Syllabus

The syllabus can be found at the syllabus tab.

Notes

Course notes can be found in the tab to the left.

Previous semester

This course is a continuation of the Fall 2021 course Derived Categories I.

Me

If you are interested in a bit about who I am and what I do visit my website.

Quasi-coherent sheaves on projective varieties

Let $R$ be a $\mathbb{N}$-graded commutative ring.

We constructed a functor $$\widetilde{(-)}:\mathrm{GrMod}R\to \mathrm{Mod}X$$ where $X=\mathrm{Proj}R$ and a functor in the other direction $$\begin{array}{rl}\underline{\Gamma }(X,-):\mathrm{Mod}X& \to \mathrm{GrMod}R\\ \mathcal{F}& \mapsto \underset{i\in \mathbb{Z}}{⨁}\Gamma (X,\mathcal{F}(i))\end{array}$$

Let's think a little more about their relationsip. For any $m\in M_l$, we get a map $$\begin{array}{rl}m:R& \to M(l)\\ r& \mapsto rm\end{array}$$ Applying $\widetilde{(-)}$, we get a map $${\mathcal{O}}_X\to \tilde{M}(l)$$ and hence an element of $\Gamma (X,\tilde{M}(l))$. This gives a natural transformation $${\eta }_M:M\to \underline{\Gamma }(X,\tilde{M})$$

Similarly, for any ${\mathcal{O}}_X$-module $\mathcal{F}$, there is a natural transformation $${ϵ}_{\mathcal{F}}:\widetilde{\underline{\Gamma }(\mathcal{F})}\to \mathcal{F}$$ which we describe on the principal opens $U_x$ for $x\in R_i$ with $i>0$. We have $$\widetilde{\underline{\Gamma }(\mathcal{F})}(U_x)=(\underline{\Gamma }(\mathcal{F}{)}_x{)}_0$$ Each element of $(\underline{\Gamma }(\mathcal{F})\_x)\_0$ can be written as $$\frac{f}{x^s}$$ where $f\in \Gamma (X,\mathcal{F}(si))$ and $x\in R_i$. Restricting to $U_x$ we get an element $f{|}_{U_x}\in \Gamma (U_x,\mathcal{F}(si))$.

On $U_x$, we have an isomorphism $1/x^s:\mathcal{F}(is)\to \mathcal{F}$. Thus, we get an element $$\frac{1}{x^s}f{|}_{U_x}\in \Gamma (U_x,\mathcal{F})$$ giving an element of $\mathcal{F}(U_x)$.

Proposition. The natural transformations $\eta$ and $ϵ$ induce an adjunction $\widetilde{(-)}⊣\underline{\Gamma }$.

Under some mild assumptions, we see that $\widetilde{(-)}$ hits all quasi-coherent sheaves on $X$.

Lemma. If $R$ is finitely-generated in degree $1$ as a graded $k$-algebra and $\mathcal{F}$ is quasi-coherent, then ${ϵ}_{\mathcal{F}}$ is an isomorphism.

Recall that for a graded $R$-module $M$, we have its torsion submodule $$\tau (M):=\{m\in M\mid \exists n_0>0,(R_{+}{)}^{n}m=0\forall n\ge n_0\}$$ and we say a $M$ is torsion if $$\tau (M)=M$$ The module is torsion-free if $$\tau (M)=0$$

There is a short exact sequence $$0\to \tau (M)\to M\to M/\tau (M)\to 0$$ Since $$\widetilde{\tau (M)}=0$$ there is a natural map $${\alpha }_M:M/\tau (M)\to \underline{\Gamma }(\tilde{M})$$

Lemma. Assume $R$ is finitely-generated over $k$ in degree $1$ by $x0,\dots ,xn$ and that each $xi$ is not a zero-divisor. Then $\alpha$M is injective

We can describe $\tau (M)$ as $$\tau (M)\cong {\mathrm{colim}}_s\underline{\mathrm{Hom}}(R/(R_{+}{)}^s,M)$$ where $$\underline{\mathrm{Hom}}(M,N):=\underset{i\in \mathbb{Z}}{⨁}\mathrm{Hom}(M,N(i))$$ is the internal Hom for graded modules.

