embedding.Toric rings and discrete convex geometry – p.21/118

For M normal the standard embedding has an important property:

Proposition 1.15. M positive affine monoid. Then the following are

equivalent:

(a) M is normal;

(b) σ maps M isomorphically onto Zs ∩σ(gp(M)).

M pure submonoid of N ⇐⇒ M = N ∩gp(M).

Corollary 1.16. M affine, positive, normal ⇐⇒ M isomorphic to a

pure submonoid of Zs+ for some s.


Normality and purity of K[M]

An integral domain R is normal if R integrally closed in QF(R).

R pure subring of S ⇐⇒ S = R⊕T as an R-module.

Theorem 1.17. M positive affine monoid.

(a) M normal ⇐⇒ K[M] normal

(b) M ⊂ Zs+ pure submonoid ⇐⇒ K[M] pure subalgebra of

K[Zs+] = K[Y1, . . . ,Ys]


Proof. (a) x ∈ gp(M), mx ∈ M for some m > 0

⇒ Xx ∈ K[gp(M)] ⊂ QF(K[M]), (Xx)m ∈ K[M].

Thus: K[M] normal ⇒ Xx ∈ K[M] ⇒ x ∈ M.

Conversely, let M be normal, gp(M) = Zr, C = R+M

⇒ M = Zr ∩C

C = H+1 ∩·· ·∩H+

s , H+i rational closed halfspace ⇒

M= N1 ∩·· ·∩Ns, Ni = Zr ∩H+i

⇒ K[M]= K[N1]∩·· ·∩K[Ns]

Ni discrete halfspace


Consider hyperplane Hi bounding H+i . Then Hi ∩Zs direct summand

of Zs.

⇒ Zs has basis u1, . . . ,ur with u1, . . . ,ur−1 ∈ Hi, ur ∈ H+i

⇒ K[Ni] ∼= K[Zr−1 ⊕Z+] ∼= K[Y±11 , . . . ,Y±1

r−1,Z]

Thus K[M] intersection of factorial (hence normal) domains ⇒ K[M]normal

(b) T = K{Xx : x ∈ Zs \M}⇒ K[Y1, . . . ,Ys] = K[M]⊕T as K-vector

space

M pure submonoid ⇒ T is K[M]-submodule

Converse not difficult.


A grading on M induces a grading on K[M]:

Proposition 1.18. Let M be an affine monoid with a grading deg.

Then

K[M] =⊕k∈Z

K{Xx : degx = k}

is a grading on K[M].

If deg is positive, then K[M] is positively graded.


The class group

R normal Noetherian domain (or a Krull domain).

I ⊂ QF(R) fractional ideal ⇐⇒ there exists x ∈ R such that xI is a

non-zero ideal

I is divisorial ⇐⇒ (I−1)−1 = I where

I−1 = {x ∈ QF(R) : xI ⊂ R}.

(I,J) → ((IJ)−1)−1 defines a group structure on

Div(R) = {div. ideals}Fact: Div(R) free abelian group with basis Zdiv(p), p height 1 prime

ideal ( div(I) denotes I as an element of Div(I) )

Princ(R) = {xR : x ∈ QF(R)} is a subgroup


Cl(R) =Div(R)

Princ(R)

is called the (divisor) class group.

It parametrizes the isomorphism classes of divisorial ideals.

R is factorial ⇐⇒ Cl(R) = 0.


Let M be a positive normal affine monoid, R = K[M]. Choose

x ∈ int(M) = {y ∈ M : σF(y) > 0 for all facets F}.

⇒ M[−x] = gp(M)

⇒ R[(Xx)−1] = K[gp(M)] = L = Laurent polynomial ring

Nagata’s theorem ⇒ exact sequence

0 →U → Cl(R) → Cl(L) → 0,

U generated by classes [p] of minimal prime overideals of Xx

L factorial ⇒ Cl(L) = 0 ⇒ Cl(R) = U .


For a facet F of R+M set

pF = R{x ∈ M : σF(x) ≥ 1}.

⇒ pF is a prime ideal since R/pF∼= K[M∩F].

Evidently

Rx = R{

Xy : σF(y) ≥ σF(x) for all F}

=⋂F

p(σF (x))F

p(k)F = R{Xy : σF(y) ≥ k} is the k-th symbolic power of pF .

⇒ Cl(R) = ∑F Z[pF ].


[⋂F

p(kF )F

]= ∑

FkF [pF ]

⇒ every divisorial ideal is isomorphic to an ideal⋂

F p(kF )F

H1

H2

0

p(2)1 ∩p

(−2)2


Monomial ideal I principal

⇐⇒ there exists a monomial Xy with I = XyR

Enumerate the facets F1, . . . ,Fs, pi = pFi , σi = σFi

Theorem 1.19 (Chouinard).

Cl(R) ∼=⊕s

i=1 Zdiv(pi){div(RXy) : y ∈ gp(M)}

∼= Zs

σ(gp(M)

where σ is the standard embedding.


