用户: Scavenger/Simplicial Commutative Rings

Convention 0.1. We use the term “category” to mean $1$-categories, and the term “$\infty$-category” to mean $(\infty ,1)$-categories.

1Preliminaries on $1$-Colimits

Definition 1.1 ([12] tag 04E6). Let $H:I\to J$ be a functor between categories. We say that $H$ is $1$-right cofinal if for every object $x\in J$ the category $I{\times }_J{J}_{X/}$ is connected (as a simplicial set) (so in particular it is nonempty).

Remark 1.2. The terminology follows from [9], and in particular differs from the one of [12].

Warning 1.3. $1$-right cofinal functors usually don’t coincide with right cofinal functors of [9] tag 02N1.

Example 1.4. Let ${\Delta }_{\le 1}$ be the full subcategory of $\Delta$ spanned by the objects $[0]$ and $[1]$, where $\Delta$ is the simplex category of [9] tag 000A. Then the inclusion ${\Delta }_{\le 1}^{\mathrm{op}}\to {\Delta }^{\mathrm{op}}$ is $1$-right cofinal.

Proposition 1.5 ([12] tag 002R). Let $\mathcal{C}$ be a category, let $H:I\to J$ be a $1$-right cofinal functor of categories, and let $\overline{f}:J^{▹}\to \mathcal{C}$ be a functor. Then $\overline{f}$ is a colimit diagram if and only if $\overline{f}\circ H^{▹}$ is a colimit diagram.

Definition 1.6. A category $\mathcal{C}$ is called $1$-sifted if it is nonempty and the diagonal functor $\mathcal{C}\to \mathcal{C}\times \mathcal{C}$ is $1$-right cofinal.

Proposition 1.7. Let ${\mathrm{Set}}_{\Delta }$ be the category of simplicial sets, let $\mathrm{Cat}$ be the category of categories, then the homotopy category functor ${\mathrm{Set}}_{\Delta }\to \mathrm{Cat}$ of [9] tag 004N commutes with finite products.

Proof. This follows from the explicit construction of homotopy category given in the proof of [9] tag 004M.

$\square$

Corollary 1.8. Let $K$ be a simplicial set, $\mathcal{C}$ be a category, then the map $\mathrm{Fun}(N_{\bullet }(\text{h}K),\mathcal{C})\to \mathrm{Fun}(K,\mathcal{C})$ is an isomorphism of simplicia sets.

Proposition 1.9. Let $K$ be a simplicial set, then the map $K^{▹}\to N_{\bullet }(\text{h}K{)}^{▹}\cong N_{\bullet }((\text{h}K{)}^{▹})$ induces an isomorphism $\text{h}(K^{▹})\to (\text{h}K{)}^{▹}$.

Proof. This follows from the explicit construction of homotopy category given in the proof of [9] tag 004M.

$\square$

Corollary 1.10. Let $K$ be a simplicial set, $\mathcal{C}$ be a category. Then a diagram $K^{▹}\to \mathcal{C}$ is a colimit diagram if and only if the induced diagram $\text{h}(K^{▹})\cong (\text{h}K{)}^{▹}\to \mathcal{C}$ is a colimit diagram.

Proof. This follows from (the opposite of) [9] tag 02WA.

$\square$

Corollary 1.11. Let $\mathcal{C}$ be a category, then $\mathcal{C}$ admits (co)limits indexed by small simplicial sets if and only if $\mathcal{C}$ admits (co)limits indexed by small categories.

Notion 1.12. Let $\mathcal{C}$ be a small ($1$-)category. We use $1-sInd(\mathcal{C})$ to denote the category obtained by freely adjoining small 1-sifted colimits to $\mathcal{C}$. It is defined as the full subcategory of $\mathrm{Fun}({\mathcal{C}}^{\mathrm{op}},\mathrm{Set})$ spanned by those presheaves which can be written as a 1-sifted colimit of representable presheaves in $\mathrm{Fun}({\mathcal{C}}^{\mathrm{op}},\mathrm{Set})$ (See [1] Corollary 2.7). When $\mathcal{C}$ has finite coproducts these are precisely the functors that preserves finite products by the proof of [1] Corollary 2.8. This category satisfies a universal property similar to that of cocompletion of categories (See [1] Definition 2.2).

Proposition 1.13. Let $\mathrm{CRing}$ denote the category of ordinary commutative rings, let $\mathrm{Poly}\subseteq \mathrm{CRing}$ denote the full subcategory spanned by those rings that are finitely generated polynomial algebras over $\mathbb{Z}$. Then the natural map $1-sInd(\mathrm{Poly})\to \mathrm{CRing}$ induced by the universal property of $1-sInd$ and the inclusion $\mathrm{Poly}\to \mathrm{CRing}$ is an equivalence. An inverse of this functor is given by the following formula:

$A\to (R\to {\mathrm{Hom}}_{\mathrm{CRing}}(R,A))$

where $A\in \mathrm{CRing}$ and $R\in \mathrm{Poly}$.

Proof. See the discussion of [2] 5.1.1.

$\square$

2Sifted Colimits

Definition 2.1 ([8] Definition 5.5.8.1). A simplicial set $K$ is called sifted if it is nonempty and the diagonal map $K\to K\times K$ is right cofinal.

Example 2.2 ([8] Example 5.5.8.3, Lemma 5.5.8.4). A filtered $\infty$-category is sifted. $N_{\bullet }({\Delta }^{\mathrm{op}})$ is sifted.

Definition 2.3 ([8] Definition 5.5.8.8). Let $\mathcal{C}$ be a small $\infty$-category that admits finite coproducts. We let $P_{\Sigma }(\mathcal{C})$ denote the full subcategory of $P(\mathcal{C})$ spanned by those functors ${\mathcal{C}}^{\mathrm{op}}\to \mathcal{S}$ that presereve finite products.

Proposition 2.4. Let $\mathcal{C}$ be a small $\infty$-category that admits finite coproducts. Then $P_{\Sigma }(\mathcal{C})$ is presentable and is stable under sifted colimits as a full subcategory of $P(\mathcal{C})$. Moreover, the Yoneda embedding $\mathcal{C}\to P(\mathcal{C})$ factors through $P_{\Sigma }(\mathcal{C})$ and brings finite coproducts in $\mathcal{C}$ to finite coproducts in $P_{\Sigma }(\mathcal{C})$.

Proof. See [8] Proposition 5.5.8.10.

$\square$

Proposition 2.5 ([8] Proposition 5.5.8.15). Let $\mathcal{C}$ be a small $\infty$-category that admits finite coproducts, let $\mathcal{D}$ be an $\infty$-category that admits small filtered colimits and geometric realizations. Let ${\mathrm{Fun}}_{\Sigma }(P_{\Sigma }(\mathcal{C}),\mathcal{D})$ denote the full subcategory of $\mathrm{Fun}(P_{\Sigma }(\mathcal{C}),\mathcal{D})$ spanned by those functors $P_{\Sigma }(\mathcal{C})\to \mathcal{D}$ that preserve small filtered colimits and geometric realizations. Then

(i) Composition with the Yoneda embedding $\mathcal{C}\to P_{\Sigma }(\mathcal{C})$ induces an equivalence ${\mathrm{Fun}}_{\Sigma }(P_{\Sigma }(\mathcal{C}),\mathcal{D})\to \mathrm{Fun}(\mathcal{C},\mathcal{D})$.

(ii) Any functor $g\in {\mathrm{Fun}}_{\Sigma }(P_{\Sigma }(\mathcal{C}),\mathcal{D})$ preserves sifted colimits.

(iii) Assume that $\mathcal{D}$ admits finite coproducts, then a functor $g\in {\mathrm{Fun}}_{\Sigma }(P_{\Sigma }(\mathcal{C}),\mathcal{D})$ preserves small colimits if and only if the composition $\mathcal{C}\to P_{\Sigma }(\mathcal{C})\to \mathcal{D}$ preserves finite coproducts.

Proposition 2.6 ([8] Corollary 5.5.8.17). Let $f:\mathcal{C}\to \mathcal{D}$ be a functor between $\infty$-categories, where $\mathcal{C}$ admits small colimits. Then $f$ preserves small sifted colimits if and only if $f$ preserves small filtered colimits and geometric realizations.