There are also the exact sequences $$0\to (R_{+}{)}^s\to R\to R/(R_{+}{)}^s$$ which gives an exact sequence $$0\to {\mathrm{colim}}_s\underline{\mathrm{Hom}}(R/(R_{+}{)}^s,M)\to M\to {\mathrm{colim}}_s\underline{\mathrm{Hom}}((R_{+}{)}^s,M)$$ similar to the sequence $$0\to \tau (M)\to M\to \underline{\Gamma }(\tilde{M})$$ which leads us to compare $\underline{\Gamma }(\tilde{M})$ and $\mathrm{colim}s\underline{\mathrm{Hom}}((R+{)}^s,M)$

Since $$\cdots \to \widetilde{(R_{+}{)}^{s+1}}\to \widetilde{(R_{+}{)}^s}\to \cdots \to \widetilde{R_{+}}\to \tilde{R}$$ are all isomorphisms, we see that any $$\varphi \in \mathrm{colim}\mathrm{Hom}((R_{+}{)}^s,M(l))$$ induces a well-defined $$\tilde{\varphi }:\mathcal{O}\to \tilde{M}(l)$$ and hence an element of $\underline{\Gamma }(\tilde{M})$.

Let's denote this map by $${\beta }_M:{\mathrm{colim}}_s\underline{\mathrm{Hom}}((R_{+}{)}^s,M)\to \underline{\Gamma }(\tilde{M})$$

Lemma. Assume that $R$ is finitely-generated over $k$ by elements of degree $1$. Then ${\beta }_M$ is an isomorphism for all $M$.

As a corollary, we get short exact sequences.

Corollary. Assume that $R$ is finitely-generated by elements of degree $1$ over $k$. Then, for each injective graded $R$-module $I$ we have a short exact sequence $$0\to \tau (I)\to I\to \underline{\Gamma }(\tilde{I})\to 0$$

Note the map $M\to \underline{\Gamma }(\tilde{M})$ is not always onto.

Example. Take $k[x]$ with $|x|=1$. Then, $$\underline{\Gamma }(\widetilde{k[x]})=k[x,x^{-1}]$$

As consequence, we get a semi-orthogonal decomposition relating graded $R$-modules and quasi-coherent sheaves on $\mathrm{Proj}R$.

Proposition. Assume that $R$ is finitely-generated by elements of degree $1$ over $k$. Then there are semi-orthogonal decompositions $$D^{†}(\mathrm{GrMod}R)=⟨D^{†}({\mathrm{GrMod}}_{\mathrm{Tors}}R),D^{†}(\mathrm{Qcoh}\mathrm{Proj}R)⟩$$ where $†\in \{b,+,-,\varnothing \}$.

Here ${\mathrm{GrMod}}_{\mathrm{Tors}}R$ denotes the abelian subcategory of torsion graded $R$-modules.

Remark.

• One can remove the condition that the finite set of generators of $R$ are of degree $1$ if we work with the quotient stack $[U/\mathbb{G}\_m]$ with the quasi-affine $U=\mathrm{Spec}R\setminus \{R_{+}\}$ instead of $\mathrm{Proj}R$.
• One would like to replace $\mathrm{GrMod}$ and $\mathrm{Qcoh}$ with their finite versions $\mathrm{grmod}$ and $\mathrm{coh}$. This can be done but one needs to truncate the grading.

Exceptional collections and Čech cohomology

We have developed most of the background necessary to describe the derived category $D^b(\mathrm{coh}{\mathbb{P}}_k^n)$ but we require one more important notion.

Definition. Let $\mathcal{T}$ be a $k$-linear triangulated category. An object $E$ is ($k$-)exceptional if $$\mathrm{Hom}(E,E[l])=\{\begin{array}{ll}0& l\ne 0\\ k\cdot 1_E& l=0\end{array}$$

A sequence of objects $E_1,\dots ,E_s$ is called an exceptional collection if

• each $E_i$ is exceptional
• and $$\mathrm{Hom}(E_i,E_j[l])=0$$ for all $l$ and $j<i$.

We say an exceptional colletion is strong if $$\mathrm{Hom}(E_i,E_j[l])=0$$ for all $l\ne 0$.

