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.

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:?

• \(T_6\):?

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!

Recall the definition of a poset.

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.

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 \(R_Y \twoheadrightarrow R_X\).

Some properties:

• For every \(\iota: V \subseteq U \implies\) there is a restriction ma \begin{align*} {\mathcal{F}}(\iota): {\mathcal{F}}(U) &\to {\mathcal{F}}(V) \\ f &\mapsto { \left.{{f}} \right|_{{V}} } .\end{align*}

• \({\mathcal{F}}({ \mathscr \emptyset^{\scriptscriptstyle \downarrow} }) = { \mathscr{1}_{\scriptscriptstyle \uparrow} }\), so e.g. for rings \({ \mathscr{1}_{\scriptscriptstyle \uparrow} }= \left\{{0}\right\}\) is the zero ring.

• (Sheaf 1, uniqueness): if \({\mathcal{U}}\rightrightarrows U\) and \(s_1, s_2 \in {\mathcal{F}}({\mathcal{U}})\), then \({ \left.{{s_1}} \right|_{{U_i}} } = { \left.{{s_2}} \right|_{{U_i}} } \implies s_1 = s_2\).

• (Sheaf 2, existence): if \(s_i\in {\mathcal{F}}(U_{ij})\) and \({ \left.{{s_1}} \right|_{{U_{ij}}} } = { \left.{{s_2}} \right|_{{U_{ij}}} }\), then there is a global section \(s\in {\mathcal{F}}(U_1 \cup U_2)\).

Recall the sheaf properties:

• \(U\to F(U)\) and \(\iota_{U, V} \mapsto \mathop{\mathrm{Res}}{F(V),F(U)}\).
• \({ \mathscr \emptyset^{\scriptscriptstyle \downarrow} }\mapsto F({ \mathscr \emptyset^{\scriptscriptstyle \downarrow} }) = { \mathscr{1}_{\scriptscriptstyle \uparrow} }\).
• Sheaf conditions:
  • Unique gluing: \({\mathcal{U}}\rightrightarrows X\) with \(\mathop{\mathrm{Res}}_{X, U_i} s = \mathop{\mathrm{Res}}_{X, U_i} t \implies s=t\in F(X)\)
  • Existence of gluing: \(\left\{{s_i\in F(U_i)}\right\}\) with \(\mathop{\mathrm{Res}}_{U_i, U_{ij}} s_i = \mathop{\mathrm{Res}}_{U_j, U_{ij}} s_j\) implies \(\exists ! s\in F(X)\) with \(\mathop{\mathrm{Res}}_{X, U_i} s = s_i\) for all \(i\).

Some upcoming standard notions:

• Stalks \(F_x\)
• Sheafification \(F \mapsto F^+\)

A less standard topic:

• The espace etale or “flat space” of \(F\).

Recall that morphisms of (pre)sheaves are natural transformations of functors. There is a forgetful functor \(\mathop{\mathrm{Forget}}: {\mathsf{Sh}}(X) \to \underset{ \mathsf{pre} } {\mathsf{Sh} }(X)\), which has a left adjoint \(({-})^+: \underset{ \mathsf{pre} } {\mathsf{Sh} }(X) \to {\mathsf{Sh}}(X)\). There is a description of \(F^+(U)\) as collections of local compatible sections of \(F\) modulo equivalence – compatibility here means that if \({\mathcal{U}}\rightrightarrows X\), then writing \(U_{ij} = \cup V_k\) we have \(\mathop{\mathrm{Res}}_{X, V_k}s_i = \mathop{\mathrm{Res}}_{X, V_k}s_j\) for all \(i, j\).

Last time: definitions of presheaves and sheaves. There is an adjunction \begin{align*} \adjunction{({-})^+}{\mathop{\mathrm{Forget}}}{ \underset{ \mathsf{pre} } {\mathsf{Sh} }(X)}{{\mathsf{Sh}}(X)} .\end{align*}

Recall that constant sheaves for \(A\in \mathsf{D}\) are defined as \(\underline{A}({-}) \mathrel{\vcenter{:}}=\mathop{\mathrm{Hom}}_{\mathsf{Top}}({-}, A)\) where \(A\) is equipped with the discrete topology.

These two definitions of sheaf coincide: for new to old, given \(Y \xrightarrow{\pi} X\) apply \(\mathrm{ContSec}_\pi \subseteq \mathop{\mathrm{Hom}}_{\mathsf{Top}}(X, Y)\). In the other direction, define \(Y\mathrel{\vcenter{:}}=\displaystyle\coprod_{x\in X} F_x\) and prove it is a local homeomorphism.

Next time: direct/inverse image, shriek functors, sheaves of modules.

Recall the definitions of presheaves and sheaves, and sheafification as an adjoint to \(\mathop{\mathrm{Forget}}: {\mathsf{Sh}}(X)\to \underset{ \mathsf{pre} } {\mathsf{Sh} }(X)\). For \(F\in \underset{ \mathsf{pre} } {\mathsf{Sh} }(X)\) we concretely construct its sheafification \(F^+\) using the sheaf space \(\pi: Y\mathrel{\vcenter{:}}=\displaystyle\coprod_{x\in X} F_x \to X\).

What are the sections of \(\pi\)? For a basic open \(U \subseteq X \ni x\), the fiber is \(\pi^{-1}(x) = F_x \mathrel{\vcenter{:}}=\colim_{V\ni x} F(V)\), which receives a map \(\mathop{\mathrm{Res}}_{U, x}: F(U) \to F_x\). Writing \(s\in F(U)\), define \(s_x \mathrel{\vcenter{:}}=\mathop{\mathrm{Res}}_{U, x}(s)\), and set \(W_{s, U} \mathrel{\vcenter{:}}=\left\{{s_x {~\mathrel{\Big\vert}~}x\in U}\right\}\) to be \(\pi^{-1}(U)\). Then define \(F^+\) to be the continuous sections of \(Y \xrightarrow{\pi} X\). What does such a section look like? For \(t:U\to \pi^{-1}(U)\) and \(x\in U\), the vertical fiber is \(F_x\). For a basic open \(V\ni X\) in the base, there is a basic open \(W_{s, V}\) in \(Y\) for \(s\in F(V)\):

There are maps \(s_{ij}: U_{ij}\to \pi^{-1}(U_{ij})\), but note that \(\mathop{\mathrm{Res}}(U_i, U_{ij}) s_i\) does not necessarily equal \(\mathop{\mathrm{Res}}(U_j,U_{ij}) s_j\) in \(F(U_{ij})\) – instead, there is an open cover \(U_{ij} = \displaystyle\bigcup V_{\alpha}\) with \(\mathop{\mathrm{Res}}(U_i, V_\alpha) s_i = \mathop{\mathrm{Res}}(U_j, V_\alpha) s_j\) for each \(\alpha\).

Todo

For \(f\in {\mathsf{Top}}(X, Y)\) we have the following constructions:

• The direct image \(f_*: {\mathsf{Sh}}(X) \to {\mathsf{Sh}}(Y)\), which is easy with the sheaf definition, and
• The inverse image \(f^{-1}: {\mathsf{Sh}}(Y)\to {\mathsf{Sh}}(X)\) which is easier with the sheaf space definition.

Recall the definition of a morphism of sheaves as a natural transformation.

For sheaves of abelian groups and \(\phi: F\to G\) a morphism of sheaves, there are notions of \(\ker \phi, \operatorname{coker}\phi, \operatorname{im}\phi\), and extension of a sheaf by zero.

To show these exist as presheaves, one only has to show existence of the following blue morphisms of abelian groups:

Link to Diagram

Write \((\operatorname{coker}\phi)^-\) and \((\operatorname{im}\phi)^-\) for these presheaves.

Recall last time: presheaf vs sheaf properties, images, kernel, cokernel. We can state the uniqueness sheaf axiom as the following: if \(s\in F(U)\) with \({ \left.{{s}} \right|_{{U_i}} } = 0\) for \({\mathcal{U}}\rightrightarrows U\), then \(s = 0\) in \(F(U)\).

• \({\mathcal{F}}\mathrel{\vcenter{:}}=(\operatorname{im}\phi)^-\) satisfies uniqueness.
• \({\mathcal{G}}\mathrel{\vcenter{:}}=(\operatorname{coker}\phi)^-\) satisfies existence.
• \({\mathcal{F}}\) fails existence \(\iff {\mathcal{G}}\) fails uniqueness
• \({\mathcal{F}}\) fails uniqueness iff \({\mathcal{G}}\) fails existence.

The presheaf image and cokernel can sometimes fail to be a sheaf: use \(\mathop{\mathrm{Hol}}(X) \xrightarrow{\exp} \mathop{\mathrm{Hol}}(X)^{\times}\). The kernel presheaf \((\ker \phi)^-\) is already a sheaf.

Hints:

• Suppose \(s\neq 0\) in \(F(U)\), does there exist a \(p\) with \(s_p = 0\)?
• Use that \(s_p \in (\ker \phi)_p\) can be regarded as \(s\in \ker(F(V) \to G(V))\) mod equivalence.

Note that for \(\phi: {\mathcal{F}}\to {\mathcal{G}}\), the image \(\operatorname{im}\phi\) is a secondary notion in additive categories, and can instead be defined as either

• \(\operatorname{coker}(\ker \phi \to {\mathcal{F}})\)
• \(\ker({\mathcal{G}}\to \operatorname{coker}\phi)\)

These need not coincide in general.

Defining the direct image: easier using the sheaf axioms. For \(f\in {\mathsf{Top}}(X, Y)\), define \(f_*: {\mathsf{Sh}}(X) \to {\mathsf{Sh}}(Y)\) by \begin{align*} f_* {\mathcal{F}}(U) \mathrel{\vcenter{:}}={\mathcal{F}}(f^{-1}(U)), \qquad \in {\mathsf{Sh}}(Y) \text{ for } {\mathcal{F}}\in {\mathsf{Sh}}(X) .\end{align*}

For the preimage: easier to use the espace étalé. As a special case, consider \(\iota: S \hookrightarrow Y\) where \(S\) is a subspace of \(Y\) (with the subspace topology). Then for \({\mathcal{F}}\in {\mathsf{Sh}}(Y)\), we can now define sections not only on open subsets \(U\) but arbitrary subsets \(S\) as \begin{align*} {\mathcal{F}}(S) \mathrel{\vcenter{:}}=(\iota^{-1}{\mathcal{F}})(S) .\end{align*}

Last time:

• Morphisms of sheaves \(\phi\),
• \(\ker \phi\) (already a sheaf),
• \((\operatorname{im}\phi)^-, (\operatorname{coker}\phi)^-\) (need to sheafify),
  • All defined to commute with taking stalks: \((\ker \phi)_p = \ker(\phi_p)\), etc
• \((\operatorname{im}\phi)^-\) may fail the existence axioms for sheaves, using \(\exp: {\mathcal{O}}^{\mathrm{an}}\to ({\mathcal{O}}^{\mathrm{an}})^{\times}\) for \(X\) a complex analytic space,
• \((\operatorname{coker}\phi)^-\) may fail the uniqueness axioms for sheaves,
• \((\operatorname{im}\phi)^-\) satisfies existence \(\iff (\operatorname{coker}\phi)^-\) satisfies uniqueness,
• For \({\mathcal{F}}\hookrightarrow{\mathcal{G}}\) injective, the presheaf quotient \(({\mathcal{G}}/{\mathcal{F}})^-\) may fail to be a sheaf.

Note that \({\operatorname{Pic}}(X)\) is also the group of invertible sheaves on \(X\), and for irreducible algebraic varieties these coincide. Use the SES \(0\to {\mathcal{O}}^{\times}\to K^{\times}\to K^{\times}/{\mathcal{O}}^{\times}\to 0\) to obtain \begin{align*} 1 \to H^0({\mathcal{O}}^{\times}) \to H^0(K^{\times}) \to \mathop{\mathrm{Prin}}\mathop{\mathrm{Cart}}\operatorname{Div}(X) \to H^1({\mathcal{O}}^{\times})\cong \text{invertible sheaves}/\sim \to 0, ,\end{align*} where \(H^1(K^{\times})\) vanishes since it’s a constant sheaf on an irreducible scheme in the Zariski topology.

Direct image sheaf: for \({\mathcal{F}}\in {\mathsf{Sh}}(X), {\mathcal{G}}\in {\mathsf{Sh}}(Y), f\in {\mathsf{Top}}(X, Y)\), the \(f_* \in [{\mathsf{Sh}}(X), {\mathsf{Sh}}(Y)]\) is defined by \(f_* {\mathcal{F}}(U) \mathrel{\vcenter{:}}={\mathcal{F}}(f^{-1}U)\). The inverse image functor \(f^{-1}\in [{\mathsf{Sh}}(Y), {\mathsf{Sh}}(X)]\) is slightly more complicated. An easy case: if \(\iota: S \hookrightarrow Y\) is a subspace, then it is just restriction: \((\iota^{-1}G)(S) \mathrel{\vcenter{:}}= G(S)\).

