*Note:
These are notes live-tex’d from a graduate course in Sheaf Cohomology taught by Valery Alexeev at the University of Georgia in Spring 2022. As such, any errors or inaccuracies are almost certainly my own.
*

dzackgarza@gmail.com

Last updated: 2022-05-29

Topic: cohomology of sheaves and derived categories. The plan:

- Sheaves (see ELC notes)
- Derived functors and coherent sheaves (see ELC notes)
- Derived categories (Gelfand-Manin, Tohoku)

References:

- Valery’s notes (see ELC)
- Gelfand-Manin,
*Methods of Homological Algebra*.

Compare (genus \(g\)) Riemann surfaces in the classical topology to (genus \(g\), projective) algebraic curves over \({\mathbb{C}}\) in the Zariski topology. Recall that \begin{align*} H^*(\Sigma_g; {\mathbb{Z}}) = \begin{cases} {\mathbb{Z}}& *=0, 2 \\ {\mathbb{Z}}^{2g} & *=1 \\ 0 & \text{else}. \end{cases} \end{align*} Note that this is a linear invariant in the sense that the constituents are free abelian groups, and we can extract a numerical invariant. For surfaces up to homeomorphism, this distinguishes them completely.

For algebraic curves, note that the topology is very different: the only closed sets are finite. In this topology, \begin{align*} H^*(X; {\mathbb{Z}})= \begin{cases} {\mathbb{Z}}& *=0 \\ 0 & \text{else}, \end{cases} \end{align*} which doesn’t see the genus at all. In fact all such curves are homeomorphic in this topology, witnessed by picking any bijection and noting that it sends closed sets to closed sets. The linear replacement: \(H^*(X; {\mathcal{O}}_X)\) for \({\mathcal{O}}_X\) the structure sheaf, which yields \begin{align*} H^*(X; {\mathcal{O}}_X) = \begin{cases} {\mathbb{C}}& *=0 \\ {\mathbb{C}}^g & *=1 \\ 0 & \text{else}. \end{cases} \end{align*} These surfaces can be parameterized by the moduli space \({ \mathcal{M}_{g} }\), which is dimension \(3g-3\) for \(g \geq 2\).

The POV in classical topology is to fix the coefficients: \({\mathbb{Z}}, {\mathbb{R}}, {\mathbb{C}}, {\mathbb{Z}}/n\), or \(R\) a general ring. A minor variation is to consider a local system \({\mathcal{L}}\), which are locally constant but may have nontrivial monodromy around loops. For example, one might have \({\mathbb{R}}\) locally, but traversing a loop induces an automorphism \(f\in \mathop{\mathrm{Aut}}({\mathbb{R}}) = {\mathbb{R}}^{\times}\). In this setting, we have a functor \(F({-}) = H({-}; R)\). For sheaf cohomology, instead fix \(X\) and take \(G({-}) = H(X; {-})\). In general, one can take sheaves of abelian groups, \({\mathcal{O}}_X{\hbox{-}}\)modules, quasicoherent sheaves, or coherent sheaves: \begin{align*} {\mathsf{Sh}}(X, {\mathsf{Ab}}{\mathsf{Grp}}) \hookrightarrow{\mathsf{{\mathcal{O}}_X}{\hbox{-}}\mathsf{Mod}}\hookrightarrow{\mathsf{QCoh}}(X) \hookrightarrow{\mathsf{Coh}}(X) .\end{align*}

We’ll be looking at three kinds of topologies:

The order topology: start with a poset and define the open sets to be the

*decreasing/lower sets*, i.e. subsets \(U_{x_0}\) that contain every element below a point \(x_0\). In other words, if \(x\in U\) and \(y\leq x\), then \(y\in U\).The Zariski topology: let \(R\) be a DVR, so \(\operatorname{Spec}R = \left\{{ \left\langle{0}\right\rangle, {\mathfrak{m}}}\right\}\). E.g. for \(R \mathrel{\vcenter{:}}={\mathbb{C}} { \left[ {t} \right] }\), \({\mathfrak{m}}= \left\langle{t}\right\rangle\), and the open sets are \(\left\{{\left\langle{0}\right\rangle}\right\}, \operatorname{Spec}R\), corresponding to the poset \({\operatorname{pt}}\to{\operatorname{pt}}\).

The classical topology, usually paracompact and Hausdorff.

One can define sheaves in all three cases, which have different properties. For posets, e.g. one can take \(C^0({-}, R)\) for \(R = {\mathbb{R}}, {\mathbb{C}}, { {\mathbb{Z}}_{\widehat{p}} }\).

