用户: Scavenger/Derived Completeness

We discuss the notion of derived complete (or simply complete) modules over an $E_{\infty }$-ring, following [1] Chapter 4.

Let $A$ be an ordinary ring. There exists an (essentially unique) equivalence between the $\infty$-category of modules over the $E_{\infty }$-ring $A$ and the derived $\infty$-category $\mathcal{D}(A)$ that respects symmetric monoidal structures, $t$-structures, and whose restriction to the hearts of both $\infty$-categories is equivalent to ${\mathrm{id}}_{{\mathrm{dMod}}_A}$, where ${\mathrm{dMod}}_A$ is the $1$-category of ordinary $A$-modules (identified with the hearts of both two $\infty$-categories through the natural functors). See [2] Subsection 7.1.2 for more details. Using this equivalence, we can transfer the theory of complete modules over the $E_{\infty }$-ring $A$ to a theory of derived complete complexes over $A$. We will show that this theory coincides with that of [3] tag 091N.

The $\infty$-categorical approach has many advantages compared with [3] tag 091N. For example, it is not so obvious that the derived completion of an $A$-algebra $B$ admits the structure of an $A$-algebra in view of the approach of [3] tag 091N. However, this is rather immediate using the $\infty$-categorical approach.

1Nilpotent Modules

Definition 1.1 ([1] Definition 4.1.1, Definition 4.1.3). Let $R$ be an ordinary ring, let $M$ be a discrete $R$-module, let $x\in M$ be an element. We let $\mathrm{Supp}(x)$ denote the set $\{a\in R:(\exists n\in {\mathbb{Z}}_{\ge 1})a^{n}x=0\}$. We refer to $\mathrm{Supp}(x)$ as the support of $x$.

Let $A$ be an $E_{\infty }$-ring, let $I\subseteq {\pi }_{0}A$ be an ideal. We say that an $A$-module $M$ is $I$-nilpotent if for every $k\in \mathbb{Z}$ and every object $x\in {\pi }_{k}M$, we have $I\subseteq \mathrm{Supp}(x)\subseteq {\pi }_{0}A$. We let ${\mathrm{Mod}}_A^{I\text{-nil}}$ denote the full subcategory of ${\mathrm{Mod}}_A$ spanned by the $I$-nilpotent modules.

Proposition 1.2 ([1] Proposition 4.1.6). Let $A$ be an $E_{\infty }$-ring, let $I\subseteq {\pi }_{0}A$ be an ideal. Then ${\mathrm{Mod}}_A^{I\text{-nil}}\subseteq {\mathrm{Mod}}_A$ is stable under translations and small colimits.

Proposition 1.3 ([1] Corollary 4.1.7). Let $A$ be an $E_{\infty }$-ring, let $I\subseteq {\pi }_{0}A$ be an ideal. Let $M,N$ be $A$-modules such that $M$ is $I$-nilpotent, then $M{\otimes }_{A}N$ is $I$-nilpotent.

2Local Modules

Definition 2.1 ([1] Definition 4.1.9). Let $A$ be an $E_{\infty }$-ring, let $I\subseteq {\pi }_{0}A$ be an ideal. An $A$-module $M$ is said to be $I$-local if ${\mathrm{Hom}}_{{\mathrm{Mod}}_A}(N,M)$ is contractible for every $I$-nilpotent $A$-module $N$. We let ${\mathrm{Mod}}_A^{I\text{-loc}}$ denote the full subcategory of ${\mathrm{Mod}}_A$ spanned by the $I$-local modules.

Notation 2.2 ([1] Notation 4.1.10). Let $A$ be an $E_{\infty }$-ring, let $a\in {\pi }_{0}A$. We let $A[a^{-1}]$ denote the (essentially unique) $E_{\infty }$-ring over $A$ that is an etale $A$-algebra and ${\pi }_0(A[a^{-1}])\simeq ({\pi }_{0}A)[a^{-1}]$ as ${\pi }_{0}A$-algebras. Let $M$ be an $A$-module, we let $M[1/a]=A[1/a]{\otimes }_{A}M$.

