A naive constructible sheaf

In this post we describe a constructible sheaf over $X=\Ran^{\leqslant n}(M)\times \R_{>0}$ valued in simplicial complexes, for a compact, smooth, connected manifold $M$. We note however that it does not capture all the information about the underlying space. Thanks to Joe Berner for helpful ideas.

Recall the category $SC$ of simplicial complexes and simplicial maps, as well as the full subcategories $SC_n$ of simplicial complexes with $n$ vertices (the vertices are unordered). Let $A = \bigcup_{k=1}^n SC_n$ with the ordering $\leqslant_A$ as in a previous post ("Ordering simplicial complexes with unlabeled vertices," 2017-12-03), and $f:X\to A$ the stratifying map. Let $\{A_k\}_{k=1}^N$ be a cover of $X$ by nested open sets of the type $f^{-1}(U_S) = f^{-1}(\{T\in A\ :\ S\leqslant_A T\})$, whose existence is guaranteed as $A$ is finite. Note that $f(A_1)$ is a singleton containg the complete simplex on $n$ vertices.

Remark: For every simplicial complex $S\in A$, there is a locally constant sheaf over $f^{-1}(S)\subseteq X$. Given the cover $\{A_k\}$ of $X$, denote this sheaf by $\mathcal F_k \in \Shv(A_k\setminus A_{k-1})$ and its value by $S_k\in SC$.

Let $i^1:A_1\hookrightarrow A_2$ and $j^2:A_2\setminus A_1 \hookrightarrow A_2$ be the natural inclusion maps . Note that $A_1$ is open and $A_2\setminus A_1$ is closed in $A_2$. The maps $i^1,j^2$ induce direct image functors on the sheaf categories\[i^1_*:\Shv(A_1) \to \Shv(A_2),
\hspace{1cm}
j^2_*:\Shv(A_2\setminus A_1) \to \Shv(A_2).\]The induced sheaves in $\Shv(A_2)$ are extended by 0 on the complement of the domain from where they come. Note that since $A_2\setminus A_1\subseteq A_2$ is closed, $j^2_*$ is the same as $j^2_!$, the direct image with compact support. We then have the direct sum sheaf $i^1_*\mathcal F_1 \oplus j_*^2\mathcal F_2 \in \Shv(A_2)$, which we interpret as the disjoint union in $SC$. Then\[\left(i_*^1\mathcal F_1 \oplus j_2^*\mathcal F_2\right)(U) = \begin{cases}
S_1 & \text{ if }U\subseteq A_1, \\
S_2 & \text{ if }U\subseteq A_2\setminus A_1, \\
S_1\sqcup S_2 & \text{ else,}
\end{cases}
\hspace{1cm}
\left(i_*^1\mathcal F_1 \oplus j_2^*\mathcal F_2\right)_{(P,t)} = \begin{cases}
S_1 & \text{ if } (P,t)\in A_1, \\
S_2 & \text{ if }(P,t)\in \text{int}(A_2\setminus A_1), \\
S_1\sqcup S_2 & \text{ else,}
\end{cases}\]for $U\subseteq A_2$ open and $(P,t)\in A_2$. Generalizing this process, we get a sheaf on $X$. The diagram

may be helpful to keep in mind. We use the fact that direct sums commute with colimits (used in the definition of the direct image sheaf) to simplify notation. We then get sheaves\[\begin{array}{r c l}
\mathcal F^1 & \in & \Shv(A_1), \\
i_*^1\mathcal F^1 \oplus j_*^2 \mathcal F^2 & \in & \Shv(A_2), \\
i_*^2i_*^1\mathcal F^1 \oplus i_*^2j_*^2 \mathcal F^2 \oplus j_*^3 \mathcal F^3 & \in & \Shv(A_3), \\
i_*^3i_*^2i_*^1\mathcal F^1 \oplus i_*^3i_*^2j_*^2 \mathcal F^2 \oplus i_*^3j_*^3 \mathcal F^3 \oplus j_*^4 \mathcal F^4 & \in & \Shv(A_4),
\end{array}\]and finally\[i_*^{N-1\cdots 1}\mathcal F^1 \oplus \left(\bigoplus_{k=2}^{N-1} i_*^{N-1\cdots k}j_*^k \mathcal F^k \right) \oplus j_*^N \mathcal F^N \in \Shv(A_N=X),\]where $i_*^{N-1\cdots k}$ is the composition $i_*^{N-1} \circ i_*^{N-2} \circ \cdots \circ i_*^k$ of direct image functors. Call this last sheaf simply $\mathcal F \in \Shv(X)$. Each $i_*^k$ extends the sheaf by 0 on an ever larger domain, so every summand in $\mathcal F$ is non-zero on exactly one stratum as defined by $f:X\to A$. We now have a functor $\mathcal F:Op(X) \to SC$ defined by\[\mathcal F(U) = \bigsqcup_{k=1}^N S_k \delta_{U,A_K\setminus A_{k-1}},
\hspace{1cm}
\mathcal F_{(P,t)} = \bigsqcup_{k=1}^N S_k \delta_{(P,t),\text{cl}(,A_K\setminus A_{k-1})},\]where $\delta_{U,V}$ is the Kronecker delta that evaluates to the identity if $U\cap V \neq \emptyset$ and zero otherwise.

Remark: The sheaf $\mathcal F$ is $A$-constructible, as $\mathcal F|_{f^{-1}(S)}$ is a constant sheaf evaluating to the simplicial complex $S\in A$. However, if we want the cohomology groups to capture how the simplicial complexes change between strata, then we must use a different approach - all groups die when leaving a stratum because of the extension by zero construction.

References: nLab (article "Simplicial complexes")

Sheaves, derived and perverse

Let $X,Y$ be topological spaces and $f:X\to Y$ a continuous map. We let $\Shv(X)$ be the category of sheaves on $X$, $D(\Shv(X))$ the derived category of sheaves on $X$, and $D_b(\Shv(X))$ the bounded variant. Recall that $D(\mathcal A)$ for an abelian category $\mathcal A$ is constructed first by taking $C(\mathcal A)$, the category of cochains of elements of $\mathcal A$, quotienting by chain homotopy, then quotienting by all acylic chains.

Remark: Let $\mathcal F\in \Shv(X)$. Recall:

• a section of $\mathcal F$ is an element of $\mathcal F(U)$ for some $U\subseteq X$,
• a germ of $\mathcal F$ at $x\in X$ is an equivalence class in $\{s\in \mathcal F(U)\ :\ U\owns x\}/\sim_x$,
• $s\sim_x t$ iff every neighborhood $W$ of $x$ in $U\cap V$ has $s|_W = t|_W$, for $s\in \mathcal F(U)$, $t\in \mathcal F(V)$,
• the support of the section $s\in\mathcal F(U)$ is $\supp(s) = \{x\in U\ :\ s \nsim_x 0\}$,
• the support of the sheaf $\mathcal F$ is $\supp(\mathcal F) = \{x\in X\ :\ \mathcal F_x\neq 0\}$.

Definition: The map $f$ induces functors between categories of sheaves, called
\[\begin{array}{r r c l}
\text{direct image} & f_*\ :\ \Shv(X) & \to & \Shv(Y), \\
& (U\mapsto \mathcal F(U)) & \mapsto & (V\mapsto \mathcal F(f^{-1}(V))),\\[15pt]
\text{inverse image} & f^*\ :\ \Shv(Y) & \to & \Shv(X), \\
& (V\mapsto \mathcal G(V)) & \mapsto & \text{sh}\left(U\mapsto \text{colim}_{V\supseteq f(U)} \mathcal G(V)\right),\\[15pt]
\text{direct image with compact support} & f_!\ :\ \Shv(X) & \to & \Shv(Y), \\
& (U\mapsto \mathcal F(U)) & \mapsto & \left(V\mapsto \left\{ s\in\mathcal F(f^{-1}(V))\ :\ f|_{\supp(s)} \text{ is proper}\right\}\right).
\end{array}\]

Above we used that $f:X\to Y$ is proper if $f^{-1}(K)\subseteq X$ is compact, for every $K\subseteq Y$ compact. Next, recall that a functor $\varphi:\mathcal A\to \mathcal B$ induces a functor $R\varphi:D(\mathcal A)\to D(\mathcal B)$, called the (first) derived functor of $\varphi$, given by $R\varphi(A^\bullet) = H^1(\varphi(A)^\bullet)$.