Some computational tools:

- Vanishing theorems
- Riemann-Roch

Some topological notions to recall:

- \(T_0\), Kolmogorov spaces: distinct points don’t have the exact same neighborhoods, i.e. there exists a neighborhood of \(x\) not containing \(y\)
**or**a neighborhood of \(y\) not containing \(x\). - \(T_1\), Frechet spaces: points are separated, so replace “or” with “and” above.
- \(T_2\), Hausdorff spaces: points are separated by disjoint neighborhoods.
- Alexandrov spaces: arbitrary intersections of opens are open.
- Metrizability
- Paracompactness

Recall that a topology \(\tau\) on \(X\) satisfies

- \(\emptyset, X\in \tau\)
- \(A,B\in \tau \implies A \cap B \in \tau\)
- \(\displaystyle\bigcup_{j\in J} A_j \in \tau\) if \(A_j\in \tau\) for all \(j\).

Equivalently one can specify the closed sets and require closure under finite unions and arbitrary intersections.

Running examples:

- Any subset \(S \subseteq {\mathbb{R}}^n\) is Hausdorff and paracompact.
- Order topologies on posets
- Zariski topologies on varieties over \(k= \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu\), e.g. \(\operatorname{mSpec}A\) for \(A\in {\mathsf{Alg}}^{\mathrm{fg}}_{/ {k}}\) or affine schemes \(\operatorname{Spec}A\).
- The discrete/initial topology \(\tau = 2^X\).
- The indiscrete topology \(\tau = \left\{{\emptyset, X}\right\}\).

Recall the separation axioms:

\(T_0\): points can be topologically distinguished. Note that the indiscrete topology s not \(T_0\) if \({\sharp}X\geq 2\).

\(T_1\): points can be separated by (not necessarily disjoint) neighborhoods. Equivalently, points are closed.

\(T_2\)/Hausdorff: points can be separated by disjoint neighborhoods.

\(T_{3.5}\)/Tychonoff:?

\(T_6\):?

Show that points are closed in \(X\) iff \(X\) is \(T_1\).

A space \(X\) is **paracompact** iff every open cover \({\mathcal{U}}\rightrightarrows X\) admits a *locally* finite refinement \({\mathcal{V}}\rightrightarrows X\), i.e. any \(x\in X\) is in only finitely many \(V_i\).

Show that any \(S \subseteq {\mathbb{R}}^n\) is paracompact, and indeed any metric space is paracompact.

Let \({\mathcal{U}}\rightrightarrows X \mathrel{\vcenter{:}}={\mathbb{R}}^d\) be an open cover and define a proposed locally open refinement in the following way:

- Write \({\mathcal{U}}\mathrel{\vcenter{:}}=\left\{{U_\alpha {~\mathrel{\Big\vert}~}\alpha\in A}\right\}\) for some index set.
- Use that \(W_n \mathrel{\vcenter{:}}={ \operatorname{cl}} _{X}({\mathbb{B}}_n(\mathbf{0}))\) is compact, and since \({\mathcal{U}}\rightrightarrows W_n\) there is a finite subcover \({\mathcal{V}}_n \mathrel{\vcenter{:}}=\left\{{U_{n, 1}, \cdots, U_{n, m}}\right\}\rightrightarrows{ \operatorname{cl}} _X({\mathbb{B}}^n(\mathbf{0}))^c\).
- Show that \({\mathcal{V}}\mathrel{\vcenter{:}}=\left\{{{\mathcal{V}}}\right\}_{n\in {\mathbb{Z}}_{\geq 0}}\) is an open refinement of \({\mathcal{U}}\).
- Why: it is a subcollection, and every \(x\in X\) is in a ball of radius \(R\approx N\mathrel{\vcenter{:}}=\ceil{{\left\lVert {x} \right\rVert}}\). So \(x\in {\mathbb{B}}_N(0)\), thus \(x\in U_{N, k}\) for some \(k\).

- Show that \({\mathcal{V}}\) is locally finite.
- Why: each \({\mathcal{V}}_n\) misses the \({\mathbb{B}}_{k<n}(0)\), so each \(x\not\in \displaystyle\bigcup_{k\geq N} {\mathcal{V}}_n\) if \(N\) is defined as above. So \(x\) is in only finitely many \({\mathcal{V}}_n\).

Paracompact spaces admit a POU – for \({\mathcal{U}}\rightrightarrows X\), a collection \(A\) of function \(f_\alpha: X\to {\mathbb{R}}\) where for all \(\alpha \in \mathop{\mathrm{supp}}f_\alpha = { \operatorname{cl}} \qty{\left\{{f\neq 0}\right\}}\), for all \(x\in X\), there exists a \(V\ni x\) such that for only finitely many \(\alpha, { \left.{{f_\alpha}} \right|_{{V}} } \not\equiv 0\), and \(\sum_{\alpha\in A} f_{ \alpha}(x) = 1\).