3Classical Homotopy Theory of Simplicial Commutative Rings

Definition 3.1. A simplicial commutative ring is a simplicial object in the category of commutative rings. We use $\mathrm{SCR}$ to denote the category of simplicial commutative rings.

Notice that the data of a simplicial commutative ring is equivalent to that of a ring object in the category of simplicial sets, so we have a natural notion of the underlying simplicial set of a simplicial commutative ring. We often view the underlying simplicial set of a simplicial commutative ring as a pointed simplicial set with 0 the specified point.

Construction 3.2. Let $R$ be an ordinary ring, then we can view $R$ as a simplicial commutative ring whose underlying simplicial set is the discrete simplicial set associated to the underlying set of $R$.

Proposition 3.3. The underlying simplicial set of a simplicial commutative ring is a Kan complex.

Proof. Immediate consequence of [9] tag 00MG or [12] tag 08NZ.

$\square$

Proposition 3.4. Let $R$ be a simplicial commutatice ring, which we regard as a pointed Kan complex with the specified point 0. The homotopy groups ${\pi }_{\ast }R$ have the structure of a graded commutative ring such that the additive group structure on ${\pi }_{n}R$ for $n\ge 1$ coincides with the homotopy group structure.

Proof. The proof is not so hard so we only give a sketch. Let $S^n:=S^1\wedge \cdots \wedge S^1$ be the $n$-dimensional sphere, so that ${\pi }_{n}X\simeq {\pi }_0({\mathrm{Fun}}_{\ast /}(S^n,X))$ for every pointed Kan complex $X$. (For relevant notions, see [3] 3.8.8). Let $f,g\in {\pi }_{n}R$, define $f+g$ to be $S^n\stackrel{(f,g)}{\to }R\times R\stackrel{+}{\to }R$. To check that this coincides with the homotopy group structure when $n\ge 1$ one use the Eckmann-Hilton argument. For $f\in {\pi }_{m}R$ and $g\in {\pi }_{n}R$ we define $f\times g$ to be $S^{m+n}\simeq S^m\wedge S^n\stackrel{f\wedge g}{\to }R\wedge R\stackrel{\times }{\to }R$. It is easy to verify that this indeed gives a graded commutative ring structure. Notice that the graded commutativity instead of strict commutativity results from the fact that the flip automorphism $S^1\wedge S^1\to S^1\wedge S^1$ creates a sign $-1$.

$\square$

The Simplicial Model Structure

We use [4] and [5] as our references.

Construction 3.5. Let $\mathcal{C}$ be a category with coproducts. Let $sC$ be the category of simplicial objects of $\mathcal{C}$. We define a functor $\otimes :\mathrm{sSet}\times s\mathcal{C}\to s\mathcal{C}$ given by $(X\otimes C{)}_n={\coprod }_{X_n}{C}_n$ for $X\in \mathrm{sSet}$ and $C\in s\mathcal{C}$. We give $s\mathcal{C}$ an enrichment over $\mathrm{sSet}$ by defining ${\mathrm{Mor}}_{s\mathcal{C}}(X,Y)$ to satisfy the universal property ${\mathrm{Hom}}_{\mathrm{sSet}}(K,{\mathrm{Hom}}_{s\mathcal{C}}(X,Y))\cong {\mathrm{Hom}}_{s\mathcal{C}}(K\otimes X,Y)$. It is not hard to check that this indeed gives a simplicial enrichment.

Choosing $\mathcal{C}$ to be the category of commutative rings we obtain a simplicial enrichment on $\mathrm{SCR}$.

Proposition 3.6. There exists a combinatorial simplicial model category structure on $\mathrm{SCR}$ such that a morphism $f:X\to Y$ is

(i) a weak equivalence if and only if $f$ is a homotopy equivalence on underlying Kan complexes.

(ii) a fibration if and only if $f$ is a Kan fibration on underlying Kan complexes.

Proof. Combine [5] Theorem 4.17 and Proposition 1.23. For the property of being combinatorial, we refer to [11].

$\square$

Example 3.7. For $n\in {\mathbb{Z}}_{\ge 0}$, $\mathbb{Z}[x_1,...,x_n]$ is cofibrant by [5] Proposition 4.21.

4Animated Rings

Definition 4.1. Let $\mathrm{Poly}$ denote the full subcategory of the category of ordinary commutative rings spanned by those rings $R$ that are finitely generated polynomial algebras over $\mathbb{Z}$. We call the $\infty$-category $P_{\Sigma }(\mathrm{Poly})$ the $\infty$-category of animated rings, and denote it by $\mathrm{AniRing}$. Elements of $\mathrm{AniRing}$ are called animated rings. ($\mathrm{Poly}$ is essentially small so there’re no size problems.) $\mathrm{AniRing}$ is presentable by Proposition 2.4.

Relation to Simplicial Commutative Rings

Construction 4.2. Let $\mathcal{C}$ be a small ($1$-)category, the category $\mathrm{Fun}(\mathcal{C},{\mathrm{Set}}_{\Delta })$ admits a simplicial enrichment defined by the tensor product

$\otimes :{\mathrm{Set}}_{\Delta }\times \mathcal{C}\to \mathcal{C},(K\otimes F)(C)=K\times F(C)$

where $K\in {\mathrm{Set}}_{\Delta },F\in \mathrm{Fun}(\mathcal{C},{\mathrm{Set}}_{\Delta }),C\in \mathcal{C}$. (i.e. ${\mathrm{Mor}}_{{\mathrm{Set}}_{\Delta }}(K,\mathrm{Mor}(A,B))=\mathrm{Mor}(K\otimes A,B)$, where $K$ is a simplicial set and $A,B\in \mathrm{Fun}(\mathcal{C},{\mathrm{Set}}_{\Delta })$.)

If $\mathcal{C}$ admits finite products, let $A$ denote the full subcategory of $\mathrm{Fun}(\mathcal{C},{\mathrm{Set}}_{\Delta })$ spanned by those functors which preserve finite products. $A$ inherits a simplicial enrichment as a full subcategory of a simplicial category.

We define a model structure on $A$ by the following rules:

(i) A natural transformation of functors $F\to F^{\prime }$ is a weak equivalence if and only if $F(C)\to F^{\prime }(C)$ is a weak homotopy equivalence for each $C\in \mathcal{C}$.

(ii) A natural transformation of functors $F\to F^{\prime }$ is a fibration if and only if $F(C)\to F^{\prime }(C)$ is a Kan fibration for each $C\in \mathcal{C}$.

This model structure is well defined and compatible with the simplicial enrichment by [8] Proposition 5.5.9.1, and thus defines a simplicial model structure on $A$.

Now the natural functor $\mathcal{C}\times A\to {\mathrm{Set}}_{\Delta }$ is a simplicially enriched functor and thus induces a functor $N_{\bullet }(C)\times N_{\mathrm{hc}}(A)\to N_{\mathrm{hc}}({\mathrm{Set}}_{\Delta })$, which induces a functor $N_{\mathrm{hc}}(A)\to \mathrm{Fun}(\mathcal{C},N_{\mathrm{hc}}({\mathrm{Set}}_{\Delta }))$. This functor restricts to a functor $N_{\mathrm{hc}}(A^{\circ })\to \mathrm{Fun}(\mathcal{C},\mathcal{S})$, where $A^{\circ }$ means the full subcategory of $A$ spanned by the fibrant-cofibrant objects. It is easy to see that this functor is given on objects by taking homotopy coherent nerve, thus restricts to a functor $N_{\mathrm{hc}}(A^{\circ })\to P_{\Sigma }({\mathcal{C}}^{\mathrm{op}})$. This functor is an equivalence of $\infty$-categories by [8] Corollary 5.5.9.3.

4.3. Now let $\mathcal{C}={\mathrm{Poly}}^{\mathrm{op}}$ in Construction 4.2. We have an equivalence of $\infty$-categories $N_{\mathrm{hc}}(A^{\circ })\to P_{\Sigma }({\mathcal{C}}^{\mathrm{op}})=\mathrm{AniRing}$, where $A$ is the simplicial model category consisting of functors $\mathcal{C}\to {\mathrm{Set}}_{\Delta }$ which preserve finite products. We now explore what $A$ is.

