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\begin{document}
Informal Seminar on Stanley-Reisner Theory, UMN, Fall 2002 \\
17 October 2002
{\bf Introduction and motivation for Stanley-Reisner rings, I} \\
Speaker: Vic Reiner
Scribe notes by Jeremy Martin \\
\hrule
\section{Definitions}
\begin{defn} Let $V$ be a finite set of vertices. An \defterm{abstract simplicial complex}
$\Delta$ on $V$ is a subset of the power set $2^V$ which is closed under inclusion, that is,
$$F \in \Delta, ~ G \subset F \quad\implies\quad G \in \Delta.$$
\end{defn}
The elements of $\Delta$ are called \defterm{faces}. The \defterm{dimension} of a face, $\dim
F$, is defined as $|F|-1$.
We can often represent $\Delta$ pictorially. For instance, if $V = \{a,b,c,d\}$ and $\Delta$
is the abstract simplicial complex
\begin{equation} \label{ex1}
\Delta = \left\{ \emptyset, a, b, c, d, ab, bc, cd \right\}
\end{equation}
(abbreviating the face $\{a\}$ by $a$, $\{a,b\}$ by $ab$, etc.), then the corresponding figure
is
\begin{center}
\begin{picture}(50,50)
\putdot{10,10} \puttext{ 2,10}{$a$}
\putdot{10,40} \puttext{ 2,40}{$b$}
\putdot{40,40} \puttext{48,40}{$c$}
\putdot{40,10} \puttext{48,10}{$d$}
\putline{10,10}{0,1}{30}
\putline{10,40}{1,0}{30}
\putline{40,40}{0,-1}{30}
\end{picture}
\end{center}
One can think of $a,b,c,d$ as orthonormal basis vectors in $|V|$-space, so that the face $ab$
(which has dimension $1$) represents the affine span of the vectors $a,b$ (which is a line
segment), etc.
Fix a field $\fld$, and let
$$S = \fld[x_v ~:~ v \in V],$$
the (commutative) polynomial ring in variables corresponding to the vertices.
\begin{defn} The \defterm{Stanley-Reisner ideal} of $\Delta$ is
$$I_{\Delta} := \left( \prod_{j=1}^r x_{v_{i_j}} ~:~ \left\{v_{i_1}, \dots,
v_{i_r}\right\} \not\in \Delta \right).$$
\end{defn}
Note that $I_\Delta$ is a \defterm{monomial ideal} (that is, it is generated by monomials) and
that the generators are \defterm{squarefree} (they are not divisible by the square of any
variable). A minimal set of generators is given by the minimal nonfaces of $\Delta$.
\begin{defn} The \defterm{Stanley-Reisner ring} (or \defterm{face ring}) of $\Delta$ is
$$\fld[\Delta] := S/I_\Delta.$$
\end{defn}
Note that the set of monomials
$$\left\{ x_{v_1}^{e_1} \dots x_{v_r}^{e_r} ~:~ \{v_1, \dots, v_r\} \in \Delta, ~~
e_1, \dots, e_r > 0 \right\}$$
is a basis for $\fld[\Delta]$ as a $\fld$-vector space. In particular, $\fld[\Delta]$ is a
graded ring.
For example, if $\Delta$ is the simplicial complex given in (\ref{ex1}), then
$$I_\Delta = (ac,ad,bd)$$
(the minimal nonfaces of $\Delta$) and $\fld[\Delta]$ is the $k$-linear span of
\begin{equation} \label{bas}
\left\{ \begin{array}{ll}
1, a, a^2, a^3, \dots, & ab, a^2b, ab^2, \dots, \\
1, b, b^2, b^3, \dots, & bc, b^2c, bc^2, \dots, \\
1, c, c^2, c^3, \dots, & cd, c^2d, cd^2, \dots, \\
1, d, d^2, d^3, \dots
\end{array} \right\}~.
\end{equation}
This construction actually gives a bijection between simplicial complexes on $V$ and ideals of
$S$ generated by squarefree monomials. The simplicial complex corresponding to such an ideal
is its \defterm{Stanley-Reisner complex}.
\section{Motivations}
1. In algebraic geometry, one wants to study rings of the form $R=S/I$, where $S$ is a
polynomial ring over a field $\fld$ and $I$ is an ideal of $S$. That is, $R$ is the
coordinate ring of the affine algebraic variety defined by $I$. To study $R$ using
Stanley-Reisner rings, we may proceed as follows:
First, ``deform'' $I$ as follows. Fix some monomial order $<$ on $S$ and compute the
\defterm{initial ideal} $\ini_<(I) \subset S$ (this is equivalent to computing a Gr\"obner
basis of $I$). By definition, $\ini_<(I)$ is generated by monomials, However, the generators
need not be squarefree, so a second step, \defterm{polarization}, may be required. The idea
of this step is to get rid of high powers of variables by the following trick: if one of the
generators of $\ini_<(I)$ is, say, $x^2$, then we adjoin a new variable $x'$, replace $x^2$
with $xx'$, and mod out by $x-x'$. The result is a squarefree monomial ideal of some
polynomial ring $S' \supset S$, which we may regard as the Stanley-Reisner ideal $I_\Delta$ of
a simplicial complex $\Delta$ on the variables of $S'$. The ideal $\ini_<(I)$ and its
polarization are {\it very\/} closely related, so we don't have to worry too much about this
second step.
The passage from $I$ to $\ini_<(I)$ does not preserve all structure, but it is pretty good (in
the language of algebraic geometry, it is a \defterm{flat deformation}). Lots of
geometric/ring invariants of $R$ are closely related---often equal---to those of $\fld[\Delta]
= S'/I_{\Delta}$. For instance, the dimension, degree and Hilbert series of $R$ are the same
as for $\fld[\Delta]$, and these can be computed combinatorially from $\Delta$. In addition,
some homological-type properties, such as Cohen-Macaulayness, can only get worse--e.g., if
$\fld[\Delta]$ is Cohen-Macaulay then so is $R$. (For an example, see the scribe's Ph.D.
thesis.)
Here's an elementary example. Let $S = \fld[a,b,c,d]$, $I = (ac-b^2, bd-c^2, ad-bc)$, and $R
= S/I$. (In fact, $\Proj(R)$ is the twisted cubic, the image of the degree-$3$ Veronese
embedding $\PP^1 \to \PP^3$ mapping $[s:t]$ to $[s^3 : s^2t : st^2 : t^3]$ in homogeneous
coordinates.) The given generators of $I$ form a Gr\"obner basis with respect to
lexicographic order on the monomials of $S$, with $a