An exceptional collection is full if it generates $\mathcal{T}$.

Assume we have an abelian category $\mathcal{A}$ and a full exceptional collection for $D^{†}(\mathcal{A})$.

Since full exceptional collections generate, we can apply the results from last semester to get an equivalence $$D(\mathcal{A})\cong \mathrm{perf}A$$ where $$A=\mathbf{R}\mathrm{End}\left(⨁E_i\right)$$

If, in addition, the collection is strong then we can take $$A=\mathrm{End}\left(⨁E_i\right)$$ and $D^{†}(\mathcal{A})$ is equivalent to the category of perfect complexes of modules over an honest $k$-algebra. This algebra is usually non-commutative.

We can now state a fundamental result of Beilinson.

Theorem. For any $i\in \mathbb{Z}$, the sheaves $$\mathcal{O}(i-n),\dots ,\mathcal{O}(i-1),\mathcal{O}(i)$$ form a full strong exceptional collection for $D^b(\mathrm{coh}{\mathbb{P}}_k^n)$.

Showing that something is a full (strong) exceptional collection breaks up into two parts:

• a computation to check that vanishing of the appropriate Hom spaces and
• a generation check.

We handle the generation statement now and the cohomology computation after introducing Čech cohomology.

Lemma. We have $$\mathcal{O}(j)\in ⟨\mathcal{O}(i-n),\dots ,\mathcal{O}(i-1),\mathcal{O}(i){⟩}_{\infty }$$

Lemma. We have $$\mathcal{F}\in ⟨\mathcal{O}(i-n),\dots ,\mathcal{O}(i-1),\mathcal{O}(i){⟩}_{\infty }$$ for any coherent sheaf $\mathcal{F}$.

Lemma. The collection $$\mathcal{O}(i-n),\dots ,\mathcal{O}(i-1),\mathcal{O}(i)$$ generates $D^b(\mathrm{coh}{\mathbb{P}}_k^n)$.

To finish the proof of Beilinson's result, we need to compute $${\mathrm{Ext}}^l(\mathcal{O}(i),\mathcal{O}(j))\cong H^l({\mathbb{P}}_k^n,\mathcal{O}(j-i))$$ We will do this next.

Čech cohomology

The universal way to "compute" $H^i(X,\mathcal{F})$ is to

• replace $\mathcal{F}$ with an injective resolution,
• apply $\Gamma (X,-)$, and
• compute cohomology of the resulting complex of $k$-modules.

Unfortunately, projective resolutions are hard enough to handle and we have some very explicit projective modules. Injective objects are generally more complicated. Having enough control over their explicit structure to compute cohomology of a sequence is a big ask.

However, there is a bit more flexibility in computing derived using adapated objects

If we can construct a resolution $$0\to \mathcal{F}\to {\mathcal{A}}^0\to {\mathcal{A}}^1\to \cdots$$ with $$H^i(X,{\mathcal{A}}^j)=0,\forall i>0$$ then $$H^i(X,\mathcal{F})\cong H^i(\Gamma (X,{\mathcal{A}}^{\bullet }))$$

Recall that $\Gamma (X,-)$ is yields an equivalence between $\mathrm{Qcoh}X$ and $\mathrm{Mod}R$ if $X=\mathrm{Spec}R$. Consequently, if $\mathcal{F}$ is a quasi-coherent sheaf supported on affine patch, then $$H^i(X,\mathcal{F})=0,\forall i>0$$ for general $\mathcal{F}$.

For a general $\mathcal{F}$ and open cover $U_1,\dots ,U_n$ by affines, each $\mathcal{F}{|}_{U_i}$ is adapted and we have the obvious map $$\mathcal{F}\to \underset{i=1}{\overset{n}{⨁}}\mathcal{F}{|}_{U_i}$$ Since $\mathcal{F}$ is a sheaf, this is injective but it is rarely surjective.

