用户: Scavenger/Deformation Theory

Our goal in these notes is to present the deformation theory for $E_{\infty }$-rings. We will prove Elkik’s algebraization theorem as an illustrating application.

1Relative Adjunctions

Let $\mathcal{C}$ be an $\infty$-category. The isofibrations $K\to \mathcal{C}$ over $\mathcal{C}$ form a locally Kan simplicial category, so we have the corresponding notion of adjunctions of isofibrations over $\mathcal{C}$.

Proposition 1.1. Let $\mathcal{C}$ be an $\infty$-category, let $p_D:\mathcal{D}\to \mathcal{C},p_E:\mathcal{E}\to \mathcal{C}$ be isofibrations over $\mathcal{C}$, let $g:\mathcal{E}\to \mathcal{D}$ be a map such that $p_D\circ g=p_E$. Assume that for each object $d\in \mathcal{D}$, there exists an object $e\in \mathcal{E}$ and an edge $\alpha :d\to g(e)$ in $\mathcal{D}$ such that $p_D(\alpha )$ is an isomorphism in $\mathcal{C}$, and for each object $e^{\prime }\in \mathcal{E}$, the composition ${\mathrm{Hom}}_{\mathcal{E}}(e,e^{\prime })\to {\mathrm{Hom}}_{\mathcal{D}}(g(e),g(e^{\prime }))\stackrel{\circ \alpha }{\to }{\mathrm{Hom}}_{\mathcal{D}}(d,g(e^{\prime }))$ is an equivalence, then there exists a map $f:\mathcal{D}\to \mathcal{E}$ that is a left adjoint of $g$ as maps of isofibrations over $\mathcal{C}$.

Proof. Just as in the case of ordinary $\infty$-categories, one can construct $f$ (and the other data of an adjunction) first on points, then on edges, and then on $2$-simplexes and so on. Nothing will go wrong if one proceeds correctly since everything should be determined up to contractible space of choices. Details omitted.

$\square$

Proposition 1.2 ([1] Proposition 1.20). Let $\mathcal{C}$ be an $\infty$-category, let $p_D:\mathcal{D}\to \mathcal{C},p_E:\mathcal{E}\to \mathcal{C}$ be locally cartesian isofibrations over $\mathcal{C}$, let $g:\mathcal{E}\to \mathcal{D}$ be a map such that $p_D\circ g=p_E$. Then $g$ admits a left adjoint relative to $\mathcal{C}$ if and only if the following conditions are satisfied:

(i) For every object $C\in \mathcal{C}$, the map of fibers ${\mathcal{E}}_C\to {\mathcal{D}}_C$ admits a left adjoint.

(ii) The map $g$ carries locally $p_E$-cartesian morphisms in $\mathcal{E}$ to locally $p_D$-cartesian morphisms in $\mathcal{D}$.

2Stable Envelopes and Tangent bundles

Definition 2.1 ([1] Definition 1.1). Let $\mathcal{C}$ be a presentable $\infty$-category. A stable envelope of $\mathcal{C}$ is an isofibration of $\infty$-categories $u:{\mathcal{C}}^{\prime }\to \mathcal{C}$ with the following properties:

(i) ${\mathcal{C}}^{\prime }$ is stable and presentable.

(ii) $u$ admits a left adjoint.

(iii) For every presentable stable $\infty$-category $\mathcal{D}$, composition with $u$ induces an equivalence of $\infty$-categories ${\mathrm{Fun}}^R(\mathcal{D},{\mathcal{C}}^{\prime })\to {\mathrm{Fun}}^R(\mathcal{D},\mathcal{C})$. Here ${\mathrm{Fun}}^R(\mathcal{D},\mathcal{C})$ is the full subcategory of $\mathrm{Fun}(\mathcal{D},\mathcal{C})$ spanned by those functors that admit left adjoints, and ${\mathrm{Fun}}^R(\mathcal{D},{\mathcal{C}}^{\prime })$ is defined similarly.

More generally, suppose that $p:\mathcal{D}\to \mathcal{C}$ is a presentable fibration of $\infty$-categories. A stable envelope of $p$ is an isofibration of $\infty$-categories $u:{\mathcal{C}}^{\prime }\to \mathcal{D}$ with the following properties:

(i) The composition $p\circ u$ is a presentable fibration.

(ii) $u$ carries $(p\circ u)$-cartesian morphisms to $p$-cartesian morphisms.

(iii) For every object $C\in \mathcal{C}$, the map of fibers ${\mathcal{C}}_C^{\prime }\to {\mathcal{D}}_C$ is a stable envelopes of ${\mathcal{D}}_C$.