Proposition 2.3 ([1] Proposition 4.1.12). Let $A$ be an $E_{\infty }$-ring, let $I\subseteq {\pi }_{0}A$ be a finitely generated ideal. Then there exists an $I$-nilpotent $A$-module $V$ and a map of $A$-modules $V\to A$ such that the functor $M\to V{\otimes }_{A}M$ is a right adjoint to the inclusion ${\mathrm{Mod}}_A^{I\text{-nil}}\to {\mathrm{Mod}}_A$.

Moreover, the module $V$ can be explicitly described as follows: Let $f_1,...,f_n$ be a set of generators of $I$. For $1\le i\le n$, let $V_i=\mathrm{fib}(A\to A[1/x_i])$, then $V={⨂}_{1\le i\le n}{V}_i$.

Notation 2.4 ([1] Notation 4.1.13). Let $A$ be an $E_{\infty }$-ring, let $I\subseteq {\pi }_{0}A$ be a finitely generated ideal. We let ${\Gamma }_I:{\mathrm{Mod}}_A\to {\mathrm{Mod}}_A^{I\text{-nil}}$ be a right adjoint to the inclusion ${\mathrm{Mod}}_A^{I\text{-nil}}\to {\mathrm{Mod}}_A$.

Proposition 2.5 ([1] Proposition 4.1.15). Let $A$ be an $E_{\infty }$-ring, let $I\subseteq {\pi }_{0}A$ be a finitely generated ideal. Then the $\infty$-category ${\mathrm{Mod}}_A^{I\text{-nil}}$ is compactly generated, and the inclusion ${\mathrm{Mod}}_A^{I\text{-nil}}\to {\mathrm{Mod}}_A$ carries compact objects to compact objects.

2.6 ([1] Remark 4.1.20). Let $A$ be an $E_{\infty }$-ring, let $I\subseteq {\pi }_{0}A$ be a finitely generated ideal. There exists an accessible $t$-structure on ${\mathrm{Mod}}_A$ such that $({\mathrm{Mod}}_A{)}_{\ge 0}={\mathrm{Mod}}_A^{I\text{-nil}}$ and $({\mathrm{Mod}}_A{)}_{\le 0}={\mathrm{Mod}}_A^{I\text{-loc}}$. In particular the inclusion ${\mathrm{Mod}}_A^{I\text{-loc}}\to {\mathrm{Mod}}_A$ admits a left adjoint, which we will denote by $L_I:{\mathrm{Mod}}_A\to {\mathrm{Mod}}_A^{I\text{-loc}}$, and we have a fiber sequence of functors ${\Gamma }_I\to \mathrm{id}\to L_I$ from ${\mathrm{Mod}}_A$ to itself. Since the functor ${\Gamma }_I$ preserves small colimits, so does the functor $L_I:{\mathrm{Mod}}_A\to {\mathrm{Mod}}_A$. Consequently the full subcategory ${\mathrm{Mod}}_A^{I\text{-loc}}\subseteq {\mathrm{Mod}}_A$ is stable under small colimits.

Proposition 2.7 ([1] Remark 4.1.23). Let $A$ be an $E_{\infty }$-ring, let $I\subseteq {\pi }_{0}A$ be a finitely generated ideal. Let $M$ be an $I$-local $A$-module, then for any $A$-module $N$, the $A$-module $M{\otimes }_{A}N$ is $I$-local.

Proposition 2.8 ([1] Proposition 4.1.25). Let $A$ be an $E_{\infty }$-ring, let $I,J\subseteq {\pi }_{0}A$ be finitely generated ideals. Then ${\mathrm{Mod}}_A^{I+J\text{-loc}}$ is generated under extenstions by the full subcategories ${\mathrm{Mod}}_A^{I\text{-loc}},{\mathrm{Mod}}_A^{J\text{-loc}}$ of ${\mathrm{Mod}}_A$.

3Complete Modules

Definition 3.1 ([1] Definition 4.2.1). Let $A$ be an $E_{\infty }$-ring, let $I\subseteq {\pi }_{0}A$ be a finitely generated ideal. We say that an $A$-module $M$ is $I$-complete if ${\mathrm{Hom}}_{{\mathrm{Mod}}_A}(N,M)$ is contractible for every $I$-local $A$-module $N$. We let ${\mathrm{Mod}}_A^{I\text{-com}}$ denote the full subcategory of ${\mathrm{Mod}}_A$ spanned by the $I$-complete modules.