To fix it and make a resolution, let $${\check{C}}^s(\mathcal{F}):=\underset{i_0<\cdots <i_s}{⨁}\mathcal{F}{|}_{U_{i_0}\cap \cdots \cap U_{i_s}}$$ and $${\delta }^s:{\check{C}}^s(\mathcal{F})\to {\check{C}}^{s+1}(\mathcal{F})$$ where the map $${\delta }^s:F{|}_{U_{i_0}\cap \cdots \widehat{U_{i_j}}\cdots \cap U_{i_s}}\to F{|}_{U_{i_0}\cap \cdots \cap U_{i_s}}$$ is restricting scaled by $(-1{)}^j$.

Lemma. ${\check{C}}^{\bullet }(\mathcal{F})$ is a resolution.

Lemma. If each $U_{i_0}\cap \cdots \cap U_{i_s}$ is affine, then $$H^i(X,\mathcal{F})\cong H^i(\Gamma (X,{\check{C}}^{\bullet }(\mathcal{F})))$$

It would be nice to know that given two affines $U,V$ in $X$ the intersection $U\cap V$ is also affine. Such a scheme is called semi-separated.

A separated scheme is one where the diagonal morphism is a closed immersion. This implies semi-separated.

We will sidestep the generalities here because we will use very specific affines: principal opens in $\mathrm{Proj}R$. Here the intersection is another principal open so the conditions of the previous lemma are immediate.

In our example of ${\mathbb{P}}_k^n$ we use the principal opens $U_{x_i}$. Then, $\Gamma (X,\mathcal{O}(l))=$ $$0\to ⨁k[x_0,\dots ,x_n,x_i^{-1}{]}_l\to \cdots \to k[x_0,\dots ,x_n,(x_0\dots x_n{)}^{-1}{]}_l\to 0$$

It makes sense to sum over $l$ and consider the complex $$0\to ⨁k[x_0,\dots ,x_n,x_i^{-1}]\to \cdots \to k[x_0,\dots ,x_n,(x_0\dots x_n{)}^{-1}]\to 0$$ of graded $k[x_0,\dots ,x_n]$ modules all at once.

Computing sheaf cohomology on ${\mathbb{P}}^n$

Last time we saw that computing $$H^i({\mathbb{P}}_k^n,\mathcal{O}(j)):={\mathbf{R}}^i\Gamma ({\mathbb{P}}_k^n,\mathcal{O}(j))$$ could be achieved by computing the cohomology of the (augmented) Čech complex $$0\to k[x_0,\dots ,x_n]\to \underset{i=0}{\overset{n}{⨁}}k[x_0,\dots ,x_n{]}_{x_i}\to \cdots \to \underset{|I|=n}{⨁}k[x_0,\dots ,x_n{]}_{x_I}\to k[x_0,\dots ,x_n{]}_{x_0\cdots x_n}\to 0$$

Let's look at the case $n=0$ to get a foothold. This reduces to $$0\to k[x_0]\to k[x_0,x_0^{-1}]\to 0$$ Since $x_0$ is regular, there is no $H^0$. We have $$H^1=k[x_0,x_0^{-1}]/k[x_0]$$ which as a $k[x_0]$-module has a filtration $$0\subset \mathrm{Ann}(x_0)\subset \cdots \subset \mathrm{Ann}(x_0^m)\subset \mathrm{Ann}(x_0^{m+1})\subset \cdots$$ whose associated graded pieces are $$\frac{\mathrm{Ann}(x_0^{m+1})}{\mathrm{Ann}(x_0^m)}\cong \frac{k}{x_0^{m+1}}$$ In particular, $H^1$ is a free $k$-module. It is also graded with $|x_0^{-1}|=-1$.

The general complex is isomorphic to the tensor product of complexes $$\underset{i=0}{\overset{n}{⨂}}\left(k[x_i]\to k[x_i{]}_{x_i}\right)$$ over $k$. Since the homologies of the complexes are flat over $k$, we have $$H^s\left(\underset{i=0}{\overset{n}{⨂}}\left(k[x_i]\to k[x_i{]}_{x_i}\right)\right)\cong \underset{i_0+\cdots +i_n=s}{⨁}{H}^{i_j}(k[x_j]\to k[x_j{]}_{x_j})$$ From our computation above we see that $$H^s=\{\begin{array}{ll}0& s\ne n\\ \frac{k[x_0,\dots ,x_n{]}_{x_0\cdots x_n}}{{\sum }_{|I|=n}k[x_0,\dots ,x_n{]}_{x_I}}& s=n\end{array}$$