Idea for sheaf space: there are strictly horizontal neighborhoods as the homeomorphic preimages of small opens in the base. So for \(\text{Ét}_{{\mathcal{G}}} \xrightarrow{\pi} Y\) the sheaf space of \({\mathcal{G}}\), define the inverse image as \begin{align*} \text{Ét}_{\iota^{-1}{\mathcal{G}}} \mathrel{\vcenter{:}}=\pi^{-1}(S) \subseteq \text{Ét}_{{\mathcal{G}}} ,\end{align*} and define a basis of sections in the following way: for \(s\in {\mathcal{G}}(U)\), set \(t(U) \mathrel{\vcenter{:}}= s(U) \cap\pi^{-1}(S) \in \text{Ét}_{{\mathcal{G}}}\) to be sections of \(\text{Ét}_{\iota^{-1}{\mathcal{G}}}\). Declare these to be a basis of opens, i.e. take the subspace topology for \(\pi^{-1}(S) \subseteq \text{Ét}_{{\mathcal{G}}}\) in the sheaf topology on the total space. More generally, for \(f\in {\mathsf{Top}}(X, Y)\), set \begin{align*} \text{Ét}_{f^{-1}{\mathcal{G}}} \mathrel{\vcenter{:}}=\text{Ét}_{\mathcal{G}}{ \underset{\scriptscriptstyle {Y} }{\times} } X .\end{align*}

The fibers are identical:

Link to Diagram

The topology on \(\text{Ét}_{f^{-1}{\mathcal{G}}}\) is the coarsest topology for which \(\pi^*\) and \(f^*\) are continuous. This is generated by \(\qty{ f^{-1}(s)(f^{-1}U)} \cap(\pi^*)^{-1}(W)\) for \(W \subseteq X\) open. Define \(f^{-1}(s) \in f^{-1}{\mathcal{G}}(f^{-1}(U)) \mathrel{\vcenter{:}}=(f^{-1}U){ \underset{\scriptscriptstyle {U} }{\times} } s(U)\). This makes the pullback continuous both vertically and horizontally.

How to prove this coincides with the previous definition:

• Show the stalks are isomorphic,
• Show that there is a map of presheaves \((f^{-1}{\mathcal{G}}) \to f^{-1}{\mathcal{G}}\),
• Show that the map induces an isomorphism on stalks, and lift using the universal property of sheafification.

Recall that \(K^{\times}/{\mathcal{O}}^{\times}\cong \bigoplus _{x\in X} (\iota_*)_* \iota_*^{-1}\underline{{\mathbb{Z}}}\) which had stalks \({\mathbb{Z}}\) but was not constant – check that the local sections differ.

Extending by zero: for \(i: U \hookrightarrow X\) an open subspace and \({\mathcal{F}}\in {\mathsf{Sh}}(U)\), define \(i_!{\mathcal{F}}\in {\mathsf{Sh}}(X)\). If the target category has a zero object, define this in the sheaf space by extending the zero section:

Thus \(\text{Ét}_{i_! {\mathcal{F}}} = \text{Ét}_{{\mathcal{F}}} {\textstyle\coprod}\left\{{s_0}\right\}\) for \(s_0\) the zero section.

Consider \(X = U {\textstyle\coprod}Z\) with \(U\) open and \(Z\) closed. Let \(U \xhookrightarrow{i} X\) and \(Z\xhookrightarrow{j} X\), and consider \(i_* { \left.{{{\mathcal{F}}}} \right|_{{U}} }\) and \(j_* { \left.{{{\mathcal{F}}}} \right|_{{U}} }\). There is a SES \begin{align*} 0 \to i_! { \left.{{{\mathcal{F}}}} \right|_{{U}} } \to {\mathcal{F}}\to j_* { \left.{{{\mathcal{F}}}} \right|_{{Z}} }\to 0 .\end{align*}

Note that we have adjunctions \begin{align*} \adjunction{f^{-1}}{f_*}{{\mathsf{Sh}}X}{{\mathsf{Sh}}Y} \\ \adjunction{i_!}{{ \left.{{{-}}} \right|_{{U}} }}{{\mathsf{Sh}}?}{{\mathsf{Sh}}?} \\ \adjunction{j_*}{{ \left.{{{-}}} \right|_{{Z}} } }{{\mathsf{Sh}}?}{{\mathsf{Sh}}?} .\end{align*}

Last time: extension by zero, inverse image, pushforward on closed sets and adjunctions. \begin{align*} f\in \mathop{\mathrm{Hom}}_{{\mathsf{Top}}}(X, Y) \leadsto \mathop{\mathrm{Hom}}_{{\mathsf{Sh}}(X)}(f^{-1}{\mathcal{G}}, {\mathcal{F}}) \cong \mathop{\mathrm{Hom}}_{{\mathsf{Sh}}(Y)}({\mathcal{G}}, f_* {\mathcal{F}}) .\end{align*}

Consider \(U \xhookrightarrow{i} X\) with \(U\) open and \(Z \xhookrightarrow{j} X\) with \(Z\) closed, then for \({\mathcal{F}}\in {\mathsf{Sh}}(X), {\mathcal{H}}\in {\mathsf{Sh}}(U), {\mathcal{G}}\in {\mathsf{Sh}}(Z)\), \begin{align*} \mathop{\mathrm{Hom}}_{{\mathsf{Sh}}(Z)}( { \left.{{{\mathcal{F}}}} \right|_{{Z}} }, {\mathcal{G}}) & { \, \xrightarrow{\sim}\, }\mathop{\mathrm{Hom}}_{{\mathsf{Sh}}(X)}({\mathcal{F}}, j_* {\mathcal{G}}) \\ \mathop{\mathrm{Hom}}_{{\mathsf{Sh}}(U)}({\mathcal{H}}, { \left.{{{\mathcal{F}}}} \right|_{{U}} } ) & { \, \xrightarrow{\sim}\, }\mathop{\mathrm{Hom}}_{{\mathsf{Sh}}(X)}(i_! {\mathcal{H}}, {\mathcal{F}}) .\end{align*}

We’ll consider \((X, {\mathcal{O}}_X) \in \mathsf{Loc}\mathsf{RingSp}_{/ {\mathsf{CRing}}}\) with sheaves of reduced commutative rings – note that noncommutative rings are also important, e.g. \(\operatorname{GL}_n\) or \({\mathfrak{gl}}_n\).

Define algebraic varieties as locally ringed spaces \((X, {\mathcal{O}}_X)\) that

1. \(X\) is covered by finitely affine algebraic varieties, so \(X = \cup U_i\) with \((U_i, {\mathcal{O}}_{U_i})\) affine algebraic, and
2. \(X\) is separated, i.e. \(X \xrightarrow{\Delta_X} X{ {}^{ \scriptscriptstyle{ \underset{\scriptscriptstyle {X} }{\times} }^{2} } }\) is closed.

Note that affine and even quasiprojective schemes are automatically separated. We require the separated condition here to rule out things like \({\mathbb{A}}^1\) with two origins, i.e. \(X \mathrel{\vcenter{:}}={\mathbb{A}}^1{ \displaystyle\coprod_{{\mathbb{A}}^1\setminus\left\{{0}\right\}} }{\mathbb{A}}^1\).

Next time: locally free, invertible, tensor, and hom.

Examples of sheaves:

• \({\mathcal{O}}_X^\text{cts}\) for \(X\in {\mathsf{Top}}\), where \({\mathcal{O}}_X^\text{cts}({-}) = {\mathsf{Top}}({-}, {\mathbb{R}})\)
• \({\mathcal{O}}_X^{{\mathsf{sm}}}({-}) = C^\infty({-}, {\mathbb{R}})\).
• \({\mathcal{O}}_X^{\text{hol}}({-}) = \mathop{\mathrm{Hol}}({-}, {\mathbb{C}})\)
• \({\mathcal{O}}_X^{{\mathrm{an}}}({-}) \subseteq {\mathsf{Top}}({-}, {\mathbb{R}})\) the sheaf of analytic functions, those locally equal to power series.
• For \(X\in {\mathsf{Alg}}{\mathsf{Var}}_{/ {k}}\), \({\mathcal{O}}_X({-}) = {\mathsf{Top}}(({-})^{\mathrm{zar}}, k)\) the Zariski-regular \(k{\hbox{-}}\)valued functions.

In all cases, \({\mathcal{O}}_X\) can be regarded as sheaves of regular sections to \(X\times {\mathbb{A}}^1_{/ {k}} \xrightarrow{\pi} X\). Note that this doesn’t necessarily coincide with sections of the espace etale, since e.g. the fibers are \({\mathbb{A}}^1\) and not necessarily the stalks. For \({\mathcal{O}}{ {}^{ \scriptscriptstyle\oplus^{d} } }\), one instead takes \(X\times {\mathbb{A}}^d_{/ {k}} \to X\).

Maps between bundles are linear in the second coordinate.

Note that there is a correspondence between vector bundles and locally free sheaves. Consider the rank 1 case, matching invertible sheaves and line bundles. The necessary data:

• An open cover \({\mathcal{U}}\rightrightarrows X\), where \({\mathcal{U}}= \left\{{U_i}\right\}_{i\in I}\)
• For all \(i, j\in I\), transition functions \(\phi_{ij}\in {\mathcal{O}}^{\times}(U_{ij}) = \mathop{\mathrm{Aut}}_{{\mathsf{{\mathcal{O}}_X}{\hbox{-}}\mathsf{Mod}}}(U_{ij})\).
• A cocycle condition: \(\phi_{ii} = \operatorname{id}, \phi_{ij} = \phi_{ji}^{-1}\), and \(\phi_{ij} \phi_{jk} \phi_{ki} =\operatorname{id}\in {\mathcal{O}}^{\times}(U_{ijk})\)

Note that any morphism of sheaves \({\mathcal{O}}_V \to {\mathcal{O}}_V\) induces a morphism of \({\mathcal{O}}_V{\hbox{-}}\)modules on global sections \begin{align*} {\mathcal{O}}_V(V) & { \, \xrightarrow{\sim}\, }{\mathcal{O}}_V(V) \in {\mathsf{{\mathcal{O}}_V}{\hbox{-}}\mathsf{Mod}} \\ 1 &\mapsto \phi ,\end{align*} and this being an isomorphism manes \(\phi\) is invertible. Note that these are not isomorphic as rings.

Write \(Z_1({\mathcal{U}}; {\mathcal{O}}^{\times}) = \left\{{\phi_{ij} \in {\mathcal{O}}^{\times}(U_{ij}) {~\mathrel{\Big\vert}~}\cdots }\right\}\) for those \(\phi_{ij}\) satisfying the conditions above, and \(B_1({\mathcal{U}}; {\mathcal{O}}^{\times}) = \left\{{\phi_{ij} \in {\mathcal{O}}^{\times}(U_{ij}) {~\mathrel{\Big\vert}~}\phi_{ij} \sim \phi_{ij}{\psi_j \over \psi_i} }\right\}\) for any \({\psi_i \over \psi_j} \in \operatorname{GL}_1({\mathcal{O}}) \cong {\mathcal{O}}^{\times}\). More generally, we let \(\phi_{ij} = \psi_j \phi_{ij} \psi_i^{-1}\) for \(\psi_i, \psi_j \in \operatorname{GL}_n({\mathcal{O}})\).

Recall that for a given space \(X\), the open covers of \(X\) form a poset under refinement, where \({\mathcal{U}}\geq {\mathcal{V}}\) iff for every \(U_i\in {\mathcal{U}}\) there is some \(V_j \in {\mathcal{V}}\) with \(U_i \supseteq V_j\). This yields a system of maps \(Z^1({\mathcal{U}}; {\mathcal{O}}^{\times}) \to Z^1({\mathcal{V}}; {\mathcal{O}}^{\times})\) compatible with transition maps, so we define \begin{align*} {\check{H}}^1(X; {\mathcal{O}}_X^{\times}) \mathrel{\vcenter{:}}=\colim_{{\mathcal{U}}\rightrightarrows X} {\check{H}}^1({\mathcal{U}}; {\mathcal{O}}^{\times}) .\end{align*}

Plan:

• \(\mathop{\mathrm{Hom}}_{{\mathsf{Sh}}(X)}({-}, {-})\)
• \(\mathop{\mathrm{Hom}}_{{\mathsf{ {\mathcal{O}}_X}{\hbox{-}}\mathsf{Mod}}}({-}, {-})\)
• \(({-}) \otimes_{{\mathcal{O}}_X}({-})\)

For \((X, {\mathcal{O}}_X)\in \mathsf{Loc}\mathsf{RingSp}\) and \({\mathcal{F}}, {\mathcal{G}}\in {\mathsf{Sh}}(X; {\mathsf{Ab}}{\mathsf{Grp}})\), define \(\mathop{\mathrm{Hom}}_{{\mathcal{O}}_X}({\mathcal{F}}, {\mathcal{G}})\) to be natural transformations which are \({\mathcal{O}}_X{\hbox{-}}\)linear. This forms an abelian group under pointwise operations, and more generally an \({\mathcal{O}}_X{\hbox{-}}\)module since one can act on morphisms by global sections. There is a sheaf version, the local hom \(\underline{\mathop{\mathrm{Hom}}}_{{\mathcal{O}}_X}({\mathcal{F}}, {\mathcal{G}})(U) \mathrel{\vcenter{:}}=\mathop{\mathrm{Hom}}_{{\mathcal{O}}_U}({\mathcal{F}}_U, {\mathcal{G}}_U)\) where we write \({\mathcal{F}}_U \mathrel{\vcenter{:}}={ \left.{{{\mathcal{F}}}} \right|_{{U}} }\).

