Generators in formal deformations of categories
Anthony Blanc, Ludmil Katzarkov, Pranav Pandit
TL;DR
This paper applies Lurie's formal moduli theory to analyze deformations of $k$-linear $$-categories, showing under certain conditions that deformations have zero curvature and possess compact generators.
Contribution
It establishes conditions under which formal deformations of $k$-linear $$-categories have zero curvature and include compact generators, extending deformation theory in higher categories.
Findings
Deformations have zero curvature under specified conditions.
Deformations admit compact generators.
The theory connects formal moduli problems with categorical deformations.
Abstract
In this paper we use the theory of formal moduli problems developed by Lurie in order to study the space of formal deformations of a -linear -category for a field . Our main result states that if is a -linear -category which has a compact generator whose groups of self extensions vanish for sufficiently high positive degrees, then every formal deformation of has zero curvature and moreover admits a compact generator.
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Generators in formal deformations of categories
Anthony Blanc
Ludmil Katzarkov
Pranav Pandit
Abstract
In this paper we use the theory of formal moduli problems developed by Lurie in order to study the space of formal deformations of a -linear -category for a field . Our main result states that if is a -linear -category which has a compact generator whose groups of self extensions vanish for sufficiently high positive degrees, then every formal deformation of has zero curvature and moreover admits a compact generator.
Contents
1 Introduction
Let be a field of characteristic zero and let be an associative -algebra. It is known since the work of Gerstenhaber that the formal deformations of up to isomorphism are in bijection with the set of solutions of the Maurer–Cartan equation in Hochschild cohomology up to gauge equivalence. In their seminal work [KS05] Kontsevich and Soibelman have pionereed the approach of studying deformations of dg-algebras and dg-categories using differential graded Lie algebras. They show that if is a connective differential graded algebra over , its deformations as a dg-algebra up to quasi-isomorphism are controlled by the truncated Hochschild cohomology complex viewed as a differential graded Lie algebra.
Given a -linear dg-category , it has been a long standing problem to give the precise relation between the dg-category deformations of up to Morita equivalence and its Hochschild cohomology complex. In the case of first order infinitesimal deformations, each Hochschild -cocycle corresponds to a curved -deformation of (see [Low08]). In [KL09], B. Keller and W. Lowen gave examples of curved deformations which are not Morita equivalent to uncurved deformations. In [LB15], W. Lowen and M. Van den Bergh study deformations of a dg-algebra up to torsion Morita equivalence and show that when is cohomologically bounded above, the set of formal deformations of up to torsion Morita equivalence is in bijection with the set of Maurer–Cartan solutions up to gauge equivalence. In order to study the whole space of deformations of a category up to Morita equivalence and not just merely its set of connected components, and also to exhibit the precise relation between this space and Hochschild cohomology, it is useful to place ourselves in the setting of derived algebraic geometry and more precisely in the language of formal moduli problems developed by Jacob Lurie in [Lurb].
Formal moduli problems. Deformation theory was first formalized through the notion of functors on artinian rings satisfying the Schlessinger conditions. Formal moduli problems form a derived and higher analog of these functors. We refer the reader to [Lurb] for an introduction to formal moduli problems and their classification. Roughly they are functors from certain artinian -algebrasaaaIn characteristic zero, we can also think of these as functors on artinian commutative differential graded algebras. to spaces satisfying the derived analogs of the Schlessinger conditions. The main result states that formal moduli problems are classified by differential graded Lie algebras (over a field of characteristic zero). In loc. cit. Lurie studies noncommutative analogs of formal moduli problems known as formal -moduli problems for , which are functors from artinian -algebras to spaces satisfying the derived Schlessinger conditions, where is the -operad of little -cubes. In this context, the formal -moduli problems are classified by augmented -algebras. It is often useful to find the minimum for which a moduli problem is defined, thereby obtaining the true algebraic structure on the tangent complex (see [Fra13] for a discussion of these ideas).
