%%%%%%%%%% the following setup maximizes number of characters per page--- %%%%%%%%%% good for solution sets, etc. Don't use for articles. \documentclass{amsart} \raggedbottom \oddsidemargin=0in \evensidemargin=0in \textwidth=6.5in \textheight=9in \topmargin=0in \headheight=0in \headsep=0in \footskip=0in \parskip=10bp \parindent=0bp %\footheight=0in \pagestyle{empty} \usepackage{amsfonts,amssymb} \newtheorem{thm}{Theorem} \newtheorem{defn}[thm]{Definition} \newtheorem{lemma}[thm]{Lemma} \renewcommand{\star}{\mathop{\rm st}\nolimits} \newcommand{\link}{\mathop{\rm lk}\nolimits} \newcommand{\sm}{\setminus} \newcommand{\st}{~\mid~} \newcommand{\0}{\emptyset} \newcommand{\ini}{\mathop{\rm in}\nolimits} \newcommand{\CE}{\check{C}} \newcommand{\Proj}{\mathop{\rm Proj}\nolimits} \newcommand{\Hilb}{\mathop{\rm Hilb}\nolimits} \newcommand{\supp}{\mathop{\rm supp}\nolimits} \newcommand{\depth}{\mathop{\rm depth}\nolimits} \newcommand{\Ann}{\mathop{\rm Ann}\nolimits} \newcommand{\Ass}{\mathop{\rm Ass}\nolimits} \newcommand{\fld}{{\bf k}} \newcommand{\NN}{{\mathbb N}} \newcommand{\ZZ}{{\mathbb Z}} \newcommand{\PP}{{\mathbb P}} \newcommand{\isom}{\cong} \newcommand{\defterm}[1] {{\bf #1}} \newcommand{\puttext}[2] {\put(#1){\makebox(0,0){#2}}} \newcommand{\putdot}[1] {\put(#1){\makebox(0,0){$\bullet$}}} \newcommand{\putline}[3] {\put(#1){\line(#2){#3}}} \begin{document} Informal Seminar on Stanley-Reisner Theory, UMN, Fall 2002 \\ 5 December 2002 {\bf Local cohomology} \\ Speaker: Anton Leykin Scribe notes by Jeremy Martin \\ \hrule We work with the following objects throughout. Let $\fld$ be a field and $R$ a finitely generated, $\NN^n$-graded $\fld$-algebra, i.e., $R = \bigoplus_{\alpha \in \NN^n} R_\alpha$ with $R_\alpha R_\beta \subset R_{\alpha+\beta}$. The motivating example is $R = \fld[\Delta]$, the Stanley-Reisner ring of a simplicial complex $\Delta$. Define $$R_+ := \bigoplus_{\alpha \neq 0} R_\alpha$$ the so-called \defterm{irrelevant ideal}. Finally, $M$ will be a $\ZZ^n$-graded $R$-module, i.e., $M = \bigoplus_{\alpha \in \ZZ^n} M_\alpha$ with $R_\alpha M_\beta \subset M_{\alpha+\beta}$. %-------------------------------------------------------------------------------------- \section{Depth and Cohen-Macaulayness} \begin{defn} An element $a \in R$ is called a \defterm{nonzerodivisor} (or NZD) on $M$ if $m \in M$, $am=0$ implies $m=0$. Equivalently, the map $$M \xrightarrow{\cdot a} M$$ given by multiplication by $r$ is one-to-one. \end{defn} \begin{defn} A sequence of homogeneous elements $\theta_1, \dots, \theta_s$ is a \defterm{regular $M$-sequence}, or \defterm{$M$-sequence} for short, if $\theta_{i+1}$ is a NZD on $M/(\theta_1, \dots, \theta_i)M$ for $i=0, \dots, s-1$. \end{defn} \begin{defn} The \defterm{dimension} of $M$, denoted $\dim_R M$ or $\dim M$, is the Krull dimension of $R/\Ann_R M$. The \defterm{depth} of $M$, denoted $\depth_R M$ or $\depth M$, is the length of a maximal $M$-sequence. It can be shown that every maximal $M$-sequence has the same