Toric rings and discrete convex geometry Lectures for the School on Commutative Algebra and Interactions with Algebraic Geometry and Combinatorics Trieste, May/June 2004 Winfried Bruns FB Mathematik/Informatik Universit ¨ at Osnabr ¨ uck Toric rings and discrete convex geometry – p.1/118
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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
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).
Toric rings and discrete convex geometry – p.2/118
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.
Toric rings and discrete convex geometry – p.3/118
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
Toric rings and discrete convex geometry – p.52/118
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.
Toric rings and discrete convex geometry – p.53/118
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.
Toric rings and discrete convex geometry – p.54/118
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.
Toric rings and discrete convex geometry – p.55/118
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 .
Toric rings and discrete convex geometry – p.56/118
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 !
Toric rings and discrete convex geometry – p.57/118
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.
Toric rings and discrete convex geometry – p.58/118
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 .
Toric rings and discrete convex geometry – p.59/118
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
Toric rings and discrete convex geometry – p.60/118
⇒ 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.
Toric rings and discrete convex geometry – p.61/118
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
Toric rings and discrete convex geometry – p.62/118
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.
Toric rings and discrete convex geometry – p.63/118
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�
Toric rings and discrete convex geometry – p.64/118
Lecture 3
Unimodular covers and triangulations
Toric rings and discrete convex geometry – p.65/118
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.
Toric rings and discrete convex geometry – p.68/118
d = 3: There exist empty simplices of arbitrary multiplicity!
Toric rings and discrete convex geometry – p.69/118
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).
Toric rings and discrete convex geometry – p.70/118
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}.
Toric rings and discrete convex geometry – p.71/118
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.
Toric rings and discrete convex geometry – p.72/118
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
Toric rings and discrete convex geometry – p.73/118
Multiples of polytopes
For c → ∞ (c ∈ N) the lattice points cP∩Zd approximate the
continuous structure of cP ∼ P better and better.
Toric rings and discrete convex geometry – p.74/118
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.
Toric rings and discrete convex geometry – p.75/118
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 !
Toric rings and discrete convex geometry – p.76/118
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
Toric rings and discrete convex geometry – p.77/118
Theorem 3.5. C has a triangulation into unimodular subcones.
Proof: Start with arbitrary triangulation. Refine by iterated stellar
subdivision to reduce multiplicities.
Toric rings and discrete convex geometry – p.78/118
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 .
Toric rings and discrete convex geometry – p.79/118
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.
Toric rings and discrete convex geometry – p.80/118
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 ?
Toric rings and discrete convex geometry – p.82/118
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,