Stable envelopes satisfy a universal property:

Proposition 2.2 ([1] Proposition 1.9). Let $p:\mathcal{D}\to \mathcal{C}$ be a presentable fibration of $\infty$-categories. Then there exists a functor of $\infty$-categories $u:{\mathcal{C}}^{\prime }\to \mathcal{D}$ with the following properties:

(i) $u$ is a stable envelope of $p$.

(ii) Let $q:\mathcal{E}\to \mathcal{C}$ be a presentable fibration, such that each fiber of $q$ is a stable $\infty$-category. Then composition with $u$ induces an equivalence ${\mathrm{Fun}}_{\mathcal{C}}^R(\mathcal{E},{\mathcal{C}}^{\prime }{)}^{\simeq }\to {\mathrm{Fun}}_{\mathcal{C}}^R(\mathcal{E},\mathcal{D}{)}^{\simeq }$, where ${\mathrm{Fun}}_{\mathcal{C}}^R(\mathcal{E},\mathcal{D})$ is the full subcategory of ${\mathrm{Fun}}_{\mathcal{C}}(\mathcal{E},\mathcal{D})$ spanned by those functors that admits a left adjoint relative to $\mathcal{C}$, and ${\mathrm{Fun}}_{\mathcal{C}}^R(\mathcal{E},{\mathcal{C}}^{\prime })$ is defined similarly.

(iii) Let $v:\mathcal{E}\to \mathcal{D}$ be another stable envelope of $p$, then there exists an equivalence of $\infty$-categories $\overline{v}:\mathcal{E}\to {\mathcal{C}}^{\prime }$ relative to $\mathcal{C}$ such that $v=u\circ \overline{v}$.

Definition 2.3 ([1] Definition 1.12). Let $\mathcal{C}$ be a presentable $\infty$-category. A tangent bundle to $\mathcal{C}$ is a functor $T_{\mathcal{C}}\to \mathrm{Fun}({\Delta }^1,\mathcal{C})$ which exhibits $T_{\mathcal{C}}$ as a stable envelope of the presentable fibration $\mathrm{Fun}({\Delta }^1,\mathcal{C})\to \mathrm{Fun}(\{0\},\mathcal{C})\simeq \mathcal{C}$.

Proposition 2.4 ([1] Proposition 1.13). Let $\mathcal{C}$ be a presentable $\infty$-category, then the tangent bundle $T_{\mathcal{C}}$ is presentable.

Proposition 2.5 ([1] Proposition 1.14). Let $\mathcal{C}$ be a presentable $\infty$-category, let $T_{\mathcal{C}}$ be a tangent bundle to $\mathcal{C}$, let $p$ denote the composition $T_{\mathcal{C}}\to \mathrm{Fun}({\Delta }^1,\mathcal{C})\to \mathrm{Fun}(\{1\},\mathcal{C})\simeq \mathcal{C}$, let $K$ be a small simplicial set, then:

(i) A diagram $\overline{q}:K^{▹}\to T_{\mathcal{C}}$ is a colimit diagram if and only if $\overline{q}$ is a $p$-colimit diagram, and $p\circ \overline{q}$ is a colimit diagram in $\mathcal{C}$.

(ii) A diagram $\overline{q}:K^{◃}\to T_{\mathcal{C}}$ is a limit diagram if and only if $\overline{q}$ is a $p$-limit diagram, and $p\circ \overline{q}$ is a limit diagram in $\mathcal{C}$.

Construction 2.6 ([1] Definition 1.22). Let $\mathcal{C}$ be a presentable $\infty$-category, using Proposition 1.2, we see that the map $T_{\mathcal{C}}\to \mathrm{Fun}({\Delta }^1,\mathcal{C})$ admits a left adjoint relative to $\mathcal{C}$, which we denote by $F$. The absolute cotangent complex functor $L:\mathcal{C}\to T_{\mathcal{C}}$ is defined to be the composition $\mathcal{C}\to \mathrm{Fun}({\Delta }^1,\mathcal{C})\stackrel{F}{\to }{T}_{\mathcal{C}}$, where the first map is the diagonal embedding. We will denote the value of $L$ on an object $A\in \mathcal{C}$ by $L_A$ (usually viewed as an object in the fiber of $T_{\mathcal{C}}$ over $A$), and we refer to $L_A$ as the cotangent complex of $A$.