$A$ is a full subcategory of $\mathrm{Fun}(\mathcal{C},\mathrm{Fun}({\Delta }^{\mathrm{op}},\mathrm{Set}))\cong \mathrm{Fun}({\Delta }^{\mathrm{op}},\mathrm{Fun}(\mathcal{C},\mathrm{Set}))$. It is easy to see that this induces an isomorphism $A\cong \mathrm{Fun}({\Delta }^{\mathrm{op}},1-sInd({\mathcal{C}}^{\mathrm{op}}))$, which is equivalent to $\mathrm{Fun}({\Delta }^{\mathrm{op}},\mathrm{CRing})$. It is not hard to check that this equivalence is compatible with the simplicial model structures given in Construction 4.2 and Section 3.

Thus we have an equivalence $N_{\mathrm{hc}}({\mathrm{SCR}}^{\circ })\to \mathrm{AniRing}$, where ${\mathrm{SCR}}^{\circ }$ is the full subcategory of $\mathrm{SCR}$ spanned by cofibrant simplicial commutative rings. This equivalence carries a simplicial commutative ring $A$ to the homotopy coherent nerve of the functor $R\to {\mathrm{Hom}}_{\mathrm{SCR}}(R,A)$, where $R\in \mathrm{Poly}$.

We now explore further properties. For a simplicial category $\mathcal{C}$ we use $d\mathcal{C}$ to denote the underlying $1$-category, so for example $\mathrm{dSCR}$ means the $1$-category of simplicial commutative rings, and $\mathrm{SCR}$ is $\mathrm{dSCR}$ with the simplicial enrichment.

Since $\mathrm{dSCR}$ is a combinatorial model category and every object of it is fibrant, there exists a cofibrant replacement functor $r:\mathrm{dSCR}\to {\mathrm{dSCR}}^{\circ }$ and a natural transformation $\alpha :i\circ r\to {\mathrm{id}}_{\mathrm{dSCR}}$, where $i:{\mathrm{dSCR}}^{\circ }\mathrm{dSCR}$ is the inclusion, such that $\alpha (x)$ is a weak equivalence for every $x\in \mathrm{dSCR}$. The functor $N_{\bullet }({\mathrm{dSCR}}^{\circ })\to N_{\mathrm{hc}}({\mathrm{SCR}}^{\circ })$ exhibits $N_{\mathrm{hc}}({\mathrm{SCR}}^{\circ })$ as the ($\infty$-categorical) localization of $N_{\bullet }({\mathrm{dSCR}}^{\circ })$ at the weak equivalences by [7] Theorem 1.3.4.20, thus the composite $N_{\bullet }(\mathrm{dSCR})\stackrel{r}{\to }{N}_{\bullet }({\mathrm{dSCR}}^{\circ })\to N_{\mathrm{hc}}({\mathrm{SCR}}^{\circ })$ exhibits $N_{\mathrm{hc}}({\mathrm{SCR}}^{\circ })$ as the localization of $N_{\bullet }(\mathrm{dSCR})$ at the weak equivalences. Composing with the equivalence $N_{\mathrm{hc}}({\mathrm{SCR}}^{\circ })\to \mathrm{AniRing}$ obtained above, we get the following result:

Proposition 4.4. The hom functor $\mathrm{Poly}\times \mathrm{dSCR}\to \mathrm{Kan}$ (where we view $\mathrm{Poly}$ as a full subcategory of $\mathrm{dSCR}$) gives rise to a functor $N_{\bullet }(\mathrm{Poly})\times N_{\bullet }(\mathrm{dSCR})\to \mathcal{S}$, which gives a functor $N_{\bullet }(\mathrm{dSCR})\to \mathrm{AniRing}$ by currying. This functor exhibits $\mathrm{AniRing}$ as the ($\infty$-categorical) localization of $\mathrm{dSCR}$ at the weak equivalences and in particular it is essentially surjective. Its restriction to the subcategory $\mathrm{CRing}\subseteq \mathrm{dSCR}$ is fully faithful (see Proposition 1.13, in particular its essential image is $1-sInd(\mathrm{Poly})\subseteq \mathrm{AniRing}$). Further restricting to $\mathrm{Poly}$ gives the Yoneda embedding $\mathrm{Poly}\to P_{\Sigma }(\mathrm{Poly})$.

Relation to Connective $E_{\infty }$-Rings

Notation 4.5. We use $E_{\infty }\text{-}\mathrm{Alg}$ to denote the $\infty$-category of $E_{\infty }$-rings, and use $E_{\infty }\text{-}{\mathrm{Alg}}^{\mathrm{cn}}$ to denote the full subcategory of connective $E_{\infty }$-rings. Let ${\mathrm{Sp}}^{\mathrm{cn}}$ denote the $\infty$-category of connective spectra.

Notion 4.6. Since $E_{\infty }\text{-}{\mathrm{Alg}}_{\mathbb{Z}/}^{\mathrm{cn}}$ admits colimits, using the universal property of $P_{\Sigma }$ and the natural inclusion $\mathrm{Poly}\to E_{\infty }\text{-}{\mathrm{Alg}}_{\mathbb{Z}/}^{\mathrm{cn}}$ we obtain a functor $\theta :\mathrm{AniRing}\to E_{\infty }\text{-}{\mathrm{Alg}}_{\mathbb{Z}/}^{\mathrm{cn}}$, which is well defined up to equivalence. $\theta$ preserves small limits and colimits by [10] Proposition 25.1.2.2 (1) (and consequently it admits left and right adjoints since the source and target $\infty$-categories are presentable), and the two functors $\mathrm{AniRing}\stackrel{\theta }{\to }{E}_{\infty }\text{-}{\mathrm{Alg}}_{\mathbb{Z}/}^{\mathrm{cn}}\to \mathrm{Sp}\stackrel{{\Omega }^{\infty }}{\to }\mathcal{S}$ and $\mathrm{AniRing}\stackrel{{\mathrm{ev}}_{\mathbb{Z}[x]}}{\to }\mathcal{S}$ are equivalent and preserves sifted colimits by the proof of [10] Proposition 25.1.2.2 (2) and consequently it is conservative. As a consequence, $\mathrm{AniRing}\to E_{\infty }\text{-}{\mathrm{Alg}}^{\mathrm{cn}}$ is conservative, and preserves small sifted colimits.

Now we have two ways to define homotopy groups of an animated ring: the first is through the functor $\theta$ constructed above, and the second through the model of simplicial commutative rings of Proposition 4.4 and the homotopy groups of simplicial commutative rings of Proposition 3.4. We next roughly prove that the two ways are equivalent (as functors $\mathrm{AniRing}\to \mathrm{GCR}$, where $\mathrm{GCR}$ is the category of graded commutative rings), with lots of details omitted.

Our strategy is as follows: the map $E_{\infty }\text{-}{\mathrm{Alg}}^{\mathrm{cn}}\to {\mathcal{S}}_{\ast }$ extends to a “multiplication law”, which is roughly a functor bringing a connective $E_{\infty }$-ring $A$ to a map $A\wedge A\to A$ in ${\mathcal{S}}_{\ast }$, where we identify $A$ with its underlying space ${\Omega }^{\infty }A$. The multiplication law of the graded commutative ring ${\pi }_{\bullet }(A)$ is induced from this map by taking homotopy groups. There are also “addition law” and “unit law”. Let’s consider only multiplication law for example, and the general case can be treated similarly.

Now there are two multiplication laws for $\mathrm{AniRing}$: the first one is obtained by composing with $\mathrm{AniRing}\to E_{\infty }\text{-}{\mathrm{Alg}}^{\mathrm{cn}}$, and the second one is obtained by using the model of simplicial rings. Our aim is to prove that this two functors are equivalent. For this purpose it suffice to prove that they coincide when being restricted to $\mathrm{Poly}$, and they both preserve sifted colimits. The first claim is easy, and it remains to deal with the second (and some other subtleties).