Recall that \({\mathsf{Sh}}(X)^{{\mathrm{lf}}, \operatorname{rank}=1} \cong {\mathsf{Bun}}_{\operatorname{GL}}^{\operatorname{rank}= 1}\), i.e. we identify rank 1 locally free sheaves with line bundles. We can write \(\mathop{\mathrm{Hom}}_{\mathcal{O}}({\mathcal{F}},{\mathcal{G}}) = \left\{{ {\phi_{ij} \over \psi_{ij} } {~\mathrel{\Big\vert}~}\phi_{ij} \in {\mathcal{O}}_X^{\times}(U_{ij}) \text{ satisfies the cocycle condition} }\right\}\). What are the transition functions?

We also define \(\mathop{\mathrm{Hom}}_{\mathcal{O}}({\mathcal{F}}, {\mathcal{O}}) \mathrel{\vcenter{:}}={\mathcal{F}} {}^{ \vee }\), and there is a relation to \({\operatorname{Pic}}(X)\).

Note also that \(\underline{\mathop{\mathrm{Hom}}}_{{\mathcal{O}}}({\mathcal{O}}, {\mathcal{F}}) \cong {\mathcal{F}}\), so global sections coincide with homs. This will be useful later when defining \(H^*\) in terms of derived functors.

Recall that \(f\in {\mathsf{Top}}(X, Y)\) for \(X, Y\in {\mathsf{Aff}}{\mathsf{Sch}}\) induces \(f^{-1}\in {\mathsf{Sh}}(X)(f^{-1}{\mathcal{O}}_Y, {\mathcal{O}}_X)\). For varieties, this is just given by pullback of regular functions. More generally, for \(X, Y\in \mathsf{Loc}\mathsf{RingSp}\), define the full pullback \(f^*\) as \begin{align*} f^*{\mathcal{F}}= f^{-1}{\mathcal{F}}\otimes_{f^{-1}{\mathcal{O}}_Y} {\mathcal{O}}_X .\end{align*}

Consider \(f\in {\mathsf{Alg}_{/k} }(S, R)\) for \(k=\mkern 1.5mu\overline{\mkern-1.5muk\mkern-1.5mu}\mkern 1.5mu\) where we only consider reduced algebra (no nonzero nilpotents). This induces maps \(\tilde f: \operatorname{Spec}R\to \operatorname{Spec}S\) and \(\tilde f': \operatorname{mSpec}R\to \operatorname{mSpec}S\). If \({\mathcal{A}}\in {\mathsf{Sh}}(X; {{{\mathcal{O}}_X}{\hbox{-}}\mathsf{Alg}})\), there are induced maps \({\mathcal{O}}_X(U) \to {\mathcal{A}}(U)\) and thus affine morphism \(\pi: \operatorname{Spec}{\mathcal{A}}(U) \to U\) covering the affine open \(U\).

Recall the definitions of:

• Cochain complexes,
• Boundaries,
• Cycles,
• Homology as cycles mod boundaries,
• Morphisms of chain complexes
• Chain homotopies
• Nullhomotopic morphisms
• Homotopic morphisms of chain complexes
• Short exact sequences of complexes:

Link to Diagram

• Small categories
  • Sets of objects, sets of morphisms, a pairing \(\mathop{\mathrm{Mor}}(A, B)\times \mathop{\mathrm{Mor}}(B, C)\to \mathop{\mathrm{Mor}}(A, C)\).
• Universes

Recall the definitions of:

• Categories
• Functors
• Diagram/index categories
  • \(\bullet \to \bullet \leftarrow\bullet\)
  • \(\bullet \leftarrow\bullet \to \bullet\)
  • \(\bullet \rightleftharpoons\bullet\)
  • \({\mathbb{N}}: \bullet \to \bullet \to \cdots\)
  • \(\bullet \rightrightarrows\bullet\)
• Sets and posets as categories
• Collections of objects \(\mathsf{C}\) as functors \(F\in [\mathsf{I}, \mathsf{C}]\) for \(\mathsf{I}\) an index category
• Products and coproducts (via their universal properties). Useful mnemonic diagram:

Link to Diagram

• Algebraic cats over sets (concrete categories) will be closed under products, i.e. \(\prod A_i\) will admit the same algebraic structure by taking pointwise operations.
• Examples of (co)products in common categories:
  • \({\mathsf{Set}}\): direct cartesian product and disjoint union.
  • \({\mathsf{Ab}}{\mathsf{Grp}}\): direct cartesian product and direct sum \(\oplus\).
  • \(\mathsf{Ring}\): \(\prod\) and \(\otimes_{\mathbb{Z}}\)
  • \({\mathsf{Top}}\): \(\prod\) whose underlying set is the cartesian product with the product topology and \(\coprod\) as the disjoint union with the union of topologies Note the difference between the box and product topologies.
• A diagram in \(\mathsf{C}\) defined as a functor.
• (co)filtered diagram categories \(\mathsf{I}\): for any pair \(i,j\), \({\sharp}\mathop{\mathrm{Mor}}_{\mathsf{I}}(i, j) \leq 1\) and there exists a \(k\) with \(i, j \to k\). Reverse arrows for cofiltered.
  • This allows for distinct but isomorphic objects, useful e.g. in \({ \mathsf{Vect} }_{/ {k}}\) where abstractly \(V\cong V {}^{ \vee }\) but it’s useful to distinguish.
• Limits (injective, cones that live above) and colimits (projective, cocones that live below):

Link to Diagram

• Fiber products/pullbacks and pushouts
• Equalizers/difference kernels \(K\) and coequalizers/difference cokernels \(C\) fitting into \(K \to A_1 \rightrightarrows A_2\to C\).
• Computing cofiltered colimits in \({\mathsf{Ab}}{\mathsf{Grp}}\): for the cofiltered set \(\left\{{A_i, \phi_{ij}: A_i\to A_j}\right\}_{i, j}\), can construct as \(\colim A_i = {\textstyle\coprod}A_i/\sim\) here \(a_i \sim \phi_{ik}(a_i)\) for any \(k\) with \(i\to k\).
  • For filtered limits, one generally gets \(\cocolim A_i = \bigoplus A_i/\sim\) where \(a_i \sim \phi_{ik}(a_i)\)
• Example: \({\textstyle\coprod}A_i \in {\mathsf{Ab}}{\mathsf{Grp}}\) is not a cofiltered colimit, since the diagram category is discrete.
  • Claim: the underlying set is not \({\textstyle\coprod}A_i\).
• For fixed objects \(A\in \mathsf{C}\), the functors \(\mathop{\mathrm{Mor}}_{\mathsf{C}}(A, {-}): \mathsf{C}\to {\mathsf{Set}}\) and \(\mathop{\mathrm{Mor}}_{\mathsf{C}}({-}, A): \mathsf{C} \to { {{\mathsf{Set}}}^{\operatorname{op}}}\).
  • More generally the target can be \({\mathsf{Ab}}{\mathsf{Grp}}, \mathsf{CRing}\), etc.

Next time: additive and abelian categories, why \({\mathsf{Sh}}(X; {\mathsf{Ab}}{\mathsf{Grp}})\) is an abelian category.

Notes:

• \(\ker f \to A \rightrightarrows^f_0 B\to \operatorname{coker}f\).
• \(B\xhookrightarrow{h} X\) is injective iff \(A\rightrightarrows_g^f B \to X\)
• \(X \xrightarrow[]{h} { \mathrel{\mkern-16mu}\rightarrow }\, A\) is surjective iff \(X \xrightarrow{h} A \rightrightarrows^f_g B\)
• Iso = mono and epi

For \(\mathsf{C} = {\mathsf{Ab}}{\mathsf{Grp}}\), \(\mathop{\mathrm{Hom}}_{\mathsf{C}}\) form abelian groups under pointwise operations. For morphisms \(\mathsf{C} = {\mathsf{Sh}}(X; {\mathsf{Ab}}{\mathsf{Grp}})\) and \(f,g\in \mathsf{C}({\mathcal{F}}, {\mathcal{G}})\), writing \(f = \left\{{f_U}\right\}, g = \left\{{g_U}\right\}\) in components, one can set \(f+g = \left\{{f_U + g_U}\right\}\) to make \(\mathop{\mathrm{Hom}}_{\mathsf{C}}\) an abelian group. Images will be isomorphic to coimages in \(\mathsf{C}\) since the induced maps will be isomorphisms on stalks, using that \({\mathsf{Ab}}{\mathsf{Grp}}\) is abelian.

If \({\mathcal{A}}\in {\mathsf{Ab}}\mathsf{Cat}\), then \({\mathsf{Sh}}(X; {\mathcal{A}})\in {\mathsf{Ab}}\mathsf{Cat}\).

Last time: abelian categories \(\mathsf{C}\).

1. Existence of kernels, cokernels, and biproducts: \(\exists A\times B \iff \exists A \oplus B\).
2. Existence of isomorphisms \(\operatorname{coim}\phi \to \operatorname{im}\phi\) for all \(\phi\in \mathsf{C}(A, B)\)

The main technical tool: every SES induces a LES in cohomology. The proof used for \(\mathsf{C} = {\mathsf{Ab}}{\mathsf{Grp}}\) works nearly identically in an arbitrary abelian category using either

• generalized elements, c/o MacLane, or
• the full Freyd-Mitchell embedding.

MacLane’s idea: define a functor \begin{align*} F: \mathsf{A} \to {\mathsf{Set}}_{{\operatorname{pt}}} \\ A &\mapsto \left\{{X\in \mathsf{A} {~\mathrel{\Big\vert}~}X\hookrightarrow A}\right\}/\sim ,\end{align*} sending \(A\) to the set of its subobjects (equivalence classes of monomorphisms), and on morphisms \(A \xrightarrow{f} B\) sending \(X\hookrightarrow A\) to its image \(f(X)\hookrightarrow B\), so \(F(f)(X) = \operatorname{im}_B(X)\). The point in the pointed set is the subobject \(0_A \to A\). One then proves

• \(f = 0 \iff F(f) = 0\),
• \(f\) is mono/epi \(\iff F(f)\) is mono/epi,
• Thus \(F\) is exact.

So one can reduce checking exactness of \(f\) (where \(\mathsf{A}\) may not have sets of elements) to checking exactness of \(F(f)\), where the source/target are sets.

Some additional axioms that hold in \({\mathsf{Ab}}{\mathsf{Grp}}\) which we could ask \(\mathsf{A}\in {\mathsf{Ab}}\mathsf{Cat}\) to have:

• AB3: existence of arbitrary sums \(\bigoplus_i A_i\).

• AB4: AB3 and if \(A_i\hookrightarrow B_i\) for all \(i\), then \(\bigoplus _i A_i \hookrightarrow\bigoplus _i B_i\) is again injective.

• The dual of AB4, with products replaced by coproducts and injectives replaced by surjections.

• AB5: AB3 and for all filtered system of subobjects \(A_i \subseteq A\) and a subobject \(B \subseteq A\), \begin{align*} (\sum A_i) \cap B \cong \sum (A_i \cap B) .\end{align*}

• AB6: AB3 and for all filtered systems \(B_{i}^j \subseteq B^j \subseteq A\), \begin{align*} \cap_{j\in J} \qty{ \sum_{i\in I_j} B_i^j } = \sum_{i\in {\textstyle\coprod}I_j}\qty{\cap_{j\in J} B_i^j } .\end{align*}

• AB: AB6 and AB4\({}^{ \vee }\), the dual conditions for AB4.

The categories \({\mathsf{Sh}}_X({\mathsf{Ab}}{\mathsf{Grp}})\) and \({\mathsf{Sh}}_X({\mathsf{{\mathcal{O}}_X}{\hbox{-}}\mathsf{Mod}})\) satisfy AB5 and AB3\({}^{ \vee }\)

Recall the definitions of \(\cocolim F\) and \(\colim F\) for \(F \in [\mathsf{I}, \mathsf{C}] = \mathsf{C}^{\mathsf{I}}\) with \(\mathsf{I}\) a small index category. Note that if \(\mathsf{N} \mathrel{\vcenter{:}}={ { {\mathsf{Open}}(X)}^{\operatorname{op}}}\), the functor category \(\mathsf{C}^{\mathsf{N}} = \underset{ \mathsf{pre} } {\mathsf{Sh} }(X; \mathsf{C})\) consists of presheaves on \(X\).

In \({\mathsf{Ab}}{\mathsf{Grp}}\), we have \(\prod,\coprod = \bigoplus, \colim, \cocolim\).

Link to Diagram

Note that the inner diamond doesn’t necessarily commute. The same diagram holds in \({\mathsf{R}{\hbox{-}}\mathsf{Mod}}\).