Formal moduli of a category after Lurie. Let be a field of arbitrary characteristic. In loc. cit. Lurie studies deformations of a presentable stable -linear -categorybbbAs a matter of facility for citations, from now on we study deformations of linear -categories instead of differential graded categories. See the work of L. Cohn [Coh13] for a comparison between the two theories, recalled in §3.1. up to equivalencecccRecall that the notion of equivalence between presentable -categories corresponds to the notion of Morita equivalence between the corresponding small subcategories of compact objects. See §3.1. by looking at the functor
[TABLE]
from the -category of artinian -algebras to the -category of spaces, classifying deformations of . That is, for an artinian -algebra , the space classifies pairs with a right -linear -category and an equivalence. In general, the functor does not satisfy the derived Schlessinger conditions, the moral reason for this being the existence of curved deformations (see Example 4.11). When is compactly generateddddAn -category is compactly generated if it is generated under filtered colimits by its subcategory of compact objects (as for example the derived -category of quasi-coherent sheaves on a finite type scheme)., the subfunctor spanned by compactly generated deformations is better behaved and closer to satisfying the Schlessinger conditions. We denote by the -algebra given by the center of whose homotopy groups are the Hochschild cohomology groups of . Lurie constructs a natural transformation
[TABLE]
from to the formal moduli problem controlled by the augmented -algebra . He shows that under certain reasonable boundedness assumptions on the map is close to being a homotopy equivalence: it induces an isomorphism on for (at any basepoint) and an injection on . The formal moduli can be regarded as a substitute for the space of curved -deformations and the map as the embedding from uncurved deformations to curved deformations. By the universal property of the center, for each artinian -algebra , the space is given by the space of linear actions of the -algebra on the -category , where is the -Koszul dual of .
1.1 Main results
Our main results provide a new understanding of the behavior of the map in the case of formal deformations, that is, over the adic ring of formal power series. Namely, we give conditions on under which the map induces a homotopy equivalence on formal deformations, therefore providing the missing step to relate deformations of categories to Hochschild cohomology. It can also be viewed as a solution to the curvature problem of [LB15], where similar computations are performed.
Let be a field and let be compactly generated -linear -category. The space of compactly generated deformations of is the space
[TABLE]
That is, a point in this space is essentially given by a formal family where each is a compactly generated deformation of over and is an equivalence, plus higher coherences. In this case, the space is given by the space of linear actions of the -algebra on the -category , where is the free graded commutative algebra on a generator in cohomological degree viewed as an -algebra.
Theorem 1.1**.**
(See Theorem 4.27).* — Let be a field and let be a compactly generated -linear -category. Suppose that admits a single compact generator such that for . Then the natural transformation induces a homotopy equivalence*
[TABLE]
The idea of the proof is to exhibit another compact generator of on which acts trivially, which implies that it is unobstructed. For this it suffices to kill the multiplication by on the generator by forming its cocone. As a by product of the proof of Theorem 4.27 we obtain the following existence result for compact generators.
Theorem 1.2**.**
(See Theorem 4.29).* — In the situation of Theorem 1.1, let be any compactly generated formal deformation of . Then there exists a formal family of compact generators of the family which satisfies for .*
We emphasize that our method in the proof of Theorems 1.1 and 1.2 relies on the fact that the -Koszul dual of the algebra of formal power series (viewed as an -algebra) is given by the free graded associative (also commutative) algebra with in degree (viewed as an -algebra). The argument cannot be adapted verbatim to first order infinitesimal deformations where the -algebra is given by the more complicated free -algebra on a generator in degree .
F. Petit has proven in [Pet12] the existence of compact generators in derived categories of -modules. We expect to give an alternative proof of this fact using Theorem 1.2.
On a more fundamental level, we prove a new result concerning the loop functor on the -category of formal moduli problems over a deformation context (Proposition 2.15). Moreover it implies a new description of the formal moduli associated to any -proximate formal moduli (Corollary 2.19). As a consequence, we obtain the following description of the formal moduli in terms of -group actions (which also appeared in [Pre12, Prop 5.3.3.4]).
