用户: 遗忘的左伴随/Derived Algebraic Geometry/Lecture 1

This lecture is an overview of derived algebraic geometry.

1$\infty$-Categories

I have written the theory of $\infty$-categories in my note 讲义: 六函子理论.

2The idea of DAG

fact 2.1. Functors in algebra are often not exact. e.g. $R$ be commutative algebra, $M$ be $R$-module. $(-){\otimes }_{R}M$ is not exact.

In homological algebra, the solution is construct an approximation of non-exact functors.

例 2.2. $(-){\otimes }_R^{\mathbb{L}}(-)$ can be computed via projective resolutions.

In homotopical algebra we approximate functors that do not preserves of weak equivalences.

$$\mathrm{Spec}(A{\otimes }_R^{\mathbb{L}}B)\mathrm{Spec}B\mathrm{Spec}A\mathrm{Spec}R┘$$

DAG idea:

•	Schemes : Zariski-locally affine scheme,
•	DM-stacks : étale-locally affine scheme,
•	Artin-stacks : smooth-locally affine scheme.

定义 2.3. $\mathsf{Ring}:=\{\text{animated rings}\}$, $\mathsf{Aff}:={\mathsf{Ring}}^{\mathrm{op}}$ be affine derived schemes,

look at presheaves on $\mathsf{Aff}$: $${\mathsf{Aff}}^{\mathrm{op}}\to \mathsf{Ani}$$Inpose a descent condtion: the derived stacks $$\mathsf{St}={\mathsf{Sh}}_{\text{eˊt}}(\mathsf{Aff})$$Note that one can also define derived schemes as locally ringed spaces. But this is more convenient.

To Do:

•	Quasi-coherent modules;
•	perfect complexes;
•	cotangenet complex;
•	properties of morphisms of derived stacks ∘ smooth ∘ étale ∘ fp ∘ $\cdots$

Weil Restrictions

Idea: Let $f:W\to X$ be a morphism of derived stacks, then we have adjunction$${\mathsf{St}}_{/W}{\mathsf{St}}_{/X}{f}_{♯}\perp f^{\ast }$$where $f_{♯}(S\to W):=S\to W\to X$, and $f^{\ast }=(-){\times }_{X}W$.

Then the Weil restriction is the adjunction$$f^{\ast }⊣f_{\ast }$$so is a functor $f_{\ast }:{\mathsf{St}}_{/W}\to {\mathsf{St}}_{/X}$ such that $$f^{\ast }TTUWX{f}_{\ast }U.f$$

Question: What is $f_{\ast }U$ Artin stack? (or scheme,...)?

Answer: Under sufficient finiteness assumptions(on $f$ and on $U\to X$), $f_{\ast }U$ is $n$-Artin.

It will turn out in DAG we have hievarchs of “$n$-Artin” stacks.

例 2.4. $$\mathrm{Spec}(\mathbb{Z}\oplus \mathbb{Z}[1])\{0\}{\mathbb{A}}^1\{0\}{\mathbb{A}}^1{f}_{\ast }{\mathbb{A}}^1┘$$${\mathbb{A}}^1(\mathrm{Spec}(\mathbb{Z}\oplus \mathbb{Z}[1]))=\mathbb{Z}\oplus \mathbb{Z}[1]$

Blow-ups

Application of algebraicits of Weil restrictions will be algebraicits of “deforation space”.

“Definition”: $W\stackrel{h}{\to }X$ be a map of derived stacks, we can define a derived stack $D_{W/X}$ such that $$D_{W/X}(T)=(\ast )\left\{DTWXV\subset D\right\}$$where $V\subset D$ means: locally cut out by one element. In particular, $D\to T$ locally look like $\mathrm{Spec}(A//(f))\to \mathrm{Spec}A$.

Suppose that $W\to X$ is closed immersion of classical schemes. If $D,T$ are classical and $\ast$ is Cartesian. The $D\to T$ is an effective Cartier divisor. So this classifies a point $${\mathsf{Bl}}_{W/X}^{\text{pt}}TX.$$

注 2.5. $D_{W/X}$ can be constructed via Weil constructions.

We will define the blow up of $X$ in $W$ for a closed immersion $W\to X$ of derived Artin stacks as a subobject $${\mathsf{Bl}}_{W/X}\subset D_{W/X}$$such that$$D_{W/X}(T)=\left\{DTWX┘V\subset D\right\}$$

定理 2.6. The assignment $(h:W\to X)\mapsto D_{W/X}$ is natural.

Classically Blow-up can be defined via Rees algebra as $W\hookrightarrow X$ : classical scheme with ideal $I$, Then the classical Rees algebra is $$R_{W/X}^{\text{cl}}:={\mathcal{O}}_X[It]=⨁U^n{t}^n\subseteq {\mathcal{O}}_X[t]$$[不想记了 (摆烂)]