Recall the definition of an additive category:

• \(\mathop{\mathrm{Mor}}_{\mathsf{C}}({-}, {-})\) are abelian groups,
• Compositions distribute
• A zero object
• Finite products \(A\times B \iff\) finite coproducts \(A \oplus B \iff\) finite biproducts:

Link to Diagram

where we require

• \(p_j i_j = \operatorname{id}\)

• \(p_j i_k = 0\) for \(i\neq j\)

• \(i_1 p_1 + i_2 p_2 = \operatorname{id}_{A \oplus B}\).

• Abelian cats: additive, plus existence of kernels, cokernels, images.

Recall the definition of exactness for chain complexes over abelian categories: \(\operatorname{im}d^{n-1} = \ker d^n\). Note that one can use epi-mono factorization to splice:

Link to Diagram

This yields collections of SESs, \begin{align*} 0\to Z^{n-1} \to C^n\to Z^{n}\to 0 .\end{align*}

Recall the definition of right/left/middle exactness: for \(0\to A\to B\to C\to 0\) and covariant functors \(F\):

• Right exact: \(FA \to FB \to FC\to 0\),
• Long exact: \(0\to FA \to FB \to FC\),
• Middle exact: \(FA \to FB \to FC\).

For contravariant functors, e.g. left exactness means \(0\to FC\to FB \to FA\), so injectivity is preserved. Equivalently, \(F: \mathsf{A}\to \mathsf{B}\) is left exact iff the covariant \(F: { {\mathsf{A}}^{\operatorname{op}}} \to \mathsf{B}\) is left-exact.

A nice proof that \({{\Gamma}\qty{{-}} }\) is left-exact: realize \({{\Gamma}\qty{X; {-}} } \cong \mathop{\mathrm{Hom}}_{{\mathsf{Sh}}(X)}(\underline{{\mathbb{Z}}}, {-})\), which is left-exact for free. Use that the map \(\underline{{\mathbb{Z}}} \to {\mathcal{F}}(X)\) is determined by \(1\mapsto s\) and extend using \(n = n\cdot 1\).

Consider the setup: \begin{align*} \adjunction F G { \mathsf{A}}{ \mathsf{B} } .\end{align*} We say \(F\) is a left adjoint and \(G\) is a right adjoint, so \(F\) has a right adjoint and \(G\) has a left adjoint, if there are natural isomorphisms \begin{align*} [FA, B]_{\mathsf{B}} { \, \xrightarrow{\sim}\, }[A, GB]_{\mathsf{A}} ,\end{align*} i.e. there is a natural isomorphism of functors \([A, G({-})] { \, \xrightarrow{\sim}\, }[FA, ({-})]\). For a fixed object \(B\), there is a natural transformation \({\varepsilon}_B: FG\to \operatorname{id}_B\) which we call the counit and \(\eta_A: \operatorname{id}_A\to GF\) called the unit:

Link to Diagram

Note that \(f_*\) is fully exact, as we knew before by checking on stalks. Also note that \({~\mathrel{\Big\vert}~}_x\) for \({\mathcal{F}}\in {\mathsf{Sh}}(X)\) is \(f^{-1}{\mathcal{F}}\) for \(f:\left\{{x}\right\} \hookrightarrow X\).

Recall \(\mathsf{R}{\hbox{-}}\mathsf{Mod} = \mathsf{Mod}{\hbox{-}}\mathsf{R} = ({R}, {R}){\hbox{-}}\mathsf{biMod}\) for \(R \in \mathsf{CRing}\) associative, but for noncommutative rings these may differ.

• The tensor product is a bifunctor \begin{align*} ({-})\otimes_R ({-}): \mathsf{Mod}{\hbox{-}}\mathsf{R} \times \mathsf{R}{\hbox{-}}\mathsf{Mod} &\to {\mathsf{Ab}}{\mathsf{Grp}}\\ M_R \times _{R}N &\mapsto M_R \otimes_R {}_R N = { F(M\times N) \over (m_1 + m_2) \otimes n - m_1 \otimes n - m_2\otimes n, ma\otimes n - m\otimes an, \cdots} ,\end{align*} satisfying the usual universal property.

• This generalizes: \begin{align*} ({-})\otimes_R ({-}): ({R}, {R}){\hbox{-}}\mathsf{biMod} \times \mathsf{R}{\hbox{-}}\mathsf{Mod} &\to \mathsf{R}{\hbox{-}}\mathsf{Mod} .\end{align*}

• If \(\phi\in \mathsf{CRing}(R, S)\), then \begin{align*} ({-})\otimes_R S: \mathsf{Mod}{\hbox{-}}\mathsf{R} \to \mathsf{S}{\hbox{-}}\mathsf{Mod} .\end{align*}

• This extends to algebras: \begin{align*} ({-})\otimes_R ({-}): \mathsf{Alg}{R}\times \mathsf{Alg}{R} \to \mathsf{Alg}{R} ,\end{align*} with multiplication given by \((s_1\otimes s_2)\cdot(t_1\otimes t_2) \mathrel{\vcenter{:}}=(s_1t_1)\otimes(s_2t_2)\).

• There is an adjunction: \begin{align*} [A\otimes_R C, B]_{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}} { \, \xrightarrow{\sim}\, }[A, B^C]_{{\mathsf{R}{\hbox{-}}\mathsf{Mod}}} .\end{align*}

Recall that one can compute \(H_*(S^2; {\mathbb{Z}})\) in several ways.

Method 1: triangulation.

This yields \begin{align*} 0 \leftarrow{\mathbb{Z}}{ {}^{ \scriptscriptstyle\times^{4} } } \leftarrow{\mathbb{Z}}{ {}^{ \scriptscriptstyle\times^{6} } }\leftarrow{\mathbb{Z}}{ {}^{ \scriptscriptstyle\times^{4} } } \leftarrow 0 \leadsto 0\leftarrow{\mathbb{Z}}\leftarrow 0 \leftarrow{\mathbb{Z}}\leftarrow 0 .\end{align*}

Method 2: cell complexes.

This directly yields \begin{align*} 0\leftarrow{\mathbb{Z}}\leftarrow 0 \leftarrow{\mathbb{Z}}\leftarrow 0 .\end{align*}

More generally, for \(F\in {\mathsf{Ab}}\mathsf{Cat}(\mathsf{A},\mathsf{B})\), we’ll want to resolve by acyclic objects. Injectives and projectives will be universal such objects, but are often hard to work with, so we’ll work on finding more economical acyclic resolutions. Next time: injectives/projectives and derived functors.

For \(F\in {\mathsf{Ab}}\mathsf{Cat}(\mathsf{A}, \mathsf{B})\) left exact, assuming \(\mathsf{A}\) has enough injectives, there is a right derived functor \({\mathbb{R}}F\) so that a SES \(0\to A\to B\to C\to 0\) admits a LES with a connecting morphism \(\delta\): \begin{align*} 0\to {\mathbb{R}}FA\to {\mathbb{R}}F B\to {\mathbb{R}}F C \xrightarrow{\delta} \Sigma^1 {\mathbb{R}}F A \to \cdots .\end{align*} Note that \(\delta\) depends on the triple appearing in the SES.

Injectives will be acyclic and homology will measure how things are glued. Analogy: simplicial or cellular homology uses contractible objects (with trivial homology) to measure how spaces are glued from simplices or spheres.

Recall the definitions of projective and injective objects, which require existence (but not uniqueness) of certain lifts. In \({\mathsf{R}{\hbox{-}}\mathsf{Mod}}\), free implies projective, so free resolutions usually suffice and one can study generators, relations, syzygies, etc.

We’ll show that \(\mathsf{A} \mathrel{\vcenter{:}}={\mathsf{Sh}}(X; {\mathsf{Ab}}{\mathsf{Grp}})\) has enough injectives, but usually won’t have enough projectives. Recall that this means that every \(A\in \mathsf{A}\) admits a monomorphism \(A\hookrightarrow I\) for \(I\) an injective object. If there are enough injectives, every object admits an injective resolution, and any two such resolutions are homotopy equivalent.

Recall that \begin{align*} { {{\mathbb{R}}}^{\scriptscriptstyle \bullet}} F(X) = { {H}_{\scriptscriptstyle \bullet}} ( F( X \leftleftarrows { {I}^{\scriptscriptstyle \bullet}} )) \end{align*} and \({\mathbb{R}}^{i\geq 1} F(I) = 0\) if \(I\) is itself injective.

Recall the Horseshoe lemma:

Link to Diagram

Note that the complex in the middle is not the direct sum of the two outer complexes, just the terms – the differential \(d_B\) on \({ {I}^{\scriptscriptstyle \bullet}} _B\) will be of the form \begin{align*} d_B = { \begin{bmatrix} {d_A} & {*} \\ {0} & {d_C} \end{bmatrix} } .\end{align*}

Idea: regard \(A\) as a chain complex supported in degree zero and \(A\underset{\leftleftarrows}{\eta} { {I}^{\scriptscriptstyle \bullet}}\) an injective resolution, then the induced map \(\eta^*: H^*(A)\to H^*( { {I}^{\scriptscriptstyle \bullet}} )\) is an isomorphism, so \(A\) and \({ {I}^{\scriptscriptstyle \bullet}}\) are quasi-isomorphic.

Hints:

Link to Diagram

Given \begin{align*} f^{n-1} - g^{n-1} = h^n d^{n-1} - d^{n-2} h^{n-1} ,\end{align*} construct \(h^{n+1} d_I^n\) such that \begin{align*} (f^n - g^n) d_I^n = ( h^{n+1} d_I^n + d_J^{n-1} h^n ) d_I^n \end{align*} and extend arbitrarily to \(h^{n+1}: I^{n+1} \to J^n\).

Call the definition of the derived functor \({\mathbb{R}}F\) for a left-exact functor \(F\in{\mathsf{Ab}}\mathsf{Cat}(\mathsf{C}, \mathsf{D})\) where \(\mathsf{A}\) has enough injectives. These satisfy \({\mathbb{R}}^0 F = F\), and for a SES \(0\to A\to B\to C\to 0\) there is an induced LES \({\mathbb{R}}F A\to {\mathbb{R}}F B \to {\mathbb{R}}F C \to {\mathbb{R}}F A[1]\) which is functorial in the triple \((A,B,C)\). Next: Grothendieck’s universality theorem.

Setup from last time: \(F\in {\mathsf{Add}}\mathsf{Cat}(\mathsf{A}, \mathsf{B})\) left-exact, \(\left\{{(S^n, \phi_S^n)}\right\}_{n\geq 0}\) exact \(\delta{\hbox{-}}\)functors where for \(n > 0\) the \(S^n\) are effaceable. Then it is universal: for all \(\delta{\hbox{-}}\)functors \(\left\{{(T^n, \phi_T^n)}\right\}_{n\geq 0}\) with a natural transformation \(S^0 \to T^0\) there exist unique morphisms \((S^n, \phi_S^n) \to (T^n, \phi_T^n)\), i.e. natural transformations \(S^n\to T^n\) commuting with the \(\phi^n\).

Take an effacement \(0\to A \xhookrightarrow{i} M\) for \(S^{n+1}\) and extend to a SES \(0\to A\to M\to Q\to 0\). We’ll define the ladder of morphisms inductively using the following commutative diagram:

Link to Diagram

We need to show

• \(f^{n+1}(A, i)\) only depends on \(A\)
• \(f^{n+1}\) is functorial in \(A\)
• \(f^{n+1}\) commutes with \(\phi_S, \phi_T\).

Given effacements:

Link to Diagram

There exists an effacement extending \(g\). Use

Link to Diagram

There is a factorization:

Link to Diagram

?? Concludes theorem from last time.z

Recall that \(\mathop{\mathrm{Hom}}(C, {-})\) is left exact covariant and \(\mathop{\mathrm{Hom}}({-}, C)\) is left exact contravariant. For left exact functors,

• Right derived functors are computed with injective resolutions.
• \(\mathsf{C}\) needs enough injectives

For right exact functors,

• Left derived functors are computed with projective resolutions.
• \(\mathsf{C}\) needs enough projectives

Projective sheaves are locally free.

Recall free \(\implies\) projective and \({\mathsf{R}{\hbox{-}}\mathsf{Mod}}\) has enough projectives and enough injectives.

On adjoint functors: \begin{align*} \adjunction F G {\mathsf{A}} {\mathsf{B}} \implies \mathsf{B}(FX, Y) { \, \xrightarrow{\sim}\, }\mathsf{A}(X, GY) .\end{align*} Here \(F\) is a left adjoint hence right exact, and \(G\) is a right adjoint and is left exact.

