Let k be a field. In this section, we will use the Koszul duality functor D(1) : (Algaugk )op → Algaugk to construct an equivalence of∞-categories Algaugk 'Moduli(1)k , and thereby obtain a proof of Theorem 3.0.4.
The main point is to show thatD(1) is a deformation theory (in the sense of Definition 1.3.9). We begin by introducing a variation on Example 1.1.4:
Construction 3.2.1. Let k be a field. Theorem A.7.3.5.14 gives an equivalence of the stabilization Stab(Algaugk ) with the ∞-category kBModk(Modk) ' Modk of k-module spectra. Let E ∈ Stab(Algaugk ) correspond to the unit object k∈Modk under this identification (so we have Ω∞−nE 'k⊕k[n] for every integern). We regard (Algaugk ,{E}) as a deformation context (see Definition 1.1.3).
Our first goal in this section is to show that the deformation context (Algaugk ,{E}) of Construction 3.2.1 allows us to recover the notion of smallk-algebra and formalE1 moduli problem via the general formalism laid out in§1.1.
Proposition 3.2.2. Letkbe a field and let (Algaugk ,{E})be the deformation context of Construction 3.2.1.
Then an objectA∈Algaugk is small (in the sense of Definition 1.1.8) if and only if its image inAlgk is small (in the sense of Definition 3.0.1). That is, Ais small if and only if it satisfies the following conditions:
(a) The algebra Ais connective: that is, πiA'0 fori <0.
(b) The algebra Ais truncated: that is, we haveπiA'0 fori0.
(c) Each of the homotopy groups πiA is finite dimensional when regarded as a vector space over fieldk.
(d) Let n denote the radical of the ring π0A (which is a finite-dimensional associative algebra over k).
Then the canonical map k→(π0A)/nis an isomorphism.
Proof. Suppose first that there there exists a finite sequence of maps A=A0→A1→ · · · →An'k
where eachAi is a square-zero extension ofAi+1 byk[ni], for someni ≥0. We prove that eachAi satisfies conditions (a) through (d) using descending induction oni. The casei=nis obvious, so let us assume that i < n and thatAi+1 is known to satisfy conditions (a) through (d). We have a fiber sequence ofk-module spectra
k[ni]→Ai→Ai+1
which immediately implies thatAi satisfies (a), (b), and (c). To prove (d), we note that the mapφ:π0Ai→ π0Ai+1 is surjective and ker(φ)2= 0, so that the quotient ofπ0Aiby its radical agrees with the quotient of π0Ai+1 by its radical.
Now suppose that Asatisfies conditions (a) through (d). We will prove thatAis small by induction on the dimension of thek-vector space π∗A. Letn be the largest integer for whichπnA does not vanish. We first treat the casen= 0. We will abuse notation by identifyingAwith the underlying associative ringπ0A.
Letndenote the radical ofA. Ifn= 0, then condition (d) implies that A'k so there is nothing to prove.
Otherwise, we can view nas a nonzero module over the associative algebraA⊗kAop. It follows that there exists a nonzero elementx∈nwhich is annihilated byn⊗kn. Using (d) again, we deduce that the subspace kx⊆A is a two-sided ideal ofA. Let A0 denote the quotient ringA/kx. Theorem A.7.4.1.23 implies that Ais a square-zero extension ofA0 byk. The inductive hypothesis implies thatA0 is small, so thatAis also small.
Now suppose that n >0 and letM =πnA. Then M is a nonzero bimodule over the finite dimensional k-algebraπ0A. It follows that there is a nonzero elementx∈M which is annihilated (on both sides) by the action of the radicaln⊆π0A. LetM0 denote the quotient ofM by the bimodule generated by x(which, by virtue of (d), coincides withkx), and letA00 =τ≤n−1A. It follows from Theorem A.7.4.1.23 that there is a pullback diagram
A //
k
A00 //k⊕M[n+ 1].
Set A0 =A00×k⊕M0[n+1]k. Then A 'A0×k⊕k[n+1]k so we have an elementary map A → A0. Using the inductive hypothesis we deduce thatA0 is small, so thatAis also small.
We will also need a noncommutative analogue of Lemma 1.1.20:
Proposition 3.2.3. Let k be a field and let f : A→ B be a morphism in Algsmk . Then f is small (when regarded as a morphism in Algaugk ) if and only if it induces a surjection of associative rings π0A→π0B.
Proof. The “only if” direction is obvious. For the converse, suppose thatπ0A→π0B is surjective, so that the fiberI= fib(f) is connective. We will prove thatf is small by induction on the dimension of the graded vector space π∗I. If this dimension is zero, then I ' 0 and f is an equivalence. Assume therefore that π∗I 6= 0, and let n be the smallest integer such that πnI 6= 0. Let LB/A denote the relative cotangent complex of B over A in the setting of E1-algebras, regarded as an object of BBModB(Modk). Remark A.7.4.1.12 supplies a fiber sequence
LB/A→B⊗AB →B.