Proposition 3.2 ([1] Lemma 4.2.2). Let $A$ be an $E_{\infty }$-ring, let $I\subseteq {\pi }_{0}A$ be a finitely generated ideal. Then there exists an accessible $t$-structure on the $\infty$-category ${\mathrm{Mod}}_A$ such that $({\mathrm{Mod}}_A{)}_{\ge 0}={\mathrm{Mod}}_A^{I\text{-loc}}$ and $({\mathrm{Mod}}_A{)}_{\le 0}={\mathrm{Mod}}_A^{I\text{-com}}$. In particular the inclusion functor ${\mathrm{Mod}}_A^{I\text{-com}}\to {\mathrm{Mod}}_A$ admits a left adjoint, which we will denote by $M\to M_I^{\wedge }$.

Proposition 3.3 ([1] Remark 4.2.4). Let $A$ be an $E_{\infty }$-ring, let $I\subseteq {\pi }_{0}A$ be a finitely generated ideal. Let $\alpha :M\to M^{\prime }$ be a morphism of $A$-modules which induces an equivalence $M_I^{\wedge }\to M_I^{\prime \wedge }$. Let $N$ be an $A$-module, then the map $(N{\otimes }_{A}M{)}_I^{\wedge }\to (N{\otimes }_A{M}^{\prime }{)}_I^{\wedge }$ is an equivalence.

Proposition 3.4 ([1] Proposition 4.2.5). Let $A$ be an $E_{\infty }$-ring, let $I\subseteq {\pi }_{0}A$ be a finitely generated ideal. Then the $I$-completion functor induces an equivalence of $\infty$-categories ${\mathrm{Mod}}_A^{I\text{-nil}}\to {\mathrm{Mod}}_A^{I\text{-com}}$, whose inverse is given by the restriction of ${\Gamma }_I$ to ${\mathrm{Mod}}_A^{I\text{-com}}$.

Proposition 3.5 ([1] Proposition 4.2.7). Let $A$ be an $E_{\infty }$-ring, let $x\in {\pi }_{0}A$ be an element. Let $M$ be an $A$-module, let $T(M)=\lim (...\stackrel{r_x}{\to }M\stackrel{r_x}{\to }M\stackrel{r_x}{\to }M)$, where $r_x$ is multiplication by $x$. Then $T(M)$ is $(x)$-local, and $\mathrm{cofib}(T(M)\to M)$ is $(x)$-complete.

Proposition 3.6 ([1] Corollary 4.2.10). Let $A$ be an $E_{\infty }$-ring, let $x\in {\pi }_{0}A$ be an element, let $I\subseteq {\pi }_{0}A$ be a finitely generated ideal. Then the $(x)$-completion functor carries $I$-complete modules to $I$-complete modules.

Proposition 3.7 ([1] Proposition 4.2.11). Let $A$ be an $E_{\infty }$-ring, let $x\in {\pi }_{0}A$ be an element, let $I\subseteq {\pi }_{0}A$ be a finitely generated ideal, let $I^{\prime }=I+(x)$. For every $A$-module $M$, the composition $M\to M_I^{\wedge }\to (M_I^{\wedge }{)}_{(x)}^{\wedge }$ exhibits $(M_I^{\wedge }{)}_{(x)}^{\wedge }$ as the $I^{\prime }$ completion of $M$.

Proposition 3.8 ([1] Corollary 4.2.9). Let $A$ be an $E_{\infty }$-ring, let $I\subseteq {\pi }_{0}A$ be a finitely generated ideal. Let $M$ be an $A$-module such that for $k\in {\mathbb{Z}}_{<0}$, ${\pi }_{k}M\simeq 0$, then the same condition holds for $M_I^{\wedge }$.

Proposition 3.9 ([1] Theorem 4.2.13). Let $A$ be a connective $E_{\infty }$-ring, let $I\subseteq {\pi }_{0}A$ be a finitely generated ideal. An $A$-module $M$ is $I$-complete if and only if for each $k\in \mathbb{Z}$, the $A$-module ${\pi }_{k}M$ is $I$-complete.