Theorem. We have $${\mathrm{Ext}}_{{\mathbb{P}}_k}^l(\mathcal{O}(i),\mathcal{O}(j))\cong H^l({\mathbb{P}}_k^n,\mathcal{O}(j-i))\cong \{\begin{array}{ll}k[x_0,\dots ,x_n{]}_{j-i}& l=0\\ k[x_0,\dots ,x_n{]}_{i-j-n-1}& l=n\\ 0& \text{otherwise}\end{array}$$

Fixing the basis vector $1/x_0\cdots x_n$ gives an isomorphism $$H^n({\mathbb{P}}_k^n,\mathcal{O}(-n-1))\cong k$$ From our computations, we see that composition $${\mathrm{Ext}}^i(\mathcal{O}(i),\mathcal{O}(j)){\otimes }_k{\mathrm{Ext}}^{n-i}(\mathcal{O}(j),\mathcal{O}(i-n-1))\to {\mathrm{Ext}}^n(\mathcal{O}(i),\mathcal{O}(i-n-1))\cong H^n({\mathbb{P}}_k^n,\mathcal{O}(-n-1))$$ is given by multiplication $$f,\frac{1}{g}\mapsto \frac{f}{g}$$ up to isomorphism.

Corollary. Composition $${\mathrm{Ext}}^i(\mathcal{O}(i),\mathcal{O}(j)){\otimes }_k{\mathrm{Ext}}^{n-i}(\mathcal{O}(j),\mathcal{O}(i-n-1))\to H^n({\mathbb{P}}_k^n,\mathcal{O}(-n-1))$$ is a perfect pairing. In other words, this induces an isomorphism $${\mathrm{Ext}}^i(\mathcal{O}(i),\mathcal{O}(j){)}^{\vee }\cong {\mathrm{Ext}}^{n-i}(\mathcal{O}(j),\mathcal{O}(i-n-1))$$

For a scheme $X\to \mathrm{Spec}k$, we can ask if the functor $$\mathbf{R}{\mathrm{Hom}}_k(\mathbf{R}{\mathrm{Hom}}_X(E,F),k)$$ is quasi-isomorphic to $$\mathbf{R}{\mathrm{Hom}}_X(F,S(E))$$ for some $k$-linear auto-equivalence $S$. In the case $k$ is simply a field, then this is just a Serre functor. In this statement, it is an enhanced version of a Serre functor.

Proposition. For any objects $E,F\in D^b(\mathrm{coh}{\mathbb{P}}_k^n)$ there are natural quasi-isomorphisms $$\mathbf{R}{\mathrm{Hom}}_k(\mathbf{R}\mathrm{Hom}(E,F),k)\cong \mathbf{R}\mathrm{Hom}(F,S(E))$$ where $S(E)=E(-n-1)[n]$.

Compact generation and Grothendieck duality

We saw last semester that compact generation of a triangulated category was a very useful property. Here we establish that $D(\mathrm{Qcoh}X)$ is compactly generated for a quasi-projective schemes $X$ over an affine base. In addition, we provide some useful replacements for deriving common functors.

Lemma. Let $R$ be a $\mathbb{Z}$-graded ring. Then $R(i)$ for $i\in \mathbb{Z}$ is a set of compact generators for $D(\mathrm{GrMod}R)$.

Proposition. If $R$ is finitely-generated in degree $1$ over $R_0$, then $D(\mathrm{Qcoh}\mathrm{Proj}R)$ is compactly generated.

Lemma. If $D(\mathrm{Qcoh}X)$ is compactly-generated and $U$ is an open subscheme, then so is $U$.

Corollary. Let $X$ be quasi-projective over an affine base. Then any compact object is quasi-isomorphic to a bounded complex of locally-free sheaves.

We saw last time that $E\mapsto E(-n-1)[n]$ is the Serre functor on ${\mathbb{P}}_k^n$. In fact, this is a consequence of a more general fact known as Grothendieck duality. One formulation of Grothendieck duality is that for a map $f:X\to Y$ of schemes possesses a right adjoint $$f^{!}:D(\mathrm{Qcoh}Y\to \mathrm{Qcoh}X)$$ to $\mathbf{R}{f}_{\ast }$. This is not automatic in full generality. But it is true with mild conditions on $X$ and $Y$.

