Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
Cohen-Macaulay toric rings arising from finite
graphs
Augustine O’Keefe
Department of Mathematics
Tulane University
New Orleans, LA 70118
[email protected]
October 15, 2011
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
Study homological properties of toric rings.
depth?
Cohen-Macaulay?
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
Study homological properties of toric rings.
depth?
Cohen-Macaulay?
Restrict our focus to toric rings arising from discrete graphs.
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
Study homological properties of toric rings.
depth?
Cohen-Macaulay?
Restrict our focus to toric rings arising from discrete graphs.
For a simple graph G, we denote its associated toric ring by
K[G].
Villarreal (1995) studied these rings as fibers of the Rees
algebra over the edge ideal of a graph.
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
Study homological properties of toric rings.
depth?
Cohen-Macaulay?
Restrict our focus to toric rings arising from discrete graphs.
For a simple graph G, we denote its associated toric ring by
K[G].
Villarreal (1995) studied these rings as fibers of the Rees
algebra over the edge ideal of a graph.
Proved all bipartite graphs yield a normal toric ring.
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
Study homological properties of toric rings.
depth?
Cohen-Macaulay?
Restrict our focus to toric rings arising from discrete graphs.
For a simple graph G, we denote its associated toric ring by
K[G].
Villarreal (1995) studied these rings as fibers of the Rees
algebra over the edge ideal of a graph.
Proved all bipartite graphs yield a normal toric ring.
Hibi and Ohsugi (1999) studied K[G] as the image of a
monomial map.
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
Study homological properties of toric rings.
depth?
Cohen-Macaulay?
Restrict our focus to toric rings arising from discrete graphs.
For a simple graph G, we denote its associated toric ring by
K[G].
Villarreal (1995) studied these rings as fibers of the Rees
algebra over the edge ideal of a graph.
Proved all bipartite graphs yield a normal toric ring.
Hibi and Ohsugi (1999) studied K[G] as the image of a
monomial map.
Classified all graphs G such that K[G] is normal.
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
Study homological properties of toric rings.
depth?
Cohen-Macaulay?
Restrict our focus to toric rings arising from discrete graphs.
For a simple graph G, we denote its associated toric ring by
K[G].
Villarreal (1995) studied these rings as fibers of the Rees
algebra over the edge ideal of a graph.
Proved all bipartite graphs yield a normal toric ring.
Hibi and Ohsugi (1999) studied K[G] as the image of a
monomial map.
Classified all graphs G such that K[G] is normal.
Hochster (1972) proved that all normal semigroup rings are
Cohen-Macaulay.
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
Study homological properties of toric rings.
depth?
Cohen-Macaulay?
Restrict our focus to toric rings arising from discrete graphs.
For a simple graph G, we denote its associated toric ring by
K[G].
Villarreal (1995) studied these rings as fibers of the Rees
algebra over the edge ideal of a graph.
Proved all bipartite graphs yield a normal toric ring.
Hibi and Ohsugi (1999) studied K[G] as the image of a
monomial map.
Classified all graphs G such that K[G] is normal.
Hochster (1972) proved that all normal semigroup rings are
Cohen-Macaulay.
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
The Goal:
Classify all graphs, G, such that K[G] is Cohen-Macaulay.
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
The Goal:
Classify all graphs, G, such that K[G] is Cohen-Macaulay.
Classify all graphs, G, such that K[G] is NOT Cohen-Macaulay.
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
The Goal:
Classify all graphs, G, such that K[G] is Cohen-Macaulay.
Classify all graphs, G, such that K[G] is NOT Cohen-Macaulay.
Recall that K[G] is Cohen-Macaulay if
depth K[G] = dim K[G].
In general depth K[G] ≤ dim K[G].
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
The Goal:
Classify all graphs, G, such that K[G] is Cohen-Macaulay.
Classify all graphs, G, such that K[G] is NOT Cohen-Macaulay.
Recall that K[G] is Cohen-Macaulay if
depth K[G] = dim K[G].
In general depth K[G] ≤ dim K[G].