Lecture 2

Homological properties andcombinatorial applications


Magic Squares

In 1966 H. Anand, V. C. Dumir, and H. Gupta investigated a

combinatorial problem:

Suppose that n distinct objects, each available in k identical

copies, are distributed among n persons in such a way that

each person receives exactly k objects.

What can be said about the number H(n,k) of such distributions?


They formulated some conjectures:

(ADG-1) there exists a polynomial Pn(k) of degree (n−1)2 such that

H(n,k) = Pn(k) for all k � 0;

(ADG-2) H(n,k) = Pr(n) for all k > −n; in particular Pn(−k) = 0,

k = 1, . . . ,n−1;

(ADG-3) Pn(−k) = (−1)(n−1)2Pn(k−n) for all k ∈ Z.

These conjectures were proved and extended by R. P. Stanley using

methods of commutative algebra.

The weaker version (ADG-1) of (ADG-2) has been included for

didactical purposes.


Reformulation:

ai j = number of copies of object i that person j receives

⇒ A = (ai j) ∈ Zn×n+ such that

n

∑k=1

aik =n

∑l=1

al j = k, i, j = 1, . . . ,n.

H(n,k) is the number of such matrices A.

The system of equations is part of the definition of magic squares. In

combinatorics the matrices A are called magic squares, though the

usually requires further properties for those.


Two famous magic squares:

8 1 6

3 5 7

4 9 2

16 3 2 13

5 10 11 8

9 6 7 12

4 15 14 1

The 3×3 square goes back to the Chinese emperor Loh-shu (about

2800 B.C.) and the 4×4 appears in Albrecht Durer’s engraving

Melancholia (1514). It has remarkable symmetries and shows the

year of its creation.


A step towards algebra

Let Mn be the set of all matrices A = (ai j) ∈ Zn×n+ such that

n

∑k=1

aik =n

∑l=1

al j, i, j = 1, . . . ,n.

By Gordan’s lemma Mn is an affine, normal monoid, and A → magic

sum k = ∑nk=1 a1k is a positive grading on M .

It is even a pure submonoid of Zn×n+ .

rankMn = (n−1)2 +1

Theorem 2.1 (Birkhoff-von Neumann). Mn is generated by the its

degree 1 elements, namely the permutation matrices.


Translation into commutative algebra

We choose a field K and form the algebra

R = K[Mn].

It is a normal affine monoid algebra, graded by the “magic sum”, and

generated in degree 1. dimR = rankMn = (n−1)2 +1.

⇒H(n,k) = dimK Rk = H(R,k) is the Hilbert function of R!

⇒ (ADG-1): there exists a polynomial Pn of degree (n−1)2 such

that H(n,k) = Pn(k) for k � 0.

In fact, take Pn as the Hilbert polynomial of R.Toric rings and discrete convex geometry – p.39/118

A recap of Hilbert functions

Let K be a field, and R = ⊕∞k=0Rk a graded K-algebra generated by

homogeneous elements x1, . . . ,xn of degrees g1, . . . ,gn > 0.

Let M be a non-zero, finitely generated graded R-module.

Then H(M,k) = dimK Mk < ∞ for all k ∈ Z.

H(M, ) : Z → Z is the Hilbert function of M.

We form the Hilbert (or Poincare) series

HM(t) = ∑k∈Z

H(M.k)tk.


Fundamental fact:

Theorem 2.2 (Hilbert-Serre). Then there exists a Laurent

polynomial Q ∈ Z[t, t−1] such that

HM(t) =Q(t)

∏ni=1(1− tgi)

.

More precisely: HM(t)is the Laurent expansion at 0 of the rational

function on the right hand side.

Refinement: M is finitely generated over a graded Noether

normalization K[y1, . . . ,yd ], d = dimM, of R/AnnM.

Special case: g1, . . . ,gn = 1. Then y1, . . . ,yd can be chosen of

degree 1 (after an extension of K).


Theorem 2.3. Suppose that g1 = · · · = gn = 1 and let d = dimM.

Then

HM(t) =Q(t)

(1− t)d .

Moreover:

There exists a polynomial PM ∈ Q[X ] such that

H(M, i) = PM(i), i > degHM,

H(M, i) = PM(i), i = degHM.

e(M) = Q(1) > 0, and if d ≥ 1, then

PM =e(M)

(d −1)!Xd−1 + terms of lower degree.


The general case is not much worse: PM must be allowed to be a

quasi-polynomial, i. e. a “polynomial” with periodic coefficients

(instead of constant ones).

Theorem 2.4. Suppose R/AnnM has a graded Noether

normalization generated by elements of degrees e1, . . . ,ed . Then

HM(t) =Q(t)

∏di=1(1− tei)

, Q(1) > 0.

Moreover there exists a quasi-polynomial PM, whose period divides

lcm(e1, . . . ,ed), such that

H(M, i) = PM(i), i > degHM,

H(M, i) = PM(i), i = degHM.