For the projection \pi : \mathbb{P}^n_k \to \operatorname{Spec} k} we have $${\pi }^{!}{\mathcal{O}}_{\mathrm{Spec}k}\cong \mathcal{O}(-n-1)[n]$$

Originally proof of the existence of a right adjoint was arduous. However, [Neeman](https://www.ams.org/journals/jams/1996-9-01/S0894-0347-96-00174- 9/S0894-0347-96-00174-9.pdf) demonstrated how Brown Representability can be used to give a clean proof.

Thanks to Brown Representability, we know that for $\mathbf{R}{f}_{\ast }$ to have a right adjoint is equivalent to the natural map $$\underset{i\in I}{⨁}\mathbf{R}{f}_{\ast }{E}_i\to \mathbf{R}{f}_{\ast }\left(⨁E_i\right)$$ being an isomorphism for all sums. This question is local on $Y$ so we reduce to asking -- when does taking cohomology commute with direct sums?

One useful fact: formation of the Čech complex for an affine cover where all intersections are affine commutes with direct sums.

Proposition. For any map $f:X\to Y$ between quasi-projective schemes over an affine base, there is a right adjoint $f^{!}$ to $\mathbf{R}{f}_{\ast }$.

Remark. Neeman proves the existence of $f^{!}$ whenver $X$ and $Y$ are quasi-compact and quasi-separated.

Deriving tensor and sheaf Hom

We have implicitly used both the tensor product $$E{\otimes }_{{\mathcal{O}}_X}F$$ and sheaf Hom $$\mathcal{H}o{m}_{{\mathcal{O}}_X}(E,F)$$ of ${\mathcal{O}}_X$-modules.

Recall that $E$ is locally-free if each $p\in X$ has a neighborhood $U$ with $$E{|}_U\cong {\mathcal{O}}_U^{\oplus I}$$ Since exactness can be checked locally, $E{\otimes }_{{\mathcal{O}}_X}-$ preserves exactness of complexes of ${\mathcal{O}}_X$-modules if $E$ is locally-free. Thus, if we can replace any complex with one whose components are locally-free, we can use these adapted objects to derive ${\otimes }_{{\mathcal{O}}_X}$.

Proposition. For any quasi-projective schemes $X$ and any chain complex $E$ of quasi-coherent sheaves, there is complex $P$ with locally-free components and a quasi-isomorphism $P\to E$.

Note that despite the notation we cannot find enough K-projective complexes in $D(\mathrm{Qcoh}X)$.

Using this result, we take $$E\stackrel{\mathbf{L}}{\otimes }-:=P\otimes -$$ for some such $P$.

Similarly, we can derive $\mathcal{H}om(E,F)$ by replacing $E$ by a complex of locally-frees or $F$ by a K-injective. The choices are naturally quasi-isomorphic so we denote either by $$\mathbf{R}\mathcal{H}om(E,F).$$

The projection formula and standard adjunctions

The main functors in play for sheaves of modules are

• $f_{\ast }$ for a map $f:X\to Y$
• $f^{\ast }$ for $f:X\to Y$
• ${\otimes }_{{\mathcal{O}}_X}$ and
• $\mathcal{H}o{m}_{{\mathcal{O}}_X}$

We will also use the notation $$E^{\vee }:=\mathcal{H}o{m}_{{\mathcal{O}}_X}(E,{\mathcal{O}}_X)$$ for both the underived and derived duals over ${\mathcal{O}}_X$.

We record some back facts about them and we omit the details for the most part.

Lemma. We have an adjunction $f^{\ast }⊣f_{\ast }$.

Lemma. We also have an adjunction $${\otimes }_{{\mathcal{O}}_X}E⊣\mathcal{H}o{m}_{{\mathcal{O}}_X}(E,-)$$

Lemma. Tensor products commute with pullbacks $$f^{\ast }(E{\otimes }_{{\mathcal{O}}_X}F)\cong f^{\ast }E{\otimes }_{{\mathcal{O}}_X}{f}^{\ast }F$$

From these, we can derive some less immediate ones.

