toric rings and discrete convex geometrytoric rings and discrete convex geometry lectures for the...
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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
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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).
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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.
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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
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Lecture 1
Affine monoids and their algebras
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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).
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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.
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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.
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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.
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Polytopal monoids
Definition 1.3. The convex hull conv(x1, . . . ,xm) of points xi ∈ Zn is
called a lattice polytope.
P
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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.
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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.
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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 ].
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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.
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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:
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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 .
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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+.
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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.
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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.
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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.
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(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!
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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.
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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]
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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
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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.
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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.
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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
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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.
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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 .
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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 ].
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[⋂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
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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.
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Lecture 2
Homological properties andcombinatorial applications
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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?
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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.
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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.
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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.
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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.
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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.
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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.
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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).
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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.
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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.
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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.
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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
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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.
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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).
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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) ?
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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).
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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.
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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))
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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).
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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.
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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.
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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.
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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 .
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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 !
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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.
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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 .
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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
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⇒ 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.
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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
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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.
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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�
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Lecture 3
Unimodular covers and triangulations
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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
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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
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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.
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d = 3: There exist empty simplices of arbitrary multiplicity!
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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).
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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}.
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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.
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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
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Multiples of polytopes
For c → ∞ (c ∈ N) the lattice points cP∩Zd approximate the
continuous structure of cP ∼ P better and better.
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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.
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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 !
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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
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Theorem 3.5. C has a triangulation into unimodular subcones.
Proof: Start with arbitrary triangulation. Refine by iterated stellar
subdivision to reduce multiplicities.
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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 .
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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.
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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
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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 ?
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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
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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 ?
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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 ?
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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.
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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
.
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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.
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Sketch of proof of Theorem 3.10: May assume P = ∆ is an
(empty) simplex. Consider corner cones of ∆:
v0
v2
v1
∆
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Apply Theorem 3.11 to corner cone:
v0
v2
v1
∆
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Must multiply P by factor c′ from Theorem 3.11 to get basic
unimodular corner simplices into P
v0
v2
v1
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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.
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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
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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 ?
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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
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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.
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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.
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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?
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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.
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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.
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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.
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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]).
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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].
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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).
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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.
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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).
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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).
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Balanced polytopes
A polytope is balanced if
σF(v) ≤ 1 for all v ∈ Col(P), F = Pw.
A nonbalanced polytope
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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)!
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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−
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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−}.
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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
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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.
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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.
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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)
).
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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
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{u,v,w}w = uv K2 ⊕K2
{v : Pv = F} K2
{u,v} K2 ⊕K2Toric rings and discrete convex geometry – p.117/118
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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