Stanley-Reisner Rings (10/24/02)

Speaker: Vic Reiner

k(Δ) associated a simplicial complex Δ on vertex set V = k[x_{v} : v ∈ V]/I_{Δ}, where

I_{Δ} = {x_{v_{1}}, …, x_{v_{r}} : {v_{1}, …, v_{r}} ∉ Δ}

= arbitrary square-free monimial ideal

Motivation (i)

Arbitrary graded rings deform to k[Δ]’s, leaving many properties (Knull dimension, Hilbert series, degree of projection embedding) unchanged; and having many homological invariants only increasing.

Motivation (ii)

For k[d], almost any (ring-theoretic) homological invariant (e.g., T or^{s}(k[Δ],), H_{m}(k[Δ]) local cohomology) are computed via simplicial (co-) homology of Δ. E.g., dependence on the characteristic of the field k can be subtle for these ring invariants, but comes down to torsion for H(Δ, k).

[figure: •; •; •; •; •; •; •; •; •; •; •; •; 1; 3; 2; 1; 3; 2; 4; 6; 5; Δ =]

= IRP^{2} has k[Δ] = k[x_{1}, x_{2}, …, x_{6}]/(x_{1}x_{2}x_{3}, x_{1}x_{2}x_{6}, …) with most of its homological invariants depending upon whether char(k) = 2 or not, since

H̃_{i}(Δ; k) = \begin{cases}0 & i > 2 \\ k & i = 2 \\ k & i = 1 \\ 0 & i = 0 \end{cases} if char(k) = 2

= 0 ∀i,  if char(k) ≠ 2.

Motivation (iii)

For some combinatorial problems about simplicial complexes Δ, the approach via k[Δ] is the easy way or the \underline{only} way. E.g., The upper bound conjecture (UBC) for simplicial polytopes and spheres (Motzkin 1957?) CONJ: Δ a simplicial (d − 1)-dimensional sphere (e.g., boundary of a simplicial convex polytope)

⇒ \underbrace{f_{i}(Δ)}_{# i-dim’l simplicies} ≤ f_{i}(Δ_{c(n,d)}) where

Δ_{c(n,d)} = boundary of the cyclic d-polytope C(n, d) with n vertices
= convex hull of any n points on the moment curve {(t, t^{2}, …, t^{d}) : t ∈ IR} ⊂ IR^{d}

e.g. n = 6 UBC is proven for convex polytopes by Peter Mcmullen in 1970 (?) using key observations about the n-vectors ...

f(Δ)(f_{−1}, f_{0}, f_{1}, f_{2}) = (1, 5, 9, 6)
h(Δ)(h_{0}, h_{1}, h_{2}, h_{3}) = (1, 2, 2, 1)

1

1  5
1  4  9
1  3  5  6
(1  2  2  1 )= h(Δ)

So

Hilb(k[Δ], t) = f_{−1} + f_{0}(\frac{t}{1 − t}) + f_{1}(\frac{t}{1 − t})^{2} + f_{2}(\frac{t}{1 − t})^{3}

= 1 + 5(\frac{t}{1 − t}) + 9(\frac{t}{1 − t})^{2} + 6(\frac{t}{1 − t})^{3}

= \frac{h_{0} + h_{1}t + h_{2}t^{2} + f_{3}t^{3}}{(1 − t)^{3}}

= \frac{1 + 2t + 2t^{2} + t^{3}}{(1 − t)^{3}}

McMullen’s observation 1

UBC follows from

h_{i}(Δ) ≤ \binom{n − d + i − 1}{i}

where n = f_{0} = # of vertices.

(follows from explicit knowledge of f_{i} for boundary of C(n, d) and a little mucking around... )

McMullen’s observation 2

h_{i}(Δ) ≤ \binom{n − d + i − 1}{i} is easy to prove by induction on f_{d−1} = # of facets (=maximal faces) for Δ which are pure

shellable simplicial complies (of dimension d − 1 with n vertices)

Δ is shellable if it can be built up by ordering facets F_{1}, F_{2}, … so that ∀i ≥ 2,

F_{i} ∩ \underbrace{(\overline{∪_{j<i}F_{j}})}_{sub complex gen’d by F_{1}, F_{2}, …, F_{i−1}}

is pure of codimension inside F_{i} When d = 3, d − 1 = 2,

Brngesser & Mani (1969?), Boundary of convex polytopes are shellable (this proves UBC)

McMullen’s observation 3

For Δ shellable, h_{i}(Δ) counts something: it is equal to the number of facets F_{i} is shelling having d − i new walls, i old walls, where d − i new walls are not in \overline{∪_{j<i}F_{i}}.

e.g.,

facets new walls d: # new walls
F_{1} 123 12, 13, 23 0 }h_{0} = 1
F_{2} 134  14, 34  1 }h_{1} = 2
F_{3} 145  15, 45  1
F_{4} 345  35  2 }h_{2} = 2
F_{5} 235  25  2
F_{6} 125  ∅  3 }h_{3} = 1

For shellable Δ, Cor 1: h_{i}(Δ) ≥ 0 Cor 2: h_{i}(Δ) = h_{d−i}(Δ) (provided Δ is the boundary of a d-dimensional polytope, or more generally has a shelling

order whose reverse is also a shelling order).

\overbrace{Dehn}^{1905} − \overbrace{Sommerville}^{1927} equations. (The reverse of a Barg-Mani shelling is still a shelling, and “old” ↔ “new”)