4Coincidence with [3] tag 091N

4.1. Let $A$ be an $E_{\infty }$-ring, let $x\in {\pi }_{0}A$ be an element, and write $I=(x)$. Since $L_I:{\mathrm{Mod}}_A\to {\mathrm{Mod}}_A^{I\text{-loc}}$ preserves small colimits and ${\mathrm{Mod}}_A$ is generated under small colimits by $\{A[k]{\}}_{k\in \mathbb{Z}}$ ([2] Proposition 1.4.4.11), ${\mathrm{Mod}}_A^{I\text{-loc}}$ is generated under small colimits by $L_I(A[k])=(A[1/x])[k]$, where $k\in \mathbb{Z}$. Thus an $A$-module $M$ is $I$-complete if and only if ${\mathrm{Ext}}_A^k(A[1/x],M)=0$ for all $k\in \mathbb{Z}$.

Now assume $A$ is discrete, so we identify $A$ with ${\pi }_{0}A$. Let $J\subseteq A$ be a finitely generated ideal. It follows from the above discussion that a complex of $A$-modules is $J$-complete in our sense if and only if it is derived $J$-complete in the sense of [3] tag 091S.

5Miscellany

Remark 5.1. Enlargement of universes has no effect on the notions of nilpotence, locality, and completeness defined above.

5.2. Let $A$ be an ordinary ring, let ${\mathrm{dMod}}_A$ denote the category of (discrete) $A$-modules. Let $I\subseteq A$ be a finitely generated ideal, a discrete $A$-module $M$ is called $I$-complete if it is $I$-complete as an module over the $E_{\infty }$-ring $A$. We use ${\mathrm{dMod}}_A^{I\text{-com}}$ to denote the full subcategory of ${\mathrm{dMod}}_A$ spanned by the $I$-complete modules. Let $B$ be a (discrete) $A$-algebra. $B$ is called $I$-complete if it is $I$-complete as a discrete $A$-module. Let ${\mathrm{dAlg}}_A$ denote the category of (discrete) $A$-algebras, let ${\mathrm{dAlg}}_A^{I\text{-com}}\subseteq {\mathrm{dAlg}}_A$ be the full subcategory spanned by the $I$-complete $A$-algebras.

Using Proposition 3.8 we see that the inclusion ${\mathrm{dMod}}_A^{I\text{-com}}\to {\mathrm{dMod}}_A$ admits a left adjoint, given by $M\to {\pi }_0(M_I^{\wedge })$. Using Proposition 3.3 we see that this localization functor is compatible with the natural symmetric monoidal structure on ${\mathrm{dMod}}_A$, in the sense of [2] Definition 2.2.1.6. Applying [2] Proposition 2.2.1.9 we see that the inclusion ${\mathrm{dAlg}}_A^{I\text{-com}}\to {\mathrm{dAlg}}_A$ admits a left adjoint. Moreover, let $B\to C$ be a morphism in ${\mathrm{dAlg}}_A$ such that $C\in {\mathrm{dAlg}}_A^{I\text{-com}}$, then $B\to C$ exhibits $C$ as a ${\mathrm{dAlg}}_A^{I\text{-com}}$-localization of $B$ if and only if the underlying map of $B\to C$ in ${\mathrm{dMod}}_A$ exhibits $C$ as a ${\mathrm{dMod}}_A^{I\text{-com}}$-localization of $B$.

Construction 5.3 (Koszul Complexes). Let $A$ be an ordinary ring, let $f\in A$ be an element. We are going to construct a simplicial commutative $A$-algebra, denoted $\mathrm{Kos}(A;f)$, called the Koszul complex associated to the element $f$. It should have the property that its underlying simplicial $A$-module corresponds to, under Dold-Kan correspondence, the complex $A\stackrel{f}{\to }A$ concentrated in (homological) degree $0,1$.

We construct $\mathrm{Kos}(A;f)$ explicitly as follows: Let $n\in {\mathbb{Z}}_{\ge 0}$, let $(\mathrm{Kos}(A;f){)}_n={⨁}_{0\le i\le n}{A}_i^{(n)}$ as $A$-module, where each $A_i^{(n)}:=A$ as $A$-modules, and inview of the explicit formula of Dold-Kan correspondence ([2] Construction 1.2.3.5), $A_0^{(n)}$ corresponds to the unique surjection $[n]\to [0]$, and for $1\le i\le n$, $A_i^{(n)}$ corresponds to the unique surjection $\alpha :[n]\to [1]$ such that $\alpha (i-1)=0$ and $\alpha (i)=1$.