length. \end{defn} In general $\depth_R M \leq \dim_R M$ (since any $M$-sequence of length $s$ generates a height-$s$ ideal of $R/\Ann M$). Equality is an important ``niceness'' condition which gets its own name: \begin{defn} $M$ is \defterm{Cohen-Macaulay} if $\depth_R M = \dim_R M$. \end{defn} %-------------------------------------------------------------------------------------- \section{Local Cohomology} Define the \defterm{torsion functor} $\Gamma$ by $$\Gamma(M) := \left\{u \in M \st R_+^n u = 0 ~\text{for}~ n \gg 0 \right\}.$$ It is routine to check that $\Gamma$ is a covariant, left-exact functor. That is, a map $f: M \to N$ of graded $R$-modules induces a map $\Gamma(f): \Gamma(M) \to \Gamma(N)$, and if $f$ is injective then $\Gamma(f)$ is injective. The $i$th local cohomology functor $H^i$ (more precisely, $H^i_{R_+}$) can now be defined as the $i$th right derived functor of $\Gamma$: $$H^i(M) = R^i \Gamma (M).$$ That is, one may calculate $H^i(M)$ by taking an injective resolution $$I^\bullet: \quad 0 ~\to~ M ~\to~ I^0 ~\to~ I^1 ~\to~ \dots,$$ applying $\Gamma$, and defining $H^i(M) := H^i(\Gamma I^\bullet)$. (See a textbook on homological algebra for more details.) \begin{lemma} \label{torsion} For all $i$, the modules $H^i(M)$ are \defterm{$R_+$-torsion}, i.e., they are killed by some power of $R_+$. \end{lemma} \begin{proof} By definition of $\Gamma$, every module of the form $\Gamma(N)$ is $R_+$-torsion. In particular, if $I^\bullet$ is an injective resolution, then every $\Gamma(I^i)$ is $R_+$-torsion, so the same is true of the cohomology modules of $\Gamma(I^\bullet)$. \end{proof} The local cohomology functors are useful because they detect depth and dimension. Specifically, we have the following fact. \begin{thm} Let $e = \depth_R M$ and $d = \dim_R M$. Then: \noindent (1) \quad $H^i(M)=0$ unless $e \leq i \leq d$. \noindent (2) \quad $H^e(M) \neq 0$ and $H^d(M) \neq 0$. \end{thm} \begin{proof} We'll prove only (1), which is the case we really need. We proceed by induction on $e$. If $H^0(M) \neq 0$, then every element of $R_+$ is a zerodivisor on $M$, which is exactly the statement that $e=0$. If $e=0$, then $$R_+ \subset \bigcup_{P \in \Ass M} P,$$ so $R_+$ is itself an associated prime. It is immediate that $\Gamma(M) = H^0(M) \neq 0$ as desired. For the inductive step, assume that (1) is true for all $R$-modules $N$ with $\depth N < e$. Let $a \in R_+$ be a homogeneous NZD on $M$ and let $N = M/aM$. Then $\depth_R N = e-1$, so by induction $H^i(N)=0$ for $i 0\}, \\ \supp(\alpha) &=& F(\alpha) \;\cup\; G(\alpha) ~=~ \{x_i \st \alpha_i \neq 0\}. \end{eqnarray*} \begin{thm}[Hochster] We have $$\Hilb(H^i(\fld[\Delta]);\;q) ~=~ \sum_{F \in \Delta} \dim_\fld \tilde H_{i-|F|-1}(\link_\Delta F;\;\fld) \prod_{x_i \in F} \frac{q_i^{-1}}{1-q_i^{-1}}.