Proposition 1.3**.**
(See Corollary A.11).* — Let be a field and let be a -linear -category. Then the formal -moduli problem is given by the formula*
[TABLE]
where is the -category of Ind-coherent sheaves on endowed with the convolution monoidal structure.
Acknowledgments
A special acknowledgment and many thanks are due to Bertrand Toën for generously sharing his ideas on the subject matter of this paper and for many enlightening discussions. We would like to thank Mohammed Abouzaid, Alexander Efimov, David Gepner, Benjamin Hennion, Julian Holstein, Maxim Kontsevich, Wendy Lowen, Tony Pantev, François Petit, Marco Robalo, Pavel Safronov, Carlos Simpson, Michel Van den Bergh and Gabriele Vezzosi for useful discussions related to the subject matter of this paper.
The second and third named authors were partially supported by ERC Gemis grant. The second named author was supported by Simons research grant, NSF DMS 150908, ERC Gemis, DMS-1265230, DMS-1201475 OISE-1242272 PASI. Simons collaborative Grant - HMS, and partially supported by the Laboratory of Mirror Symmetry NRU HSE, RF government grant, ag. 14.641.31.000.
1.2 General conventions
- •
Let be a Grothendieck universe, with satisfying the axiom of infinity. The -small mathematical objects will be called only small. We assume the axiom of Universes. Some arguments in this article will require to enlarge the universe , which is always possible by assuming the axiom of Universes. If is such an enlargement in which is small, the -small mathematical objects will be called not necessarily small.
- •
We work within the theory of -categories in the sense of Lurie [Lur09], a.k.a quasicategories. We follow the terminology and the conventions of loc.cit. regarding the theory of -categories. We denote by the -category of small -categories.
- •
For two -categories and , we denote by the -category of functors from to .
- •
If is an -category admitting a final object . The -category of pointed objects in is by definition the full subcategory of consisting of morphisms such that is a final object of .
- •
We denote by the -category of small spaces in the sense of [Lur09, Def 1.2.16.1]. It is therefore defined as the simplicial nerve of the simplicial category of small Kan complexes. We denote by the -category of not necessarily small spaces. The word space means an object of the -category . The expression not necessarily small space means an object of for some appropriate universe . We denote by the final object of .
- •
We follow the terminology of [Lurd] concerning -connective spaces. Let be an integer. A space is -connected if for every and every vertex . Every space is declared to be -connective.
- •
We follow the terminology of [Lurd] concerning -truncated spaces. Let be an integer. A space is -truncated if for every and every vertex . By convention a space is -truncated if it is either empty or contractible, and is -truncated if it is contractible. A map of pointed spaces is called -truncated if its fiber is -truncated.
- •
We denote by the stable -category of spectra (see [Lurd, Def 1.4.3.1]).
- •
Let be an integer. We consider the usual t-structure on the stable -category . A spectrum is -connective (resp. -truncated) if for every (resp. ).
2 Generalities on formal moduli problems
We start by recalling from [Lurb, §4] the notion of formal moduli problems defined over -algebras and their classification. Then we recall the setting of axiomatic deformation theory from [Lurb, §1], in which our results from the next subsection §2.2 concerning the approximation of formal moduli via loop spaces are proven.
2.1 Reminders on formal moduli problems
Reminders on formal -moduli problems
Let be a field of arbitrary characteristic. Let denote the -category of -module spectra. Let be an integer and let denote the -category of -algebras over in the sense of [Lurd, Def 7.1.3.5]. Let denote the -category of augmented -algebras. We call an -algebra over artinian if the following conditions are satisfied
- •
for and for .
- •
For every , the -vector space is finite dimensional.
- •
Let denote the radical of the algebra , then the natural map is an isomorphism.
We denote by the full subcategory spanned by artinian -algebras. For every integer , we denote by the shifted square zero extension viewed as -algebras over , which are examples of artinian -algebras.