We first rigorously define the map of multiplication law. We only treat the case of $E_{\infty }$-rings for the sake of simplicity. The $\infty$-category ${\mathcal{S}}_{\ast }$ has a canonical symmetric monoidal structure, given by the smash product, which preserves colimits separately in each variable, and the map ${\Sigma }^{\infty }:{\mathcal{S}}_{\ast }\to \mathrm{Sp}$ is symmetric monoidal. We thus obtain a map $f:{\mathcal{S}}_{\ast }\stackrel{\Delta }{\to }{\mathcal{S}}_{\ast }\times {\mathcal{S}}_{\ast }\stackrel{\wedge }{\to }{\mathcal{S}}_{\ast }$. Consider the map $f^{\prime }:{\mathcal{S}}_{\ast }\to {\mathcal{S}}_{\ast }\times {\mathcal{S}}_{\ast }$ defined by coordinates $(f,\mathrm{id})$. The target $\infty$-category of the multiplication law functor is defined to be $M:=\mathrm{Fun}({\Delta }^1,{\mathcal{S}}_{\ast }){\times }_{\mathrm{Fun}(\partial {\Delta }^1,{\mathcal{S}}_{\ast })}{\mathcal{S}}_{\ast }$. Let $A$ be a connective $E_{\infty }$-ring, then we have a natural map of spectra $A\otimes A\to A$, and this map is functorial in $A$. Using adjunctions this induces a map $A\wedge A\to A$ in ${\mathcal{S}}_{\ast }$ functorial in $A$, and thus can be promoted to a functor $E_{\infty }\text{-}{\mathrm{Alg}}^{\mathrm{cn}}\to M$.

We next prove that $M$ admits sifted colimits and that $E_{\infty }\text{-}{\mathrm{Alg}}^{\mathrm{cn}}\to M$ preserves sifted colimits. Using [8] Proposition 5.5.8.6 we see that $f$ and hence $f^{\prime }$ preserves sifted colimits. Using [9] tag 02X6, tag 02KQ and tag 030M we see that $M$ admits sifted colimits and the functor $E_{\infty }\text{-}{\mathrm{Alg}}^{\mathrm{cn}}\to M$ preserves sifted colimits since $E_{\infty }\text{-}{\mathrm{Alg}}^{\mathrm{cn}}\to {\mathcal{S}}_{\ast }$ preserves sifted colimits.

Summarizing all we’ve done above we obtain the following

Proposition 4.7. The two ways of defining ${\pi }_{\bullet }:\mathrm{AniRing}\to \mathrm{GCR}$ yields equivalent functors, where $\mathrm{GCR}$ is the category of graded commutative rings.

Truncations of Animated Rings

Proposition 4.8. Let $n\in {\mathbb{Z}}_{\ge 0}$. An animated ring $A$ is $n$-truncated as an object of $\mathrm{AniRing}$ if and only if its underlying space is $n$-truncated.

Proof. Omitted.

$\square$

4.9. Let $n\in {\mathbb{Z}}_{\ge 0}$. Since the truncation functor ${\tau }_{\le n}:\mathcal{S}\to \mathcal{S}$ preserves finite product, the $n$-truncation functor of $\mathrm{AniRing}$ is induced by the $n$-truncation functor of $\mathcal{S}$.

Proposition 4.10. Let $n\in {\mathbb{Z}}_{\ge 0}$, let $A$ be an animated ring, let $A\to {\tau }_{\le n}A$ be a map of animated rings that exhibits ${\tau }_{\le n}A$ as an $n$-truncation of $A$, then the map $\theta (A)\to \theta ({\tau }_{\le n}A)$ exhibits $\theta ({\tau }_{\le n}A)$ as the $n$-truncation of $\theta (A)$.

Proof. Consider homotopy groups.

$\square$

Modules over Animated Rings

Notion 4.11. We let $\mathrm{SCRMod}$ denote the fiber product ${\mathrm{Mod}}^{\mathrm{Comm}}(\mathrm{Sp}){\times }_{\mathrm{CAlg}(\mathrm{Sp})}\mathrm{AniRing}$. Let $A\in \mathrm{AniRing}$ be an animated , we let ${\mathrm{SCRMod}}_A$ denote the fiber product $\mathrm{SCRMod}{\times }_{\mathrm{AniRing}}\{A\}$, so there is a canonical isomorphism ${\mathrm{SCRMod}}_A\cong {\mathrm{Mod}}_{\theta (A)}$. We use ${\mathrm{SCRMod}}^{\mathrm{cn}}$ and ${\mathrm{Mod}}^{\mathrm{cn}}$ to denote the full subcategories spanned by the connective modules.

4.12. Let $A$ be an animated ring, then the forgetful functor $E_{\infty }\text{-}{\mathrm{Alg}}_{\theta (A)/}^{\mathrm{cn}}\to {\mathrm{Mod}}_{\theta (A)}^{\mathrm{cn}}$ has a left adjoint by [7] Example 3.1.3.14. The functor ${\mathrm{AniRing}}_{A/}\to E_{\infty }\text{-}{\mathrm{Alg}}_{\theta (A)/}^{\mathrm{cn}}$ also has a left adjoint by [8] Proposition 5.2.5.1. So we see that the forgetful functor ${\mathrm{AniRing}}_{A/}\to {\mathrm{SCRMod}}_A^{\mathrm{cn}}$ has a left adjoint.

4.13. [7] Corollary 3.2.3.5 (2) shows that ${\mathrm{Alg}}_{\mathrm{LMod}}({\mathrm{Sp}}^{\mathrm{cn}})$ and ${\mathrm{Alg}}_{\mathrm{Assoc}}({\mathrm{Sp}}^{\mathrm{cn}})$ are presentable. The opposite of [7] Proposition 7.3.2.6 in the special case where $\mathcal{C}=\mathcal{E}$ and $q={\mathrm{id}}_{\mathcal{C}}$ yields the following:

Proposition 4.14. Let $p:\mathcal{D}\to \mathcal{C}$ be a cocartesian fibration of $\infty$-categories such that for each $c\in \mathcal{C}$ the fiber $\mathcal{D}{\times }_{\mathcal{C}}\{c\}$ admits a final object, then $p$ has a right adjoint and in particular preserves all colimits which exist in $\mathcal{D}$.

Applying the above proposition to the presentable fibration $f:{\mathrm{Alg}}_{\mathrm{LMod}}({\mathrm{Sp}}^{\mathrm{cn}})\to {\mathrm{Alg}}_{\mathrm{Assoc}}({\mathrm{Sp}}^{\mathrm{cn}})$ ($f$ is indeed a presentable fibration by virtue of [7] Corollary 4.2.3.7) we see that $f$ preserves small colimits. Using [7] Proposition 3.2.3.1 (4) we see that the functor $\mathrm{CAlg}({\mathrm{Sp}}^{\mathrm{cn}})\to {\mathrm{Alg}}_{\mathrm{Assoc}}({\mathrm{Sp}}^{\mathrm{cn}})$ preserves sifted colimits and in particular is accessible. Using [8] Lemma 5.4.5.5, Proposition 5.4.6.6 and [7] Proposition 4.5.1.4 we see that the $\infty$-category $\mathrm{Mod}({\mathrm{Sp}}^{\mathrm{cn}})$ is accessible and the functor $\mathrm{Mod}({\mathrm{Sp}}^{\mathrm{cn}})\to {\mathrm{Alg}}_{\mathrm{LMod}}({\mathrm{Sp}}^{\mathrm{cn}})$ preserves sifted colimits, and the map $\mathrm{Mod}({\mathrm{Sp}}^{\mathrm{cn}})\to \mathrm{CAlg}({\mathrm{Sp}}^{\mathrm{cn}})$ is a presentable fibration. Using [9] tag 02X6 and tag 031R we see that $\mathrm{Mod}({\mathrm{Sp}}^{\mathrm{cn}})$ admits small colimits and the map $\mathrm{Mod}({\mathrm{Sp}}^{\mathrm{cn}})\to \mathrm{CAlg}({\mathrm{Sp}}^{\mathrm{cn}})$ preserves small colimits, in particular $\mathrm{Mod}({\mathrm{Sp}}^{\mathrm{cn}})$ is presentable. Using the above argument again we see that ${\mathrm{SCRMod}}^{\mathrm{cn}}$ is presentable and the forgetful functor ${\mathrm{SCRMod}}^{\mathrm{cn}}\to \mathrm{AniRing}$ preservs small colimits.

Using [7] Proposition 3.2.3.1 (4) we see that the forgerful functor ${\mathrm{Alg}}_{\mathrm{LMod}}({\mathrm{Sp}}^{\mathrm{cn}})\to {\mathrm{Sp}}^{\mathrm{cn}}$ and hence ${\mathrm{SCRMod}}^{\mathrm{cn}}\to {\mathrm{Sp}}^{\mathrm{cn}}$ preserves sifted colimits.