Remark: Each of the maps $f_*,f^*,f_!$ have their derived analogues $Rf_*, Rf^*,Rf_!$, respectively. For reasons unclear, $Rf_!$ has a right adjoint, denoted $Rf^!:D(\Shv(Y))\to D(\Shv(X))$. This is called the exceptional inverse image.

We are now ready to define perverse sheaves.

Definition: Let $A^\bullet \in D(\Shv(X))$. Then:

• the $i$th cohomology sheaf of $A^\bullet$ is $H^i(A^\bullet) = \ker(d^i)/\im(d^i)$,
• $A^\bullet$ is a constructible complex if $H^i(A^\bullet)$ is a constructible sheaf for all $i$,
• $A^\bullet$ is a perverse sheaf if $A^\bullet\in D_b(\Shv(X))$ is constructible and $\dim(\supp(H^{-i}(P))) \leqslant i$ for all $i\in \Z$ and for $P=A^\bullet$ and $P=(A^\bullet)^\vee = (A^\vee)^\bullet$ the dual complex of sheaves.

We finish off with an example.

Example: Let $X = \R$ be a stratified space, with $X_0=0$ the origin and $X_1 = \R\setminus 0$. Let $\mathcal F\in \Shv(X)$ be an $\R$-valued sheaf given by $\mathcal F(U) = \inf_{x\in U} |x|$, and define a chain complex $A^\bullet$ in the following way:
\[0 \longrightarrow A^{-1} = \mathcal F \xrightarrow{ d^{-1}=\text{id} } A^0 = \mathcal F \xrightarrow{ d^0=0 } 0.\]
Note that for any $U\subseteq \R$, we have $H^{-1}(A^\bullet)(U) = \ker(d^{-1})(U) = \ker(\id:\mathcal F(U)\to \mathcal F(U)) = \emptyset$ if $0\not\in U$, and $0$ otherwise. Hence $\supp(H^{-1}(A^\bullet)) = \R\setminus 0$, whose dimension is 1. Next, $H^0(A^\bullet)(U) = \ker(d^0)(U)/\im(d^{-1})(U) = \ker(0:\mathcal F(U)\to 0)/\im(\id:\mathcal F(U)\to \mathcal F(U)) = \mathcal F(U)/\mathcal F(U) = 0$, and so $\dim(\supp(H^0(A^\bullet))) = 0$. Note that $A^\bullet$ is self-dual and constructible, as the cohomology sheaves are locally constant. Hence $A^\bullet$ is a perverse sheaf.

References: Bredon (Sheaf theory, Chapter II.1), de Catalado and Migliorini (What is... a perverse sheaf?), Stacks project (Articles "Supports of modules and sections" and "Complexes with constructible cohomology")

Constructible sheaves

Let $X$ be a topological space with an open cover $\mathcal U = \{U_i\}$, and category $Op(X)$ of open sets of $X$. The goal is to define constructible sheaves and consider some applications. Thanks to Joe Berner for helpful pointers in this area.

Definition: Constructible subsets of $X$ are the smallest family $F$ of subsets of $X$ such that

• $Op(X)\subset F$,
• $F$ is closed under finite intersections, and
• $F$ is closed under complements.

This idea can be applied to sheaves. Recall that a locally closed subset of $X$ is the intersection of an open set and a closed set.

Definition: A sheaf $\mathcal F$ over $X$ is constructible if there exists, equivalently,

• a filtration $\emptyset=U_0\subset \cdots \subset U_n=X$ of $X$ by opens such that $\mathcal F|_{U_{i+1}\setminus U_i}$ is constant for all $i$, or
• a cover $\{V_i\}$ of locally closed subsets of $X$ such that $\mathcal F|_{V_i}$ is constant for all $i$.

Since the category of abelian sheaves over a topological space has enough injectives, we may consider an injective resolution of a sheaf $\mathcal F$ rather than the sheaf itself. The resolution may be considered as living inside the derived category of sheaves on $X$.

Definition: Let $A$ be an abelian category.

• $C(A)$ is the category of cochain complexes of $A$, 
• $K(A) = C(A)$ modulo cochain homotopy, and
• $D(A) = K(A)$ modulo $F\in K(A)$ such that $H^n(F)=0$ for all $n$, called the derived category of $A$.

Next we consider an example. Recall the Ran space $\Ran(M) = \{X\subset M\ :\ 0<|X|<\infty\}$ of non-empty finite subsets of a manifold $M$ and the Čech complex of radius $t>0$ of $P\in \Ran(M)$, a simplicial complex with $n$-cells for every $P'\subset P$ of size $n+1$ such that $d(P'_1,P'_2)<t$ for all $P'_1,P'_2\in P'$.

Example: Consider the subset $\Ran^{\leqslant 2}(M) = \{X\subset M\ :\ 1\leqslant |X|\leqslant 2\}$ of the Ran space. Decompose $X=\Ran^{\leqslant 2}(M)\times \R_+$ into disjoint sets $U_\alpha\cup U_\beta$, where
\[
U_\alpha = \underbrace{\left(\Ran^1(M)\times \R_+\right)}_{U_{\alpha,1}} \cup \underbrace{\bigcup_{P\in \Ran^2(M)}\{P\}\times (d_M(P_1,P_2),\infty)}_{U_{\alpha,2}},
\hspace{1cm}
U_\beta = \bigcup_{P\in \Ran^2(M)} \{P\} \times (0,d_M(P_1,P_2)],
\]
with $d_M$ the distance on the manifold $M$. The idea is that for every $(P,t)\in U_\alpha$, the Čech complex of radius $t$ on $P$ has the homotopy type of a point, whereas on $U_\beta$ has the homotopy type of two points. With this in mind, define a constructible sheaf $F\in\text{Shv}(\Ran^{\leqslant 2}(M)\times \R_+)$ valued in simplicial complexes, with $F|_{U_\alpha}$ and $F|_{U_\beta}$ constant sheaves. Set
\[
F_{(P,t)\in U_\alpha} = F(U_\alpha) = \left(0\to \{*\} \to 0\right),
\hspace{1cm}
F_{(P,t)\in U_\beta} = F(U_\beta) = \left(0\to \{*,*\}\to 0\right).
\]
Note that the chain complex $F(U_\alpha)$ is chain homotopic to $0\to \{-\}\to \{*,*\}\to 0$, where $-$ is a single 1-cell with endpoints $*,*$. To show that this is a constructible sheaf, we need to filter $\Ran^{\leqslant 2}(M)\times \R_+$ into an increasing sequence of opens. For this we use a distance on $\Ran^{\leqslant 2}(M)\times \R_+$, given by $d((P,t),(P',t'))=d_{\Ran(M)}(P,P')+d_\R(t,t'),$ where $d_\R(t,t')=|t-t'|$ and
\[
d_{\Ran(M)}(P,P')=\max_{p\in P}\left\{\min_{p'\in P'}\left\{d_M(p,p')\right\}\right\} + \max_{p'\in P'}\left\{\min_{p\in P}\left\{d_M(p,p')\right\}\right\}.
\]
Note that $U_\alpha$ is open. Indeed, for $(P,t)\in U_{\alpha,1}$, every other $P'\in \Ran^1(M)$ close to $P$ is also in $U_{\alpha,1}$, and if $P'\in \Ran^2(M)$ is close to $P$, then the non-zero component $t\in\R_+$ still guarantees the same homotopy type. The set $U_{\alpha,2}$ is open as well, so $U_\alpha$ is open. The whole space is open, so a filtration $\emptyset\subset U_\alpha\subset X$ works for us.

References: Hartshorne (Algebraic geometry, Section II.3), Hartshorne (Residues and Duality, Chapter IV.1), Kashiwara and Schapira (Sheaves on manifolds, Chapters 2 and 8), Lurie (Higher algebra, Section 5.5.1)

Sheaves and cosheaves

Let $X$ be a topological space with an open cover $\mathcal U = \{U_i\}$, and category $Op(X)$ of open sets of $X$. Let $C$ be any abelian category, most often groups.

Definition: A presheaf $\mathcal F$ over $X$ is a functor $Op(X)^{op}\to D$, and a sheaf if it satisfies the gluing axiom. A precosheaf $\widehat{\mathcal F}$ over $X$ is a functor $Op(X)\to D$, and a cosheaf if it satisfies the cutting axiom.