In the∞-category LModB, this sequence splits; we therefore obtain an equivalence of leftB-modules LB/A'cofib(B→B⊗AB)'B⊗Acofib(A→B)'B⊗AI[1].
The kernel of the map π0A → π0B is contained in the radical of π0A and is therefore a nilpotent ideal.
It follows that πn+1LB/A ' Torπ00B(π0A, πnI) is a nonzero quotient of of πnI. Let us regard πn+1LB/A
as a bimodule over π0B, and let n be the radical of π0B. Since n is nilpotent, the two-sided submodule n(πn+1LB/A) + (πn+1LB/A)ndoes not coincide withπn+1LB/A. It follows that there exists a map ofπ0 B-bimodulesπn+1LB/A →k, which determines a map LB/A→k[n+ 1] in the∞-category BBModB(Modk).
We can interpret this map as a derivation B →B⊕k[n+ 1]. Let B0 =B×B⊕k[n+1]k be the associated square-zero extension ofB byk[n]. Thenf factors as a composition
A f
0
→B0 f
00
→B.
Since the map f00 is elementary, it will suffice to show that f0 is small, which follows from the inductive hypothesis.
Corollary 3.2.4. Letkbe a field and letX: Algsmk →Sbe a functor. ThenX belongs to the full subcategory Moduli(1)k of Definition 3.0.3 if and only if it is a formal moduli problem in the sense of Definition 1.1.14.
Proof. The “if” direction follows immediately from Proposition 3.2.3. For the converse, suppose that X satisfies the conditions of Definition 3.0.3; we wish to show thatX is a formal moduli problem. According to Proposition 1.1.15, it will suffice to show that for every pullback diagram in Algsmk
A //
B
k //k⊕k[n]
satisfyingn >0, the associated diagram of spaces
X(A) //
X(B)
X(k) //X(k⊕k[n])
is also a pullback square. This follows immediately from condition (2) of Definition 3.0.3.
We can now state the main result of this section:
Theorem 3.2.5. Letk be a field. Then the Koszul duality functor D(1): (Algaugk )op→Algaugk
is a deformation theory (on the deformation context(Algaugk ,{E})of Construction 3.2.1).
Proof of Theorem 3.0.4. Letk be a field of characteristic zero, and let Ψ : Algaugk →Fun(Algsmk ,S) be the functor given on objects by the formula Ψ(A)(R) = MapAlgaug
k (D(1)(R), A). Combining Theorems 3.2.5 and 1.3.12, we deduce that Ψ is a fully faithful embedding whose essential image is the full subcategory Moduli(1)k ⊆Fun(Algsmk ,S) spanned by the formal E1 moduli problems.
Proof of Theorem 3.2.5. The ∞-category Algaugk is presentable by Corollary A.3.2.3.5, and D(1) admits a left adjoint by Remark 3.1.7. Let Ξ0 ⊆ Algaugk be the full subcategory spanned by those algebras which are coconnective and locally finite (see Definition 3.1.13). We will complete the proof that D(1) is a weak deformation theory by showing that the subcategory Ξ0 satisfies conditions (a) thorugh (d) of Definition 1.3.1:
(a) For every objectA∈Ξ0, the unit mapA→D(1)D(1)(A) is an equivalence. This follows from Corollary 3.1.15.
(b) The full subcategory Ξ0 contains the initial objectk∈Algaugk . This is clear from the definitions.
(c) For each n≥1, there exists an objectKn ∈Ξ0 and an equivalence k⊕k[n]'D(1)(Kn). In fact, we can takeKn to be the free algebraL
m≥0V⊗m generated byV =k[−n−1] (this is a consequence of Proposition 4.5.6, but is also not difficult to verify by direct calculation).