For \(f\in \mathsf{CRing}(S\to R)\), there is an adjunction \begin{align*} \adjunction {M_R\mapsto M_S} { {\mathsf{S}{\hbox{-}}\mathsf{Mod}}(R, {-}) } {{\mathsf{R}{\hbox{-}}\mathsf{Mod}}}{{\mathsf{S}{\hbox{-}}\mathsf{Mod}}} \end{align*} where \({\mathsf{S}{\hbox{-}}\mathsf{Mod}}(R, N) \in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}\) via the action \((rf)(x) \mathrel{\vcenter{:}}= f(rx)\), sometimes called the induced \(R{\hbox{-}}\)module. Note that \({\mathsf{R}{\hbox{-}}\mathsf{Mod}}(R, N) { \, \xrightarrow{\sim}\, }N\) by \(1_R \mapsto n\), and there is an iso \begin{align*} {\mathsf{S}{\hbox{-}}\mathsf{Mod}}(M_S, N) &\rightleftharpoons{\mathsf{R}{\hbox{-}}\mathsf{Mod}}(M_R, {\mathsf{S}{\hbox{-}}\mathsf{Mod}}(R, N)) \\ \qty{ m\mapsto \psi(m)(1) } &\mapsfrom \psi \\ \phi &\mapsto \qty{m \mapsto \psi(m)(i) \mathrel{\vcenter{:}}=\psi(im) \mathrel{\vcenter{:}}=\phi(im) } .\end{align*}

Proving \({\mathsf{R}{\hbox{-}}\mathsf{Mod}}\) has enough injectives if \({\mathsf{S}{\hbox{-}}\mathsf{Mod}}\) has enough injectives: use \(M_R \cong {\mathsf{R}{\hbox{-}}\mathsf{Mod}}(R, M)\hookrightarrow{\mathsf{S}{\hbox{-}}\mathsf{Mod}}(R, M_S) \hookrightarrow{\mathsf{S}{\hbox{-}}\mathsf{Mod}}(R, I)\) where \(M_S \hookrightarrow I\) embeds into some injective. Take \(R\) arbitrary and \(S={\mathbb{Z}}\) to conclude any \({\mathsf{R}{\hbox{-}}\mathsf{Mod}}\) has enough injectives.

Any \(M\in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}\) admits a minimal injective hull \(M\hookrightarrow I\).

Induced and coinduced modules:

Link to Diagram

Note that coinduction sends injective to injectives, and induction sends projectives to projectives. Recall that \({\mathsf{Sh}}(X; {\mathsf{Ab}}{\mathsf{Grp}})\) and \({\mathsf{{\mathcal{O}}_X}{\hbox{-}}\mathsf{Mod}}\) have enough injectives, so left exact covariant functors \(F\) admit right-derived functors \({\mathbb{R}}F\), and similarly right exact contravariant functors \(F\) admit left-derived functors \({\mathbb{L}}F\).

Consider \(F = \mathsf{A}(A, {-})\) (covariant) or \(\mathsf{A}({-}, A)\) (contravariant), so \(F\in \mathsf{Cat}(\mathsf{A}, {\mathsf{Ab}}{\mathsf{Grp}})\). Note that acyclic objects for \(F\) are exactly injectives: take \(0\to A\to B\to C\to 0\) to obtain \(0\to [C, I] \to [B, I]\to [A, I] \to \operatorname{Ext} ^1(C, I) = 0\) by acyclicity of \(I\), meaning that \([B,I] \twoheadrightarrow[A, I]\) and thus there exist lifts:

Link to Diagram

Note that fine \(\implies\) soft and flasque \(\implies\) soft. Fine sheaves are best for paracompact Hausdorff spaces, and flasque are better for e.g. the order topology.

Important classes of sheaves:

• Universal: flasque or flabby.
• Classical topologies (Hausdorff, paracompact): fine \(\implies\) soft.
• AG: quasicoherent sheaves on affine sets and covers.

There is a canonical flasque resolution:

Link to Diagram

This is useful e.g. for finite sets with the order topology, but less useful if \({\left\lvert {X} \right\rvert}\) is infinite and there are non-closed points.

Recall that a sheaf is fine iff it satisfies the POU property.

• Classically: there is an open cover \({\mathcal{U}}\rightrightarrows X\) and \(\phi_i: U_i \to {\mathbb{R}}\) with \(\mathop{\mathrm{supp}}\phi_i \subseteq U_i\) where \(\sum \phi_i = 1\) and locally there are only finitely many nonzero \(\phi\).
• For sheaves: there is an open cover \({\mathcal{U}}\rightrightarrows X\) and \(\phi_i: {\mathcal{F}}\to {\mathcal{F}}\) with \(\mathop{\mathrm{supp}}\phi\) a closed set \(Z_i\) where \(\sum \phi_i = \operatorname{id}_{\mathcal{F}}\) and locally there are only finitely many \(i\) with \(\phi({\mathcal{F}})\neq 0\).

In this case, fine implies soft.

Let \(X\) be a smooth manifold over \({\mathbb{R}}\). Note that \(\underline{{\mathbb{R}}}\) is not fine and not soft, and not even an \({\mathcal{O}}_X{\hbox{-}}\)module. However it admits a resolution \(0 \leftarrow\underline{{\mathbb{R}}} \leftleftarrows { {\Omega_{X}}^{\scriptscriptstyle \bullet}}\) where \(\Omega^0_{X} \mathrel{\vcenter{:}}={\mathcal{O}}_X^{{\mathsf{sm}}}\), and this resolution computes the sheaf cohomology \({ {H}^{\scriptscriptstyle \bullet}} (X; \underline{{\mathbb{R}}})\).

Similarly, \(0 \to \underline{{\mathbb{C}}} \leftleftarrows_{{ \mkern 1.5mu\overline{\mkern-1.5mu{\partial}\mkern-1.5mu}\mkern 1.5mu}} \Omega^{0, \bullet}\) where \({ \mkern 1.5mu\overline{\mkern-1.5mu{\partial}\mkern-1.5mu}\mkern 1.5mu}= \sum {\frac{\partial }{\partial \mkern 1.5mu\overline{\mkern-1.5muz\mkern-1.5mu}\mkern 1.5mu_i }\,} dz_i\).

Upcoming topics related to \({ {H}^{\scriptscriptstyle \bullet}} (X; {\mathcal{F}})\):

• General vanishing theorems
• Čech cohomology
• Riemann-Roch

• Note that the theorem statement uses the Zariski topology, and so doesn’t contradict that \({H}^{2d}_{{\mathrm{sing}}}(X; {\mathbb{Z}}) \neq 0\) for (say) \(X\) a compact complex manifold.
  • The theorem uses algebraic dimension \(d \mathrel{\vcenter{:}}=\dim_{\mathbb{C}}X\), which is generally twice the real dimension.
• Recall that \(X\) is Noetherian iff \(X\) satisfies the DCC on closed sets.
• Algebraic varieties with the Zariski topology are Noetherian, since dimension strictly decreases on proper closed subsets.
• Affine schemes over Noetherian rings are Noetherian, since closed subsets corresponds to radical ideals, which satisfy the ACC.
• \(\dim X\) is defined as \(\sup \left\{{d {~\mathrel{\Big\vert}~}Z_0 \subsetneq Z_1 \subsetneq \cdots \subsetneq Z_d }\right\}\).
• Noetherian spaces can have infinite dimension (see examples by Nagata)
• Schemes are nonsingular if the completions of local rings are formal power series.
• Smallest class of nice rings in AG: referred to as “Japanese rings” in the literature, finitely generated rings over DVRs, plus localizations, completions, direct sums, etc.

Analogy:

• Quasicoherent: arbitrary modules \(M\)
• Coherent: finitely presented modules \(M\).

Affine schemes correspond to general rings, and projective schemes correspond to graded rings. In the second statement, coherence is used as a kind of finiteness.

Why \({\partial}^2 = 0\): if \(k < \ell\), forget \(\ell\) first and then \(k\) to get a sign \((-1)^\ell (-1)^k\), or forget \(k\) first then \(\ell\) to get \((-1)^k (-1)^{\ell - 1}\) due to the shift. So these contributions cancel.

Topics:

• General vanishing (Serre 1 and 2)
• Čech cohomology
• Riemann-Roch and Serre duality
• Advanced vanishing (e.g. Kodaira vanishing)

Setup: \(X\) and \({\mathcal{F}}\in {\mathsf{Sh}}(X; {\mathsf{Ab}}{\mathsf{Grp}})\), an open cover \({\mathcal{U}}\rightrightarrows X\). We defined the Čech complex: \begin{align*} {\check{C}}^p({\mathcal{U}}; {\mathcal{F}}) = \bigoplus _{i_1 < \cdots < i_p} {\mathcal{F}}(U_{i_1, \cdots, i_p}) ,\end{align*} which had certain differentials.

More generally, we can just assume that all intersections of affines are affine, and instead there is a spectral sequence. This can fail if \(X\) is not separated, e.g. \(X \mathrel{\vcenter{:}}={\mathbb{A}}^2 { \displaystyle\coprod_{{\mathbb{A}}^2\setminus\left\{{0}\right\}} } {\mathbb{A}}^2\) where the intersection \({\mathbb{A}}^2\setminus\left\{{0}\right\}\) is not affine. Recall that \(X\) is separated iff \(X \xhookrightarrow{\Delta_X} X{ {}^{ \scriptscriptstyle{ \underset{\scriptscriptstyle {X} }{\times} }^{2} } }\) is closed.

A combinatorial device to keep track of monomials: let \(X={\mathbb{P}}^2\), and build simplices which track which monomials are allowed to be negative.

See Hartshorne for a description of how to encode this as a simplicial set:

As a result, we can compute \begin{align*} \dim H^0({\mathbb{P}}^n; {\mathcal{O}}_{{\mathbb{P}}^n}(d)) = {n+d\choose n} = {n+d\choose d} \end{align*} by counting monomials using a stars and bars argument. Moreover \begin{align*} \dim H^n({\mathbb{P}}^n; {\mathcal{O}}(d)) = \dim H^0({\mathbb{P}}^n; {\mathcal{O}}(n-1-d)) = \dim H^0({\mathbb{P}}^n; {\mathcal{O}}(K) \otimes{\mathcal{O}}(d)^{-1}) \end{align*} where the canonical class of \({\mathbb{P}}^n\) is given by \({\mathcal{O}}(K_{{\mathbb{P}}^n}) = {\mathcal{O}}(-n-1)\).

Recall that this LES arises from \begin{align*} 0 \to C^n(\tilde X) \to C^n(X)\to C^n(X, \tilde X)\to 0 .\end{align*}

When does this trick work? For any pair \((X, L)\) with \(L\in {\operatorname{Pic}}X\) where the sections are \({\mathbb{Z}}^{n+1}{\hbox{-}}\)graded where each graded piece is dimension at most 1. These are referred to as multiplicity-free. Examples: toric varieties:

Let \(X\in \mathop{\mathrm{Proj}}{\mathsf{Var}}_{/ {k}}\) and \(F \in {\mathsf{Coh}}({\mathsf{{\mathcal{O}}_X}{\hbox{-}}\mathsf{Mod}})\). By Grothendieck, \({ {H}^{\scriptscriptstyle \bullet}} (X; F)\) is supported in degrees \(0 \leq d \leq \dim X\) and \(h^i = \dim_k H^d(X; F) < \infty\) for all \(d\).

What this formula means: for \(X\) smooth projective, there is a Chow ring \(A^*(X) = \bigoplus _{0\leq i \leq \dim X} A^i(X)\) where \(A^i\) is analogous to \(H^{2i}_{\mathrm{sing}}(X; {\mathbb{C}})\). These are often different, but sometimes coincide (which can only happen if odd cohomology vanishes). For curves, these differ, and \(A^1(X) \cong {\operatorname{Pic}}(X)\) which breaks up as a discrete part (degree) and continuous part (Jacobian). Define \(A^i(X) \mathrel{\vcenter{:}}={\mathbb{Z}}[C_i]\sim\) where \(C_i\) are codimension \(i\) algebraic cycles (subvarieties) and we quotient by linear equivalence. Recall that for divisors, \(D_1\sim D_2\) if \(D_1-D_2\) is the divisor of zeros/poles of a rational functions. More generally, for \(Z\) of codimension \(i\) and \(Z \xrightarrow{f} X\), consider \(f_* D_1 \sim f_* D_2\) in order to define linear equivalence.

The integration map: \begin{align*} \int_X: A^{\dim X}(X) &\to {\mathbb{Z}}\\ \sum n_i p_i &\to \sum n_i .\end{align*}

There are two non-homogeneous polynomials \({\mathrm{ch}}(F)\) and \(\operatorname{Td} ({\mathbf{T}}_X)\) in \(A^*(X)\otimes_{\mathbb{Z}}{\mathbb{Q}}\), and the formula for Riemann-Roch says to multiply and extract only the top-dimensional component, i.e. take \(\deg({\mathrm{ch}}(F) \operatorname{Td} ({\mathbf{T}}_X))_{\dim X}\). This is very computable!

The Chern character of \(F\) is additive on SESs, i.e. \(0\to A\to B\to C\to 0\) yields \({\mathrm{ch}}(B) = {\mathrm{ch}}(A) + {\mathrm{ch}}(C)\).