$$ \end{thm} \begin{proof} We compute $H^i(R)$ explicitly as the $i$th cohomology of the \v{C}ech complex $\CE^\bullet = \CE^\bullet(x_1, \dots, x_n ;\; M)$. If $F \not\in \Delta$, then the ring $R_F$ is zero, because $x_F=0$ in $R$. On the other hand, if $F \in \Delta$, then the variables in $F$ become units in $R_F$, and those not in $\star_\Delta F$ get killed (since they annihilate the unit $x_F$). That is, $$R_F ~=~ \fld\left[ \left\{x_i,x_i^{-1} ~:~ i \in F\right\} ~\cup~ \left\{x_j ~:~ x_j \in \link F \right\} \right] \otimes R.$$ Let $\alpha \in \ZZ^n$. We will compute the $\alpha$th graded piece $\CE^\bullet_\alpha$ of the \v{C}ech complex. If $\supp(\alpha) \not\in \Delta$, then $\CE^\bullet_\alpha = 0$, because $R_\alpha=0$ and adjoining inverses doesn't change this. So suppose that $\supp(\alpha) \in \Delta$. Let $F = F(\alpha)$, $j=|F|$, and $G = G(\alpha)$. A priori, we have \begin{equation} \CE^r_\alpha ~=~ \left[ \bigoplus_{|F'|=r} R_{F'} \right]_\alpha ~=~ \bigoplus_{|F'|=r} [R_{F'}]_\alpha. \end{equation} A whole bunch of these summands are zero. Specifically, for $R_{F'}$ to be nonzero, we must have $F' \in \Delta$ (as previously noted), $F' \supset F$ (since the variables in $F$ must be units in the $\alpha$th graded piece of the \v{C}ech complex), and $F' \cup G \in \Delta$ (so that $x^\alpha$ itself is nonzero). This is all equivalent to the condition that $F'' = F' \sm F$ belong to $\link_{\star G} F$, so we may write \begin{equation} \CE^r_\alpha ~=~ \left[ \bigoplus_{\substack{F'' \in \link_{\star G} F \\ |F''|=r-j}} R_{F \cup F''} \right]_\alpha. \end{equation} The maps in $\CE^r_\alpha$ correspond to the usual coboundary maps of the simplicial cochain complex of $\link_{\star G} F$, shifted by $j+1$. That is, \begin{eqnarray} \left[H^i(R)\right]_\alpha &=& \tilde H^{i-j-1}(\link_{\star G} F;\; \fld) \\ &\isom& \tilde H_{i-j-1}(\link_{\star G} F;\; \fld) \end{eqnarray} since this is a finite-dimensional $\fld$-vector space, hence isomorphic to its dual. (The isomorphism is not canonical, but we don't care because we're really only interested in its dimension.) If $G \neq \emptyset$, then $\link_{\star G} F$ is a cone over $G$. In particular it is contractible, so $\tilde H_\bullet(\link_{\star G} F;\; \fld)=0$. Therefore we only have nonzero terms when $G = \emptyset$, so $\link_{\star G} F = \link_\Delta F$ and the last equation becomes \begin{equation} \left[H^i(R)\right]_\alpha ~\isom~ \tilde H_{i-j-1}(\link F;\; \fld). \end{equation} Therefore \begin{eqnarray} \Hilb(H^i(\fld[\Delta]);\;q) &=& \sum_{F \in \Delta} \sum_{\substack{\alpha: \\ \supp(\alpha)=F(\alpha)=F}} \dim_\fld \tilde H_{i-|F|-1}(\link F;\;\fld) q^\alpha \\ &=& \sum_{F \in \Delta} \dim_\fld \tilde H_{i-|F|-1}(\link F;\;\fld) \sum_{\substack{\alpha: \\ \supp(\alpha)=F(\alpha)=F}} q^\alpha \\ &=& \sum_{F \in \Delta} \dim_\fld \tilde H_{i-|F|-1}(\link F;\;\fld) \prod_{x_i \in F} \frac{q_i^{-1}}{1-q_i^{-1}} \end{eqnarray} as desired. \end{proof} Here's an example (from Stanley) of computing a \v{C}ech complex. Let $\Delta$ be the complex $\langle 12,13,23,4 \rangle =$ \begin{equation} \label{delta-ex} \begin{picture}(110,60) \putdot{10,10} \puttext{10, 2}{$1$} \putdot{40,50} \puttext{40,58}{$2$} \putdot{70,10} \puttext{70, 