Hochster’s theorem

Theorem 2.5. Let M be an affine normal monoid. Then K[M] is

Cohen-Macaulay for every field K.

There is no easy proof of this powerful theorem. For example, it can

be derived from the Hochster-Roberts theorem, using that

K[M] ⊂ K[X1, . . . ,Xs] can be chosen as a pure embedding.

A special case is rather simple:

Definition 2.6. An affine monoid M is simplicial if the cone R+M is

generated by rankM elements.

Proposition 2.7. Let M be a simplical affine normal monoid. Then

K[M] is Cohen-Macaulay for every field K.


Proof. Let d = rankM. Choose elements x1, . . . ,xd ∈ M generating

R+M, and set

par(x1, . . . ,xd) ={ d

∑i=1

qixi : qi ∈ [0,1)}

.

0x1

x2


B =par(x1, . . . ,xd)∩Zd

N = Z+x1 · · ·+Z+xd.

The arguments in the proof Gordan’s lemma and the linear

independence of x1, . . . ,xd imply:

M =⋂

z∈B z+N

N is free.

The union is disjoint.

In commutative algebra terms:

K[M] is finite over K[N].K[N] ∼= K[X1, . . . ,Xd].K[M] is a free module over K[N].

⇒ K[M] is Cohen-Macaulay.


Combinatorial consequence of the Cohen-Macaulay property:

Theorem 2.8. Let M be a graded Cohen-Macaulay module over the

positively graded K-algebra R and x1, . . . ,xd a h.s.o.p. for M,

ei = degxi. Let

HM(t) =hata + · · ·+hbtb

∏di=1(1− tei)

, ha,hb = 0.

Then hi ≥ 0 for all i.

If M = R = K[R1], then hi > 0 for all i = 0, . . . ,b.

Proof. hata + · · ·+hbtb is the Hilbert series of

M/(x1M + · · ·+ xdM).


Reciprocity

(ADG-3) compares values PR(k) of the Hilbert polynomial of

R = K[Mn] for all values of k:

(ADG-3) Pn(−k) = (−1)(n−1)2Pn(k−n) for all k ∈ Z.

According to (ADG-2) the shift −n in Pn(k−n) is the degree of HR(t)(not yet proved).

What identity for HR(t) is encoded in (ADG-3) ?


Lemma 2.9. Let P : Z → C be a quasi-polynomial. Set

H(t) =∞

∑k=0

P(k)tk and G(t) = −∞

∑k=1

P(−k)tk.

Then H and G are rational functions. Moreover

H(t) = G(t−1).

Corollary 2.10. Let R be a positively graded, finitely generated

K-algebra, dimR = d, with Hilbert quasi-polynomial P. Suppose

degHR(t) = g < 0. Then the following are equivalent:

P(−k) = (−1)d−1P(k +g) for all k ∈ Z;

(−1)dHR(t−1) = t−gHR(t).


Strategy:

Find an R-module ω with Hω(t) = (−1)dHR(t−1)

This is possible for R Cohen-Macaulay: ω is the canonical module of

R.

Compute ω for R = K[Mn] and show that ω ∼= R(g),g = degHR(t).

R(g) free module of rank 1 with generator in degree −g. Thus

HR(g)(t) = t−gHR(t).

More generally:

Compute ω for R = K[M] with M affine, normal.


The canonical module

In the following: R positively graded Cohen-Macaulay K-algebra,

x1, . . . ,xd h.s.o.p., degxi = gi

First R = S = K[X1, . . . ,Xd ]:

(−1)dHS(t−1) =(−1)d

∏di=1(1− t−gi)

=tg1+···+gd

∏di=1(1− t−gi)

= Hω(t)

with ω = ωS = S(−(g1 + · · ·+gd))


The general case: R free over K[x1, . . . ,xd] ∼= K[X1, . . . ,Xd], say with

homogeneous basis y1, . . . ,ym:

R ∼=m⊕

j=1

Sy j∼=

u⊕i=1

S(−i)hi , hi = #{ j : degy j = i},

HR(t)= (h0 +h1t + · · ·+hutu)HS(t) = Q(t)HS(t)

Set ωR = HomS(R,ωS). Then, with s = g1 + · · ·+gd

ωR∼=

u⊕i=1

HomS(S(−i)hi ,S(−s)) ∼=u⊕

i=1

S(−i+ s)hi ,

HωR(t)= (h0ts + · · ·+huts−u)HS(t) = Q(t−1)tsHS(t)

= (−1)dQ(t−1)HS(t−1) = (−1)dHR(t−1).


Multiplication in the first component makes ωR an R-module:

a ·ϕ( ) = ϕ(a · ).

But: Is ωR independent of S ?

Theorem 2.11. ωR depends only on R (up to isomorphism of graded

modules).

The proof requires homological algebra, after reduction from the

graded to the local case.


Gorenstein rings

Definition 2.12. A positively graded Cohen-Macaulay K-algebra is

Gorenstein if ωR∼= R(h) for some h ∈ Z.