Proposition 4.15 ([10] 25.2.1.2). Let $\mathcal{C}\subseteq {\mathrm{SCRMod}}^{\mathrm{cn}}$ denote the full subcategory spanned by those pairs $(A,M)$ where $A$ is a discrete animated ring equivalent to $\mathbb{Z}[x_1,...,x_n]$ for some $n\in {\mathbb{Z}}_{\ge 0}$, and $M$ is a discrete $A$-module equivalent to $A^m$ for some $m\in {\mathbb{Z}}_{\ge 0}$. Then the inclusion $\mathcal{C}\to {\mathrm{SCRMod}}^{\mathrm{cn}}$ extends to an equivalence $P_{\Sigma }(\mathcal{C})\to {\mathrm{SCRMod}}^{\mathrm{cn}}$.

4.16. Let $\mathrm{dMod}$ denote the ($1$-)category of pairs $(A,M)$ where $A$ is an ordinary commutative ring and $M$ is a discrete $A$-module. Let $\mathrm{sdMod}$ denote the category of simplicial objects of $\mathrm{dMod}$. $\mathrm{sdMod}$ admits a natural combinatorial simplicial model structure ([8] Proposition 5.5.9.1) and ${\mathrm{SCRMod}}^{\mathrm{cn}}$ is equivalent to the localization of $\mathrm{sdMod}$ at the equivalences ( $(A,M)\to (B,N)$ is an equivalence if $A\to B$ and $M\to N$ are equivalences of Kan complexes) by Proposition 4.15.

5Localizations of Animated Rings

Local Objects

Definition 5.1 ([8] Definition 5.5.4.1). Let $\mathcal{C}$ be an $\infty$-category and $S$ an collection of morphisms of $\mathcal{C}$. We say that an object $Z$ of $\mathcal{C}$ is $S$-local if, for every morphism $s:X\to Y$ belonging to $S$, composition with $s$ induces an equivalence ${\mathrm{Hom}}_{\mathcal{C}}(Y,Z)\to {\mathrm{Hom}}_{\mathcal{C}}(X,Z)$ of spaces.

A morphism $f:X\to Y$ of $\mathcal{C}$ is an $S$-equivalence if, for every $S$-local object $Z$, composition with $f$ induces an equivalence ${\mathrm{Hom}}_{\mathcal{C}}(Y,Z)\to {\mathrm{Hom}}_{\mathcal{C}}(X,Z)$ of spaces.

Proposition 5.2 ([8] Proposition 5.5.4.15). Let $\mathcal{C}$ be a presentable $\infty$-category and let $S$ be a small set of morphisms of $\mathcal{C}$. Let ${\mathcal{C}}^{\prime }\subseteq \mathcal{C}$ denote the full subcategory spanned by the $S$-local objects, then ${\mathcal{C}}^{\prime }$ is a reflective subcategory of $\mathcal{C}$.

Now we turn to localizations of animated rings.

Proposition 5.3. Let $A$ be an animated ring, let $F\subseteq {\pi }_0(A)\simeq {\pi }_0({\mathrm{Hom}}_{{\mathrm{SCRMod}}_A}(A,A))$ be a subset. Then there exists a morphism of animated rings $A\to A[F^{-1}]$ such that for all $B\in {\mathrm{AniRing}}_{A/}$ the Kan complex ${\mathrm{Hom}}_{{\mathrm{AniRing}}_{A/}}(A[F^{-1}],B)$ is nonempty if and only if the image of all $f\in F$ under the map ${\pi }_0(A)\to {\pi }_0(B)$ is invertible. Further, if it is nonempty, then it is contractible.

Proof. Let $\mathrm{Sym}:{\mathrm{SCRMod}}_A^{\mathrm{cn}}\to {\mathrm{AniRing}}_{A/}$ be a left adjoint to the forgerful functor ${\mathrm{AniRing}}_{A/}\to {\mathrm{SCRMod}}_A^{\mathrm{cn}}$. Consider the set $S:=\{\mathrm{Sym}(f):\mathrm{Sym}(A)\to \mathrm{Sym}(A)\mid f\in F\}$. Set $A\to A[F^{-1}]$ to be a $S$-equivalence such that $A[F^{-1}]$ is $S$-local as an object of ${\mathrm{AniRing}}_{A/}$.

An object $B\in {\mathrm{AniRing}}_{/A}$ is $S$-local if and only if for every $f\in F$, the map ${\mathrm{Hom}}_{{\mathrm{SCRMod}}_A}(A,B)\to {\mathrm{Hom}}_{{\mathrm{SCRMod}}_A}(A,B)$ induced by $f:A\to A$ is an equivalence. This condition is equivalent to the multiplication of $f$ on ${\pi }_i(B)$ (via the map $f\in {\pi }_0(A)\to {\pi }_0(B)$) being an equivalence for all $i\in {\mathbb{Z}}_{\ge 0}$. Therefore an $A$-algebra $B$ is $S$-local if and only if the image every $f\in F$ under the map ${\pi }_0(A)\to {\pi }_0(B)$ is invertible.

Now let $B$ be an $A$-algebra. If $B$ is $S$-local, then ${\mathrm{Hom}}_{{\mathrm{AniRing}}_{A/}}(A[F^{-1}],B)$ is contractible hence nonempty. Conversely, if ${\mathrm{Hom}}_{{\mathrm{AniRing}}_{A/}}(A[F^{-1}],B)$ is nonempty, then it is easy to see that $B$ is $S$-local.

$\square$

6Cotangent Complexes of Animated Rings

Construction 6.1. Let $\mathcal{C}$ be the category of Proposition 4.15, the construction $(A,M)\to A\oplus M$ determines a functor $\mathcal{C}\to \mathrm{CRing}\to \mathrm{AniRing}$, which extends to a functor ${\mathrm{SCRMod}}^{\mathrm{cn}}\to \mathrm{AniRing}$ preserving small sifted colimits, which we denote by $\mathrm{sze}$. Notice that we have a commutative diagram

so $\mathrm{sze}$ extends to a functor $\overline{sze}:{\mathrm{SCRMod}}^{\mathrm{cn}}\to \mathrm{Fun}({\Delta }^2,\mathrm{AniRing})$, whose restriction to $\{1\}$ is equal to $\mathrm{sze}$, and the restriction to $\{0,2\}$ is equal to the composite ${\mathrm{SCRMod}}^{\mathrm{cn}}\to \mathrm{AniRing}\to \mathrm{Fun}({\Delta }^1,\mathrm{AniRing})$, where the first functor is the forgerful functor, and the second functor is the diagonal map.

The functor ${\mathrm{SCRMod}}^{\mathrm{cn}}\stackrel{\mathrm{sze}}{\to }\mathrm{AniRing}\to {\mathrm{Sp}}^{\mathrm{cn}}$ is equivalent to the functor ${\mathrm{SCRMod}}^{\mathrm{cn}}\to {\mathrm{Sp}}^{\mathrm{cn}}\times {\mathrm{Sp}}^{\mathrm{cn}}\to {\mathrm{Sp}}^{\mathrm{cn}}$, where the first functor is given by $(A,M)\to A$ and $(A,M)\to M$, and the second functor is given by $(X,Y)\to X\times Y$.

We take the simplicial model structure of 4.16 into account. In this case $(A,M)\to A\oplus M$ gives a functor $\mathrm{sdMod}\to \mathrm{dSCR}$. Localizing at the equivalences we obtain a functor ${\mathrm{SCRMod}}^{\mathrm{cn}}\to \mathrm{AniRing}$. It is easy to see that this functor is induced by the functor $\mathrm{Poly}\to \mathcal{C},A\to (A,{\Omega }_A)$, where $\mathcal{C}$ is the category in Proposition 4.15, under the identifications $\mathrm{AniRing}\simeq P_{\Sigma }(\mathrm{Poly})$ and ${\mathrm{SCRMod}}^{\mathrm{cn}}\simeq P_{\Sigma }(\mathcal{C})$, and in particular it preserves sifted colimits, and consequently coincides with $\mathrm{sze}$.