The gluing axiom may be interpreted as a colimit condition and the cutting axiom (thanks to Keaton Quinn for suggesting the name) may be interpreted as a limit condition. The components of sheaves and cosheaves are compared in the table below.
\[
\begin{array}{r|c|c}
& \text{sheaf} & \text{cosheaf} \\\hline
&&\\[-5pt]
\text{functoriality} & \begin{array}{r c l}
Op(S)^{op} & \to & D \\
U & \mapsto & \mathcal F(U)\\
(V\hookrightarrow U)^{op} & \mapsto & (\rho_{UV}:\mathcal F(U)\to \mathcal F(V))
\end{array}
&
\begin{array}{r c l}
Op(S) & \to & D \\
U & \mapsto & \widehat{\mathcal F}(U)\\
(V\hookrightarrow U) & \mapsto & (\varepsilon_{VU}:\widehat{\mathcal F}(V)\to \widehat{\mathcal F}(U))
\end{array}
\\&&\\
\text{gluing / cutting} &
\begin{array}{r l}
\text{if} &  s_i|_{U_i\cap U_j}=s_j|_{U_i\cap U_j},\\[5pt]
\text{then} & \begin{array}{c}\exists s\in \mathcal F(U_i\cup U_j) \text{ s.t.}\\ s|_{U_i}=s_i,s|_{U_j}=s_j. \end{array}
\end{array}
&
\begin{array}{r l}
\text{if} & s_i|^{U_i\cup U_j}=s_j|^{U_i\cup U_j},\\[5pt]
\text{then} & \begin{array}{c}\exists s\in \widehat{\mathcal F}(U_i\cap U_j) \text{ s.t.}\\ s|^{U_i}=s_i,s|^{U_j}=s_j. \end{array}
\end{array}
\\&&\\
\text{colimit / limit cond.} &
\mathcal F(U)\tov\cong \displaystyle\varprojlim_{V\subseteq U} \mathcal F(V)
&
\widehat{\mathcal F}(U)\xleftarrow{\hspace{3pt}\cong\hspace{3pt}} \displaystyle\varinjlim_{V\subseteq U} \widehat{\mathcal F}(V)
\end{array}
\]
The maps $\rho_{UV}$ are called restrictions and $\varepsilon_{VU}$ are called extensions. Above, $s_i$ is a (co)section over $U_i$ and $s_j$ is a (co)section over $U_j$. For $s$ a (co)section of $U$ with $V\subset U\subset W$, write $s|_V$ for $\rho_{UV}(s)$ and $s|^W$ for $\varepsilon_{UW}(s)$. The isomorphisms with the colimits and limits are the natural maps from the respective colimit and limit diagrams.

Now we relate sheaves to persistent homology. All cohomology is be taken over a field $k$.

Remark: Suppose we have a finite point sample $P$ and some $t>0$, for which we can construct the nerve $N_{t,P}$, a cellular complex, of the union of balls of radius $t$ around the points of $P$. If $t'<t$, then there is a natural inclusion $N_{t',P}\hookrightarrow N_{t,P}$, which induces a map $H_\ell(N_{t',P})\to H_\ell(N_{t,P})$ on degree $\ell$ homology groups. Define a sheaf $\mathcal F^\ell$ over $\R$ for which
\[
\mathcal F^\ell(U) = H^\ell(N_{\inf(U),P}),
\hspace{1cm}
\mathcal F^\ell_t = H^\ell(N_{t,P}).
\]
This is indeed a sheaf, as $V\subseteq U$ implies that $\inf(U)\leqslant \inf(V)$, giving a natural map $\mathcal F^\ell(U)\to \mathcal F^\ell(V)$. The gluing axiom is also satisfied: assume without loss of generality that $\inf(U_i)\leqslant \inf(U_j)$ and take $s_i\in \mathcal F^\ell(U_i)$, $s_j\in \mathcal F^\ell(U_j)$ with the assumptions as above. Then $\inf(U_i)=\inf(U_i\cup U_j)$ and $\inf(U_j) = \inf(U_i\cap U_j)$, so
\[
\mathcal F^\ell(U_i) = \mathcal F^\ell(U_i\cup U_j),
\hspace{1cm}
\mathcal F^\ell(U_j) = \mathcal F^\ell(U_i\cap U_j),
\]
hence $s_i=s\in \mathcal F^\ell(U_i\cup U_j)$ and $s|_{U_j} = s_i|_{U_j} = s_i|_{U_i\cap U_j} = s_j|_{U_i\cap U_j} = s_j|_{U_j} = s_j$. Therefore sheaves capture all the persistent homology data. Note we do not take the sheaf cohomology of $\mathcal F^\ell$, instead the usual sequence of homology groups is induced by any increasing sequence in $\R$.

References: Bredon (Sheaf theory, Section VI.4), Bott and Tu (Differential forms in algebraic topology, Section 10)

Čech (co)homology

In this post we briefly recall the construction of Čech cohomology as well as compute a few examples. Let $X$ be a topological space with a cover $\mathcal U = \{U_i\}$, $\mathcal F$ a $C$-valued sheaf on $X$, and $\widehat{\mathcal F}$ a $C$-valued cosheaf on $X$, for some category $C$ (usually abelian groups).

Definition: The nerve $N$ of $\mathcal U$ is the simplicial complex that has an $r$-simplex $\rho$ for every non-empty intersection of $r+1$ opens of $\mathcal U$. The support $U_\rho$ of $\rho$ is this non-empty intersection. The $r$-skeleton $N_r$ of $N$ is the collection of all $r$-simplices.

Remark: The sheaf $\mathcal F$ and cosheaf $\widehat {\mathcal F}$ may be viewed as being defined either on the opens of $\mathcal U$ over $X$, or on the nerve $N$ of $\mathcal U$. Indeed, the inclusion map $V\hookrightarrow U$ on opens is given by the forgetful map $\partial$. That is, $\partial_i:N_r\to N_{r-1}$ forgets the $i$th open defining $\rho\in N_r$, so if $U_\rho = U_0\cap \cdots \cap U_r$, then $U_{\partial_0\rho} = U_1\cap\cdots \cap U_r$.

The Čech (co)homology will be defined as the (co)homology of a particular complex, whose boundary maps will be induced by, equivalently, the inclusion map on opens or $\partial_i$ on simplices.

Definition: In the context above:

• a $p$-chain is a finite formal sum of elements $a_{\sigma_i}\in \widehat{\mathcal F}(U_{\sigma_i})$, for every $\sigma_i$ a $p$-simplex,
• a $q$-cochain is a finite formal sum of elements $b_{\tau_j}\in \mathcal F(U_{\tau_j})$, for every $\tau_j$ a $q$-simplex,
• the $p$-differential is the map $d_p:\check C_p(\mathcal U,\mathcal F) \to \check C_{p-1}(\mathcal U,\mathcal F)$ given by

\[
d_p(a_\sigma) = \sum_{i=0}^p (-1)^i \widehat{\mathcal F}(\partial_i)(a_\sigma),\]

• the $q$-codifferential is the map $\delta^q:\check C^q(\mathcal U,\mathcal F) \to \check C^{q+1}(\mathcal U,\mathcal F)$ given by

\[
\delta^q(b_\tau) = \sum_{j=0}^{q+1} (-1)^j \mathcal F(\partial_j)(b_\tau).\]The collection of $p$-chains form a group $\check C_p(\mathcal U,\mathcal F)$ and the collection of $q$-cochains also form a group $\check C^q(\mathcal U,\mathcal F)$, both under the respective group operation in each coordinate. The Čech homology $H_*(\mathcal U,\mathcal F)$ is the homology of the chain complex of $\check C_p$ groups, and the Čech cohomology $H^*(\mathcal U,\mathcal F)$ is the cohomology of the cochain complex of $\check C^q$ groups.

Example: Let $X=S^1$ with a cover $\mathcal U = \{U,V,W\}$ and associated nerve $N_{\mathcal U}$ as below.