Setup: \(X\in \mathop{\mathrm{Proj}}{\mathsf{Var}}_{/ {k}} , {\mathcal{F}}\in {\mathsf{Sh}}(X; {\mathsf{Ab}}{\mathsf{Grp}})\). What is \(H^0(X; {\mathcal{F}})\)? Note that if \begin{align*} \chi(X; {\mathcal{F}}) \mathrel{\vcenter{:}}=\sum_k (-1)^k h^k(X; {\mathcal{F}}) ,\end{align*} if \(\tau_{\geq 1} { {H}^{\scriptscriptstyle \bullet}} (X;{\mathcal{F}}) = 0\) then this \(\chi(X;{\mathcal{F}}) = h^0(X; {\mathcal{F}})\). By Serre duality, \(h^n(X;{\mathcal{F}}) = h^0(\omega_X \otimes{\mathcal{F}}^{-1})\) which holds if \(X\) is Gorenstein, e.g. a locally complete intersection.

Recall that \({\mathcal{O}}_X(D)(U) = \left\{{\phi {~\mathrel{\Big\vert}~}(\phi) + D \geq 0}\right\}\). Note that if \({\mathcal{F}}= {\mathcal{O}}(D)\) then \(h^0(X;{\mathcal{F}})\neq 0 \iff D\sim D'\) where \(D' > 0\) is effective.

If \(D\sim D'\) where \(-D' > 0\) is effective, then \(h^0(X; {\mathcal{O}}(D)) = 0\). Note that if \(D \subseteq X \subseteq {\mathbb{P}}^N\) is projective, take \(H \subseteq {\mathbb{P}}^N\) and \({\mathcal{O}}_{{\mathbb{P}}^N}(1) = {\mathcal{O}}_{{\mathbb{P}}^N}(H)\) and intersect to obtain \(D \cdot H^{n-1} = \deg D\).

A note on the proof: uses Deligne-Illusie and liftability from Witt vectors. This liftability holds for all curves, all K3s, and some Calabi-Yau threefolds.

For curves, \(h^1(X; \omega_X(L)) = h^0(-L)\).

Say \(X\) has klt singularities (Kawamata log terminal) iff there exists a projective morphism \(Y \xrightarrow{f} X\) with \(Y\supseteq\cup_i D_i\) with each \(D_i\) snc, and \(f^* K_X = K_Y + \Delta\). Generally \(Y\) is smooth and \(f\) is a resolution.

A note on the MMP: take \(X_0\) a variety, produce a variety \(X\) with \(K_X\) nice, e.g. \(-K_X > 0\) or \(K_X \geq 0\) numerically. At each stage, contract a curve (the result is a \(-1\) curve) are perform a flip. So if \(C \in X_0\), produce \(X_0 \to X_1\) with \(CK_X < 0\).

This recovers the Leray spectral sequence via \({\mathsf{Sh}}(X; {\mathsf{Ab}}{\mathsf{Grp}}) \xrightarrow{f_*} {\mathsf{Sh}}(Y; {\mathsf{Ab}}{\mathsf{Grp}}) \xrightarrow{{{\Gamma}\qty{Y; {-}} }} {\mathsf{Ab}}{\mathsf{Grp}}\), where the composition is \({{\Gamma}\qty{X; {-}} }\). Note that injective sheaves are flasque, and pushforwards of flasque sheaves are again flasque. Why flasque implies injective:

Link to Diagram

Recall that cohomology vanishes above the dimension of a Noetherian space. The analog for pushforward involves the relative dimension.

General setup:

• \(d_r: E_r^{p, q} \to E_r^{p+r, q-r+1}\) (down and to the right) moves between diagonals.

  • For a fixed \(p, q\), all differentials out of \(E_{p, q}\) land on the same diagonal.

• \(E_{r+1} = H(E_r, d_r)\).

• Letting \(E_n = \bigoplus_{p+q=n} E_\infty^{p, q}\), there is a descending filtration \({ {{\operatorname{Fil}}}_{\scriptscriptstyle \bullet}} E_n\) such that \({\mathsf{gr}\,}_p { {{\operatorname{Fil}}}_{\scriptscriptstyle \bullet}} E_n \mathrel{\vcenter{:}}={\operatorname{Fil}}_p E_n/ {\operatorname{Fil}}_{p+1} E_n = E_\infty^{p, n-p}\).

• Extension problem: \({\mathbb{Z}}\supseteq 2{\mathbb{Z}}\supseteq 0\) where \({\mathsf{gr}\,}_1 = C_2\) and \({\mathsf{gr}\,}_1 \cong {\mathbb{Z}}\), but another group and filtration may have the same associated graded, e.g. \({\mathbb{Z}}\oplus C_2 \supseteq{\mathbb{Z}}\supseteq 0\).

• Double complexes naturally arise by taking an injective resolution \(A \leftleftarrows { {J}^{\scriptscriptstyle \bullet}}\) and individually resolving the pieces by \(J^n \leftleftarrows C^{n, \bullet}\). Writing \({ \operatorname{Tot} }(C^{p, q})_n \mathrel{\vcenter{:}}=\oplus_{p+q=n} C^{p, q}\), there are maps \(A\to C^{0,0} \to (C^{0,1} \oplus C^{1, 0}) \to \cdots\) by summing horizontal and vertical differentials. Using the sign trick makes this a differential (multiply the vertical differentials in every even column by \(-1\)).

• There are spectral sequences \begin{align*} E_1{p, q} &= H^p(C^{\bullet, q}, d_h) \Rightarrow H^{p+q}( { \operatorname{Tot} }( C^{\bullet, \bullet} ) ) \\ E_1{p, q} &= H^q(C^{p, \bullet}, d_v) \Rightarrow H^{p+q}( { \operatorname{Tot} }( C^{\bullet, \bullet} ) ) .\end{align*}

  • Why this is useful: resolve \(A\) by \(J\) which are not necessarily injective, and resolve each \(J^n\) by injectives, then \({ \operatorname{Tot} }\) is now an injective resolution.

Recall that for spectral sequences, the diagonal entries \(p+q=n\) are the successive quotients in a filtration on \(E^n\mathrel{\vcenter{:}}={ \operatorname{Tot} }(E_\infty^{\bullet, \bullet})_n\). Kodaira vanishing: for the original argument, go to characteristic \(p\) and look at liftability.

Recall the definition of a double complex: \((C^{\bullet, \bullet}, d_h, d_v)\) where each row is a complex for \(d_h\) and each column for \(d_v\), and each square skew-commutes. Note that the sign trick does not change the cohomology. The totalized complex is is \(({ \operatorname{Tot} }(C), {{\partial}})\) where \(C^n \mathrel{\vcenter{:}}=\bigoplus _{p+q=n} C^{p, q} \xrightarrow{{{\partial}}} C^{n+1} \mathrel{\vcenter{:}}=\bigoplus_{p+q=n+1} C^{p, q}\) and the differential is constructed from \(C^{p, q} \xrightarrow{d_h \oplus d_v} C^{p+1, q} \oplus C^{p, q+1}\). There is a descending filtration \({ {{\operatorname{Fil}}}_{\scriptscriptstyle \bullet}} { \operatorname{Tot} }(C)\) where \({\operatorname{Fil}}_n { \operatorname{Tot} }(C) = \tau_{\geq n, \bullet} { \operatorname{Tot} }(C) = \bigoplus _{p\geq n} C^{p, q}\), which is the double complex obtained by truncating all columns to the left of column \(n\).

Given \({\operatorname{Fil}}A\) a descending filtration, define \({\mathsf{gr}\,}_i A \mathrel{\vcenter{:}}={\operatorname{Fil}}_i A/ {\operatorname{Fil}}_{i+1} A\). Convention: everywhere we’ll set \(p+q\mathrel{\vcenter{:}}= n, p = n-q\), etc.

This results in a collection of short exact sequences: \begin{align*} 0 \to {\operatorname{Fil}}_{i+1} A\to {\operatorname{Fil}}_i A \to {\mathsf{gr}\,}_i A \to 0 .\end{align*}

Our main example: a double complex \({ {C}^{\scriptscriptstyle \bullet, \bullet}}\) with \({ {A}^{\scriptscriptstyle \bullet}} \mathrel{\vcenter{:}}= { {{ \operatorname{Tot} }}^{\scriptscriptstyle \bullet}} { {C}^{\scriptscriptstyle \bullet, \bullet}}\) with \(A^n \mathrel{\vcenter{:}}=\oplus _{p+q=n} C^{p ,q}\) and differentials \({{\partial}}= (d_v, d_h)\) producing skew-commuting squares. The main question is computing \(H^*(A)\).

Each \(A^n\) is a filtration \({\operatorname{Fil}}A^n\) where \({{\partial}}{\operatorname{Fil}}^i A^n \subseteq {\operatorname{Fil}}^{i+1} A^n\). The filtration is defined by \begin{align*} {\operatorname{Fil}}^{p_0} A^n = \bigoplus _{p+q=n, p\geq p_0} C^{p, q} ,\end{align*} taking everything to the right of column \(p_0\). The claim is that this induces a filtrations on \(Z^n(A), B^n(A), H^n(A)\) (cycles, boundaries, and homology). One can restrict the differential on \({ {A}^{\scriptscriptstyle \bullet}}\) to \({\operatorname{Fil}} { {A}^{\scriptscriptstyle \bullet}}\); note that cycles \(Z_n\mapsto 0\) and boundaries are the image and we’re taking cycles mod boundaries. Writing \({\operatorname{Fil}}^p Z^n \mathrel{\vcenter{:}}={\operatorname{Fil}}^p A^n \cap Z_n\) and similarly for \(B^n, H^n\), one gets a filtration \({\operatorname{Fil}}H({\operatorname{Fil}}^p A)\) on \(H({\operatorname{Fil}}^p A)\). This yields \begin{align*} E_\infty^{p, q} = {\mathsf{gr}\,}_p H^n = {\operatorname{Fil}}^p H^n / {\operatorname{Fil}}^{p+1} H^n .\end{align*} If all of the SESs split, then \(H^n = \bigoplus _{p+q=n} E^{p, q}_\infty\).

Set

• \(E_0^{p, q} \mathrel{\vcenter{:}}= C^{p, q}\)
• \(E_1^{p, q} = H^n( C^{p, \bullet}, d_v )\).
• \(E_2^{p, q} = H^n( E_1^{p, q}, d_v) = H^*(\cdots\to H^{n-1}C^{p, \bullet} \to H^n(C^{p, \bullet}) \to H^{n+1} C^{p+1,\bullet}\to \cdots )\).

What are the cycles in \(E_0\)? To map to zero under the total differential \({\partial}\), things emanating from column \(p\) must go to zero, and for the columns \(p+k\), images under \(d_h^{p+k, \ell}\) must cancel with images under \(d_h^{p+k+1, \ell-1}\). Define the approximate homology \begin{align*} {\operatorname{Fil}}^p H^{\approx}_{p \pm r} = {{\partial}^{-1}({\operatorname{Fil}}^{p+r} A^{n+1} ) \over {\partial}({\operatorname{Fil}}^{p-r+1} A^{n-1} ) } .\end{align*} Note that this increases the number of allowed cycles and decreases the number of allowed boundaries. Then \(E_r^{p, q} = {\mathsf{gr}\,}_p H^n_{p \pm r}\).

Note that the statement is not the \(E_r\) is computed as \(H^*(E_{r-1})\); instead there is a formula for \(E_r^{p, q}\) for all \(r,p,q\) a priori, and it is a property that taking homology of pages computes this.

Claim: \({\mathsf{gr}\,}_p H^n_{p\pm 0} = C^{p, q}\). Check that \begin{align*} {\operatorname{Fil}}^{p_0} H^n_{p\pm 0} = {\bigoplus _{p+q=n, p\geq p_0} C^{p, q} \over d\qty{ \bigoplus _{p+q=n-1, p\geq p_0 + 1} C^{p, q} } } .\end{align*}

A filtered complex:

Link to Diagram

This yields \begin{align*} H^n_{p\pm r} &= {A^n \cap d^{-1}({\operatorname{Fil}}^{p+r} A^{n+1} ) \over A^n \cap d({\operatorname{Fil}}^{p-r+1} A^{n+1} ) } \\ \\ H^n_{p \pm \infty} &= {A^n \cap d^{-1}(0) \over A^n \cap d(A^{n+1}) } .\end{align*}

Notation: write \begin{align*} {}^n E^p_r \mathrel{\vcenter{:}}= E_{r}^{p, q} = {\mathsf{gr}\,}^p H^n_{p\pm r} = { {\operatorname{Fil}}^p A^{n} \cap d^{-1}\qty{ {\operatorname{Fil}}^{p+r} A^{n+1} } \over {\operatorname{Fil}}^{p+1} A^n \cap d^{-1}({\operatorname{Fil}}^{p+r} A^{n+1} ) + {\operatorname{Fil}}^p A^{n} \cap d^{-1}\qty{{\operatorname{Fil}}^{p-r+1} A^{n-1} } } .\end{align*}

The main properties:

• \(d_r: E_r^{p, q} \to E_r^{p+r, q-r+1}\)
• \(H( { {E_r}^{\scriptscriptstyle \bullet, \bullet}} , d_r) = { {E_{r+1}}^{\scriptscriptstyle \bullet, \bullet}}\).