Two approaches:
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
The Goal:
Classify all graphs, G, such that K[G] is Cohen-Macaulay.
Classify all graphs, G, such that K[G] is NOT Cohen-Macaulay.
Recall that K[G] is Cohen-Macaulay if
depth K[G] = dim K[G].
In general depth K[G] ≤ dim K[G].
Two approaches:
1 Explicitly calculate depth for particular families of graphs.
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
The Goal:
Classify all graphs, G, such that K[G] is Cohen-Macaulay.
Classify all graphs, G, such that K[G] is NOT Cohen-Macaulay.
Recall that K[G] is Cohen-Macaulay if
depth K[G] = dim K[G].
In general depth K[G] ≤ dim K[G].
Two approaches:
1 Explicitly calculate depth for particular families of graphs.
Construct a graph such that dim K[G] − depth K[G] is
arbitrarily large.
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
The Goal:
Classify all graphs, G, such that K[G] is Cohen-Macaulay.
Classify all graphs, G, such that K[G] is NOT Cohen-Macaulay.
Recall that K[G] is Cohen-Macaulay if
depth K[G] = dim K[G].
In general depth K[G] ≤ dim K[G].
Two approaches:
1 Explicitly calculate depth for particular families of graphs.
Construct a graph such that dim K[G] − depth K[G] is
arbitrarily large.
2
Find forbidden structures in the graph which prevent
Cohen-Macaulayness.
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
The Goal:
Classify all graphs, G, such that K[G] is Cohen-Macaulay.
Classify all graphs, G, such that K[G] is NOT Cohen-Macaulay.
Recall that K[G] is Cohen-Macaulay if
depth K[G] = dim K[G].
In general depth K[G] ≤ dim K[G].
Two approaches:
1 Explicitly calculate depth for particular families of graphs.
Construct a graph such that dim K[G] − depth K[G] is
arbitrarily large.
2
Find forbidden structures in the graph which prevent
Cohen-Macaulayness.
If H is an induced subraph of G such that K[H] is not
Cohen-Macualay, is the same true for K[G]?
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
The toric ring K[G]
G = (V, E) a finite, connected and simple graph.
E(G) = {x1 , . . . , xn }
V (G) = {t1 , . . . , td }
←→
←→
A. O’Keefe
K[x] = K[x1 , . . . , xn ]
K[t] = K[t1 , . . . , td ]
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
The toric ring K[G]
G = (V, E) a finite, connected and simple graph.
E(G) = {x1 , . . . , xn }
V (G) = {t1 , . . . , td }
←→
←→
K[x] = K[x1 , . . . , xn ]
K[t] = K[t1 , . . . , td ]
Define the homomorphism
π : K[x] → K[t],
xi %→ ti1 ti2
where xi is the edge {ti1 , ti2 }
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
The toric ring K[G]
G = (V, E) a finite, connected and simple graph.
E(G) = {x1 , . . . , xn }
V (G) = {t1 , . . . , td }
←→
←→
K[x] = K[x1 , . . . , xn ]
K[t] = K[t1 , . . . , td ]
Define the homomorphism
π : K[x] → K[t],
xi %→ ti1 ti2
where xi is the edge {ti1 , ti2 }
IG := ker π is the toric ideal of G
K[G] := K[x]/IG is the toric ring of G
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
Facts about K[G]
Hibi and Ohsugi (1999) showed K[G] is normal exactly
when G satisfies the odd cycle property, i.e. any two odd
cycles C and C ! in G must share a vertex or have an edge
between them.
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
Facts about K[G]
Hibi and Ohsugi (1999) showed K[G] is normal exactly
when G satisfies the odd cycle property, i.e. any two odd
cycles C and C ! in G must share a vertex or have an edge
between them.
dim K[G] =
!
|V (G)| − 1
|V (G)|
A. O’Keefe
if G is bipartite
otherwise
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
Facts about K[G]
Hibi and Ohsugi (1999) showed K[G] is normal exactly
when G satisfies the odd cycle property, i.e. any two odd
cycles C and C ! in G must share a vertex or have an edge
between them.
dim K[G] =
!
|V (G)| − 1
|V (G)|
if G is bipartite
otherwise
Generators of IG arise from even closed paths in G.