Since the diagonal embedding $\mathcal{C}\to \mathrm{Fun}({\Delta }^1,\mathcal{C})$ is a left adjoint to the evaluation map $\mathrm{Fun}({\Delta }^1,\mathcal{C})\to \mathrm{Fun}(\{0\},\mathcal{C})\simeq \mathcal{C}$, we deduce that the cotangent complex functor is a left adjoint to the composition $T_{\mathcal{C}}\to \mathrm{Fun}({\Delta }^1,\mathcal{C})\to \mathrm{Fun}(\{0\},\mathcal{C})\simeq \mathcal{C}$.

3Relative Cotangent Complexes

Construction 3.1 ([1] Definition 1.39). Let $\mathcal{C}$ be a presentable $\infty$-category, and let $p:T_{\mathcal{C}}\to \mathcal{C}$ be a tangent bundle to $\mathcal{C}$. A relative cofiber sequence in $T_{\mathcal{C}}$ is a diagram $\sigma$:

in $T_{\mathcal{C}}$ satisfying the following conditions:

(i) The map $p\circ \sigma$ factors through the projection ${\Delta }^1\times {\Delta }^1\to {\Delta }^1$, so the vertical arrows above become degenerate in $\mathcal{C}$.

(ii) The diagram is a pushout square in $T_{\mathcal{C}}$.

(iii) $W$ is a zero object in the corresponding fiber of $p$.

Let $\mathcal{E}$ denote the full subcategory of $\mathrm{Fun}({\Delta }^1\times {\Delta }^1,T_{\mathcal{C}}){\times }_{\mathrm{Fun}({\Delta }^1\times {\Delta }^1,\mathcal{C})}\mathrm{Fun}({\Delta }^1,\mathcal{C})$ spanned by the relative cofiber sequences. Then restriction to the upper half of the diagram above gives a trivial Kan fibration $\psi :\mathcal{E}\to \mathrm{Fun}({\Delta }^1,T_{\mathcal{C}})$. The relative cotangent complex functor is defined to be the composition $\mathrm{Fun}({\Delta }^1,\mathcal{C})\stackrel{L}{\to }\mathrm{Fun}({\Delta }^1,T_{\mathcal{C}})\stackrel{s}{\to }\mathcal{E}\stackrel{s^{\prime }}{\to }{T}_{\mathcal{C}}$, where $s$ is a section of $\psi$, and $s^{\prime }$ is given by restricting to the lower right vertex of the above diagram. We will denote the image of a morphism $A\to B$ of $\mathcal{C}$ under the relative cotangent complex functor by $L_{B/A}$.

Proposition 3.2 ([1] Proposition 1.43). Let $\mathcal{C}$ be a presentable $\infty$-category, let $T_{\mathcal{C}}$ be a tangent bundle to $\mathcal{C}$. Suppose given a commutative diagram

in $\mathcal{C}$, then the resulting diagram

is a pushout diagram in $T_{\mathcal{C}}$.

Proposition 3.3 ([1] Proposition 1.45). Let $\mathcal{C}$ be a presentable $\infty$-category, let $T_{\mathcal{C}}$ be a tangent bundle to $\mathcal{C}$, and let $p$ be the composite map $T_{\mathcal{C}}\to \mathrm{Fun}({\Delta }^1,\mathcal{C})\to \mathrm{Fun}(\{1\},\mathcal{C})\simeq \mathcal{C}$. Suppose given a pushout diagram

in $\mathcal{C}$. Then the induced map $L_{B/A}\to L_{B^{\prime }/A^{\prime }}$ is a $p$-cocartesian morphism in $T_{\mathcal{C}}$.

4The Tangent Bundle of the $\infty$-Category of $E_{\infty }$-Rings

Let $\mathrm{CAlg}=\mathrm{CAlg}(\mathrm{Sp})$ denote the $\infty$-category of $E_{\infty }$-rings. Let $\mathrm{Mod}={\mathrm{Mod}}^{\mathrm{Comm}}(\mathrm{Sp})$ denote the $\infty$-category of modules of $\mathrm{Sp}$ over the commutative operad, so there is a natural isofibration $\mathrm{Mod}\to \mathrm{CAlg}$ and a forgetful functor $\mathrm{Mod}\to \mathrm{Sp}$. Our goal in this section is to show that the tangent bundle to $\mathrm{CAlg}$ is equivalent to $\mathrm{Mod}$, and the functor $\mathrm{Mod}\to \mathrm{Fun}({\Delta }^1,\mathrm{CAlg})$ can be think of bringing the pair $(A,M)$ (where $M$ is a module over $A$) to the map $A\oplus M\to A$.