(d) Letn≥1 and suppose we are given a pushout diagramσ: Kn //
φ
k
A //A0
in Algaugk , whereKn is as in (c). We must show that ifA ∈Ξ0, thenA0 ∈Ξ0. Note that σ is also a pushout diagram in Algk. We will make use of the fact that Algk is the underlying∞-category of the model category Algdgk of differential graded algebras overk(for a different argument which does not use the theory of model categories, we refer the reader to the proof of Theorem 4.5.5). Choose a cofibrant differential graded algebra A∗ representing A, and let B∗ denote the free differential graded algebra generated by a classxin degree (−n−1). SinceB∗ is cofibrant andA∗is fibrant, the mapφ:Kn →A can be represented by a map φ0 : B∗ → A∗ of differential graded algebras, which is determined by the element x0 =φ0(x)∈A−n−1. LetB∗0 denote the free differential graded algebra generated by the chain complex E(−n)∗ (see the proof of Proposition 2.1.10): in other words, B0∗ is obtained fromB∗ by freely adjoining an element y∈B−n0 satisfyingdy=x. ThenB∗0 is quasi-isomorphic to the ground field k. Letψ0:B∗→B∗0 be the evident inclusion, and form a pushout diagramσ0:
B∗ ψ0 //
φ0
B∗0
A∗ //A0∗
in the category Algdgk . SinceA∗is cofibrant andψ0is a cofibration, the diagramσ0is also a homotopy pushout square, so that the image of σ0 in Algk is equivalent to the diagram σ. It follows that the differential graded algebra A0∗ represents A0. We can describeA0∗ explicitly as the differential graded algebra obtained from A∗ by adjoining an element y0 in degree −n satisfying dy0 = x0. As a chain complex,A0∗ can be written as a union of an increasing family of subcomplexes
A∗=A00∗ ⊆A01∗ ⊆A02∗ ⊆ · · ·,
where A0m∗ denote the graded subspace of A0∗ generated by products of the form a0ya1y· · ·am1yam. The successive quotients for this filtration are given byA0m∗ /A0m−1∗ 'A⊗m+1∗ [−nm]. It follows that the homology groups ofA0∗/A∗can be computed by means of a (convergent) spectral sequence{Erp,q, dr}r≥2 with
E2p,q'
(0 ifp≤0 (H∗(A∗)⊗p+1)q+p+np ifp≥1.
Since A is coconnective and n > 0, the groups E2p,q vanish unless p+q ≤ −np <0. It follows that each homology group Hm(A0∗/A∗) admits a finite filtration by subquotients of the vector spacesE2p,q with p+q = m, each of which is finite dimensional (since A is locally finite), and that the groups Hm(A0∗/A∗) vanish form≥0. Using the long exact seuqence
· · · →Hm(A∗)→Hm(A0∗)→Hm(A0∗/A∗)→Hm−1(A∗)→ · · ·,
we deduce that Hm(A0∗) is finite dimensional for allm and isomorphic to Hm(A∗) for m ≥ 0, from which it follows immediately that A0∈Ξ0.
We now complete the proof by showing that the weak deformation theory D(1) satisfies axiom (D4) of Definition 1.3.9. Forn≥1, andA∈Algaugk , we have a canonical homotopy equivalence
Ψ(A)(Ω∞−n(E)) = MapAlgaug
k (D(1)(k⊕k[n]), A)'MapAlgaug
k (Kn, A)'Ω∞−n−1fib(A→k).
These maps determine an equivalence from the functore : Algaugk →Sp with I[1], where I : Algaugk →Sp denotes the functor which assigns to each augmented algebra:A→kits augmentation ideal fib(). This functor is evidently conservative, and preserves sifted colimits by Proposition A.3.2.3.1.
Remark 3.2.6. Letkbe a field, let:A→kbe an object of Algaugk , and letX= Ψ(A) denote the formal E1moduli problem associated toAvia the equivalence of Theorem 3.0.4. The proof of Theorem 3.2.5 shows that the shifted tangent complexX(E)[−1]∈Sp can be identified with the augmentation ideal fib() ofA.
We close this section by proving a noncommutative analogue of Corollary 2.3.6.
Proposition 3.2.7. Let k be a field and let X : Algsmk →S be a formal E1 moduli problem over k. The following conditions are equivalent:
(1) The functor X is prorepresentable (see Definition 1.5.3).
(2) Let X(E) denote the tangent complex ofX. Then πiX(E)'0 fori >0.
(3) The functor X has the form Ψ(A), where A∈Algaugk is coconnective and Ψ : Algaugk →Moduli(1)k is the equivalence of Theorem 3.0.4.
Proof. The equivalence of (2) and (3) follows from Remark 3.2.6. We next prove that (1)⇒(2). Since the constructionX 7→X(E) commutes with filtered colimits, it will suffice to show that πiX(E)'0 fori >0 in the case whenX = SpecRis representable by an objectR∈Algsmk . In this case, we can writeX = Ψ(A) whereA=D(1)(R) belongs to the full subcategory Ξ0⊆Algaugk appearing in the proof of Theorem 3.2.5. In particular,Ais coconnective, so thatX satisfies condition (3) (and therefore also condition (2)).