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
Facts about K[G]
Hibi and Ohsugi (1999) showed K[G] is normal exactly
when G satisfies the odd cycle property, i.e. any two odd
cycles C and C ! in G must share a vertex or have an edge
between them.
dim K[G] =
!
|V (G)| − 1
|V (G)|
if G is bipartite
otherwise
Generators of IG arise from even closed paths in G.
By Auslander-Buchsbaum formula,
depth K[G] = dimK K[G] − pd K[G] = |E(G)| − pd K[G]
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
A family of graphs
t2
x1
Gk+6 :
t5
t7
x7
t1
t8
x9
x4
x8
x10
t4
x2
x5
x3
t3
x2k+5
x2k+6
tk+6
A. O’Keefe
x6
t6
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
A family of graphs
t2
x1
Gk+6 :
t5
t7
x7
t1
t8
x9
x4
x8
x10
t4
x2
x5
x3
t3
x2k+5
x2k+6
tk+6
A. O’Keefe
x6
t6
C-M toric rings
depth K[Gk+6 ] = 7
for k ≥ 1.
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
A family of graphs
t2
x1
Gk+6 :
t5
t7
x7
t1
t8
x9
x4
x8
x10
t4
x2
x5
x3
t3
x2k+5
x2k+6
x6
tk+6
depth K[Gk+6 ] = 7
for k ≥ 1.
t6
Theorem (Hibi, Higashitani, Kimura, -)
Let f, d be integers such that 7 ≤ f ≤ d. Then there exists a
graph G with |V (G)| = d such that dim K[G] = d and
depth K[G] = f .
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
Sketch of proof
Let f = 8 and d = 10.
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
Sketch of proof
Let f = 8 and d = 10.
t2
x1
G9 :
x9
x2
x3
t8
x11
Consider Gd−f +7 = G9.
x4
x8
x7
t1
t3
t5
t7
t4
x10
x5
x12
t9
x6
t6
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
Sketch of proof
Let f = 8 and d = 10.
t10
t2
x1
G:
x9
x2
t8
x11
t3
Consider Gd−f +7 = G9.
x4
x8
x7
t1
x3
t5
t7
t4
x10
x5
x12
Add path of length
x6
f-7=1
t9
t6
dim K[G] = 10
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
Sketch of proof
Let f = 8 and d = 10.
t10
t2
x1
G:
x9
x2
t8
x11
t3
Consider Gd−f +7 = G9.
x4
x8
x7
t1
x3
t5
t7
t4
x10
x5
x12
Add path of length
x6
f-7=1
t9
t6
dim K[G] = 10
pd K[G] = pd K[G9 ] = |E(G9 )| − 7 = 5
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
Sketch of proof
Let f = 8 and d = 10.
t10
t2
x1
G:
x9
x2
t8
x11
t3
Consider Gd−f +7 = G9.
x4
x8
x7
t1
x3
t5
t7
t4
x10
x5
x12
Add path of length
x6
f-7=1
t9
t6
dim K[G] = 10
pd K[G] = pd K[G9 ] = |E(G9 )| − 7 = 5
depth K[G] = |E(G)| − 5 = 8
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
Let’s look again at the family of graphs.
t2
x1
Gk+6 :
t5
t7
x7
t1
t8
x9
x4
x8
x10
t4
x2
x5
x3
x2k+5
x2k+6
tk+6
t3
x6
t6
depth K[Gk+6 ] = 7 '= k + 6 = dim K[G] for k ≥ 2.
What about these graphs prevent Cohen-Macaulayness?
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
Suppose G consists of 2 odd cycles, C1 and C2 , connected by a
path of length ≥ 2.
C1
C2
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
Suppose G consists of 2 odd cycles, C1 and C2 , connected by a
path of length ≥ 2.
C1
C2
One generator in IG since only one even closed path.
Therefore pdK[x] K[G] = 1.
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
Suppose G consists of 2 odd cycles, C1 and C2 , connected by a
path of length ≥ 2.