The multiplication law on $(\mathrm{Kos}(A;f){)}_n={⨁}_{0\le i\le n}{A}_i^{(n)}$ is defined as follows: Let $0\le i\le j\le n$, $x\in A_i^{(n)}$, $y\in A_j^{(n)}$, then

$$xy=\{\begin{array}{rlr}& (xy\in A_j^{(n)})& \text{if }i=0\\ & (fxy\in A_j^{(n)})& \text{if }i>0\end{array}$$

It is easy to check that this indeed gives $\mathrm{Kos}(A;f)$ the structure of a simplicial commutative $A$-algebra.

Digression 5.4. Dold-Kan correspondence provides an equivalence of categories ${\mathrm{sMod}}_A\to {\mathrm{Ch}}_{\ge 0}(A)$, where the former category is the category of simplicial $A$-modules, and the latter is the category of complexes of $A$-modules concerntrated in nonnegative degree. Localizing at the equivalences (or quasi-isomorphisms) we obtain an equivalence of $\infty$-categories ${\mathrm{sMod}}_A[{\mathrm{equivalences}}^{-1}]\to {\mathcal{D}}_{\ge 0}(A)$.

Let ${\mathrm{sAlg}}_A$ denote the category of simplicial $A$-algebras. The forgetful functor ${\mathrm{sAlg}}_A\to {\mathrm{sMod}}_A$ gives rise to a functor ${\mathrm{AniRing}}_{A/}\to {\mathcal{D}}_{\ge 0}(A)\simeq {\mathrm{SCRMod}}_A^{\mathrm{cn}}$ by inverting equivalences. By checking on finitely generated polynomial algebras over $A$ we see that this functor is equivalent to the natural forgetful functor.

In particular the map of $A$-modules underlying the map $A\to \mathrm{Kos}(A;f)$ of $A$-algebras can be extended to a fiber sequence $A\stackrel{f}{\to }A\to \mathrm{Kos}(A;f)$.

Now let $n\in {\mathbb{Z}}_{\ge 1}$, let $f_1,...,f_n\in A$ be elements. Let $\mathrm{Kos}(A;f_1,...,f_n)$ be the coproduct ${\coprod }_{1\le i\le n}\mathrm{Kos}(A;f_i)$ in the $\infty$-category ${\mathrm{AniRing}}_{A/}$ of animated rings over $A$. This animated $A$-algebra is called the Koszul complex associated to the elements $f_1,...,f_n$.

Let $I\subseteq A$ be the ideal generated by $f_1,...,f_n$. For $m\in {\mathbb{Z}}_{\ge 1}$ and $i\in \mathbb{Z}$, $1\le i\le n$ we denote $A_i^{(m)}:=\mathrm{Kos}(A;f_i^m)$ and $A^{(m)}:=\mathrm{Kos}(A;f_1^m,...,f_n^m)$. It is easy to see that (see the proof of [1] Proposition 4.2.7) for every module $M$ over the animated ring $A$, the $A$-module $M{\otimes }_A{A}_i^{(m)}$ is $(f_i)$-complete, thus the module $M{\otimes }_A{A}^{(m)}$ is $I$-complete.

We have a sequence of maps $...\to A_i^{(3)}\to A_i^{(2)}\to A_i^{(1)}$ of animated rings over $A$ that is naturally defined at the level of simplicial commutative $A$-algebras. We thus obtain a sequence of maps $...\to A^{(3)}\to A^{(2)}\to A^{(1)}$ of animated rings over $A$ whose underlying maps of $A$-modules is the natural maps of (classical) Koszul complexes. Let $M$ be an arbitrary module over the animated ring $A$, then we have a natural map $M\to M^{\prime }:=\lim (...\to M{\otimes }_A{A}^{(2)}\to M{\otimes }_A{A}^{(1)})$ of $A$-modules. We prove that this map exhibits $M^{\prime }$ as the $I$-completion of $M$. It follows from the above discussion that $M^{\prime }$ is $I$-complete, thus it remains to prove that $\mathrm{fib}(M\to M^{\prime })$ is $I$-local.