Lemma 6.2. Let $\mathcal{C}$ be an $\infty$-category, $e:X\to Y$ a morphism of $\mathcal{C}$. Then the functor $\mathrm{Fun}({\Delta }^2,\mathcal{C}){\times }_{\mathrm{Fun}(\{0,2\},\mathcal{C})}\{e\}\to \mathrm{Fun}({\Delta }^1,\mathcal{C}){\times }_{\mathrm{Fun}(\{1\},\mathcal{C})}\{Y\}$ is a left fibration.

Proof. Omitted.

$\square$

Construction 6.3. Consider the functor ${\mathrm{SCRMod}}^{\mathrm{cn}}\stackrel{\overline{sze}}{\to }\mathrm{Fun}({\Delta }^2,\mathrm{AniRing})\to \mathrm{Fun}({\Lambda }_2^2,\mathrm{AniRing})$, let $\mathrm{SCRDer}$ denote the $\infty$-category ${\mathrm{SCRMod}}^{\mathrm{cn}}{\times }_{\mathrm{Fun}({\Lambda }_2^2,\mathrm{AniRing})}\mathrm{Fun}({\Delta }^2,\mathrm{AniRing})$, so we have an isofibration $\mathrm{SCRDer}\to {\mathrm{SCRMod}}^{\mathrm{cn}}$. Let $A$ be an animated ring, then using Lemma 6.2, we see that $\mathrm{SCRDer}{\times }_{\mathrm{AniRing}}\{A\}\to {\mathrm{SCRMod}}^{\mathrm{cn}}{\times }_{\mathrm{AniRing}}\{A\}$ is a left fibration.

Definition 6.4. Let $A$ be an animated ring. A universal derivation for $A$ is an element $\eta$ of $\mathrm{SCRDer}{\times }_{\mathrm{AniRing}}\{A\}$ such that for each $N\in {\mathrm{SCRMod}}^{\mathrm{cn}}{\times }_{\mathrm{AniRing}}\{A\}$, the map ${\mathrm{Hom}}_{{\mathrm{SCRMod}}_A^{\mathrm{cn}}}(M,N)\to {\mathrm{Hom}}_{\mathcal{S}}(\mathrm{SCRDer}{\times }_{{\mathrm{SCRMod}}^{\mathrm{cn}}}\{M\},\mathrm{SCRDer}{\times }_{{\mathrm{SCRMod}}^{\mathrm{cn}}}\{N\})\stackrel{{\mathrm{ev}}_{\eta }}{\to }\mathrm{SCRDer}{\times }_{{\mathrm{SCRMod}}^{\mathrm{cn}}}\{N\}$ is a homotopy equivalence, where $M$ is the image of $\eta$ in ${\mathrm{SCRMod}}^{\mathrm{cn}}$. In view of [9] tag 02J8, this is equivalent to $\eta$ being an initial object of $\mathrm{SCRDer}{\times }_{\mathrm{AniRing}}\{A\}$.

Proposition 6.5 ([10] Proposition 25.3.1.5). Let $A$ be an animated ring, then there exists a universal derivation for $A$.

Lemma 6.6. The map $\mathrm{SCRDer}\to \mathrm{AniRing}$ is a cartesian fibration, and the map $\mathrm{SCRDer}\to {\mathrm{SCRMod}}^{\mathrm{cn}}$ brings $(\mathrm{SCRDer}\to \mathrm{AniRing})$-cartesian edges to $({\mathrm{SCRMod}}^{\mathrm{cn}}\to \mathrm{AniRing})$-cartesian edges.

Proof. Use the fact that whenever $(A,M)\to (B,M)$ is a $({\mathrm{SCRMod}}^{\mathrm{cn}}\to \mathrm{AniRing})$-cartesian edge of ${\mathrm{SCRMod}}^{\mathrm{cn}}$, the diagram

is a pullback square in $\mathrm{AniRing}$ to show that $\mathrm{SCRDer}\to \mathrm{AniRing}$ is a locally cartesian fibration, and then show that the composition of two locally cartesian edges is locally cartesian. Details omitted.

$\square$

Corollary 6.7. Let $A$ be an animated ring and $\eta \in \mathrm{SCRDer}{\times }_{\mathrm{AniRing}}\{A\}$. Inview of [9] tag 02X6 and Lemma 6.6, $\eta$ is a universal derivation for $A$ if and only if $\eta$ is an initial object of $\mathrm{SCRDer}{\times }_{\mathrm{AniRing}}\{A\}$ if and only if $\eta$ is a $(\mathrm{SCRDer}\to \mathrm{AniRing})$-initial object.

Construction 6.8. Let $B\subseteq {\mathrm{Fun}}_{/\mathrm{AniRing}}(\mathrm{AniRing},\mathrm{SCRDer})$ denote the full subcategory spanned by those functors $f:\mathrm{AniRing}\to \mathrm{SCRDer}$ such that for every animated ring $A$, $f(A)$ is a universal derivation for $A$. Then $B$ is a contractible Kan complex by Proposition 6.5 and [9] tag 030R. Choose an arbitrary element $\mathrm{ud}:\mathrm{AniRing}\to \mathrm{SCRDer}$ of $B$, we call it the universal derivation functor. We denote the composite $\mathrm{AniRing}\stackrel{\mathrm{ud}}{\to }\mathrm{SCRDer}\to {\mathrm{SCRMod}}^{\mathrm{cn}}$ by $L$ and call it the (algebraic) cotangent complex functor.

Example 6.9. Let $A=\mathbb{Z}[x_1,...,x_n]$, where $n\in {\mathbb{Z}}_{\ge 0}$, we prove that the cotangent complex for $A$ is equivalent to the $A$-module $A^n$.

Consider the diagram

in $\mathrm{CAlg}\subseteq \mathrm{AniRing}$ where $\eta$ is obtained form the universal derivation for $A$ in ordinary commutative algebra. This diagram determines an element $\eta \in \mathrm{SCRDer}$ whose image in ${\mathrm{SCRMod}}^{\mathrm{cn}}$ is $(A,M):=(A,A^n)$. We need to prove that $\eta$ is an initial object of $\mathrm{SCRDer}{\times }_{\mathrm{AniRing}}\{A\}$. For this purpose we only need to verify that $\eta$ satisfies the following universal property:

For every $N\in {\mathrm{SCRMod}}_A^{\mathrm{cn}}$, the map ${\pi }_0{\mathrm{Hom}}_{{\mathrm{SCRMod}}_A^{\mathrm{cn}}}(M,N)\to {\pi }_0(\mathrm{SCRDer}{\times }_{{\mathrm{SCRMod}}^{\mathrm{cn}}}\{(A,N)\})$, mapping $x:M\to N$ to $y\in \mathrm{SCRDer}{\times }_{{\mathrm{SCRMod}}^{\mathrm{cn}}}\{(A,N)\}$ such that there exists a morphism $\eta \to y$ in $\mathrm{SCRDer}$ covering $f$, is a bijection.

This is easy: Notice that ${\pi }_0{\mathrm{Hom}}_{{\mathrm{SCRMod}}_A^{\mathrm{cn}}}(M,N)\simeq ({\pi }_{0}N{)}^n$ and ${\mathrm{Hom}}_{\mathrm{AniRing}}(A,B)\simeq B$ for any animated ring $B$.

Proposition 6.10. The cotangent complex functor $L:\mathrm{AniRing}\to {\mathrm{SCRMod}}^{\mathrm{cn}}$ is a left adjoint to the functor $\mathrm{sze}:{\mathrm{SCRMod}}^{\mathrm{cn}}\to \mathrm{AniRing}$.

Proof. The construction of the cotangent complex provides a natural transformation ${\mathrm{id}}_{\mathrm{AniRing}}\to \mathrm{sze}\circ L$, bringing an animated ring to the morphism $A\to A\oplus L_A$ provided by the universal derivation. We prove that this natural transformation exhibits $L$ as a left adjoint to $\mathrm{sze}$. For this purpose we only need to prove that for animated rings $A$ and $B$ and a $B$-module $M$, the map ${\mathrm{Hom}}_{{\mathrm{SCRMod}}^{\mathrm{cn}}}((A,L_A),(B,M))\to {\mathrm{Hom}}_{\mathrm{AniRing}}(A\oplus L_A,B\oplus M)\to {\mathrm{Hom}}_{\mathrm{AniRing}}(A,B\oplus M)$ is an equivalence.