2}{$3$} \putdot{100,40} \puttext{100,32}{$4$} \putline{10,10}{3,4}{30} % 12 \putline{10,10}{1,0}{60} % 13 \putline{70,10}{-3,4}{30} % 23 \end{picture} \end{equation} and $R = \fld[\Delta] = \fld[x_1,x_2,x_3,x_4]/(x_1x_4,x_2x_4,x_3x_4,x_1x_2x_3)$. Then the \v{C}ech complex is $$0 ~\to~ \underbrace{R}_{\CE^0} ~\to~ \underbrace{R_1 \oplus R_2 \oplus R_2 \oplus R_3 \oplus R_4}_{\CE^1} ~\to~ \underbrace{R_{12} \oplus R_{13} \oplus R_{23}}_{\CE^2} ~\to~ 0,$$ where $R_1 = R[x_1^{-1}]$, $R_{12} = R[x_1^{-1},x_2^{-1}]$, etc. $\bullet$ \quad For $\alpha=(0,0,0,0)$, so $F(\alpha)=G(\alpha)=\emptyset$, we have $$\link_{\star G} F = \link_\Delta \emptyset = \Delta,$$ \quad~ so $\left[ H^i(R) \right]_\alpha = \tilde H_{i-1}(\Delta;\;\fld)$. $\bullet$ \quad For $\alpha=(-2,3,0,0)$, we have $$F = \{x_1\}, \qquad G = \{x_2\}, \qquad \link_{\star G} F = \langle x_2 \rangle ~~ \text{(i.e., a point)}$$ \quad~ so $\left[ H^i(R) \right]_\alpha = \tilde H_{i-2}($point$) = 0$. %-------------------------------------------------------------------------------------- \section{Reisner's Theorem} Let $\Delta$ be a simplicial complex and $R=-\fld[Delta]$. Let $d = \dim R = 1+\dim \Delta$. We will say that $\Delta$ satisfies \defterm{Reisner's criterion} if for all $F \in \Delta$, and $i<\dim(\link F)$. we have $$\tilde H_i(\link F;\;\fld)=0.$$ \begin{thm} $\Delta$ is Cohen-Macaulay if and only if it satisfies Reisner's criterion. \end{thm} {\bf Remark:} $\Delta$ is \defterm{Gorenstein} (a stronger condition than Cohen-Macaulayness) if in addition $\tilde H_i(\link F;\;\fld) \isom \fld$ for $i=\dim(\link F)$. \begin{proof} First, we show that a Cohen-Macaulay complex is pure (i.e., all maximal faces have the same dimension). Indeed, if $\Delta$ is Cohen-Macaulay of dimension $d-1$ and $\dim F < d-1$, then $\tilde H_{-1}(\link F)=0$ by Hochster's theorem, so $\link F \neq 0$ and $F$ is not maximal. (This can also be shown without Hochster's theorem; see Bruns and Herzog, p.~210.) Next, we show that a complex $\Delta$ satisfying Reisner's criterion is pure. If $\dim F = 0$ then there is nothing to show. Otherwise, we induct on dimension. Reisner's criterion gives $\tilde H_0(\Delta) = \tilde H_0(\link \emptyset) = 0$, so $\Delta$ is connected. Moreover, for every vertex $v$, the subcomplex $\link\{v\}$ of $\Delta$ satisfies Reisner's criterion and has dimension less than that of $\Delta$, so it is pure by induction. Now, for any maximal face $F$, let $v,w \in F$; we have $$\dim \link\{v\} = |F-v|-1 = |F-w|-1 = \dim \link\{w\};$$ by connectedness all links of vertices must have the same dimension, and the same equation implies that $\Delta$ is pure. By these two observations together, we may assume that $\Delta$ is pure, so $|F|=d$ for all maximal faces and $\dim(\link F)=d-|F|-1$ for all faces. So Cohen-Macaulayness \begin{eqnarray*} F~\text{is Cohen-Macaulay} &\iff& \fld[\Delta]~\text{is Cohen-Macaulay} \\ &\iff& \tilde H_{j-|F|-1}(\link F;\;\fld)=0 ~\text{for}~ j