Actually, there is no choice for h:

Theorem 2.13 (Stanley). Let R be Gorenstein. Then

ωR∼= R(g), g = degHR(t);

h0 = hu−i for i = 0, . . . ,u: the h-vector is palindromic;

HR(t−1) = (−1)dt−gHR(t).

Conversely, if R is a Cohen-Macaulay integral domain such that

HR(t−1) = (−1)dt−hHR(t) for some h ∈ Z, then R is Gorenstein.


Proof.

(−1)dHR(t) = (h0ts + · · ·+huts−u)HS(t)

t−hHR(t) = (hstu−h + · · ·+h−h

0 )HS(t)

Equality holds ⇐⇒

h = u− s = degHR(t) and hi = hu−i, i = 0, . . . ,u

If R is a domain, then ωR is torsionfree. Consider R → ωR, a → ax,

x ∈ ωR homogeneous, degx = −g.

This linear map is injective: R(g) ↪→ ωR. Equality of Hilbert functions

implies bijectivity.


The canonical module of K[M]

In the following M affine, normal, positive monoid. We want to find

the canonical module of R = K[M] (Cohen-Macaulay by Hochster’s

theorem).

Theorem 2.14 (Danilov, Stanley). The ideal I generated by the

monomials in the interior of R+M is the canonical module of K[M](with respect to every positive grading of M).

Note: x ∈ int(R+M) ⇐⇒ σF(x) > 0 for all facets F of R+M.

pF is generated by all monomials Xx, x ∈ M such that σF(x) > 0.

⇒ I =⋂

F facet

pF .


Choose a positive grading on M and let ω be the canonical module

of R with respect to this grading.

By definition ω is free over a Noether normalization ⇒ ω is a

Cohen-Macaulay R-module ⇒ ω is (isomorphic to) a divisorial

ideal ⇒

As discussed in Lecture 1, there exist jF ∈ Z such that

ω =⋂

p( jF )F .

Without further standardization we cannot conclude that jF = 1 for

all F .

We have to use the natural Zr-grading, Zr = gp(M) on R !


The homological property characterizing the canonical module is

Ext jR(K,ωR) =

⎧⎨⎩K, j = d,

0, j = d.d = dimR.

This is to be read as an isomorphism of graded modules: let m be

the irrelevant maximal ideal; then K = R/m lives in degree 0.

In our case R/m is a Zr-graded module, as is ω

⇒ Ext jR(K,ωR) ∼= K lives in exactly one multidegree v ∈ Zr

⇒ Ext jR(K,X−vωR) ∼= K in multidegree 0 ∈ Zr

Replace ωR by X−vωR.


Definition 2.15. Let R be a Zn-graded Cohen-Macaulay ring such

that the homogeneous non-units generate a proper ideal p of R.

⇒ p is a prime ideal; set d = dimRp.

One says that ω is a Zn-graded canonical module of R if

Ext jR(R/p,ω) =

⎧⎨⎩R/p, j = d,

0, j = d.

We have seen: R = K[M] has a Zr-graded canonical module

ω =⋂

p( jF )F

and it remains to show that jF = 1 for all facets F .


Let RF = R[(M∩F)−1]: we invert all the monomials in F .

⇒ RF is the “discrete halfspace algebra” with respect to the support

hyperplane through F .

0 F

MF M

x


⇒ pFRF is the Zr-graded canonical module of RF (easy to see since

pFRF is principal generated by a monomial Xx with σF(x) = 1)

On the other hand: ωRF = (ωR)F : the Zr-graded canonical module

“localizes” (a nontrivial fact)

⇒ jF = 1.


Back to the ADG conjectures

Recall that Mn denotes the “magic” monoid. It contains the matrix 1with all entries 1.

Let C = R+Mn. Then C is cut out from RMn by the positive orthant

⇒ int(C) = {A : ai j > 0 for all i, j}.

⇒ A−1 ∈ Mn for all A ∈ M∩ int(C)

⇒ interior ideal I is generated by X1; 1 has magic sum n

⇒ I ∼= R(−n). R = K[Mn] is a Gorenstein ring with degHR(t) =−n


degHR(t) = −n ⇒

(ADG-2) H(n,r) = Pr(n) for all r > −n; in particular Pn(−r) = 0,

r = 1, . . . ,n−1;

(ADG-2) and R Gorenstein ⇒

(ADG-3) Pn(−r) = (−1)(n−1)2Pn(r−n) for all r ∈ Z.


In terms of

HR(t) =1+h1t + · · ·+hutu

(1− t)(n−1)2+1, hu = 0,

we have seen that

u = (n−1)2 +1−n (ADG-2)

hi > 0 for i = 1, . . . ,u (R Cohen-Macaulay)

hi = hu−i for all i (ADG-3)

Very recent result, conjectured by Stanley and now proved by Ch.