Recall the order topology: for \((P, \leq )\) a poset, so

- \(x\leq y, y\leq x\implies x=y\),
- \(x\leq y\leq z\implies x\leq z\)
- \(x\leq x\)

Define

- Open sets to be increasing sets, so \(x\in U, x\leq y \implies y\in U\),
- Closed sets to be decreasing sets, so \(x\in U, x\geq y \implies y\in U\)

Note that this is a free choice!

Show that the order topology is closed under arbitrary unions *and* intersections of opens.

Show that the order topology is not \(T_1\) by showing \({ \operatorname{cl}} _P\qty{\left\{{x}\right\}} = Z_{\leq}(x) \mathrel{\vcenter{:}}=\left\{{y\in P{~\mathrel{\Big\vert}~}y\leq x}\right\}\).

For \(k\) an infinite field, \({\mathbb{A}}^1_{/ {k}}\) is the cofinite topology and thus not Hausdorff.

Recall the definition of a **poset**.

Given a polytope, one can take its face poset \(\mathrm{FP} (P) = \left\{{F \leq P}\right\}\) where \(F_1 \leq F_2\) iff \(F_1 \subseteq F_2\) for the faces \(F_i\). More generally, one can take a complex of polytopes, i.e. a collection of polytopes that only intersect at faces. An example of a complex is the fan of a toric variety.

Similarly, one can take **cones** \(\sum c_i \mathbf{v}_i \subseteq {\mathbb{R}}^d\) for some positive coefficients.

Conversely, given a poset \(I\), one can associate a simplicial complex \({ {\left\lvert {I} \right\rvert} }\), the geometric realization. Any chain \(i_{n_1} < \cdots i_{n_k}\) is sent to a face and glued.

Consider a polytope \(P\), taking the face poset \(\mathrm{FP} (P)\), and its geometric realization \({ {\left\lvert { \mathrm{FP} (P)} \right\rvert} }\). A square has

- \({\sharp}P_2 = 1\)
- \({\sharp}P_1 = 4\)
- \({\sharp}P_0 = 1\)

Note that one can take the geometric realization of a category by using the nerve to first produce a poset.

With the right choices, there exists a continuous map \({ {\left\lvert {I} \right\rvert} } \to I\) where \(I\) is given the order topology. Pulling back sheaves on the latter yields constructible sheaves on convex objects, which are locally constant on the interior components.

A first version of the Zariski topology: let \(k = \mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu \in \mathsf{Field}\) and let \(R\in{\mathsf{Alg}}^{\mathrm{fg}}_{/ {k}}\) be of the form \(R = k[x_1, \cdots, x_{n}]/\left\langle{f_a}\right\rangle\). We can consider \(X\mathrel{\vcenter{:}}=\operatorname{mSpec}R \subseteq {\mathbb{A}}^n_{/ {k}}\) as the points \(\mathbf{x}\in k{ {}^{ \scriptscriptstyle\times^{n} } }\) such that \(f_a(\mathbf{x}) = 0\). Recall Noether’s theorem – the \(f_a\) can be replaced with a finite collection. The closed subsets are of the form \(V(g_b)\). Note that this is \(T_1\) since points are closed: given \(\mathbf{p} = {\left[ {p_0,\cdots, p_n} \right]}\), take \(f(\mathbf{p}) = \prod_{i\leq n} (x-p_i)\) so that \(V(f) = \left\{{\mathbf{p}}\right\}\). These points biject with maximal ideals in \(R\).

An improved version of the Zariski topology: \(X = \operatorname{Spec}R\), including prime ideals. The points are as before, and additionally for every irreducible subvariety \(Z \subseteq X\), there is a generic point \(\eta_Z\). This adds new points which can’t be described in coordinates.

Note that this generalizes to arbitrary (associative, commutative) rings. For rings that aren’t finitely generated, one loses the coordinate interpretation. These generally won’t embed into \({\mathbb{A}}^n_{/ {k}}\) for any \(n\), but can be embedded into (say) \({\mathbb{A}}^1_{/ {R}}\). Use that a closed embedding \(X\hookrightarrow Y\) corresponds precisely to a surjection of associated rings \(R_Y \twoheadrightarrow R_X\).

Let \(U \subseteq \Omega \subseteq {\mathbb{C}}\) and consider \(C^0(\Omega, {\mathbb{C}}) \mathrel{\vcenter{:}}=\mathop{\mathrm{Hom}}_{\mathsf{Top}}(\Omega, {\mathbb{C}})\) – this forms a sheaf of abelian groups, \({\mathbb{C}}{\hbox{-}}\)algebras, rings, sets, etc.