The cover is chosen so that all intersections are contractible. Let $k$ be a field. Let $\widehat{\mathcal F}$ be a cosheaf over $N$ and $\mathcal F$ a sheaf over $N$, with $\widehat {\mathcal F}(\text{0-cell})=\mathcal F(\text{1-cell}) = (1,1)\in k^2$ and $\widehat{\mathcal F}(\text{1-cell})=\mathcal F(\text{0-cell})=1\in k$, so that the natural extension and restriction maps work. Then all the degree 0 and 1 chain and cochain groups are $k^3$. Giving a counter-clockwise orientation to $X$, we easily see that
\begin{align*}
d_1\sigma_{U\cap V} & = \sigma_V-\sigma_U, & \delta^0\sigma_U & = \sigma_{U\cap V}-\sigma_{W\cap U}, \\
d_1\sigma_{V\cap W} & = \sigma_W-\sigma_V, & \delta^0\sigma_V & = \sigma_{V\cap W}-\sigma_{U\cap V}, \\
d_1\sigma_{W\cap U} & = \sigma_U-\sigma_W, & \delta^0\sigma_W & = \sigma_{W\cap U}-\sigma_{V\cap W}.\end{align*}If we give an ordered basis of $(\sigma_{U\cap V},\sigma_{V\cap W},\sigma_{W\cap U})$ to $\check C_1(\mathcal U,\widehat{\mathcal F})$ and $\check C^1(\mathcal U,\mathcal F)$, and $(\sigma_U,\sigma_V,\sigma_W)$ to $\check C_0(\mathcal U,\widehat{\mathcal F})$ and $\check C^0(\mathcal U,\mathcal F)$, we find that
\[
d_1 = \begin{bmatrix}
-1 & 0 & 1 \\ 1 & -1 & 0 \\ 0 & 1 & -1
\end{bmatrix}
\sim
\begin{bmatrix}
1 & 0 & -1 \\ 0 & 1 & -1 \\ 0 & 0 & 0
\end{bmatrix},
\hspace{1cm}
\delta^0 = \begin{bmatrix}
-1 & 1 & 0 \\ 0 & -1 & 1 \\ 1 & 0 & -1
\end{bmatrix}
\sim
\begin{bmatrix}
1 & 0 & -1 \\ 0 & 1 & -1 \\ 0 & 0 & 0
\end{bmatrix}.
\]
The Čech chain and cochain complexes are then
\[
0 \to \check C_1(\mathcal U,\widehat{\mathcal F}) \tov{d_1} \check C_0(\mathcal U,\widehat{\mathcal F}) \to 0,
\hspace{1cm}
0 \to \check C^0(\mathcal U,\mathcal F) \tov{\delta^0} \check C^1(\mathcal U,\mathcal F) \to 0,\]for which
\begin{align*}
H_1(\mathcal U,\widehat{\mathcal F}) & = \ker(d_1) = k,
& H^0(\mathcal U,\mathcal F) & = \ker(\delta^0) = k, \\
H_0(\mathcal U,\widehat{\mathcal F}) & = k^3/\im(d_1) = k^3/k^2 = k,
& H^1(\mathcal U,\mathcal F) & = k^3/\im(\delta^0) = k^3/k^2 = k.\end{align*}By the Čech-de Rham theorem, we know that the (co)homology groups should agree with the usual groups for $S^1$, as $\mathcal U$ was a good cover, which they do. Next we compute another example with a view towards persistent homology.

Definition: Let $X$ be a topological space and $f:X\to Y$ a map with $\mathcal U$ covering $f(X)$. The Leray sheaf $L^i$ of degree $i$ over $N_{\mathcal U}$ is defined by $L^i(\sigma) = H^i(f^{-1}(U_\sigma))$ and $L^i(\sigma\hookrightarrow \tau) = H^i(f^{-1}(U_\tau)\hookrightarrow f^{-1}(U_\sigma))$, whenever $\sigma$ is a face of $\tau$.

Theorem (Curry, Theorem 8.2.21): In the context above, if $N_{\mathcal U}$ is at most 1-dimensional, then for any $t\in \R$,
\[
H^i(f^{-1}(-\infty,t])\cong H^0((-\infty,t],L^i)\oplus H^1((-\infty,t],L^{i-1}).\]
The idea is to apply this theorem in a filtration, for different values of $t$, but in the example below we will have $t$ large enough so that $X\subset f^{-1}(-\infty,t]$.

Example: Let $f:S^1\to \R$ be a projection map, and let $X = f(S^1)$ with a cover $\mathcal U = \{U,V\}$ as below.

Note that although $f^{-1}(U)\cap f^{-1}(V)$ is not contractible, $U\cap V$ is, and the Čech cohomology will be over $\mathcal U\subset \R$, so we are fine in applying the Čech-de Rham theorem. It is immediate that the only non-zero Leray sheaves are $L^0$, for which
\[
L^0(\sigma_U) = k,\hspace{1cm}
L^0(\sigma_V) = k,\hspace{1cm}
L^0(\sigma_{U\cap V}) = k^2,\]hence $\check C^0(\mathcal U,L^0)=\check C^1(\mathcal U,L^0) = k^2$. Giving $\check C^0(\mathcal U,L^0)$ the ordered basis $(\sigma_U,\sigma_V)$ and noting the homology maps $H^0(f^{-1}(U)\hookrightarrow f^{-1}(U\cap V))$ and $H^0(f^{-1}(V)\hookrightarrow f^{-1}(U\cap V))$ are simply $1\mapsto (1,1)$, the \v Cech complex is
\[
0 \to \check C^0(\mathcal U,L^0) \tov{\left[\begin{smallmatrix}-1 & -1 \\ 1 & 1 \end{smallmatrix}\right]} \check C^1(\mathcal U,L^0) \to 0.
\]
Hence $H^0(\mathcal U,L^0)=\ker(\delta^0)=k$ and $H^1(\mathcal U,L^0)=k^2/\im(\delta^0)=k^2/k=k$, allowing us to conclude, using Curry's and the Čech--de Rham theorems, that
\begin{align*}
H^0(S^1) & \cong H^0(\mathcal U,L^0) \oplus H^1(\mathcal U,L^{-1}) = k\oplus 0 = k, \\
H^1(S^1) & \cong H^0(\mathcal U,L^1) \oplus H^1(\mathcal U,L^0) = 0\oplus k = k, \\
H^2(S^1) & \cong H^0(\mathcal U,L^2) \oplus H^1(\mathcal U,L^1) = 0\oplus 0=0,\end{align*}as expected.

References: Bott and Tu (Differential forms in algebraic topology, Section 10), Bredon (Sheaf theory, Section VI.4), Curry (Sheaves, cosheaves, and applications, Section 8)

Serre duality on schemes

 Lecture topic

This post goes through the statement and proof of Serre duality for arbitrary projective schemes, as presented in Chapter III.7 of Hartshorne. Only the necessary tools and definitions to prove the statement are introduced.

Recall a scheme is a topological space $X$ and a sheaf of rings $\mathcal O_X$ such that for every open set $U\subset X$, $\mathcal O_X(U)\cong \Spec(R)$ for some ring $R$. Its dimension is its dimension as a topological space. A projective scheme is a scheme where $X\subset \P^n$. A sheaf (or scheme) over a scheme $X$ is a sheaf (or scheme) $Y$ and a morphism $Y\to X$. Recall also the sheafification $\widetilde {\mathcal F}$ of a presheaf $\mathcal F$.

Definition 1: Let $\mathcal F$ be a sheaf over a projective scheme $X$. Then $\mathcal F$ is

• proper if it is the image of a proper morphism (separated, finite type, universally closed),
• quasi-coherent if there exists a cover $\{U_i=\Spec(A_i)\}$ of $X$ such that $\mathcal F|_{U_i} = \widetilde{M_i}$ for some $A_i$-module $M_i$,
• coherent if it is quasi-coherent and each $M_i$ is finitely-generated as an $A_i$-module,
• locally free if for every $x\in X$, there exists $U\owns x$ open such that $\mathcal F|_U = \bigoplus_{i\in I} \mathcal O_X|_U$,
• very ample if there is an immersion $i:X\to \P^n$ for some $n$ such that $i^*\mathcal O(1) \cong \mathcal F$.

Often we say $\mathcal F$ is very ample if it has "enough sections," as $\P^n$ has many sections.

Remark 1: Recall some basic definitions of the $\Ext$ functor. Let $\mathcal F,\mathcal G$ be sheaves of $\mathcal O_X$-modules, and $\mathcal L$ a locally free sheaf of finite rank. Then:

1. $\Ext^i(\mathcal O_X,\mathcal F) \cong H^i(X,\mathcal F)$ for all $i\>0$ (Proposition III.6.3)
2. $\Ext^i(\mathcal F\otimes \mathcal L,\mathcal G) \cong \Ext^i(F,\mathcal L^\vee\otimes \mathcal G)$ (Proposition III.6.7)
3. $\Ext^i_{\mathcal O_X}(\mathcal F_x,\mathcal G_x) \cong \mathcal Ext (\mathcal F,\mathcal G)_x $ (Proposition III.6.8)
4. $\Ext^i(\mathcal F,\mathcal G(q)) \cong \Gamma(X,\mathcal Ext^i(\mathcal F,\mathcal G(q))$ (Proposition III.6.9)
5. $\mathcal Ext^i(\mathcal F\otimes \mathcal L,\mathcal G) \cong \mathcal Ext^i(F,\mathcal L^\vee\otimes \mathcal G) \cong \mathcal Ext^i(\mathcal F,\mathcal G)\otimes \mathcal L^\vee$
6. $\mathcal Ext^0(\mathcal O_X,\mathcal F) \cong \mathcal F$
7. $\mathcal Ext^i(\mathcal O_X,\mathcal F) \cong 0$ for all $i>0$

Recall that a local ring of a scheme $X$ is $\mathcal O_{X,x}$ for $x\in X$. It is equivalently a ring with a unique maximal left or right ideal. A regular local ring is a local ring $R$ whose maximal ideal is generated by $\dim(R)$ elements.