A functor is called a formal -moduli problem if the space is contractible and if for every pullback diagram
[TABLE]
in such that the maps are surjective, the induced square
[TABLE]
is a pullback diagram in . Let denote the full subcategory spanned by those functors such that is a contractible space. We denote by the full subcategory spanned by formal -moduli problems. The class of pullback diagrams of the form
[TABLE]
for are used to defined the tangent complex of a formal -moduli problem. Indeed if is a formal -moduli problem over , the induced map is a homotopy equivalence, and defines a spectrum called the tangent complex of (see [Lurb, Def 1.2.5]). By [Lurb, Prop 1.2.10], a map of formal -moduli problems is an equivalence if and only if the induced map is an equivalence of spectra.
By [Lurb, Thm 4.0.8] there exists a functor
[TABLE]
which is an equivalence of -categories. For an augmented -algebra and for an artinian -algebra over , there is an equivalence where is the Koszul duality functor defined in [Lurb, Prop 4.4.1]. Moreover there exists a commutative diagram of -categories
[TABLE]
where is given by taking the augmentation ideal.
Reminders on axiomatic deformation theory
We recall some definitions and constructions from [Lurb, §1].
For a presentable -category, we denote by the -category of spectra objects of in the sense of [Lurd, Def 1.4.2.8]. By [Lurd, Rem 1.4.2.25] there exists an equivalence of -categories . We denote by the evaluation on the [math]-sphere and for every integer , we denote by the composition of with the th shift functor on .
A deformation context in the sense of [Lurb, Def 1.1.3] is a pair where is a presentable -category and is a sequence of objects of the -category of spectra objects in . Given a deformation context , a map in is called elementary if there exists an index , an integer and a pullback diagram
[TABLE]
in , where is a final object of . A map in is called small if it can be written as a finite composition of elementary maps in . An object is called artinian if the map is small. We denote by the full subcategory of spanned by artinian objects.
Given a deformation context , recall that a formal moduli problem in the sense of [Lurb, Def 1.1.14] is a functor which satisfies the two conditions
The space is contractible. 2. 2.
For every pullback diagram
[TABLE]
in such that the map is small, the induced square
[TABLE]
is a pullback diagram in .
By [Lurb, Prop 1.1.15] we are allowed to test condition (2) on the smaller class of pullback diagrams of the form
[TABLE]
in where and .
We denote by the subcategory spanned functors satisfying condition (1) above, or in other words sending to a contractible space. We denote by the subcategory spanned by the formal moduli problems.
Example 2.1**.**
Let be a field. Let and be the presentable -category of augmented -algebras over . Consider the deformation context where is seen as a module over itself. The deloopings of are given by the shifted square zero extensions for . By [Lurb, Prop 4.5.1], an augmented -algebra is artinian with respect to the deformation context if and only if it is artinian in the sense of §2.1. By [Lurb, Prop 4.5.1], a morphism in is small with respect to the deformation context if and only if the induced map is surjective. Then [Lurb, Cor 4.5.4] implies that is a formal moduli problem with respect to the deformation context if and only if it is a formal -moduli problem in the sense of §2.1.
Notation 2.2**.**
Let be a deformation context. Consider the embedding into the -category of functors such that is a contractible space. The -categories and are presentable and by definition of formal moduli problems the functor preserves all small limits. By the adjoint functor theorem [Lur09, Cor 5.5.2.9], this implies that admits a left adjoint denoted by
[TABLE]
For every functor sending to a contractible space we have the unit natural transformation of this adjunction. The formal moduli problem is therefore initial among formal moduli problems having a natural transformation .
Let be a deformation context. Recall from [Lurb, Def 1.2.5] that every formal moduli problem has a tangent spectrum of at . Moreover a map of formal moduli problems is an equivalence if and only if for every the map is an equivalence of spectra.