Note that \({}^n E^p_r \xrightarrow{ d_r} {}^{n+1}E^{p+r}_r\), so \begin{align*} { {\operatorname{Fil}}^p A^{n} \cap d^{-1}\qty{ {\operatorname{Fil}}^{p+r} A^{n+1} } \over {\operatorname{Fil}}^{p+1} A^n \cap d^{-1}({\operatorname{Fil}}^{p+r} A^{n+1} ) + {\operatorname{Fil}}^p A^{n} \cap d^{-1}\qty{{\operatorname{Fil}}^{p-r+1} A^{n-1} } } \xrightarrow{d_r} { {\operatorname{Fil}}^{p+r} A^{n+1} \cap d^{-1}\qty{ {\operatorname{Fil}}^{p+2r} A^{n+2} } \over {\operatorname{Fil}}^{p+r+1} A^{n+1} \cap d^{-1}({\operatorname{Fil}}^{p+2r} A^{n+2} ) + \cdots } ,\end{align*} and \(d_r^2 = 0\) since the first denominator above appears as the next numerator.

An application: consider a 2-step resolution \(0 \to A \to J^0 \to J^1\), and take injective resolutions of each \(J^i\) to form an \(E_0\):

Link to Diagram

Then \(0 \to A\to { {{ \operatorname{Tot} }}^{\scriptscriptstyle \bullet}} ( { {A}^{\scriptscriptstyle \bullet, \bullet}} )\) is exact, i.e. this is an injective resolution of \(A\). Take vertical cohomology to get \(E_1\):

Link to Diagram

Since no functor has been applied, we obtain the follow \(E_2\) after taking horizontal cohomology:

Link to Diagram

So \(H^n({ \operatorname{Tot} }I) = A[0]\).

Let \(F\in \mathsf{Cat}(A, B)\) be additive left-exact, then \({\mathbb{R}}^n FA = H^n(F{ \operatorname{Tot} } { {I}^{\scriptscriptstyle \bullet, \bullet}} )\) for \(0\to A\to I\) a biresolution as above. Define \(E_0 = FI\), then \(E_1^{p, q} = {\mathbb{R}}^q F J^p\).

Using two spectral sequences for a single bicomplex: given \({ {C}_{\scriptscriptstyle \bullet, \bullet}}\), \begin{align*} E_2^{p, q} = H_h^{p} H_v^q C^{p, { \scriptscriptstyle \bullet} } &\Rightarrow H^n({ {{ \operatorname{Tot} }}_{\scriptscriptstyle \bullet}} { {C}_{\scriptscriptstyle \bullet, \bullet}} ) \\ E_2^{p, q} = H^q_v H^p_h C^{{ \scriptscriptstyle \bullet}, q} &\Rightarrow H^n({ {{ \operatorname{Tot} }}_{\scriptscriptstyle \bullet}} { {C}_{\scriptscriptstyle \bullet, \bullet}} ) .\end{align*}

Grothendieck spectral sequences: for \(\mathsf{A} \xrightarrow{F} \mathsf{B} \xrightarrow{G} \mathsf{C}\), form the composite \(\mathsf{A} \xrightarrow{GF} \mathsf{C}\) to obtain \begin{align*} E_2^{p, q} = {\mathbb{R}}^p G {\mathbb{R}}^q F A \Rightarrow{\mathbb{R}}^{p+q} GFA ,\end{align*} provided \(F\) sends injectives to \(G{\hbox{-}}\)acyclics. This comes from running the two spectral sequences above, where one collapses onto a single row.

Recall how to construct derived functors. It is advantageous to embed \(\mathsf{C} \hookrightarrow\mathsf{Ch}\mathsf{C}\) and resolve by nicer objects. A complex contains strictly more information than homology: e.g. \(0\to {\mathbb{Z}}\xrightarrow{\cdot 2} {\mathbb{Z}}\to 0\) and \(0 \to {\mathbb{Z}}\hookrightarrow{\mathbb{Z}}\oplus {{\mathbb{Z}}\over 2{\mathbb{Z}}}\to 0\) have isomorphic homology but aren’t isomorphic as complexes.

The basic morphisms in \({\mathbb{D}}\mathsf{C}\) are given by usual chain maps \(f:A\to B\), and if \(f\) is a quasi-isomorphism we formally add inverses \(X_f: B\to A\). A general morphism is a sequence of morphisms \(\bullet \to \bullet \to \cdots \to \bullet\) where we quotient by

• \(\bullet \xrightarrow{f} \bullet \xrightarrow{g} \bullet \sim \bullet \xrightarrow{g f} \bullet\)
• \(A \xrightarrow{f} B \xrightarrow{X_f} A \sim A \xrightarrow{\operatorname{id}} A\)
• \(B \xrightarrow{X_f} A \xrightarrow{f} B \sim B \xrightarrow{\operatorname{id}} B\).

One would like a calculus of fractions, so define:

The set \(S\) of quasi-isomorphisms in \(\mathsf{Ch}\mathsf{A}\) is localizing.

Note that we can take

• \(\mathsf{Ch}\mathsf{C}:\) all complexes,
• \(\mathsf{Ch}^+\mathsf{C}:\) complexes bounded from below,
• \(\mathsf{Ch}^-\mathsf{C}:\) complexes from above,
• \(\mathsf{Ch}^b\mathsf{C}:\) complexes from above and below.

These yield derived categories \({\mathbb{D}}\mathsf{C}, {\mathbb{D}}^+\mathsf{C}, {\mathbb{D}}^-\mathsf{C}, {\mathbb{D}}^b\mathsf{C}\). Note: frequently \({\mathbb{D}}\mathsf{C}\) actually means \({\mathbb{D}}^+\mathsf{C}\) in the literature. When \({\mathbb{D}}^b\mathsf{C}\) is used: if \({\mathcal{F}}\in {\mathsf{Coh}}(X)\) and \(X\) is projective, which corresponds to a graded module which (by Hilbert) has a finite resolution.

One can similarly define homotopy categories \({\mathsf{ho}}\mathsf{Ch}\mathsf{C}, {\mathsf{ho}}\mathsf{Ch}^+\mathsf{C}, {\mathsf{ho}}\mathsf{Ch}^-\mathsf{C}, {\mathsf{ho}}\mathsf{Ch}^b\mathsf{C}\) with \({\operatorname{Ob}}( {\mathsf{ho}}\mathsf{Ch}\mathsf{C} ) \mathrel{\vcenter{:}}={\operatorname{Ob}}(\mathsf{Ch}\mathsf{C})\) and \(\mathop{\mathrm{Mor}}({\mathsf{ho}}\mathsf{Ch}\mathsf{C}) \mathrel{\vcenter{:}}=\mathop{\mathrm{Mor}}({\mathsf{ho}}\mathsf{Cat}{C})/\sim\) where \(\sim\) denotes chain homotopy equivalence.

Generally there is a functor \({\mathsf{ho}}\mathsf{Ch}\mathsf{A} \hookrightarrow{\mathbb{D}}\mathsf{A}\) since chain homotopy equivalences induce isomorphisms on homology (where we apply the universal property of \({\mathbb{D}}\mathsf{A}\)) There is also a functor \({\mathbb{D}}\mathsf{A}\to {\mathsf{ho}}\mathsf{Ch}\mathsf{A}\) where \(A\mapsto { \operatorname{Tot} }( { {I}^{\scriptscriptstyle \bullet, \bullet}} )\) is a quasi-isomorphism.

Recall that for \(S\subseteq \mathop{\mathrm{Mor}}(\mathsf{C})\), there is a localized category \(\mathsf{C} \left[ { \scriptstyle { {S}^{-1}} } \right]\) whose morphisms are chains \(s_0^{-1}\circ f_0 \circ s_1^{-1}\circ \cdots\) modulo an equivalence, and if \(S\) is localizing then

• Morphisms are single roofs (i.e. we can collect the product fraction involving \(s_i, f_i\) into a single fraction).
  • Note that roofs can be multiplied, and roofs are equivalent when they admit a common roof:

Link to Diagram

• Morphisms are equivalent when they admit a common roof:

Link to Diagram

• If \(\mathsf{C}\in {\mathsf{Add}}\mathsf{Cat}\) then \(\mathsf{C} \left[ { \scriptstyle { {S}^{-1}} } \right] \in {\mathsf{Add}}\mathsf{Cat}\), where the calculus of fractions behaves as in ring localization: \({f_1\over s} + {f_2\over s} = {f_1+f_2\over s}\).
• If \(I \leq \mathsf{C}\) is a full subcategorya nd \(S\) is compatible, i.e. \(S \cap\mathop{\mathrm{Mor}}(I)\) is localizing, then \(I \left[ { \scriptstyle { {S}^{-1}} } \right] \leq \mathsf{C} \left[ { \scriptstyle { {S}^{-1}} } \right]\) is a full subcategory.

\(\mathsf{Ch}\mathsf{C}\) with \(S\) quasi-isomorphisms yields \({\mathbb{D}}\mathsf{A}\mathrel{\vcenter{:}}=\mathsf{C} \left[ { \scriptstyle { {S}^{-1}} } \right]\).

An application: for \(F\in {\mathsf{Ab}}\mathsf{Cat}(\mathsf{A}, \mathsf{B})\) additive (with no left/right exactness conditions), there is a derived functor \({\mathbb{D}}F\in \mathsf{{\mathbb{D}}^+ \mathsf{A}, {\mathbb{D}}^+ \mathsf{B}}\) if \(\mathsf{A}\) has enough injectives. Note that \({\mathbb{D}}\mathsf{A}\) is never abelian but admits a triangulated structure.

For \(X\mathop{\mathrm{Proj}}{\mathsf{Var}}_{/ {k}}\), recall \({\mathsf{K}}_0 X \mathrel{\vcenter{:}}={\mathsf{K}}_0 {\mathsf{Coh}}X\) where \([b] = [a] + [c]\) for \(0\to a\to b\to c\), and \({\mathsf{K}}^0 X \mathrel{\vcenter{:}}={\mathsf{K}}^0 {\mathsf{Sh}}^{{\mathsf{loc}}{\mathrm{free}}}(X)\). If \(X\) is smooth, these are isomorphic, but generally they are not if \(X\) is singular. In general, \({\mathbb{D}}X \mathrel{\vcenter{:}}={\mathbb{D}}^+ {\mathsf{Coh}}X\) replaces \({\mathsf{K}}_0(X)\), and \({\mathbb{D}}^+ {\mathsf{Sh}}^{{\mathsf{loc}}{\mathrm{free}}}(X)\) replaces \({\mathsf{K}}^0 X\).

Although these do not have SESs, there are distinguished triangles for which any morphism \(X\to Y\) can be completed to \(X\to Y\to Z\to { \Sigma^{\scriptstyle[1]} X }\). This can be accomplished using mapping cylinders/cones:

See tilting of complexes, exceptional sequences.

Any distinguished triangle \(X \xrightarrow{f} Y \to Z\to X[1]\) in \(\mathbf{D} {\mathsf{A}}\) is isomorphic to a triangle of the form \(X\to { \mathrm{Cyl} }(f) \to { \mathrm{Cone} }(f)\to X[1]\). For \(\mathsf{Ch}\mathsf{A}\), define \(T^nA \mathrel{\vcenter{:}}= A[n]\), so \((T^nA)_k = A[n]_k = A_{n+k}\), and \({{\partial}}_{TA} \mathrel{\vcenter{:}}=(-1)^n {{\partial}}_A\).

Recall that for \(X \xrightarrow{f} Y\), \(\operatorname{cone}(f) \approx X[1] \oplus Y\) and \({ \mathrm{Cyl} }(f) \approx X \oplus X[1] \oplus Y\) with differential \begin{align*} d_{{ \mathrm{Cyl} }(f)} \mathrel{\vcenter{:}}= \begin{bmatrix} d_X & -1 & \\ & d_X[1] & \\ & f[1] & d_Y \end{bmatrix} \curvearrowright{\left[ {x_i, x_{i+1}, y_i} \right]} \in { \mathrm{Cyl} }(f)^i .\end{align*}

Note: I use \(\approx\) above because these formulas hold levelwise, but the SESs they fit into may not be split exact, so \(\operatorname{cone}(f), { \mathrm{Cyl} }(f)\) may not such direct sums.

There are related exact triples, here the first and second rows:

Link to Diagram

Here \(\beta\alpha = \operatorname{id}_Y\) and \(\alpha\beta \simeq\operatorname{id}_{{ \mathrm{Cyl} }(f)}\).

Some major work in this area:

• Beilinson-Gelfand-Gelfand (BGG)
• Bendel-Kapranov
• Mukai
• Bendal-Orlov
• Orlov
• Kutznatsov
• Kontsevich, Fukaya (homological mirror symmetry)
• Beilinson-Bernstein-Gabber-Deligne on perverse sheaves
• Bridgeland

Results:

• BGG: \(\mathbf{D}^b ({\mathsf{Coh}}{\mathbb{P}}^n) \cong \mathbf{D}^b ({\mathsf{R}{\hbox{-}}\mathsf{Mod}})\) for \(R\) a certain ring.
• BK, K: same for quadrics and grassmannians.