Since $\mathrm{AniRing}$ is generated under sifted colimits by $\mathbb{Z}[x_1,...,x_n]$, for $n\in {\mathbb{Z}}_{\ge 0}$, it suffices to prove the case where $A=\mathbb{Z}[x_1,...,x_n]$. In this case $(A,L_A)$ is a compact projective object of ${\mathrm{SCRMod}}^{\mathrm{cn}}$, $A$ is a compact projective object of $\mathrm{AniRing}$, and the functor $\mathrm{sze}$ preserves sifted colimits, so it suffices to prove the situation where $B$ is a finitely generated polynomial algebra over $\mathbb{Z}$ and $M\simeq B^m$ for some $m\in {\mathbb{Z}}_{\ge 0}$, which is easy.

$\square$

Corollary 6.11. The cotangent complex functor $L$ preserves small colimits.

Relative Cotangent Complexes

Construction 6.12 ([6] Definition 1.39). Let $\mathcal{C}$ be an $\infty$-category, and let $p:T_{\mathcal{C}}\to \mathcal{C}$ be a cocartesian fibration of $\infty$-categories satisfying the following conditions:

(i) Each fiber of $p$ is a stable $\infty$-category.

(ii) $T_{\mathcal{C}}$ admits pushouts and $p$ preserves pushouts.

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 $\psi :\mathcal{E}\to \mathrm{Fun}({\Delta }^1,T_{\mathcal{C}})$.

Now let $\mathcal{C}=\mathrm{AniRing}$ and $T_{\mathcal{C}}={\mathrm{SCRMod}}^{\mathrm{cn}}$. 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 6.13. Suppose given a pushout diagram

in $\mathrm{AniRing}$, then the induced map $L_{B/A}\to L_{B^{\prime }/A^{\prime }}$ is a $({\mathrm{SCRMod}}^{\mathrm{cn}}\to \mathrm{AniRing})$-cocartesian morphism in ${\mathrm{SCRMod}}^{\mathrm{cn}}$.

Proof. The same proof of [6] Proposition 1.45 works.

$\square$

7Deformation Theory of Animated Rings

Construction 7.1. We have a natural functor $\mathrm{SCRDer}\to \mathrm{Fun}({\Lambda }_2^2,\mathrm{AniRing})$, taking a derivation $\eta :A\to M$ to a diagram

where $t$ is the trivial square zero extension. This functor extends to a functor $\mathrm{SCRDer}\to \mathrm{Fun}({\Delta }^1\times {\Delta }^1,\mathrm{AniRing})$, taking a derivation $\eta :A\to M$ to a pullback diagram

Restricting to the upper left vertex we obtain a functor $\Phi :\mathrm{SCRDer}\to \mathrm{AniRing}$.

Proposition 7.2. Let ${\mathrm{SCRDer}}^{+}$ denote the subcategory of $\mathrm{SCRDer}$ consisting of objects $x$ and morphisms $f$ of $\mathrm{SCRDer}$ such that the image $(A,M)$ of $x$ in ${\mathrm{SCRMod}}^{\mathrm{cn}}$ satisfies that ${\pi }_{0}M=0$, and the image of $f$ in ${\mathrm{SCRMod}}^{\mathrm{cn}}$ is $({\mathrm{SCRMod}}^{\mathrm{cn}}\to \mathrm{AniRing})$-cocartesian. Let $\eta \in \mathrm{SCRDer}$, then the functor $\Phi$ induces an equivalence of $\infty$-categories ${\mathrm{SCRDer}}_{\eta /}^{+}\to {\mathrm{AniRing}}_{\Phi (\eta )/}$.

Remark 7.3. Let $({\eta }_1:A\to M)\to ({\eta }_2:B\to N)$ be a morphism in ${\mathrm{SCRDer}}^{+}$, then we obtaing a commutative diagram

where the upper, bottom, left , right faces are pushout squares, and the front and back faces are pullback squares. The proof of this fact will be given below.

Proof. I’ve not found a proof in the literature, so I’ll give a rough sketch below.

We first explain how we are going to prove the proposition. Let $K$ be the nerve of the following category:

Let $\mathcal{C}\subseteq \mathrm{Fun}(K,\mathrm{AniRing})$ be the full subcategory spanned by the diagrams in $\mathrm{AniRing}$ such that the upper left square is a pullback square. Let $K_0\subseteq K$ be the nerve of the subcategory

of $K$. We have a natural map ${\mathrm{SCRMod}}^{\mathrm{cn}}\to \mathrm{Fun}(K_0,\mathrm{AniRing})$, mapping $(A,M)$ to

where $t$ is the trivial square zero extension. Let $\widetilde{\mathrm{SCRDer}}={\mathrm{SCRMod}}^{\mathrm{cn}}{\times }_{\mathrm{Fun}(K_0,\mathrm{AniRing})}\mathcal{C}$, so the natural map $\widetilde{\mathrm{SCRDer}}\to \mathrm{SCRDer}$ is a trivial Kan fibration. Let ${\widetilde{\mathrm{SCRDer}}}^{+}=\widetilde{\mathrm{SCRDer}}{\times }_{\mathrm{SCRDer}}{\mathrm{SCRDer}}^{+}$. We are going to prove that the functor ${\widetilde{\mathrm{SCRDer}}}^{+}\to \mathrm{AniRing}$ given by evaluation at the upper left vertex is a left fibration after a fibrant replacement. For this purpose, we first prove that every edge of ${\widetilde{\mathrm{SCRDer}}}^{+}$ is a $({\widetilde{\mathrm{SCRDer}}}^{+}\to \mathrm{AniRing})$-colimit diagram (see [9] tag 01UM and tag 02KL).

To prove this we claim that we only need the following two facts:

(i) Remark 7.3 is true.

(ii) Let $(A,M)\to (B,N)$ be a $({\mathrm{SCRMod}}^{\mathrm{cn}}\to \mathrm{AniRing})$-cocartesian edge of ${\mathrm{SCRMod}}^{\mathrm{cn}}$, then every square of the induced diagram

is a pushout diagram in $\mathrm{AniRing}$.

Indeed, using [7] Lemma 7.4.2.10 and (i), we see that the image of every edge of ${\widetilde{\mathrm{SCRDer}}}^{+}$ in $\mathcal{C}$ is a $(\mathcal{C}\to \mathrm{AniRing})$-colimit diagram. Using (ii) we see that the image of every edge of ${\widetilde{\mathrm{SCRDer}}}^{+}$ in ${\mathrm{SCRMod}}^{\mathrm{cn}}$ is a $({\mathrm{SCRMod}}^{\mathrm{cn}}\to \mathrm{Fun}(K_0,\mathrm{AniRing}))$-colimit diagram.

Now we start proving (i) and (ii). The proof of (ii) is rather easy, so we omit. To prove (i), notice that all our constructions above regarding animated rings (including the trivial square zero extension, the definition of a derivation, and the square zero extension corresponding to a derivation) are compatible with the corresponding constructions regarding $E_{\infty }$-rings (see [7] Subsection 7.4.2), in a suitable sense, so the claim follows from (ii) and [7] Theorem 7.4.2.7.

We next prove that ${\widetilde{\mathrm{SCRDer}}}^{+}\to \mathrm{AniRing}$ is a cocartesian fibration after a fibrant replacement (which completes the proof by virtue of [9] 02LK). Suppose we are given a diagram

as in Construction 7.1. Let $A{\oplus }_{\eta }M[-1]\to \tilde{B}$ be a map of animated rings, form a diagram

in $\mathrm{AniRing}$ by pushing out along $A{\oplus }_{\eta }M[-1]\to \tilde{B}$. By claim (ii) above we see that $B^{\prime }\simeq B\oplus (M{\otimes }_{A}B)$, so the back part of the above diagram gives an element in $\mathrm{SCRDer}$. It is easy to see that the front square is a pullback square in $\mathrm{Sp}$, hence so is the back square, thus the back square is a pullback square in $\mathrm{AniRing}$. Thus every edge in $\mathrm{AniRing}$ admits a $({\widetilde{\mathrm{SCRDer}}}^{+}\to \mathrm{AniRing})$-cocartesian lift, after a fibrant replacement, and we’ve finished the proof.