Athanasiadis:

the sequence (hi) is unimodal: h0 ≤ h1 ≤ ·· · ≤ h�u/2�


Lecture 3

Unimodular covers and triangulations


Recall: P = conv(x1, . . . ,xn) ⊂ Rd , xi ∈ Zd , is called a lattice

polytope.

P

∆1

∆2

∆ = conv(v0, . . . ,vd), v0, . . . ,vd affinely independent, is a

simplex.Toric rings and discrete convex geometry – p.66/118

Set U∆ = ∑di=0 Z(vi − v0).

µ(∆) = [Zd : U∆] = multiplicity of ∆

∆ is unimodular if µ(∆) = 1.

∆ is empty if vert(∆) = ∆∩Zd.

Lemma 3.1.

µ(∆) = d!vol(∆) = ±det

⎛⎜⎜⎝v1 − v0

...

vd − v0

⎞⎟⎟⎠

When is P covered by its unimodular subsimplices?

For short: P has UC.Toric rings and discrete convex geometry – p.67/118

Low Dimensions

d = 1: 0 1 2 3 4−1 P has a unique

unimodular triangulation.

d = 2:

Every empty lattice triangle is

unimodular ⇒ every 2-polytope

has a unimodular triangulation.


d = 3: There exist empty simplices of arbitrary multiplicity!


Polytopal cones and monoids

The cone over P is CP = R+{(x,1) ∈ Rd+1 : x ∈ P}.The monoid associated with P is MP = Z+{(x,1) : x ∈ P∩Zd}.The integral closure of MP is MP = CP ∩Zd+1.

P

CP

Proposition 3.2. P has UC ⇒ MP = MP (P is integrally closed).


P is integrally closed ⇐⇒(i) gp(MP) = Zd+1 and

(ii) MP is a normal monoid (MP = CP ∩gp(MP))

There exist non-normal 3-dimensional polytopes, for example

P = {x ∈ R3 : xi ≥ 0, 6x1 +10x2 +15x3 ≤ 30}.


Monoid algebras, toric ideals and Grobnerbases

Let K be a field. The polytopal K-algebra K[P] is the monoid

algebra

K[P] = K[MP] = K[Xx : x ∈ P∩Zd]/IP.

The toric ideal IP is generated by all binomials

∏x∈P∩Zd

Xaxx − ∏

x∈P∩Zd

Xbxx ,

∑axx = ∑bxx, ∑ax = ∑bx

expressing the affine relations between the lattice points in P.


Sturmfels:

“generic” weights for Xx −→⎧⎨⎩(i) regular triangulation Σ of P, vert(Σ) ⊂ P∩Zd

(ii) term (pre)order on K[Xx], ini(IP) monomial ideal

Theorem 3.3.

(Stanley-Reisner ideal of Σ) = Rad(ini(IP))Σ is unimodular ⇐⇒ ini(IP) squarefree


Multiples of polytopes

For c → ∞ (c ∈ N) the lattice points cP∩Zd approximate the

continuous structure of cP ∼ P better and better.


Algebraic results:

Theorem 3.4.

cP integrally closed for c ≥ dimP−1. Thus K[cP] normal for

c ≥ dimP−1.

IcP has an initial ideal generated by degree 2 monomials for

c ≥ dimP. Thus K[cP] is Koszul for c ≥ dimP.

Proof of Koszul property uses technique of Eisenbud-Reeves-Totaro.


Questions:

(i) Does cP have UC for c ≥ dimP−1 ?

(ii) Does cP have a regular unimodular triangulation of degree 2 for

c ≥ dimP ?

Positive answers: (i) dimP ≤ 3, (ii) dimP ≤ 2.

No algebraic obstructions !


Positive rational cones and Hilbert bases

C generated by finitely many v ∈ Zd , x,−x ∈C ⇒ x = 0.

Gordan’s lemma: C∩Zd is a finitely generated monoid.

Its irreducible element form the Hilbert basis Hilb(C) of C.

C is simplicial ⇐⇒ C generated by linearly independent vectors

v1, . . . ,vd

µ(C) = [Zd : Zv1 + · · ·+Zvd]

C is unimodular ⇐⇒ C generated by a Z-basis of Zd

⇐⇒ µ(C) = 1


Theorem 3.5. C has a triangulation into unimodular subcones.

Proof: Start with arbitrary triangulation. Refine by iterated stellar

subdivision to reduce multiplicities.


But: P has a unimodular triangulation ⇒ CP satisfies UHT.

UHT: C has a Unimodular Triangulation into cones generated by

subsets of Hilb(C).

UHC: C is Covered by its Unimodular subcones generated by

subsets of Hilb(C).

A condition with a more algebraic flavour:

ICP: (Integral Caratheodory Property) for every x ∈C∩Zd there

exist y1, . . . ,yd ∈ Hilb(C) with x ∈ Z+y1 + · · ·+Z+yd .