Preliminary definitions and lemmas

Let $A,B$ be abelian categories (recall this means kernels and cokernels exist).

Definition 2: A $\delta$-functor between $A$ and $B$ is a collection of functors $T^i:A\to B$ that generalize derived functors, in the sense that $R^i\mathcal F = T^i$. A $\delta$-functor is universal if for any other $\delta$-functor $U$, there is a natural transformation $f:T^0\to U^0$ that induces a unique collection of morphisms $f^{i\geqslant 0}:T^i\to U^i$ that extend $f$.

See Weibel for a more thorough definition (and Grothendieck for the original setting). These functors may be covariant or contravariant, homological or cohomological. Note that $\delta$-functors are unique up to isomorphism.

Definition 3: Let $F:A\to B$ be a functor. $F$ is effaceable if for every $X\in A$ there exists a monomorphism $u\in \Hom_A(X,Y)$ such that $F(u)=0$. Similarly, $F$ is coeffaceable if for every $X\in A$ there exists an epimorphism $v\in \Hom_A(Y,X)$ such that $F(v)=0$.

Lemma 1: If a covariant (or contravariant) cohomological $\delta$-functor is effaceable for every $i>0$, then it is universal. Similarly, if a covariant (or contravariant) homological $\delta$-functor is coeffaceable for every $i>0$, then it is universal.

This appears as Proposition II.2.2.1 in Grothendieck and Exercise 2.4.5 in Weibel. Now let $\mathcal F$ be a sheaf over a projective scheme $X$.

Lemma 2: (Theorem III.5.2 in Hartshorne) If $\mathcal F$ is coherent, there is $q\gg 0$ such that $H^i(X,\mathcal F(q))=0$ all $i>0$.

Definition 4: The dualizing sheaf of $X$ is a coherent sheaf $\omega_X^\circ$ and a trace map $t:H^n(X,\omega_X^\circ) \to k$ such that the isomorphism $\Hom(\mathcal F,\omega_X^\circ)\to H^n(X,\mathcal F)^\vee$ is induced by the natural pairing
\[\Hom(\mathcal F,\omega_X^\circ)\times H^n(X,\mathcal F) \to H^n(X,\omega_X^\circ)\]
composed with $t$.

Lemma 3: (Corollary II.5.18 in Hartshorne) If $\mathcal F$ is coherent, then it is a quotient of $\bigoplus_{i=1}^N \mathcal O_X(-q)$ for $q\gg 0$.

Next we recall some ring theory. Let $A$ be a ring and $M$ an $A$-module.

Definition 5: A sequence $a_1,\dots,a_n\in M$ is $M$-regular if $a_i$ is not a zero divisor of $M/(a_1,\dots,a_{i-1})M$ and $M\neq (a_1,\dots,a_i)M$ for all $i$. The depth of $M$ is the maximal length of an $M$-regular sequence of elements in some maximal ideal $\mathfrak m\leqslant M$. A local Noetherian ring is Cohen-Macaulay if $\text{depth}(A)=\dim(A)$, where dimension is Krull dimension (maximal length of prime ideal chains). A scheme $X$ is Cohen-Macaulay if every point $x\in X$ has a neighborhood $U$ such that the local ring $\mathcal O_X(U)$ is Cohen--Macaulay.

Lemma 4: Let $A$ be a regular local ring of dimension $n$ and $M,N$ be $A$-modules. Then:

1. $\text{pd}(M)\leqslant n$ iff $\Ext^i(M,N)=0$ for all $i>n$ (Proposition III.6.10)
2. $\text{pd}(M)+\text{depth}(M)=n$ if $M$ is f.g. (Proposition III.6.12A)

Main theorem and proof

First we state the duality theorem for $X=\P^n$, without proof. Let $\omega_X$ be the canonical sheaf of $X$.

Theorem 1: (Theorem III.7.1 in Hartshorne) For $\mathcal F$ coherent over $\P^n$, for $i\geqslant 0$ there are natural isomorphisms
\[ \Hom(\mathcal F,\omega_X) \cong H^n(X,\mathcal F)^\vee, \hspace{2cm} \Ext^i(\mathcal F,\omega) \cong H^{n-i}(X,\mathcal F)^\vee.\]

Now we give the duality theorem for an arbitrary projective scheme, going through the proof as in Hartshorne.

Theorem 2: (Theorem III.7.6 in Hartshorne) Let $X$ be a projective scheme of dimension $n$ such that $\mathcal O(1)$ is very ample. For $\mathcal F$ coherent,
\begin{align*}
\Ext^i(\mathcal F, \omega_X^\circ) \cong H^{n-i}(X,\mathcal F)^\vee &\ \iff\ H^i(X,\mathcal F(-q))=0 \text{ for all $\mathcal F$ locally free, }i<n,q\gg0,\\
&\ \iff\ X\text{ is CM and equidimensional.}
\end{align*}

Proof: Natural maps $\Ext^i(\mathcal F,\omega^\circ_X)\to H^{n-i}(X,\mathcal F)^\vee$ exist, as $\Ext^i(-,\omega_X^\circ):\text{Coh}(X)\to \text{Mod}$ is a coeffaceable $\delta$-functor for every $i>0$, hence universal by Lemma 1. Indeed, by Lemma 3, we have a surjection

\[\underbrace{\bigoplus_{j=1}^N \mathcal O_X(-q)}_{\mathcal E}\tov u \mathcal F\to 0,\]
for which
\[\Ext^i(\mathcal E,\omega_X^\circ) = \bigoplus_{j=1}^N \Ext^i(\mathcal O_X(-q),\omega_X^\circ)= \bigoplus_{j=1}^N \Ext^i(\mathcal O_X,\omega_X^\circ (q)) = 0\]
for $i>0$. The first equality was distributing $\Ext^i$ over the sum and the second was by applying Remark 1.2. Hence $\Ext^i(-,\omega_X^\circ)(u)=0$ for $i>0$, so the functor is coeffeaceable for $i>0$, and so universal. By Definition 2 there exist maps generalizing the map $\Ext^0$ from Definition 4.

First iff $\Leftarrow$: Since universal $\delta$-functors are unique (up to isomorphism), we show $H^{n-i}(X,-)^\vee:\text{Coh}(X)\to \text{Mod}$ is also universal contravariant, which follows as it is coeffaceable for $i>0$. Using the same sequence and sheaf as in the equation above, we have that
\[H^{n-i}(X,\mathcal E) = \bigoplus_{j=1}^N H^{n-i}(X,\mathcal O_X(-q)) = 0\]
whenever $n-i<n$ by hypothesis, or equivalently, when $i>0$. The dual module is then also zero for $i>0$, so we are done.

First iff $\Rightarrow$: Assume the hypothesis with index $n-i$ and a locally free sheaf $\mathcal F(-q)$ for $q\gg 0$, for which
\begin{align*}
H^i(X,\mathcal F(-q))^\vee & \cong \Ext^{n-i}(\mathcal F(-q),\omega_X^\circ) & (\text{hypothesis}) \\
& \cong \Ext^{n-i}(\mathcal O_X,\mathcal F^\vee \otimes \mathcal O_X(q) \otimes\omega_X^\circ) & (\text{Remark 1.2}) \\
& \cong H^{n-i}(X,(\mathcal F^\vee \otimes\omega_X^\circ)\otimes \mathcal O_X(q)). & (\text{Remark 1.1})
\end{align*}
Tensoring with $\mathcal O_X(q)$ is twisting by $q$, and Lemma 2 says that $H^{n-i}(X,\mathcal G(q))=0$ for $\mathcal G$ coherent, for all $n-i>0$, for $q$ large enough. So for $i<n$ and $q$ large enough $H^i(X,\mathcal F(-q))^\vee=0$, and so its dual, the original cohomology group, is also trivial.