Notice that $\mathrm{fib}(M\to M^{\prime })\simeq \lim (...\to M{\otimes }_A\mathrm{fib}(A\to A^{(2)})\to M{\otimes }_A\mathrm{fib}(A\to A^{(1)}))$. We prove that this is $I$-local by induction on $n$. For $n=1$ this follows from the proof of [1] Proposition 4.2.7. Suppose $n>1$ and the cases of $1,...,n-1$ have already been handled. Let $I^{\prime }\subseteq A$ be the ideal generated by $f_1,...,f_{n-1}$. Let $A^{\prime (m)}:=\mathrm{Kos}(A;f_1^m,...,f_{n-1}^m)$. Notice that we have a triangle

inducing a fiber sequence $\mathrm{fib}(M\to M{\otimes }_A{A}_n^{(m)})\to \mathrm{fib}(M\to M{\otimes }_A{A}_n^{(m)}{\otimes }_A{A}^{\prime (m)})\to \mathrm{fib}(M{\otimes }_A{A}_n^{(m)}\to M{\otimes }_A{A}_n^{(m)}{\otimes }_A{A}^{\prime (m)})$. Taking limit over $m$ and using inductive hypothesis, we get what we want. (To handle ${\lim }_m(\mathrm{fib}(M{\otimes }_A{A}_n^{(m)}\to M{\otimes }_A{A}_n^{(m)}{\otimes }_A{A}^{\prime (m)}))$, we need to consider a diagram of the shape of a double complex...)

In the above constructions we may replace some of the elements $f_i$ by Cartier divisors $I_i$ (that is, $I_i\subseteq A$ is an ideal which determines a line bundle over $\mathrm{Spec}A$): Notice that if $I$ is a Cartier divisor, then $\mathrm{Kos}(A;I^n)\simeq A/I^n$ as $A$-modules, where $n\in {\mathbb{Z}}_{\ge 1}$, so for every $A$-module $M$ and every element $x\in I$, $M{\otimes }_A\mathrm{Kos}(A;I^n)$ is annihilated by $x^n$, thus is $(x)$-complete. So the only nontriviality left is to prove that, for every $A$-module $M$, $\lim (...\to M{\otimes }_A{I}^2\to M{\otimes }_{A}I\to M)$ is $I$-local. Tensoring with the sequence of maps of $A$-modules $...\to I^2\to I\to A\to I^{-1}\to I^{-2}\to ...$ gives a sequence of natural transformations between functors between the $\infty$-category ${\mathrm{Mod}}_A$. Each of these functors is an equivalence since $I$ is a Cartier divisor. Let $N$ be an $I$-nilpotent $A$-module, then ${\mathrm{Hom}}_{{\mathrm{Mod}}_A}(N,\lim (...\to M{\otimes }_A{I}^2\to M{\otimes }_{A}I\to M))\simeq \lim (...\to {\mathrm{Hom}}_{{\mathrm{Mod}}_A}(N,M{\otimes }_A{I}^2)\to {\mathrm{Hom}}_{{\mathrm{Mod}}_A}(N,M{\otimes }_{A}I)\to {\mathrm{Hom}}_{{\mathrm{Mod}}_A}(N,M))\simeq \lim (...\to {\mathrm{Hom}}_{{\mathrm{Mod}}_A}(N{\otimes }_A{I}^{-2},M)\to {\mathrm{Hom}}_{{\mathrm{Mod}}_A}(N{\otimes }_A{I}^{-1},M)\to {\mathrm{Hom}}_{{\mathrm{Mod}}_A}(N,M))\simeq {\mathrm{Hom}}_{{\mathrm{Mod}}_A}(\mathrm{colim}(N\to N{\otimes }_A{I}^{-1}\to N{\otimes }_A{I}^{-2}\to ...),M)\simeq {\mathrm{Hom}}_{{\mathrm{Mod}}_A}(N{\otimes }_A\mathrm{colim}(A\to I^{-1}\to I^{-2}\to ...),M)\simeq {\mathrm{Hom}}_{{\mathrm{Mod}}_A}(N{\otimes }_{A}A[1/I],M)\simeq 0$, where $A[1/I]$ corresponds to the open subscheme of $\mathrm{Spec}A$ whose completion is the closed subset $V(I)\subseteq \mathrm{Spec}A$, thus the claim is proved.

References

[1]	Jacob Lurie, “Derived Algebraic Geometry XII: Proper Morphisms, Completions, and the Grothendieck Existence Theorem”, November 8, 2011 version, Avalible for download at https://www.math.ias.edu/ lurie/papers/DAG-XII.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