C1
C2
One generator in IG since only one even closed path.
Therefore pdK[x] K[G] = 1.
By Auslander-Buchsbaum,
depth K[G] = |E(G)| − pd K[G] = |E(G)| − 1
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
Suppose G consists of 2 odd cycles, C1 and C2 , connected by a
path of length ≥ 2.
C1
C2
One generator in IG since only one even closed path.
Therefore pdK[x] K[G] = 1.
By Auslander-Buchsbaum,
depth K[G] = |E(G)| − pd K[G] = |E(G)| − 1 = |V (G)|.
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
Suppose G consists of 2 odd cycles, C1 and C2 , connected by a
path of length ≥ 2.
C1
C2
One generator in IG since only one even closed path.
Therefore pdK[x] K[G] = 1.
By Auslander-Buchsbaum,
depth K[G] = |E(G)| − pd K[G] = |E(G)| − 1 = |V (G)|.
K[G] is not normal but is Cohen-Macaulay.
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
What happens if add another path of between C1 and C2 of
length ≥ 2?
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
What happens if add another path of between C1 and C2 of
length ≥ 2?
dim K[G] = 13
depth K[G] = 12
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
What happens if the paths intersect?
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
What happens if the paths intersect?
dim K[G] = 13
depth K[G] = 12
K[G] is not Cohen-Macaulay.
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
What happens if the paths intersect?
dim K[G] = 12
depth K[G] = 12
K[G] is Cohen-Macaulay!
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
Theorem (Hà, -)
Suppose G is comprised of two odd cycles, C1 and C2 , connected
by k ≥ 2 paths, Γ1 , . . . , Γk , each of length li ≥ 2, for
i = 1, . . . , k. Furthermore, suppose that the Γi do not share any
edges and if a vertex v ∈ V (Γi ) ∩ V (Γj ), then
v ∈ V (C1 ) ∪ V (C2 ). Then K[G] is not Cohen-Macaulay.
Γ1
Γ2
C2
C1
Γk−1
Γk
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
Sketch of Proof
By Auslander-Buchsbaum, reduced to showing
pd K[G] > dimK K[G] − dim K[G] = |E(G)| −| V (G)| = k.
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
Sketch of Proof
By Auslander-Buchsbaum, reduced to showing
pd K[G] > dimK K[G] − dim K[G] = |E(G)| −| V (G)| = k.
Let AG = {a1 , . . . , an } be the incidence matrix of G.
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
Sketch of Proof
By Auslander-Buchsbaum, reduced to showing
pd K[G] > dimK K[G] − dim K[G] = |E(G)| −| V (G)| = k.
Let AG = {a1 , . . . , an } be the incidence matrix of G.
Let SG = NAG be the semigroup generated by the columns of
AG .
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
Sketch of Proof
By Auslander-Buchsbaum, reduced to showing
pd K[G] > dimK K[G] − dim K[G] = |E(G)| −| V (G)| = k.
Let AG = {a1 , . . . , an } be the incidence matrix of G.
Let SG = NAG be the semigroup generated by the columns of
AG .
For s ∈ SG , define the simplicial complex
∆s = {F ⊂ [n] : s − nF ∈ SG },
A. O’Keefe
nF =
C-M toric rings
"
i∈F
ai
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
Sketch of Proof
By Auslander-Buchsbaum, reduced to showing
pd K[G] > dimK K[G] − dim K[G] = |E(G)| −| V (G)| = k.
Let AG = {a1 , . . . , an } be the incidence matrix of G.
Let SG = NAG be the semigroup generated by the columns of
AG .
For s ∈ SG , define the simplicial complex
∆s = {F ⊂ [n] : s − nF ∈ SG },
nF =
"
ai
i∈F
Briales, Campillo, Marijuán and Pisón (1998) showed that
βj+1,s (K[G]) = dimK H̃j (∆s ; K).
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
∆s in terms of G
Choose s = (1 + pi : pi = number of paths ti lies on).
Then each path determines 2 facets in ∆s .
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
∆s in terms of G
Choose s = (1 + pi : pi = number of paths ti lies on).