Recall that given \(\mathsf{T}\in{\mathsf{triang}}\mathsf{Cat}\) with an exceptional collection \(\left\{{{\mathcal{E}}_i}\right\}\), they generate a triangulated subcategory \(\left\langle{{\mathcal{E}}_i}\right\rangle \leq \mathsf{T}\). It turns out that \(\left\langle{{\mathcal{E}}_i}\right\rangle\cong {\mathsf{R}{\hbox{-}}\mathsf{Mod}}\) for \(R = \bigoplus \mathop{\mathrm{End}}{\mathcal{E}}_i\). Beilinson produces a collection \(\left\{{{\mathcal{O}}, \Omega^1,\cdots, \Omega^{n-1}}\right\}\), but an easier alternative is \(\left\{{{\mathcal{O}}, {\mathcal{O}}(1), \cdots, {\mathcal{O}}(n-1)}\right\}\). If the collection is full, then \(\mathsf{T} \cong \left\langle{{\mathcal{E}}_i}\right\rangle\). As an alternative to \(R\), one can take the corresponding quiver: make a directed graph \({\mathcal{E}}_1\to{\mathcal{E}}_2\to\cdots\) where each node has \(\mathop{\mathrm{End}}{\mathcal{E}}_i\) attached and each edge \({\mathcal{E}}_i\to{\mathcal{E}}_j\) is assigned \(\oplus_n \mathop{\mathrm{Hom}}({\mathcal{E}}_i, {\mathcal{E}}_j[n])\). So the derived category corresponds to representations of this quiver.

Example: for \({\mathbb{P}}^1\), one obtains the following quiver:

Link to Diagram

For \(\mathsf{C} \in {\mathsf{triang}}\mathsf{Cat}\), one can take \({\operatorname{HH}}\mathsf{C}\). For \(\mathsf{C} = \mathbf{D} (X)\), the \({\operatorname{HH}}_0 D(X) \cong {\mathbb{Z}}^n \oplus A\) as a group, for \(A\) some finite torsion group. If one has a full exceptional collection, then \(A = {\operatorname{HH}}_0( \left\langle{{\mathcal{E}}_1, \cdots, {\mathcal{E}}_n}\right\rangle^\perp )\). As a corollary, the length \(m\) of an exceptional collection satisfies \(m\leq \operatorname{rank}_{\mathbb{Z}}{\operatorname{HH}}_0 \mathbf{D} (X)\).

For surfaces of general type, a special Godeaux surface produces a counterexample. There is a much easier counterexample coming from a Burnist (?) surface – generally fake \({\mathbb{P}}^2\), fake Fanos, etc. See A-Orlov, Orlov-Gorheaise, Katgerov-?, ??

Phantoms: categories with zero \({\operatorname{HH}}\), so no full exceptional collections.

Having \(-K_X\) ample yields Fano varieties, and \(K_X\) ample yields general-type surfaces.

How to construct the equivalence \(\mathbf{D} (A) \to \mathbf{D} (A {}^{ \vee })\): take the Fourier-Mukai transform. Use the Poincare bundle \(P_A \to A\times A {}^{ \vee }\), and construct the functor as a push-pull over the span \((A \leftarrow_{p_1} A\times A {}^{ \vee }\to_{p_2} A {}^{ \vee })\), so \begin{align*} {\mathcal{F}}\mapsto (p_2)_* \qty{ (p_1)^* {\mathcal{F}}\otimes P_A) } .\end{align*}

The next in line: K3 surfaces. An easy example: take Kummer surfaces, so \(A\to A/\pm 1\) and then blow up the 16 nodes.

Recall that a collection \({\mathcal{E}}_i\) is exceptional iff \([{\mathcal{E}}_j, {\mathcal{E}}_i[n]] = 0\) if \(n\geq0\) and \(j > i\). If there exists a full exceptional collection, \(\mathbf{D} {X} \cong \mathbf{D} {(} {\mathsf{R}{\hbox{-}}\mathsf{Mod}})\) for some \(R\). Recall that a variety is Fano if \(-K_X\) is ample.

Let \begin{align*} X_3 \mathrel{\vcenter{:}}= V(f_3(x_0,\cdots, x_5)) \subseteq {\mathbb{P}}^5 ,\end{align*} then which \(X_3\) are rational? Note that \(K_{X_3} = {\mathcal{O}}(-6+3) = {\mathcal{O}}(-3)\). Kuznatsov shows that \(H^i({\mathcal{O}}_X) = {\mathbb{C}}[0]\) and \(\operatorname{Ext} ^i({\mathcal{L}}, {\mathcal{L}}) = H^i({\mathcal{O}}_X)\). One could look for exceptional collections of line bundles, so \(\operatorname{Ext} ^i({\mathcal{L}}_j, {\mathcal{L}}_i) = H^m({\mathcal{L}}_i \otimes{\mathcal{L}}_j^{-1}) = 0\) for all \(m\). On \({\mathbb{P}}^n\), take \({\mathcal{O}}(-k)\) for \(1\leq k\leq n\) since \(K_{{\mathbb{P}}^n} = {\mathcal{O}}(-n-1)\). For \(X_3\), there is enough vanishing that \({\mathcal{O}}, {\mathcal{O}}(1), {\mathcal{O}}(2)\) are exceptional (everything below the index 3 from above). Kuznatsov shows that the “Kuznatsov component” \(K = \left\langle{{\mathcal{O}}, {\mathcal{O}}(1), {\mathcal{O}}(2)}\right\rangle^\perp\) is a Calabi-Yau category of dimension 2.

If \(Y\) is a Calabi-Yau variety of dimension \(n\), so \(K_Y = 0\), there is a Serre functor \begin{align*} S: \mathbf{D}^b (Y) &\to \mathbf{D}^b (Y) \\ F &\mapsto F\otimes\omega_Y[n] .\end{align*} Then (probably) \(S = T^n\), a shift by \(n\). A category is a Calabi-Yau category of dimension \(n\) iff

• It has a Serre functor
• \(S = T^n\)

One can also define fractional dimension using \(S^q\).

A technique due to Clemens-Griffith for cubic threefolds. Let \(X \subseteq {\mathbb{P}}^4\) be a smooth non-nodal curve. Consider the intermediate Jacobian \(J_3(X)\), which is a PPAV for any smooth 3-fold. Basic operations: blowing up a point \(p\) or a curve \(C\), since blowing up a surface is the identity. Blowing up a point: \(J_3(\operatorname{Bl}_p X) = J_3(X)\), so it doesn’t change. For a curve, \(J_3(\operatorname{Bl}_C X) = J(C) \oplus J_3(X)\). As a corollary, if \(X\) is rational then \(J_3(X) = \bigoplus J(C_i)\) for some curves \(C_i\). For non-rationality, show it’s not the Jacobian of a curve by considering the theta divisor.

For 4-folds \(X\), one can now also blow up surfaces. The intermediate cohomology carries a Hodge structure. Conjecture: \(X\) is rational iff its Hodge structure looks like a K3.

Older techniques for checking rationality: see log thresholds, generally birational geometry e.g. due to Manin. E.g. groups of birational automorphism for quartic 4-folds are small. See another approach due to Mumford using torsion in cohomology.

Note that it’s possible for \(\mathsf{A},\mathsf{B}\in {\mathsf{Ab}}\mathsf{Cat}\) to satisfy \(\mathbf{D} {\mathsf{A}} { \, \xrightarrow{\sim}\, }\mathbf{D} {\mathsf{B}}\).

On \(T{\hbox{-}}\)structures: write \(D = \mathbf{D} {\mathsf{A}}\), then there are subcategories

• \(D^{\leq n} = D^{\leq 0}[n]\), complexes such that \(H^{> n} = 0\).
• \(D^{\geq n} = D^{\geq 0}[n]\), complexes such that \(H^{< n} = 0\).

Then \(D^{\geq 0} \cap D^{\leq 0} = A\) is a category equivalent to complexes supported in degree zero, since any such bounded complex is quasi-isomorphic to such a complex. Some properties:

• \(A_0\in D^{<0}\) and \(A_1 \in D^{> 1}\) satisfy \([A_0, A_1] = 0\).
• For all \(C\in \mathbf{D}^b (\mathsf{A})\), there exists \(A_0\to C\to A_1 \to A_0[1]\).

Note that there is a canonical truncation \begin{align*} \tau_{\leq 0}(\cdots \to C^{-1}\xrightarrow{d^0} C^0\to C^1\to \cdots) = (\cdots\to C^{-1}\to \ker d^0 \to 0) .\end{align*}

Take \(X\) a smooth projective curve, let \(D = \mathbf{D}^b ({\mathsf{Coh}}X) = \mathbf{D}^b { {\mathsf{Bun}}\qty{\operatorname{GL}_r} }\). There is a notion of a semistable sheaf (all subsheaves have smaller slopes \(\mu({\mathcal{F}}) \mathrel{\vcenter{:}}=\deg {\mathcal{F}}/\operatorname{rank}{\mathcal{F}}\)) and an HN filtration where the quotients are semistable and the slopes decrease. Bridgeland observed there is a central charge \begin{align*} Z: {\mathsf{Coh}}X &\to {\mathbb{C}}\\ {\mathcal{F}}&\mapsto -\deg {\mathcal{F}}+ i \operatorname{rank}{\mathcal{F}} ,\end{align*} which can be used to recovered the heart \(\mathsf{A} = {\mathsf{Coh}}X\). Idea: vary \(Z\) to get different hearts, and \(\left\{{Z_i}\right\}\) form a complex analytic variety, and one can form a new category of tilted complexes (complexes sitting in two degrees).

• Products: a collection of maps into factors \(Y\to X_i\) is the same as a map \(Y\to \prod X_i\). Products are easy to map into. Products have projections \(\prod X_i \to X_i\).

  • Products are limits.

• Coproducts: a collection of out of factors \(X_i\to Y\) is the same as a map \(\coprod X_i \to Y\). Coproducts are easy to map out of. Coproducts have injections \(X_i \to \coprod X_i\).

  • Coproducts are colimits.

• If \(\mathsf{C}\) has a zero object, there is a canonical map \(\coprod_{i\in I} X_i \to \prod_{j\in I} X_j\) given by assembling maps \(\delta_{ij}\).

• \(\colim_{i\in I}({-})\) is generally not exact, but is exact if the colimit is filtered.

  • In any case, the functor of taking stalks \(({-})_x: {\mathsf{Sh}}(X; {\mathsf{Ab}}{\mathsf{Grp}}) \to {\mathsf{Ab}}{\mathsf{Grp}}\) is always exact.

• Left adjoints/colimits are characterized by morphisms on \(F(x)\), and right adjoints/limits by morphisms into it.

• Why RAPL and LAPC: \begin{align*} [\colim_i L(x_i), {-}] \cong \cocolim_i [L(x_i), {-}] \cong \cocolim_i [x_i, R({-})] \cong [\colim_i x_i, R({-})] = [L(\colim_i x_i), {-}] .\end{align*}

• Right-derived functors are left Kan extensions.

• Colimits are quotients of coproducts and receive maps from objects (i.e. they are cocones). Taking colims is right exact. Limits send maps.

Tor:

• \(\operatorname{Tor}\) commutes with arbitrary direct sums, colimits (direct limits), localization.
• If \(M\) is flat over \(R, \operatorname{Tor}_{i}^{R}(M \otimes A, B) \cong M \otimes_{R} \operatorname{Tor}_{i}^{R}(A, B)\).
• If \(S\) is a flat \(R\)-algebra, \(S \otimes_{R} \operatorname{Tor}_{i}^{R}(A, B) \cong \operatorname{Tor}_{i}^{S}\left(S \otimes_{R} A, S \otimes_{R} B\right)\).
• \(\operatorname{Tor}_1^R(R/I, R/J) \cong {I \cap J \over IJ}\)
• If \(I\) is an \(R{\hbox{-}}\)regular sequence \(I = \left\langle{x_1,\cdots, x_n}\right\rangle\), then \(\operatorname{Tor}_n^R(R/I, M) = (0 :_M I)\) is a colon ideal.
• If \(A\in {\mathsf{R}{\hbox{-}}\mathsf{Mod}}^\flat\) then \(\operatorname{Tor}_{\geq 1}^R(A, B) = 0\).
• \(\operatorname{Tor}(A, B) \cong \operatorname{Tor}(B, A)\).

Ext:

• \(\tau_{\geq 1} { {\operatorname{Ext} }^{\scriptscriptstyle \bullet}} _R(A, B) = 0\) if either \(A\) is injective or \(B\) is projective.
• The Koszul complex for \(k[x,y]\): \(K_x \otimes K_y = 0\to k[x,y]\to k[x,y]{ {}^{ \scriptscriptstyle\times^{2} } }\to k[x,y] \to k \to 0\) where \(K_x = 0\to k[x,y] \xrightarrow{\cdot x} k[x,y] \to 0\).
• \({ {\operatorname{Ext} }^{\scriptscriptstyle \bullet}} _{k[x,y]}(k) = k \oplus \Sigma k{ {}^{ \scriptscriptstyle\times^{2} } } \oplus \Sigma^2 k\).