Dimension 3

Cones of dimension 3:

Theorem 3.6 (Sebo). dimC = 3 ⇒C has UHT

If C = CP, dimP = 2, this is easy since P has UT. General case is

somewhat tricky.


Polytopes of dimension 3:

First triangulate P into empty simplices and then use classification of

empty simplices (White):

∆pq = conv

⎛⎜⎜⎜⎜⎜⎝0 0 0

0 1 0

0 0 1

p q 1

⎞⎟⎟⎟⎟⎟⎠ , 0 ≤ q < p, gcd(p,q) = 1

µ(∆pq) = p

No classification known in dimension ≥ 4. Essential difference to

dimension 3: lattice width of ∆ may be > 1.Toric rings and discrete convex geometry – p.81/118

Lagarias & Ziegler , Kantor & Sarkaria:

Proposition 3.7. cP has UC for c ≥ 2.

Theorem 3.8.

2∆pq has UT ⇐⇒ q = 1 or q = p−1.

4P has UT for all P.

c∆pq has UT for c ≥ 4.

Question: What about 3∆pq ?


CounterexamplesP = 2∆53 integrally closed 3 polytope without UT ⇒ CP has

dimension 4 and violates UHT (first counterexample by Bouvier &

Gonzalez-Sprinberg)

C6 with Hilbert basis z1, . . . ,z10, is of form CP5 , dimP5 = 5, P5

integrally closed, and violates UHC and ICP (B & G & Henk, Martin,

Weismantel)

z1 = (0, 1, 0, 0, 0, 0), z6 = (1, 0, 2, 1, 1, 2),

z2 = (0, 0, 1, 0, 0, 0), z7 = (1, 2, 0, 2, 1, 1),

z3 = (0, 0, 0, 1, 0, 0), z8 = (1, 1, 2, 0, 2, 1),

z4 = (0, 0, 0, 0, 1, 0), z9 = (1, 1, 1, 2, 0, 2),

z5 = (0, 0, 0, 0, 0, 1), z10 = (1, 2, 1, 1, 2, 0).

⇒ P5 violates UCToric rings and discrete convex geometry – p.83/118

There exists a polytope of dimension 10 with UT, but without a

regular unimodular triangulation (Hibi & Ohsugi)

Questions:

Do all integrally closed polytopes P of dimensions 3 and 4 have

UC ?

Do all cones C of dimensions 4 and 5 have UHC ?

Does there exist C with ICP, but violating UHC ?


Triangulating cP

Theorem 3.9 (Knudsen & Mumford, Toroidal embeddings). Let P

be a lattice d-polytope. Then cP has a regular unimodular

triangulation for a some c ∈ Z+, c > 0.

Not so hard: UC of d-simplices with non-overlapping interiors

Harder: UT

Most difficult: regularity

Questions: Does cP have UT for c � 0 ? Can we bound c

uniformly in terms of dimension ? Is c ≥ dimP enough ?


Covering cP

Theorem 3.10. Let P be a d-polytope. Then there exists cd such that

cP has UC for all c ≥ cpold , and

cpold = O

(d16.5

)(94

)(ldγ(d))2

, γ(d) = (d −1)�√d −1�.

For the proof one needs a similar theorem about cones—cones allow

induction on d.


Theorem 3.11. Let C be a rational simplicial d-cone and ∆C the

simplex spanned by O and the extreme integral generators. Then

(a) (M. v. Thaden) C has a triangulation into unimodular simplicial

cones Di such that Hilb(Di) ⊂ c∆C for some

c ≤ d2

4(µ(C))7

(94

)(ld(µ(C)))2

.

(b) C has a cover by unimodular simplicial cones Di such that

Hilb(Di) ⊂ c∆C for some

c ≤ d2

4(d +1)

(γ(d)

)8(

94

)(ld(γ(d)))2

.


Sketch of proof of Theorem 3.11:

(i) d −1 → d: we can cover the “corners”] of C with unimodular

subcones.

(ii) Extend the corner covers far enough into C. To have enough

room, we must go “further up” in the cone. We loose unimodularity,

but the multiplicity remains under control: ≤ γ(d) =⌈√

d −1⌉(d−1)

(iii) Apply part (a) of theorem to restore unimodularity.

(iv) Part (a): Control the “lengths” of the vectors in iterated stellar

subdivision.


Sketch of proof of Theorem 3.10: May assume P = ∆ is an

(empty) simplex. Consider corner cones of ∆:

v0

v2

v1

∆


Apply Theorem 3.11 to corner cone:

v0

v2

v1

∆


Must multiply P by factor c′ from Theorem 3.11 to get basic

unimodular corner simplices into P

v0

v2

v1


Tile corner cones:

v0

v2

v1

Must multiply c′P by c′′ ≈ d√

d to get the tiling by unimodular corner

simplices close enough (= beyond red line) to facet opposite of v0.


Lecture 4

From vector spaces to polytopalalgebras

Every so often you should try a damn-fool experiment —

from J. Littlewood’s A MATHEMATICIAN’S MISCELLANY


The category Pol(K)

Recall from Lecture 3 that a lattice polytope P is the convex hull of

finitely many points xi ∈ Zn.