Lemma. We have an adjunction $$f^{\ast }(-{\otimes }_{{\mathcal{O}}_Y}F)⊣\mathcal{H}o{m}_{{\mathcal{O}}_Y}(F,f_{\ast }(-))$$

We also have a sheafy version of pull-push adjunction.

Lemma. There is a natural isomorphism $$\mathcal{H}o{m}_{{\mathcal{O}}_Y}(E,f_{\ast }F)\cong f_{\ast }\mathcal{H}o{m}_{{\mathcal{O}}_X}(f^{\ast }E,F)$$

One of the most useful relationships between these functors is one that is not true. Thanks to the map $$f^{\ast }(f_{\ast }E\otimes F)\cong f^{\ast }{f}_{\ast }E\otimes f^{\ast }F\to E\otimes f^{\ast }F$$ coming from the counit and adjunction we have a natural map $$f_{\ast }E\otimes F\to f_{\ast }(E\otimes f^{\ast }F)$$ However, this is not an isomorphism of quasi-coherent sheaves in general.

For what $E$ and $F$ is the map $$f_{\ast }E\otimes F\to f_{\ast }(E\otimes f^{\ast }F)$$ an isomorphism? Taking $F={\mathcal{O}}_Y$ works. Since isomorphism can be detected locally, it suffices that $F$ is locally isomorphic to ${\mathcal{O}}_Y$.

Moreover, if $$f_{\ast }E\otimes F_a\to f_{\ast }(E\otimes f^{\ast }{F}_a)$$ for some $a\in A$, then $$f_{\ast }E\otimes \underset{a\in A}{⨁}{F}_a\to f_{\ast }\left(E\otimes f^{\ast }\left(⨁F_a\right)\right)$$ will also be an isomorphism if $f_{\ast }$ commutes with coproducts which is true for quasi-projective schemes (and more generally).

Proposition. Let $f:X\to Y$ be a map of quasi-projective schemes. Then for any $E\in D(\mathrm{Qcoh}X)$ and $F\in D(\mathrm{Qcoh}Y)$, the natural map $$\mathbf{R}{f}_{\ast }E\stackrel{\mathbf{L}}{\otimes }F\to \mathbf{R}{f}_{\ast }\left(E\stackrel{\mathbf{L}}{\otimes }\mathbf{L}{f}^{\ast }F\right)$$ is a quasi-isomorphism.

The previous result is often called the projection formula.

Base change

Recall in commutative algebras given a diagram
$$\begin{array}{ccc}B& \to & R\\ \downarrow & & \\ S& & \end{array}$$ we can form a pushout (or fiber coproduct) $$\begin{array}{ccc}B& \to & R\\ \downarrow & & \downarrow \\ S& \to & S{\otimes }_{B}R\end{array}$$ using the tensor product $S{\otimes }_{B}R$.

Consequently, in the category of affine schemes, we have fiber products (or pullbacks) $$\begin{array}{ccc}X{\times }_{Z}Y& \to & Y\\ \downarrow & & \downarrow \\ X& \to & Z\end{array}$$ where, if $X=\mathrm{Spec}R$, $Y=\mathrm{Spec}S$, and $Z=\mathrm{Spec}B$, we have $$X{\times }_{Z}Y=\mathrm{Spec}(S{\otimes }_{B}R).$$

For schemes generally, we can use this to build fiber products. Given $$\begin{array}{ccc}& & Y\\ & & \downarrow g\\ X& \underset{f}{\to }& Z\end{array}$$ $X{\times }_{Z}Y$ exists. Given affine $U\subset X,V\subset Y$, and $W\subset Z$ with $f(U),g(V)\subset W$, we have an affine chart $$U{\times }_{W}V$$ and these can be glued together appropriately to give the fiber product as a scheme.

A square $$\begin{array}{ccc}W& \to & Y\\ \downarrow & & \downarrow \\ X& \to & Z\end{array}$$ isomorphic to a fiber product square is commonly called Cartesian.