In [Lurb, Def 1.3.1 and 1.3.9], Lurie defines the notion of weak deformation theory and of deformation theory. In short a weak deformation theory for consists of a functor such that is a presentable -category, has a left adjoint which behaves like an inverse equivalence to on a certain subcategory of . In this situation, [Lurb, Cor 1.3.6] asserts the existence of an -functor
[TABLE]
which is given on objects by . By applying the tangent spectrum we obtain for every a functor given on objects by . A deformation theory for is a weak deformation theory such that each preserves small sifted colimits and such that a morphism in is an equivalence if and only if each is an equivalence of spectra. The main theorem [Lurb, Thm 1.3.12] then asserts that under the existence of a deformation theory for , the functor is an equivalence of -categories.
Formal spectrum and Koszul duality
We now set up some notations and intermediate results concerning the formal spectrum, pro-objects and formal Koszul duality.
Notation 2.3**.**
Let be a deformation context and let be any object. Consider the functor which is co-represented by given by the formula . Then is a formal moduli problem which is called the* formal spectrum of *. Moreover the construction determines a functor
[TABLE]
which is fully faithful when restricted to the subcategory spanned by artinian objects.
Example 2.4**.**
Let and consider the deformation context formed by augmented -algebras over . Let be an augmented -algebra over . If we think about the -algebra as determining a noncommutative scheme , then the formal spectrum can be thought of as the functor parametrizing deformations of the point of determined by over artinian -algebras.
The following Lemma will be useful in the sequel and follows from the fact that the Koszul duality functor is a deformation theory.
Lemma 2.5**.**
Let be an integer and denote by the equivalence given by [Lurb, Thm 4.0.8]. Let be an augmented -algebra over . Then there exists a natural equivalence of augmented -algebras or equivalently an equivalence of formal moduli problems .
Proof.
Consider the natural transformation induced by the natural map
[TABLE]
for each artinian -algebra over . By adjunction the target space of is naturally equivalent to the space and the map is induced by the unit map . Let be an integer and be the trivial square zero extension. We claim that the biduality map is an equivalence. Indeed by [Lurb, Prop 4.5.6] there exists an equivalence of augmented -algebras and by [Lurb, Prop 4.1.13] the -algebra is -coconnective and locally finite whenever , which by [Lurb, Thm 4.4.5] implies that the biduality map is an equivalence for the -algebra , and thus also for . We deduce that the map is a homotopy equivalence and therefore that induces an equivalence on the tangent spectrum, and is therefore an equivalence, which proves the lemma. ∎
2.2 Approximation of formal moduli and loop spaces
As explained above, the deformation functor of a linear -category over a field does not in general satisfy the derived Schlessinger conditions, but turns out to be rather close to satisfy them. Many examples of deformation functors of some algebro-geometric objects behave similarly. In [Lurb, §5.1], Lurie developed a formalism to study these functors under the name of -proximate formal moduli problems for an integer. We will prove that the functor from Notation 2.2 admits a special description when restricted to the subcategory spanned by -proximate formal moduli problems in terms of -maps between certain loop -spaces (see Corollary 2.19). For this we will prove a result concerning the loop space functor on the -category of formal moduli problem (see Proposition 2.15).
Let be a deformation context and an integer. Recall from [Lurb, Def 5.1.5] that a functor is called an -proximate formal moduli problem if the space is contractible and if for every pullback square
[TABLE]
in such that the map is small, the induced map
[TABLE]
is -truncated (that is, all homotopy fibers of this map are -truncated). We observe that [math]-proximate formal moduli problems are exactly the formal moduli problems.
Notation 2.6**.**
We denote by the full subcategory spanned by -proximate formal moduli problems. There is a tower of -categories
[TABLE]
For example, we will see below that the deformation functor of an object in a linear -category is in general 1-proximate and that the deformation functor of a fixed linear -category itself is in general 2-proximate.
Notation 2.7**.**
Let and , and consider the deformation context of Example 2.1. We will denote by the full subcategory of -proximate formal -moduli problem.
Notation 2.8**.**
Let be a deformation context and suppose that is a pointed -category in the sense that any final object is also initial (e.g. the -category of augmented -algebras over ). Then the -category is also pointed. We denote by
[TABLE]
the loop functor in the sense of [Lurd, Rem 1.1.2.9]. The embedding commutes with small limits and the loop functor restricts to a functor . For an integer, we denote by ( times) the iterated loop functor.