$\square$

Definition 7.4. Let $B\to A$ be a morphism of animated ring. It is called a square zero extension if there exists an $A$-module $M$ and a derivation $\eta :A\to M$ such that ${\pi }_{0}M=0$ and $B\to A$ is equivalent to $A{\oplus }_{\eta }M[-1]\to A$ in ${\mathrm{AniRing}}_{/A}$.

Example 7.5. Let $A^{\prime }\to A$ be a (not necessarily trivial) square zero extension of ordinary rings. Let $I=\ker (A^{\prime }\to A)$, regarded as an $A$-module. We are going to prove that $A^{\prime }$ is a square zero extension of $A$ by $I$, in the sense of Definition 7.4.

We define a simplicial commutative ring $A\oplus I[1]$ whose underlying simplicial abelian group corresponds to the chain complex $I\stackrel{0}{\to }A$ concerntrated in degrees $0,1$ under the Dold-Kan correspondence as follows:

Let $n\in {\mathbb{Z}}_{\ge 0}$, let $(A\oplus I[1]{)}_n=A\oplus {⨁}_{1\le i\le n}{I}_i^{(n)}$ as abelian group, where each $I_i^{(n)}:=I$ as abelian groups, and inview of the explicit formula of Dold-Kan correspondence ([7] Construction 1.2.3.5), $A$ corresponds to the unique surjection $[n]\to [0]$, and for $1\le i\le n$, $I_i^{(n)}$ corresponds to the unique surjection $\alpha :[n]\to [1]$ such that $\alpha (i-1)=0$ and $\alpha (i)=1$.

The multiplication law on $(A\oplus I[1]{)}_n=A\oplus {⨁}_{1\le i\le n}{I}_i^{(n)}$ is defined as follows: Let $0\le i\le j\le n$, $x\in A$ if $i=0$, and $x\in I_i^{(n)}$ otherwise; $y\in A$ if $j=0$, and $y\in i_j^{(n)}$ otherwise, then

$$xy=\{\begin{array}{rlr}& (xy\in A)& \text{if }j=0\\ & (xy\in I_j^{(n)})& \text{if }i=0\text{ and }j>0\\ & 0& \text{if }i>0\end{array}$$

It is easy to check that this indeed gives $A\oplus I[1]$ the structure of a simplicial commutative ring.

We have a natural diagram

in $\mathrm{AniRing}$, which we can easily check to be equivalent to $\overline{\mathrm{sze}}(A,I[1])$.

Now we construct another simplicial commutative ring $\tilde{A}$ whose underlying simplicial abelian group corresponds to the chain complex $I\hookrightarrow A^{\prime }$ concerntrated in degrees $0,1$ under the Dold-Kan correspondence as follows:

Let $n\in {\mathbb{Z}}_{\ge 0}$, let $(\tilde{A}{)}_n=A^{\prime }\oplus {⨁}_{1\le i\le n}{I}_i^{(n)}$ as abelian group, where each $I_i^{(n)}:=I$ as abelian groups, and inview of the explicit formula of Dold-Kan correspondence ([7] Construction 1.2.3.5), $A^{\prime }$ corresponds to the unique surjection $[n]\to [0]$, and for $1\le i\le n$, $I_i^{(n)}$ corresponds to the unique surjection $\alpha :[n]\to [1]$ such that $\alpha (i-1)=0$ and $\alpha (i)=1$.

The multiplication law on $(\tilde{A}{)}_n=A^{\prime }\oplus {⨁}_{1\le i\le n}{I}_i^{(n)}$ is defined as follows: Let $0\le i\le j\le n$, $x\in A^{\prime }$ if $i=0$, and $x\in I_i^{(n)}$ otherwise; $y\in A^{\prime }$ if $j=0$, and $y\in i_j^{(n)}$ otherwise, then

$$xy=\{\begin{array}{rlr}& (xy\in A)& \text{if }j=0\\ & (xy\in I_j^{(n)})& \text{if }i=0\text{ and }j>0\\ & 0& \text{if }i>0\end{array}$$

It is easy to check that this indeed gives $\tilde{A}$ the structure of a simplicial commutative ring.

We have a natural map of simplicial commutative rings $\tilde{A}\to A$ which is an equivalence. We also have a natural commutative diagram of simplicial commutative rings

which becomes a pushout diagram when passing to $\mathcal{D}(\mathbb{Z})$, by [7] Corollary 1.3.2.16. Thus this is a pullback diagram of animated rings. This is the diagram that exhibits $A^{\prime }$ as a square zero extension of $A$.

Proposition 7.6. Suppose we have a pushout diagram

in $\mathrm{AniRing}$ such that $f$ is a square zero extension, then $g$ is a square zero extension.

Proof. Immediate consequence of Proposition 7.2 and Remark 7.3.

$\square$

Proposition 7.7. Let $A$ be an animated ring, let $n\in {\mathbb{Z}}_{\ge 0}$, then the map ${\tau }_{\le n+1}A\to {\tau }_{\le n}A$ is a square zero extension.

Proof. See [14] Lemma 2.52 or [13] Lemma 2.2.1.1.

$\square$

8Properties of Cotangent Complexes

Definition 8.1. Let $A$ be an animated ring. An $A$-module $M$ is called flat (projective) if it is flat (projective) as a $\theta (A)$-module.

Let $A\to B$ be a morphism of animated rings. It is called etale if $\theta (A)\to \theta (B)$ is an etale morphism of $E_{\infty }$-rings. It is called smooth if $B$ is flat as an $A$-module and ${\pi }_0(A)\to {\pi }_0(B)$ is a smooth morphism of ordinary rings.

Proposition 8.2. Let $A\to B$ be a morphism of animated rings. Then it is smooth if and only if $L_{B/A}$ is a projective $B$-module and ${\pi }_0(B)$ is of finite presentation over ${\pi }_0(A)$. It is etale if and only if $L_{B/A}$ vanishes and ${\pi }_0(A)$ is of finite presentation over ${\pi }_0(A)$.

Proof. See [14] Proposition 2.56 or [13] Theorem 2.2.2.6.

$\square$

9Miscellany

Proposition 9.1. Let $f:C\to D$ and $g:X\to Y$ be two morphisms of $\mathrm{AniRing}$, such that $g$ is a square zero extension. If $f$ is smooth, then $f$ is weakly left orthogonal to $g$. If $f$ is etale, then $f$ is left orthogonal to $g$. (See [9] tag 04MY and tag 04NN for relevant definitions.)

Proof. Suppose the square zero extension $g$ comes from a pullback diagram of animated rings as follows:

where $M$ is an $1$-connective $A$-module, $\eta$ is a derivation from $A$ to $M$, and $t$ is the trivial derivation from $A$ to $M$. According to [9] tag 04N5 and tag 04P4 we only need to prove that $f$ is weakly left orthogonal or left orthogonal to $A\stackrel{t}{\to }A\oplus M$ in the corresponding cases. Suppose we are given a diagram

then it is easy to see that ${\mathrm{Hom}}_{{\mathrm{AniRing}}_{C//A}}(D,A\oplus M)$ is equivalent to ${\mathrm{Hom}}_{{\mathrm{SCRMod}}_D^{\mathrm{cn}}}(L_{D/C},M)$. Therefore if we are given a diagram

then the homotopy fibers of the map ${\mathrm{Hom}}_{{\mathrm{AniRing}}_{C/}}(D,A\oplus M)\to {\mathrm{Hom}}_{{\mathrm{AniRing}}_{C/}}(D,A)$ are connected if $f$ is smooth, and contractible if $f$ is etale. Since the composite map ${\mathrm{Hom}}_{{\mathrm{AniRing}}_{C/}}(D,A)\to {\mathrm{Hom}}_{{\mathrm{AniRing}}_{C/}}(D,A\oplus M)\to {\mathrm{Hom}}_{{\mathrm{AniRing}}_{C/}}(D,A)$ is equivalent to the identity, we easily get what we want.

$\square$

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[13]	Bertrand Toën, Gabriele Vezzosi, “Homotopical algebraic geometry II: geometric stacks and applications”, Mem. Amer. Math. Soc., 193(902):x+224, 2008.
[14]	Can Yaylali, “Notes on Derived Algebraic Geometry”, arXiv:2208.01506v1.