Second iff $\Leftarrow$: Embed $X\hookrightarrow \P^N$. As $X$ is Cohen--Macaulay and equidimensional of dimension $n$, for $\mathcal F$ locally free on $X$, a stalk $\mathcal F_x$ of a closed point $x\in X$ has depth $n$. Also, $\mathcal F_x\subset \mathcal O_{\P^N,x}$, and $\mathcal O_{\P^n,x}$ is regular as $\P^N$ is smooth over $k$. By Lemma 4.2, we have that
\[\text{pd}(\mathcal F_x) +n \leqslant \text{pd}(\mathcal O_{\P^N,x})+n=N, \]
so Lemma 4.1 and Remark 1.3 gives us that, for $i>N-n$,
\[\Ext^i(\mathcal F_x,-)=0
\ \ \implies\ \
\mathcal Ext^i(\mathcal F_x,-)=0
\ \ \implies\ \
\mathcal Ext^i(\mathcal F,-)=0.\]
Applying Theorem 1, Remark 1.4, and letting the functor $\mathcal Ext^i(\mathcal F,-)$ act on $\omega_{\P^N}(q)$, we have
\[H^i(X,\mathcal F(-q))^\vee \cong \Ext_{\P^n}^{N-i}(\mathcal F,\omega_{\P^N}(q)) \cong \Gamma(\P^N,\mathcal Ext_{\P^N}^{N-i}(\mathcal F, \omega_{\P^n}(q))) \cong \Gamma(\P^N, 0) = 0\]
for $q\gg0$ and $N-i>N-n$, or $i<n$. Since the dual is trivial, the cohomology group $H^i(X,\mathcal F(-q))$ is also trivial.

Second iff $\Rightarrow$: Omitted (techniques are similar to previous step, but use many others not used elsewhere). $\square$

Addendum

In certain cases, Serre duality holds for the canonical sheaf instead of the dualizing sheaf.

Proposition 1: For $X$ a smooth projective variety over $k=\overline k$, $\omega_X^\circ \cong \omega_X$.

References: Grothendieck (Tohoku paper), Hartshorne (Algebraic Geometry, Section III.7), Weibel (An introduction to homological algebra, Section 2.1), Matsumura (Commutative algebra, Chapter 6)

What is a stack?

 Conference topic

This is from discussions at the 2016 West Coast Algebraic Topology Summer School (WCATSS) at The University of Oregon. Thanks to Piotr Pstragowski for explaining the material.

Definition: A groupoid is a category where all the morphisms are invertible. Alternatively, a groupoid is a set of objects $A$, a set of morphisms $\Gamma$, and a collection of maps as described by the diagram below.

To describe stacks, we compare them with sheaves. Both start out with a space $X$ and a topology on it, so that we may consider open sets $U$.

In addition to these conditions, there is a triple intersection condition for stacks that does not have an analogous one in sheaves. It is given by:

for every $U_i,U_j,U_k$ and $s_i,s_j,s_k\in \widehat{\mathcal F}(U_i), \widehat{\mathcal F}(U_j), \widehat{\mathcal F}(U_k)$, respectively, such that there exist isomorphisms $\varphi_{ij}:s_i|_{U_i\cap U_j}\to s_j|_{U_i\cap U_j}$, $\varphi_{jk}:s_j|_{U_j\cap U_k}\to s_k|_{U_j\cap U_k}$, and $\varphi_{ik}:s_i|_{U_i\cap U_k}\to s_k|_{U_i\cap U_k}$, the diagram below commutes:

Example: A Hopf algebroid may be viewed as a functor into groupoids, so that with the appropriate topology, it becomes a stack. Indeed, by definition a Hopf algebroid is a pair of $k$-algebras $(A,\Gamma)$ such that $(\Spec(A),\Spec(\Gamma))$ is a groupoid object in affine schemes, or in other words, is a functor from affine schemes into groupoids.

References: nLab (article on groupoids)

Morphisms of schemes

 Conference topic

This is from discussions at the 2016 West Coast Algebraic Topology Summer School (WCATSS) at The University of Oregon. Thanks to Zijian Yao for explaining the material.

Consider a morphism of schemes $\varphi:S'\to S$ and coherent sheaves $\mathcal F,\mathcal G$ over $S$. Consider also a map of sheaves $f:\mathcal F\to \mathcal G$ and a map $f'$ between the pullbacks of $\mathcal F$ and $\mathcal G$, as described by the diagram below.

There are two natural questions to ask.

1. When is $f' = \varphi^*f$?
2. If we start with $\mathcal G'$ over $S'$, when is $\mathcal G' = \varphi^*\mathcal G$?

To answer these questions, consider fiber products of schemes and projections from them, as given below.

Remark: If 1. is true, then $p_1^*(f') = p_2^*(f')$. If the previous statement is an equivalence, then $\varphi$ is a morphism of descent.

Remark: If 2. is true, then there exists $\alpha:p_1^*(\mathcal G') \to p_2^*(\mathcal G')$ such that $\pi_{32}^*(\alpha)\pi_{21}^*(\alpha) = \pi_{31}^*(\alpha)$ and $\pi^*(\Delta) = \alpha$. If the previous statement is an equivalence, then $\varphi$ is effective.

What is a scheme?

 Conference topic

This is from a problem session at the 2016 West Coast Algebraic Topology Summer School (WCATSS) at The University of Oregon. Thanks to Tyler Lawson for explaining the material.

Definition: Affine schemes are the category $\Ring^{op}$. An object $R\in \Ring$ becomes an object $\Spec(R)$ in affine schemes, and a ring map $R\to S$ becomes a map $\Spec(S)\to \Spec(R)$, where $\Spec$ denotes the set of prime ideals.

We try to think of $Spec(R)$ as a geometrical object.

Example: Let $k$ be a field and consider the ring
\[
R = k[x_1,\dots,x_n] / (f_1(x_1,\dots,x_n),\dots,f_r(x_1,\dots,x_n)).
\]
$\Spec(R)$ is supposed to be a substitute for the set of solutions to a system of equations
\begin{align*}
f_1(x_1,\dots,x_n) & = 0,\\
\vdots \hspace{.7cm}\\
f_r(x_1,\dots,x_n) & = 0.
\end{align*}

The scheme $\Spec(R)$ has a more precise definition. It consists of a set, a topology, and a sheaf. 

1. Set: The underlying set of the scheme $\Spec(R)$ is the set of prime ideals of $R$. For example:

• if $R = \C[x]$, then the prime ideals are $(x-\alpha)$ and $(0)$;
• if $R = \C[x,y]$, then the prime ideals are $(x-\alpha,y-\beta)$, irreducible polynomials $(f(x,y))$, and $(0)$.

2. Topology: For every ideal $I\subset R$, the set $V(I) = \{P\subset R$ prime, $P\supset I\}$ is a closed set. Note that

\[
\bigcup_{n=1}^N V(I_n) = V\left(\bigcap_{n=1}^N I_n\right)
\hspace{1cm}\text{and}\hspace{1cm}
\bigcap_{\alpha\in I} V(I_\alpha) = V\left(\sum_{\alpha\in A} I_A\right).
\]
Geometrically, the closed sets are sets of points where one or more identities (like $f(x)=0$) can hold. For example, if $R=\C[x]$, then we have three different closed set types: $\Spec(C[x])$, $\emptyset$, or a finite union of $(x-\alpha_1,\dots, x-\alpha_n)$. Solutions to equations can be one of the following types below.

3. Sheaf: Let $X$ be a set with a topology. $\mathcal O_X$ is the sheaf for which:

• to each open set $U\subseteq X$ we get a ring $\mathcal O_X(U)$;
• to each containment $V\subseteq U\subseteq X$ of open sets, there exists a restriction map $\res_{UV}:\mathcal O_X(U)\to \mathcal O_X(V)$;
• the restriction maps are compatible, in the sense that $\res_{VW}\circ \res_{UV} = \res_{UW}$.

This is called the structure sheaf of $X$.

Say $R$ is our ring, $\Spec(R)$ our set of primes, and we have some open set $U\subseteq \Spec(R)$. We like to think of it in the following way:

• elements of $R$ are functions;
• elements of $\Spec(R)$ are points where we can evaluate a function $f\in P$ (or where the function vanishes);
• subsets $S\subset R$ are the sets $\{f\in R\ :\ f$ only vanishes at points outside $U\}$.

Note that $S$ is closed under multiplication. We localize $R$ at $S$ to get a set
\[
S^{-1}R = \left\{\left[\frac fs\right]\ :\ f\in R, s\in S\right\},
\]
for which $\mathcal O_X(U) = S^{-1}R$ (good enough for today's purposes). Now we have a triple $(\Spec(R),\tau,\mathcal O_X)$, for $\tau$ the Zariski topology, which we call a locally ringed space.