We now complete the proof by showing that (3) ⇒ (1). Let A ∈ Algaugk be coconnective. Choose a representative of A by a cofibrant differential graded algebra A∗ ∈ Algdgk . Since A∗ is cofibrant, we may assume that the augmentation ofAis determined by an augmentation of A∗. We now construct a sequence of differential graded algebras
k=A(−1)∗→A(0)∗→A(1)∗→A(2)∗→ · · ·
equipped with maps φ(i) : A(i)∗ → A∗. For each n < 0, choose a graded subspace Vn ⊆ An consisting of cycles which maps isomorphically onto the homology Hn(A∗). We regard V∗ as a differential graded vector space with trivial differential (which vanishes in nonnegative degrees). Let A(0)∗ denote the free differential graded Lie algebra generated by V∗, and φ(0) : A(0)∗ → A∗ the evident map. Assume now that we have constructed a map φ(i) : A(i)∗ → A∗ extending φ(1). Since A is coconnective, the map θ: Hn(A(i)∗)→H∗(A∗) is surjective. Choose a collection of cycles xα∈A(i)nα whose images form a basis for ker(θ). Then we can writeφ(i)(xα) =dyα for someyα∈Anα+1. LetA(i+ 1)∗ be the differential graded algebra obtained from A(i)∗ by freely adjoining elements Yα (in degrees nα+ 1) satisfying dYα =xα. We letφ(i+ 1) :A(i+ 1)∗→A∗ denote the unique extension ofφ(i) satisfyingφ(i+ 1)(Yα) =yα.
We now prove the following assertion for each integer i≥0:
(∗i) The inclusionV−1 ,→A(i)−1 induces an isomorphismV−1 →H−1(A(i)∗), the unit mapk→A(i)0 is an isomorphism, and A(i)j'0 forj >0.
Assertion (∗i) is obvious wheni= 0. Let us assume that (∗i) holds, and letθ be defined as above. Then θis an isomorphism in degrees≥ −1, so thatA(i+ 1)∗is obtained fromA(i)∗by freely adjoining generators Yαin degrees≤ −1. It follows immediately thatA(i+ 1)j'0 forj >0 and that the unit mapk→A(i+ 1)0
is an isomorphism. Moreover, we can writeA(i+ 1)−1'A(i)−1⊕W, whereW is the subspace spanned by
elements of the formYαwherenα=−2. By construction, the differential onA(i+ 1)∗carriesW injectively into
A(i)−2/dA(i)−1⊆A(i+ 1)−2/dA(i)−1,
so that the differential graded algebrasA(i+ 1)∗ andA(i)∗ have the same homology in degree−1.
Let A0∗ denote the colimit of the sequence {A(i)∗}i≥0. The evident map g0∗ → g∗ is surjective on homology (since the map A(0)∗ →g∗ is surjective on homology). Ifη ∈ker(H∗(A0∗)→ H∗(A∗), thenη is represented by a classη∈ker(H∗(A(i)∗)→H∗(A∗)) fori0. By construction, the image of η vanishes in H∗(A(i+ 1)∗), so thatη= 0. It follows that the mapA0∗→A∗ is a quasi-isomorphism. Since the collection of quasi-isomorphisms in Algdgk is closed under filtered colimits, we conclude that A∗ is a homotopy colimit of the sequence{A(i)∗}i≥0 in the model category Algdgk . LetA(i)∈Algaugk be the image of the differential graded algebraA(i)∗ (equipped with the augmentation determined by the mapφ(i) :A(i)∗→A∗), so that A'lim
−→A(i) in Algaugk . SettingX(i) = Ψ(A(i)∗)∈Moduli(1)k , we deduce thatX 'lim
−→X(i). To prove that X is prorepresentable, it will suffice to show that eachX(i) is prorepresentable.
We now proceed by induction on i, the case i= −1 being trivial. To carry out the inductive step, we note that each of the Lie algebras A(i+ 1)∗ is obtained from A(i)∗ by freely adjoining a set of generators {Yα}α∈S of degrees nα+ 1≤ −1, satisfyingdYα =xα ∈A(i)nα. Choose a well-ordering of the set S. For eachα∈S, we letA<α∗ denote the Lie subalgebra of A(i+ 1)∗ generated byA(i)∗ and the elementsYβ for β < α, and letA≤α∗ be defined similarly. Set
X<α= Ψ(A<α∗ ) X≤α= Ψ(A≤α∗ ).
For each integer n, letB(n)∗ be the free differential graded algebra generated by a classxin degree nand B0(n)∗ the free differential graded algebra generated by a classxin degreen and a classy in degree n+ 1 satisfyingdy=x. For eachα∈S, we have a homotopy pushout diagram of differential graded algebras
B(n)∗ //
B0(n)∗
A<α∗ //A≤α∗ , hence a pushout diagram diagram of formalE1moduli problems
Spec(k⊕k[nα+ 1]) //
Spec(k)
X<α //X≤α.
It follows that the mapX(i)→X(i+1) satisfies the criterion of Lemma 1.5.9. SinceX(i) is prorepresentable, we conclude thatX(i+ 1) is prorepresentable.