MP submonoid of Zn+1 generated by (x,1), x ∈ P∩Zn.

For a field K we let Pol(K) be the category

with objects the graded algebras K[P] = K[MP]with morphisms the graded K-algebra homomorphisms

Main question: To what extent is Pol(K) determined by combinatorial

data ?


Pol(K) generalizes Vect(K), the category of finite-dimensional

K-vector spaces:

∆n n-dimensional unit simplex

⇒ K[∆n] = K[X1, . . . ,Xn+1]

HomK(Km,Km) ↔ gr.homK(S(Km),S(Kn))

↔ gr.homK(K[X1, . . . ,Xm],K[X1, . . . ,Xn])

↔ gr.homK(K[∆m−1],K[∆n−1]))

What properties of Vect(K) can be passed on Pol(K)?

Note: Pol(K) not abelian


Why not graded affine monoid algebras K[M] in full generality?

Proposition 4.1. Let P,Q be lattice polytopes. Then the K-algebra

homomorphisms K[P] → K[Q] correspond bijectively to K-algebra

homomorphisms K[P] → K[Q] of the normalizations.

In fact, K[P] equals K[P] in degree 1.

In the following the base field K is often replaced by a general

commutative base ring R.


Toric automorphisms and symmetries

Elementary fact of linear algebra: GLn(K) is generated by matrices

of 3 types:

diagonal matrices

permutation matrices

elementary transformations

Actually, the permutation matrices are not needed. But their

analogues in the general case cannot always be omitted.


It easy to generalize diagonal matrices and permutation matrices:

the diagonal matrices correspond to

(λ1, . . .λn+1) ∈ Tn+1 = (K∗)n+1 acting on

K[P] ⊂ K[X1, . . . ,Xn+1] via the substitution Xi → λiXi,

the permutation matrices represent symmetries of ∆n−1 and

correspond to the elements of the (affine!) symmetry group

Σ(P) of P.

How can we generalize elementary transformations?


Column structures

A column structure arranges the lattice points in P in columns:

v

F = Pv

More formally: v ∈ Zn is a column vector if their exists a facet F , the

base facet Pv = F of v, such that

x+ v ∈ P for all x ∈ P\F.

A column vector v ∈ Zn is to be identified with (v,0) ∈ Zn+1.


Elementary automorphisms

To each facet F of P there corresponds a facet of the cone R+MP,

also denoted by F .

Recall the support form σF . For F = Pv set σv = σF .

Define a map from MP to R[Zn+1] by

eλv : x → (1+λv)σvx.

σv Z-linear and v column vector ⇒ eλv homomorphism from MP into

(R[MP], ·)⇒ eλ

v extends to an endomorphism of R[MP]

Since e−λv is its inverse, eλ

v is an automorphism.


Proposition 4.2. v1, . . . ,vs pairwise different column vectors for P

with the same base facet F = Pvi . Then

ϕ : (R,+)s → gr.autR(R[P]), (λ1, . . . ,λs) → eλ1v1◦ · · · ◦ eλs

vs,

is an embedding of groups.

eλivi and e

λ jv j commute and the inverse of eλi

vi is e−λivi .

R field ⇒ ϕ is homomorphism of algebraic groups.

⇒ subgroup A(F) of gr.autR(R[P]) generated by eλv with F = Pv is

an affine space over R

Col(P) = set of column vectors of P.


The polytopal linear group

Theorem 4.3. Let K a field.

Every γ ∈ gr.autK(K[P]) has a presentation

γ = α1 ◦α2 ◦ · · · ◦αr ◦ τ ◦σ ,

σ ∈ Σ(P), τ ∈ Tn+1, and αi ∈ A(Fi).A(Fi) and Tn+1 generate conn. comp. of unity gr.autK(K[P])0.

= {γ ∈ gr.autK(K[P]) inducing id on div. class group of K[P]}.

dimgr.autK(K[P]) = #Col(P)+n+1.

Tn+1 is a maximal torus of gr.autK(K[P]).


The proof uses in a crucial way that every divisorial ideal of K[MP] is

isomorphic to a monomial ideal.

This fact allows a polytopal Gaussian algorithm.

Using elementary automorphisms it corrects an arbitrary γ to an

automorphism δ such that δ (int(K[P])) = int(K[P]).

Lemma 4.4. δ (int(K[P])) = int(K[P]) ⇒ δ = τ ◦σ ,

τ ∈ Tn+1, σ ∈ Σ(P)

Important fact: the divisor class group of K[P]} is a discrete object.

To some extent one can also classify retractions of K[P].


Milnor’s classical K2

Its construction is based on

the passage to the “stable” group of elementary automorphisms

the Steinerg relations

Construction of the stable group: En(R) subgroup generated by of

elementary matrices,

E ∈ En(R) →⎛⎝E 0

0 1

⎞⎠ ∈ En+1(R)

E(R) = lim−→ En(R).