Definition: A scheme is a space $X$ with a topology and a sheaf of rings that is locally isomorphic to $\Spec(R)$.

Since the sheaf has the space $X$ and the topology (through the open sets) encoded in it, we may think of a scheme as a special type of sheaf. Also, isomorphism is meant in the category of locally ringed spaces.

Proposition: Morphisms of schemes $\Spec(R)\to \Spec(S)$ are the same as ring maps $S\to R$.

Example: In the Zariski topology, take $U\subseteq \Spec(k[x,y])$. Locally $U$ looks like it is covered by rings, though that may not be the case globally. Indeed:

Example: Consider projective space $\P^2$, where $[x:y:z] = [\lambda x: \lambda y:\lambda z]$. We may write
\[
\begin{array}{r c c c c c c}
\P^2 & = & U_0 & \cup & U_1 & \cup & U_2. \\
& & [1:y:z] & & [x:1:z] & & [x:y:1] \\
& & \Spec(k[y,z]) & & \Spec(k[x,z]) & & \Spec(k[x,y])
\end{array}
\]
How can we express $U_0\cap U_1$? This is left as an exercise.

Connections, curvature, and Higgs bundles

Recall (from a previous post) that a Kähler manifold $M$ is a complex manifold (with natural complex structure $J$) with a Hermitian metic $g$ whose fundamental form $\omega$ is closed. In this context $M$ is Kähler. Previously we used upper-case letters $V,W$ to denote vector fields on $M$, but here we use lower-case letters $s,u,v$ and call them sections (to consider vector bundles more generally as sheaves).

Definition: A connection on $M$ is a $\C$-linear homomorphism $\nabla: A^0_M\to A^1_M$ satisfying the Leibniz rule $\nabla(fs) = (df)\wedge s + f\nabla (s)$, for $s$ a section of $TM$ and $f\in C^\infty(M)$.

For ease of notation, we often write $\nabla_us$ for $\nabla(s)(u)$, where $s,u$ are sections of $TM$. On Kähler manifolds there is a special connection that we will consider.

Proposition: On $M$ there is a unique connection $\nabla$ that is (for any $u,v\in A^0_M$)

1. Hermitian (satisfies $dg(u,v) = g(\nabla (u),v) + g(u,\nabla (v))$),
2. torsion-free (satisfies $\nabla_uv - \nabla_vu-[u,v] = 0$), and
3. compatible with the complex structure $J$ (satisfies $\nabla_uv = \nabla_{Ju}(Jv)$).

If $\nabla$ satisfies the first two conditions, it is called the Levi-Civita connection, and if it satisfies the first and third conditions, it is called the Chern connection. If $g$ is not necessarily Hermitian, $\nabla$ is called metric if it satisfies the first condition. From here on out $\nabla$ denotes the unique tensor described in the proposition above.

Definition: The curvature tensor of $M$ is defined by
\[
R(u,v) = \nabla_u\nabla_v - \nabla_v\nabla_u-\nabla_{[u,v]}.
\]
It may be viewed as a map $A^2 \to A^1$, or $A^3\to A^0$, or $A^0\to A^0$. The Ricci tensor of $M$ is defined by
\[
r(u,v) = \trace(w\mapsto R(u,v)w) = \sum_i g(R(a_i,u)v,a_i),
\]
for the $a_i$ a local orthonormal basis of $A^0 = TM$. This is a map $A^2\to A^0$. The Ricci curvature of $M$ is defined by
\[
\Ric(u,v) = r(Ju,v).
\]
This is a map $A^2\to A^0$.

Definition: An Einstein manifold is a pair $(M,g)$ that is Riemannian and for which the Ricci curvature is directly proportional to the Riemannian metric. That is, there exists a constant $\lambda\in \R$ such that $\Ric(u,v) = \lambda g(u,v)$ for any $u,v\in A^1$.

Recall that a holomorphic vector bundle $\pi:E\to M$ has complex fibers and holomorphic projection map $\pi$. Here we consider two special vector bundles (as sheaves), defined on open sets $U\subset M$ by
\begin{align*}
\End(E)(U) & = \{f:\pi^{-1}(U)\to \pi^{-1}(U)\ :\ f|_{\pi^{-1}(x)}\text{\ is a homomorphism}\}, \\
\Omega_M(U) & = \left\{\sum_{i=0}^n f_idz_1\wedge\cdots \wedge dz_i\ :\ f_i\in C^\infty(U)\right\},
\end{align*}
where $z_1,\dots,z_n$ are local coordinates on $U$. The first is the endomorphism sheaf of $E$ and the second is the sheaf of differential forms of $M$, or the holomorphic cotangent sheaf. The cotangent sheaf as defined is a presheaf, so we sheafify to get $\Omega_M$.

Definition: A Higgs vector bundle over a complex manifold $M$ is a pair $(E,\theta)$, where $\pi:E\to M$ is a holomorphic vector bundle and $\theta$ is a holomorphic section of $\text{End}(E)\otimes \Omega_M$ with $\theta\wedge\theta = 0$, called the Higgs field.

References: Huybrechts (Complex Geometry, Chapters 4.2, 4.A), Kobayashi and Nomizu (Foundations of Differential Geometry, Volume 1, Chapter 6.5)

The Hodge decomposition, diamond, and Euler characteristics

 Seminar topic

Recall the sheaf of $r$-differential forms $\Omega^r_X$ on $X$ (with $\Omega^r_X(U) = \{fdx_{i_1}\wedge \cdots \wedge dx_{i_r}\ :\ f\ $is well-defined on $U\}$ and such sums) and the structure sheaf $\mathcal O_X$ on $X$ (with $\mathcal O_X(U) = \{f/g\ :\ f,g\in k[U],\ g\neq 0\ $on$\ U\}$). Then we may consider the sheaf cohomology of $X$, with values in $\Omega^r_X$ or $\mathcal O_X$.

Definition: Let $X$ be a smooth manifold of dimension $n$. The $(p,q)$th Hodge number is $h^{p,q}=\dim(H^{p,q})$, where $H^{p,q} = H^q(X,\Omega^p_X)$. These numbers are arranged in a Hodge diamond as below.

The Hodge diamond has a lot of repetition - by complex conjugation, we get that $h^{p,q}=h^{q,p}$, so it is symmetric about its vertical axis. By the Hard Lefschetz theorem (or the Hodge star operator, or Poincare duality), we get that $h^{p,q}=h^{n-q,n-p}$, so it is symmetric about its horizontal axis.

Proposition: Let $X$ be a Kähler manifold (note that all smooth projective varieties are Kähler) of dimension $n$. Then the cohomology groups of $X$ decompose as
\[
H^k(X,\C) = \bigoplus_{p+q=k}H^{p,q}(X),
\]
for all $0\leqslant k\leqslant 2n$. This is called the Hodge decomposition of $X$.

This decomposition immediately gives all the Hodge numbers for $\P^n$, knowing its cohomology. For a manifold of complex dimension $n$, there are several numbers and polynomials that may be defined. These are:
\begin{align*}
\chi_{top}(X) & = \sum_{i=1}^{2n}(-1)^i \dim(H^i(X,\C)) & \text{the (topological) Euler characteristic} \\
\chi^p(X) & = \sum_{q=0}^{n-1}(-1)^qh^{p,q} & \text{the chi-$p$ characteristic} \\
\chi_y(X) & = \sum_{p=0}^{n-1}\chi^py^p & \text{the chi-$y$ characteristic}
\end{align*}
Note the Euler characteristic is the alternating sum of the rows of the Hodge diamond, and the chi-$p$ characteristic is the alternating sum of the left-right diagonals of the diamond.

Example: In the case $X$ is a hypersurface in projective $n$-space $\P^n$ defined by a degree $d$ polynomial,
\[
\chi_y = [z^n]\frac{1}{(1+zy)(1-z)^2}\cdot\frac{(1+zy)^d-(1-z)^d}{(1+zy)^d+y(1-z)^d}.
\]
Since every row except the middle row of the Hodge diamond of a hypersurface is known (as it comes from the Hodge diamond of $\P^n$ by the Lefschetz hyperplane theorem), this expression gives all the unknown numbers. This particular formula is a simplification  of Theorem 22.1.1 in Hirzebruch, which itself comes from the Riemann--Roch theorem.