Given a Cartesian square $$\begin{array}{ccc}W& \underset{\tilde{f}}{\to }& Y\\ \downarrow \tilde{g}& & \downarrow g\\ X& \underset{f}{\to }& Z\end{array}$$ and a $E\in D(\mathrm{Qcoh}X)$ can we compare $$\mathbf{L}{g}^{\ast }\mathbf{R}{f}_{\ast }E\text{and }\mathbf{R}{\tilde{f}}_{\ast }\mathbf{L}{\tilde{g}}^{\ast }E$$ In general, they will not be quasi-isomorphic. However, under some assumptions, this is the case.

Theorem. Assume that $$\mathcal{T}o{r}_i^{{\mathcal{O}}_Z}({\mathcal{O}}_X,{\mathcal{O}}_Y)=0$$ for $i\ne 0$. Then, there is a natural isomorphism $$\mathbf{L}{g}^{\ast }\mathbf{R}{f}_{\ast }E\cong \mathbf{R}{\tilde{f}}_{\ast }\mathbf{L}{\tilde{g}}^{\ast }E$$

In particular, if either ${\mathcal{O}}_X$ or ${\mathcal{O}}_Y$ is flat over ${\mathcal{O}}_Z$, then we can exchange pushing and pulling.

The previous result is often called (flat) base change. We won't provide more details on this result. Lipman has some very thorough notes on Grothendieck duality which include exposition on Tor-independent base change.

Along with the projection formula, base change will serve as regular tool in our future investigations of derived categories.

Kernels and integral transforms

We saw last semester that for two rings $S$ and $R$ the category of $(S,R)$-bimodules was an imporant source of functors between $D(\mathrm{Mod}S)$ and $D(\mathrm{Mod}R)$.

Here we globalize this idea. Note that an $(S,R)$-bimodule is the same data as a $S{\otimes }_{\mathbb{Z}}R$-module. Geometrically, $S\otimes R$ corresponds to the fiber product. Therefore, it is natural to look to objects on the (fiber) product for a source of functors.

For $$X\stackrel{{\pi }_X}{\leftarrow }X{\times }_{Z}Y\stackrel{{\pi }_Y}{\to }Y$$ and $Q\in D(\mathrm{Qcoh}X{\times }_{Z}Y)$, we get a functor $$\begin{array}{rl}{\Phi }_Q:D(\mathrm{Qcoh}X)& \to D(\mathrm{Qcoh}Y)\\ E& \mapsto \mathbf{R}{\pi }_{Y\ast }\left(\mathbf{L}{\pi }_X^{\ast }E\stackrel{\mathbf{L}}{\otimes }Q\right)\end{array}$$

We call ${\Phi }_Q$ an integral transform and $Q$ its kernel. If ${\Phi }_Q$ is an equivalence, we say ${\Phi }_Q$ and $Q$ are Fourier-Mukai and $X$ and $Y$ are Fourier-Mukai partners.

The language is inspired by the analogy with integral transformations in analysis. $$f(x)\mapsto {\int }_{X}Q(x,y)f(x)dx.$$

Each integral transform comes with a right adjoint thanks to Grothendieck duality.

Lemma. We have $${\Phi }_Q⊣{\Psi }_Q:=\mathbf{R}{\pi }_{X\ast }\mathbf{R}\mathcal{H}o{m}_{X\times Y}(Q,{\pi }_Y^{!}(-))$$

Identities, projections, and convolutions

Let's expand out analogy with integral transforms in analysis a bit more. There are three nice properties of such linear transformations:

1. Identity kernels. Let $$\Delta =\{(x,x)\in X\times X\}$$ be the diagonal. Then, $${\int }_X{\delta }_{\Delta }(x,y)f(x)dx=f(y)$$ for the $\delta$-function along $\Delta$.
2. Projection kernels. For a kernel of the form $K(x,y)=e(x)h(y)$, we have $${\int }_{X}K(x,y)f(x)dx=h(y){\int }_{X}e(x)f(x)dx$$ is a type of projection onto $h(y)$.
3. Composition as convolution. For $K(x,y)$ and $Q(y,z)$ the composition of $$f(x)\mapsto {\int }_{X}K(x,y)f(x)dx$$ and $$f(y)\mapsto {\int }_{Y}Q(y,z)f(y)dy$$ is also an integral transformation with kernel $$Q\star K:={\int }_{Y}K(x,y)Q(y,z)dy$$