The Steinberg relations for elementary matrices:

eλi je

µi j = eλ+µ

i j

[ei j,eµjk] = eλ µ

ik , i = k

[ei j,eµki] = e−λ µ

k j j = k

[eλi j,e

µkl] = 1 i = l, j = k

The stable Steinberg group of K is defined by

generators xλi j, i, j ∈ N, i = j, λ ∈ K representing the

elementary matrices

the (formal) Steinberg relations xλi jx

µi j = xλ+µ

i j , [xi j,xµjk] = xλ µ

ik

etc.


Set

K2(R) = Ker(St(R) → E(R)

), xλ

i j → eλi j.

Milnor’s theorem:

Theorem 4.5. The exact sequence

1 → K2(R) → St(R) → E(R) → 1

is a universal central extension and K2(R) is the center of St(R).


Products of column vectors

Let u,v,w ∈ Col(P). We say that

uv = w ⇐⇒ w = u+ v and Pw = Pu

Examples of products of column vectors:

w = uv u

v

−v

w u = w(−v)

⇒ partial, non-commutative product structure on Col(P).


Balanced polytopes

A polytope is balanced if

σF(v) ≤ 1 for all v ∈ Col(P), F = Pw.

A nonbalanced polytope


Polytopal Steinberg relations

Proposition 4.6. P balanced, u,v ∈ Col(P), u+ v = 0, λ ,µ ∈ R.

Then

eλv eµ

v = el+µv

[eλu ,eµ

v ] =

⎧⎪⎪⎨⎪⎪⎩e−λ µ

uv if uv exists,

eµλvu if vu exists,

1 if u+ v /∈ Col(P).

Note: we know nothing about [eλu ,eµ

−u] if u,−u ∈ Col(P)!


Doubling along a facet

Let F = Pv be a facet of P and choose coordinates in Rn such that

Rn−1 is the affine hyperplane spanned by F .

P− = {(x′,xn,0) : (x′,xn) ∈ P}P | = {(x′,0,xn) : (x′,xn) ∈ P}

P F = conv(P,P |) ⊂ Rn+1.

P

P |

F

δ−δ+

v−

v |

v = v− = δ+v |

v | = δ−v−


Crucial facts:

Col(P) ↪→ Col(P F )

G → conv(G−,G |), G = F

F → P |

P− = new facet

Lemma 4.7. P balanced ⇒ P F balanced and

Col(P F ) = Col(P)−∪Col(P) | ∪ {δ+,δ−}.


Doubling spectra

The chain of lattice polytopes P = (P = P0 ⊂ P1 ⊂ . . .) is called a

doubling spectrum if

for every i ∈ Z+ there exists a column vector v ⊂ Col(Pi) such

that Pi+1 = P vi ,

for every i ∈ Z+ and any v ∈ Col(Pi) there is an index j ≥ i

such that Pj+1 = P vj .

Associated to P are the ‘infinite polytopal’ algebra

R[P] = limi→∞

R[Pi]

and the filtered union

Col(P) = limi→∞

Col(Pi).Toric rings and discrete convex geometry – p.112/118

Now we can define a stable elementary group:

E(R,P) = subgroup of gr.autR(R[P]) generated by eλv

v ∈ Col(P), λ ∈ R.

Note: it depends only on P, not on the doubling spectrum.

Theorem 4.8. E(R,P) is a perfect group with trivial center.


Polytopal Steinberg groups

The group St(R,P) is defined by

generators xλv , v ∈ Col( f P), λ ∈ R representing the elementary

automorphisms eλv

the (formal) Steinberg relations between the xλv

It depends only on the partial product structure on Col(P). This

allows some functoriality in P.


Polytopal K2

In analogy with Milnor’s theorem we have

Theorem 4.9. P balanced polytope ⇒ St(R,P) → E(R,P) is a

universal central extension with kernel equal to the center of St(R,P).

Definition 4.10.

K2(R,P) = Ker(St(R,P) → E(R,P)

).


Balanced polygons

It is not difficult to classify the balanced polygons = 2-dimensional

polytopes: (K2 = classical K2)

{±u,±v,±w} K2

{±u,±v} K2 ⊕K2

{u,±v,w}w = uv

u = w(−v) K2 ⊕K2


{u,v,w}w = uv K2 ⊕K2

{v : Pv = F} K2

{u,v} K2 ⊕K2Toric rings and discrete convex geometry – p.117/118

Higher K-groups

Using Quillen’s +-construction or Volodin’s construction one can

define higher K-groups.

For certain well-behaved polytopes both constructions yield the

same result (in the classical case proved by Suslin).

Potentially difficult polytope:

uv

wToric rings and discrete convex geometry – p.118/118

Documents

Toric rings and discrete convex geometryToric rings and discrete convex geometry Lectures for the School on Commutative Algebra and Interactions with Algebraic Geometry and Combinatorics