We had this subcategory $$ \mathsf{LMod} \subseteq \mathsf{Fun}(\Delta^\mathrm{op} \times [1], \mathsf{Sp}) $$ with a forgetful functor $$ \mathsf{LMod} \to \mathsf{Alg}(\mathsf{Sp}) $$ with fiber over $A$ being $\mathsf{Mod}_ A$. This defined a functor $$ \mathsf{Alg}(\mathsf{Sp}) \to \mathsf{LinCat} \subseteq \widehat{\mathsf{Cat}}_ \infty $$ This sends $A \to B$ to $\mathsf{Mod}_ A \to \mathsf{Mod}_ B$ given by $- \otimes_A B$.
For a discrete ring $\Lambda$, we defined $$ \mathsf{CAlg}_ \Lambda = \mathsf{Ani}(\mathsf{CAlg}_ \Lambda^\heartsuit) \to \mathsf{CAlg}(\mathsf{Sp})_ {/\Lambda} \to \mathsf{Alg}(\mathsf{Sp}) \to \mathsf{LinCat}. $$
Quasi-coherent sheaves on prestacks#
Definition 1. A prestack over $\Lambda$ is an accessible functor $\mathsf{CAlg}_ \Lambda \to \mathsf{Spc}$. The category of prestacks is defined to be the full subcategory of $\mathsf{Fun}(\mathsf{CAlg}_ \Lambda, \mathsf{Spc})$.
Then we have a fully faithful embedding $$ (\mathsf{CAlg}_ \Lambda)^\mathrm{op} \hookrightarrow \mathsf{PreStk}_ \Lambda \hookrightarrow \mathsf{Fun}(\mathsf{CAlg}_ \Lambda, \mathsf{Spc}). $$ Since we know what a module over an algebra is, we can define a quasi-coherent sheaf as a right Kan extension $$ \begin{CD} \mathsf{PreStk}_ \Lambda^\mathrm{op} @>>> \mathsf{LinCat}_ \Lambda \br @AAA @| \br \mathsf{CAlg}_ \Lambda @>>> \mathsf{LinCat}_ \Lambda. \end{CD} $$ More explicitly, we can define $$ \mathsf{QCoh}(\mathscr{F}) = \varprojlim_{\mathscr{F}(A)} \mathsf{Mod}_ A. $$
Actually, every $\mathsf{Mod}_ A$ is a symmetric monoidal category, we actually have a factorization $$ \mathsf{CAlg}_ \Lambda \to \mathsf{CAlg}(\mathsf{LinCat}) \to \mathsf{LinCat}. $$ Since the last forgetful functor commute with limits, we can promote this category $\mathsf{QCoh}(\mathscr{F})$ into a symmetric monoidal $\Lambda$-linear category.
Stacks#
Definition 2. A map $A \to B$ in $\mathsf{CAlg}_ \Lambda$ is said to be flat when
- the map $\pi_0(A) \to \pi_0(B)$ is flat and
- the map $\pi_i(A) \otimes_{\pi_0(A)} \pi_0(B) \cong \pi_i(B)$ for all $i$.
We say that $A \to B$ is a Zariski localization (respectively, étale, smooth) when it is flat and $\pi_0(A) \to \pi_0(B)$ is a Zariski localization (respectively, étale, smooth).
Theorem 3. The map $A \to B$ being flat is equivalent to $M \otimes_A B$ being in degree zero whenever $M$ is in degree zero.
Example 4. For $f \in \pi_0 A$, we can identify it with a map $\Lambda[x] \to A$. Then we can write $$ A_f = A \otimes_{\Lambda[x]} \Lambda[x, x^{-1}]. $$
Let $\tau$ be either the Zariski topology or the étale topology.
Definition 5. A $\tau$-stack is a prestack $\mathfrak{X} \colon \mathsf{CAlg}_ \Lambda \to \mathsf{Spc}$ satisfying
- we have $$ \mathfrak{X}\Bigl( \prod_I A_i \Bigr) = \prod_I \mathfrak{X}(A_i) $$ for $I$ finite,
- for $A \to B$ a $\tau$-covering (i.e., a $\tau$-map that with $\Spec \pi_0(B) \to \Spec \pi_0(A)$ surjective), we have $$ \mathfrak{X}(A) = \varprojlim(\mathfrak{X}(B) \twoheadrightarrow \mathfrak{X}(B \otimes_A B) \dotsb). $$
Example 6. An example of an étale stack is $\Spec A$.
Schemes#
We define the topological space $\Spec A = \Spec \pi_0(A)$. The idea is that these higher data is like a nilpotent thickening, and so it doesn’t affect the underlying topological space.
Definition 7. The topological space attached to a prestack is defined as the left Kan extension $$ \lvert \mathfrak{X} \rvert = \varinjlim_{\Spec A \to \mathfrak{X}} \lvert \Spec(A) \rvert. $$
The functor $$ \lvert - \rvert \colon \mathsf{PreStk}_ \Lambda \to \mathsf{Top} $$ admits a right adjoint $$ T \mapsto (T^\mathrm{PreStk} \colon A \mapsto \Hom_\mathsf{Top}(\Spec A, T)). $$
Definition 8. For $\mathfrak{X}$ a prestack, an open sub-prestack is a sub-prestack of the form $$ \mathfrak{Y} = \mathfrak{X} \times_{\lvert \mathfrak{X} \rvert^\mathrm{PreStk}} U^\mathrm{PreStk} $$ for some open subset $U \subseteq \lvert \mathfrak{X} \rvert$. Explicitly, it is given by $$ \mathfrak{Y}(A) = \lbrace \Spec A \to \mathfrak{X} : \lvert \Spec A \rvert \to U \rbrace. $$
Definition 9. A (derived) scheme over $\Lambda$ is a prestack $\mathfrak{X}$ satisfying
- $\mathfrak{X}$ is a Zariski sheaf,
- there exist $U_i \subseteq \mathfrak{X}$ substacks such that $U_i \cong \Spec A_i$ and $\bigcup \lvert U_i \rvert = \lvert \mathfrak{X} \rvert$.
Example 10. Every quasi-compact separated classical scheme can be promoted to a derived scheme in the following way: we can find an open cover $\Spec A_i$ of $X$ and then then we can look at the colimit of the diagram $$ \dotsb \coprod_{i,j} \Spec A_{ij} \rightrightarrows \coprod_i \Spec A_i $$ in the category of Zariski stacks.
Theorem 11. If $f \colon X \to Y$ is a faithfully flat map of derived schemes, then $$ \mathsf{QCoh}(Y) \cong \varprojlim(\mathsf{QCoh}(X) \rightrightarrows \mathsf{QCoh}(X \times_Y X) \dotsb). $$
The proof uses a higher version of the Barr–Beck theorem and is outside the scope of this course.
Theorem 12. If $X$ is a quasi-compact and quasi-separated classical scheme, then
- the category $\mathsf{QCoh}(X)$ is compactly generated, and hence it is dualizable in $\mathsf{LinCat}_ \Lambda$,
- $\mathsf{QCoh}^\omega$ are precisely the dualizable objects objects, and these are the perfect objects.
Moreover, if $X$ and $Y$ are two such classical schemes and flat over $\Lambda$, then $$ \mathsf{QCoh}(X \times Y) \cong \mathsf{QCoh}(X) \otimes_ \Lambda \mathsf{QCoh}(Y). $$
We will prove this next time.