References: Huybrechts (Complex Geometry: An Introduction, Chapters 3.2, 3.3), Hirzebruch (Topological Methods in Algebraic Geometry, Appendix 1, Section 22)

The canonical bundle of projective space and hypersurfaces

Let $\P^n$ be projective $n$-space with coordinates $[x_0:\cdots:x_n]$. Cover $\P^n$ with affine pieces $U_i = \{x_i\neq 0\}$, each of which are $\A^n$, in coordinates $(y_1,\dots,y_n)$, where $y_j = x_j/x_i$. Recall that the canonical bundle of $\P^n$ is the $n$-fold wedge of the cotangent bundle of $\P^n$, or $\omega_{\P^n} = \bigwedge^nT^*_{\P^n}$. The canonical bundle for an arbitrary variety is defined analogously.

Definition: Let $X$ be a projective $n$-dimensional variety. The sheaf of regular functions on $X$ is $\mathcal O_X$, with $\mathcal O_X(U)=\{f/g\ :\ f,g\in k[x_1,\dots,x_n]/I(X), g\neq 0\}$, and the restriction maps are function restriction.

There is a natural grading on $\mathcal O_X$, given by $\deg(f)-\deg(g)$. A shift in the grading may be applied, called a {\it Serre twist}, to get a differently graded (but isomorphic) module: for $\varphi\in \mathcal O_X$ with $\deg(\varphi)=k$, set $\varphi\in\mathcal O_X(\ell)$ to have $\deg(\varphi) = k-\ell$.

Let $\alpha = dy_1\wedge\cdots\wedge dy_n\in \omega_{\P^n}$, which is well-defined on all of $U_i$. We claim this is well-defined on all of $\P^n$. We check this on the overlap $U_0\cap U_n$ (for nicer notation), but the approach is analogous for $U_i\cap U_j$.
\begin{align*}
U_0 & = \{(y_1,\dots,y_n)\ :\ y_i = x_i/x_0\} & y_i & = \frac{z_{i+1}}{z_i} & dy_i & = \frac{z_1dz_{i+1}-z_{i+1}dz_1}{z_1^2} \\
U_n & = \{(z_1,\dots,z_n)\ :\ z_i = x_{i-1}/x_n\} & y_n & = \frac1{z_1} & dy_n & = \frac{-dz_1}{z_1^2}
\end{align*}
Therefore
\begin{align*}
\alpha & = dy_1\wedge\cdots\wedge dy_n \\
& = \frac{z_1dz_2-z_2dz_1}{z_1^2}\wedge\cdots\wedge \frac{z_1dz_n-z_ndz_1}{z_1^2}\wedge \frac{-dz_1}{z_1^2} \\
& = \frac{dz_2}{z_1}\wedge\cdots\wedge \frac{dz_n}{z_1}\wedge \frac{-dz_1}{z_1^2} \\
& = \frac{(-1)^n}{z_1^{n+1}}dz_1\wedge\cdots \wedge dz_n.
\end{align*}
Since the transition function has a pole of order $n+1$ when $z_1 = 0$, which happens when $x_0=0$, we have that $\alpha$ has a pole of order $n+1$ at $\infty$. Therefore $\omega_{\P^n} \cong \mathcal O_{\P^n}(-n-1)$.

Let $X\subset \P^n$ be a smooth hypersurface defined by a degree $d$ equation $F(x_0,\dots,x_n)=0$. On the affine piece $U_0$ this becomes $f(y_1,\dots,y_n)=F(1,\frac{x_1}{x_0},\dots,\frac{x_n}{x_0})$ with $y_i = x_i/x_0$. The total derivative is
\[
\frac{\dy f}{\dy y_1} dy_1 + \cdots + \frac{\dy f}{\dy y_n} dy_n = \sum_{i=1}^n\frac{\dy f}{\dy y_i}dy_i = 0,
\]
and since $X$ is smooth, the terms never all vanish at the same time. Let $V_i=\{\frac{\dy f}{\dy y_i} \neq 0\}$, and set
\[
\beta_i = \frac{(-1)^{i-1}}{\dy f/\dy y_i} dy_1\wedge\cdots \wedge \widehat{d y_i}\wedge \cdots \wedge d y_n \in \omega_X,
\]
which is well-defined on all of $V_i\subset U_0$. We claim that the choice of $V_i$ does not matter, and indeed, assuming $i<j$,
\begin{align*}
\beta_j & = \frac{(-1)^{j-1}}{\dy f/\dy y_j} dy_1\wedge\cdots \wedge \widehat{d y_j}\wedge \cdots \wedge d y_n \\
& = \frac{(-1)^{j-1+i-1}dy_i}{\dy f/\dy y_j} \wedge dy_1\wedge\cdots \wedge \widehat{d y_i}\wedge \cdots \wedge \widehat{d y_j}\wedge \cdots \wedge d y_n \\
& = \frac{(-1)^{j-1+i-1}\frac{-1}{\dy f/\dy y_i}\left(\frac{\dy f}{\dy y_1}dy_1+\cdots + \widehat{\frac{\dy f}{\dy y_i}dy_i} + \cdots + \frac{\dy f}{\dy y_n}dy_n\right)}{\dy f/\dy y_j} \wedge dy_1\wedge\cdots \wedge \widehat{d y_i}\wedge \cdots \wedge \widehat{d y_j}\wedge \cdots \wedge d y_n \\
& = \frac{(-1)^{j-1+i-1+1}\frac{1}{\dy f/\dy y_i}\cdot \frac{\dy f}{\dy y_j}dy_j}{\dy f/\dy y_j} \wedge dy_1\wedge\cdots \wedge \widehat{d y_i}\wedge \cdots \wedge \widehat{d y_j}\wedge \cdots \wedge d y_n \\
& = \frac{(-1)^{j-1+i-1+1+j-2}}{\dy f/\dy y_i} dy_1\wedge\cdots \wedge \widehat{d y_i}\wedge \cdots \wedge d y_n \\
& = \frac{(-1)^{i-1}}{\dy f/\dy y_i} dy_1\wedge\cdots \wedge \widehat{d y_i}\wedge \cdots \wedge d y_n \\
& = \beta_i.
\end{align*}
Hence $\beta_i$ is well-defined on all of $U_0$, and we call it simply $\beta$. Next we claim it is well-defined on all of $X$. Again we only check on the overlap of $U_0\cap U_n$. On the affine piece $U_n$ this becomes $g(z_1,\dots,z_n)=F(\frac{x_0}{x_n},\dots,\frac{x_{n-1}}{x_n},1)=f(\frac{z_2}{z_1},\dots,\frac{z_n}{z_1},\frac1{z_1})$ with $z_i = x_{i-1}/x_n$. We employ the chain rule $\frac{\dy f}{\dy y_i}=\frac{\dy f}{\dy z_j}\frac{\dy z_j}{\dy y_i}$ and the results above to find that
\begin{align*}
\beta & = \frac{(-1)^{i-1}}{\dy f/\dy y_i} dy_1\wedge\cdots \wedge \widehat{d y_i}\wedge \cdots \wedge d y_n \\
& = \frac{(-1)^{i-1}}{\dy f/\dy z_j \cdot \dy z_j/\dy y_i} \frac{z_1dz_2-z_2dz_1}{z_1^2}\wedge \cdots \wedge \widehat{dy_i}\wedge \cdots \wedge \frac{z_1dz_n-z_ndz_1}{z_1^2}\wedge \frac{-dz_1}{z_1^2} \\
& = \frac{(-1)^{i-1}}{\dy f/\dy z_j \cdot \dy z_j/\dy y_i} \frac{(-1)^{n-1}}{z_1^n}dz_1\wedge\cdots \wedge \widehat{dz_i}\wedge \cdots \wedge dz_n \\
& = \frac{(-1)^{i+n}}{\left(\frac{1}{z_1}\right)^{d-1}\left(c+\cdots\right) z_1^n}dz_1\wedge\cdots \wedge \widehat{dz_i}\wedge \cdots \wedge dz_n \\
& = \frac{(-1)^{i+n}}{z_1^{n-d+1} \left(c+\cdots \right)}dz_1\wedge\cdots \wedge \widehat{dz_i}\wedge \cdots \wedge dz_n,
\end{align*}
for some constant $c$. This comes from expressing $f$ in terms of the $z_i$s and factoring. Since the transition function has a pole of order $n-d+1$ when $z_1 = 0$, which happens when $x_0=0$, we have that $\beta$ has a pole of order $n-d+1$ at $\infty$. Therefore $\omega_{X} \cong \mathcal O_{X}(-n+d-1)$.

References: Griffiths and Harris (Principles of Algebraic Geometry, Chapter 1.2)