Then each path determines 2 facets in ∆s .
s = (3, 1, 2, 3, 1, 2, 2, 2, 2, 2, 2, 2)
t2
x1
x8
x10
x9
t1
x4
x12
x7
x2
t4
t9
t8
t7
x11
x5
x3
x15
x14
x13
t3
∆1 :
!
t5
x6
t10
t11
t12
t6
F1,1 = {2, 4, 6, 7, 9, 11, 12, 13, 14, 15}
F1,2 = {1, 3, 5, 8, 10, 11, 12, 13, 14, 15}
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
∆s in terms of G
Choose s = (1 + pi : pi = number of paths ti lies on).
Then each path determines 2 facets in ∆s .
s = (3, 1, 2, 3, 1, 2, 2, 2, 2, 2, 2, 2)
t2
x1
x8
x10
x9
t1
x4
x12
x7
x2
t4
t9
t8
t7
x11
x5
x3
x15
x14
x13
t3
∆1 :
!
t5
x6
t10
t11
t12
t6
F1,1 = {2, 4, 6, 7, 9, 11, 12, 13, 14, 15}
F1,2 = {1, 3, 5, 8, 10, 11, 12, 13, 14, 15}
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
∆s in terms of G
Choose s = (1 + pi : pi = number of paths ti lies on).
Then each path determines 2 facets in ∆s .
s = (3, 1, 2, 3, 1, 2, 2, 2, 2, 2, 2, 2)
t2
x1
x8
x10
x9
t1
x4
x12
x7
x2
t4
t9
t8
t7
x11
x5
x3
x15
x14
x13
t3
∆1 :
!
t5
x6
t10
t11
t12
t6
F1,1 = {2, 4, 6, 7, 9, 11, 12, 13, 14, 15}
F1,2 = {1, 3, 5, 8, 10, 11, 12, 13, 14, 15}
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
Want to show pd K[G] > |E(G)| −| V (G)| = k.
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
Want to show pd K[G] > |E(G)| −| V (G)| = k.
Show βs,k+1 (K[G]) = dimK H̃k (∆s ; K) '= 0.
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
Want to show pd K[G] > |E(G)| −| V (G)| = k.
Show βs,k+1 (K[G]) = dimK H̃k (∆s ; K) '= 0.
Can express ∆s as a union of subcomplexes.
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
Want to show pd K[G] > |E(G)| −| V (G)| = k.
Show βs,k+1 (K[G]) = dimK H̃k (∆s ; K) '= 0.
Can express ∆s as a union of subcomplexes.
Apply Mayer-Vietoris recursively to get
H̃k (∆s ; K) ∼
= H̃k (F1,1 ∩ F1,2 ∩ ∆2 ∩ · · · ∩ ∆k ; K) '= (0).
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
What’s next?
Currently working on showing that if H is an induced
subgraph of G such that K[H] is not Cohen-Macaulay,
then neither is K[G]
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
What’s next?
Currently working on showing that if H is an induced
subgraph of G such that K[H] is not Cohen-Macaulay,
then neither is K[G]
Want to find an even more general class of graphs failing
the Cohen-Macaulay property.
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
What’s next?
Currently working on showing that if H is an induced
subgraph of G such that K[H] is not Cohen-Macaulay,
then neither is K[G]
Want to find an even more general class of graphs failing
the Cohen-Macaulay property.
Experimental evidence shows depth K[G] ≥ 7 when
|V (G)| ≥ 7.
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
What’s next?
Currently working on showing that if H is an induced
subgraph of G such that K[H] is not Cohen-Macaulay,
then neither is K[G]
Want to find an even more general class of graphs failing
the Cohen-Macaulay property.
Experimental evidence shows depth K[G] ≥ 7 when
|V (G)| ≥ 7.
What about other properties of K[G]? When is K[G]
Gorenstein?
A. O’Keefe
C-M toric rings
Motivation
Defining K[G]
Arbitrary depth and dimension
Obstructions to the C-M property
Open questions and future directions
Thanks!
A. O’Keefe
C-M toric rings