Proposition 4.1 ([1] Theorem 1.78). Let $T_{\mathrm{CAlg}}$ be a tangent bundle to $\mathrm{CAlg}$, then there exists a map $T_{\mathrm{CAlg}}\to \mathrm{Mod}$ which is an equivalence of presentable fibrations over $\mathrm{CAlg}$.

Remark 4.2. Of course a mere equivalence is not sufficient for our usage. In the proof of [1] Theorem 1.78 this equivalence is constructed explicitly. Let me list some properties this equivalence should possess:

(i) The two functors $T_{\mathrm{CAlg}}\to \mathrm{Fun}({\Delta }^1,\mathrm{CAlg})\to \mathrm{Fun}({\Delta }^1,\mathrm{Sp})$ and $T_{\mathrm{CAlg}}\to \mathrm{Mod}\to \mathrm{Sp}$ (where the second map is the forgetful functor) together merge into a functor $T_{\mathrm{CAlg}}\to \mathrm{Fun}({\Delta }^1{\coprod }_{\{0\}}{\Delta }^1,\mathrm{Sp})$, whose image consists of product diagrams in $\mathrm{Sp}$.

(ii) Let us denote an image of the above functor by the diagram

Then when passing to homotopy groups, the above diagram exhibits the graded commutative ring ${\pi }_{\bullet }(A\otimes M)$ as the canonical square zero extension of the graded commutative ring ${\pi }_{\bullet }(A)$ by the ${\pi }_{\bullet }(A)$-module ${\pi }_{\bullet }(M)$.

Sorry that I don’t know whether these properties are sufficient for our usage, nor whether these properties are characterizing.

From now on we fix an equivalence as in Proposition 4.1 (see the proof of [1] Theorem 1.78 for an explicit construction), and we use this equivalence to identify elements of $T_{\mathrm{CAlg}}$ with elements of $\mathrm{Mod}$.

5Properties of Cotangent Complexes of $E_{\infty }$-Rings

Proposition 5.1 ([1] Corollary 2.5). Let $f:A\to B$ be a morphism of connective $E_{\infty }$-rings. Assume that $f$ is $n$-connective, for some $n\in {\mathbb{Z}}_{\ge -1}$ (meaning that $\ker (f)$ is $n$-connective), then the relative cotangent complex $L_{B/A}$ is $(n+1)$-connective. The converse holds provided that $f$ induces an isomorphism ${\pi }_0(A)\to {\pi }_0(B)$.

Corollary 5.2 ([1] Corollary 2.6). Let $A$ be a connective $E_{\infty }$-ring, then $L_A$ is connective.

Proposition 5.3 ([1] Corollary 2.8). Let $f:A\to B$ be a map of connective $E_{\infty }$-rings. Assume that $f$ is $n$-connective for some $n\in {\mathbb{Z}}_{\ge -1}$. Then the induced map $L_A\to L_B$ is $n$-connective. In particular the canonical map ${\pi }_0{L}_A\to {\pi }_0{L}_{{\pi }_{0}A}$ is an isomorphism.

Proposition 5.4 ([1] Lemma 2.10). Let $A$ be a discrete $E_{\infty }$-ring, then ${\pi }_0{L}_A\cong {\Omega }_{{\pi }_{0}A}$ in the category of (discrete) ${\pi }_{0}A$-modules. Moreover, this isomorphism can be chosen to depend functorially on $A$.

Proposition 5.5 ([1] Proposition 2.11). Let $f:A\to B$ be a morphism of connective $E_{\infty }$-rings. Then:

(i) $L_{B/A}$ is connective.

(ii) ${\pi }_0{L}_{B/A}\cong {\Omega }_{{\pi }_{0}B/{\pi }_{0}A}$ in the category of (discrete) ${\pi }_{0}B$-modules. Moreover, this isomorphism can be chosen to depend functorially on $f$.

Lemma 5.6 ([1] Proposition 2.22). Let $A\to B$ be an etale morphism of $E_{\infty }$-rings, then $L_{B/A}$ vanishes.

References

[1]	Jacob Lurie, “Derived Algebraic Geometry IV: Deformation Theory”, October 8, 2009 version, Avalible for download at https://www.math.ias.edu/ lurie/papers/DAG-IV.pdf
[2]	Jacob Lurie, “Higher Algebra”, Avalible for download at https://www.math.ias.edu/ lurie/papers/HA.pdf
[3]	The Stacks project authors, “The Stacks project”, https://stacks.math.columbia.edu