Remark 2.9**.**
In the situation of Notation 2.8, let be any functor. Then the functor is given on objects by where the loop space is taken at the base point corresponding to the map induced by the essentially unique map .
If we imagine that the functor parametrizes deformations of an object of an -category corresponding to the connected component , then the functor parametrizes deformations of the identity of this object, or in other words infinitesimal autoequivalences of this object.
Remark 2.10**.**
Let be a Kan fibration between Kan complexes. Let be an integer and suppose that the map is -truncated, that is, all its homotopy fibers are -truncated. Then we deduce from the long exact sequence of homotopy groups for any choice of basepoint that the induced map is a homotopy equivalence, and whenever that the morphism is injective for any basepoint . Besides, we deduce from the same long exact sequence that the induced map is -truncated.
Lemma 2.11**.**
Let be a deformation context such that is a pointed -category. Let and let be an -proximate formal moduli problem. Then the natural map induced by the unit map is an equivalence. In particular the functor is a formal moduli problem.
Proof.
By [Lurb, Thm 5.1.9] the functor is -proximate if and only if the unit map is -truncated. If it is the case, we deduce from Remark 2.10 that the induced map is an equivalence. ∎
Notation 2.12**.**
Let be an -category which admits finite products. Consider the cartesian symmetric monoidal structure with monoidal product given by the cartesian product (see [Lurd, §2.4.1]). Let and consider the -category of -monoid objects in . Recall that an -monoid object in with multiplication map is called grouplike if the two maps
[TABLE]
are equivalences in , where is the th projection map. Recall that an -monoid object in is called grouplike if the underlying -monoid (for any embedding of -operads ) is grouplike. Let denote the full subcategory spanned by grouplike -monoid objects (see [Lurd, Def 5.2.6.6]). If is a presentable -category, then an -monoid object in is grouplike if and only if the monoid is a group.
If is a pointed presentable -category, by [Lurd, Not 5.2.6.11] the iterated loop functor factorizes as a functor
[TABLE]
where by convention, if we set via the equivalence .
Remark 2.13**.**
Let be an -category which admits finite products. The property of being grouplike for an -monoid object is independent of the choice of the embedding of -operads (see [Lurd, Rem 5.2.6.8]). This implies that the equivalence restricts to an equivalence .
Remark 2.14**.**
Let be an -category which admits finite products. Consider the full embedding of grouplike -monoid objects into all -monoid objects. This inclusion preserves finite product and therefore extends to a symmetric monoidal functor. We have an induced functor on the -categories of -monoid objects which is fully faithful. By the additivity theorem [Lurd, Thm 5.1.2.2] the target of this functor is equivalent to the -category of -monoid objects and by definition of grouplike objects the equivalence restricts to an equivalence . Moreover the above embedding restricts to an embedding
[TABLE]
on the subcategories of grouplike -monoid objects. This embedding is an equivalence of -categories. Indeed, it is also essentially surjective, for if is a grouplike -object in , it follows from Remark 2.13 that the underlying -monoid object of is grouplike and lies in the essential image.
The following result provides conditions on the underlying deformation context under which the iterated loop functor is an equivalence for the -category of formal moduli problems.
Proposition 2.15**.**
Let be a deformation context admitting a deformation theory and such that is a pointed -category. Then for every , the iterated loop functor is an equivalence of -categories.
Proposition 2.15 is a consequence of the following Lemma.
Lemma 2.16**.**
Let be a pointed presentable -category. Suppose that there exists a presentable stable -category and a functor which is conservative, preserves geometric realizations of simplicial objects and preserves finite limits. Then for any the functor is an equivalence of -categories.
Proof.
If the assertion follows from the assumption that is pointed, since there are equivalences , where is the -category of pointed objects. Otherwise for finite , by Remark 2.14 and ascending induction on it suffices to prove the result for . For this, consider the diagram
[TABLE]
Because preserves finite limits, there exists a natural equivalence making this diagram commutative up to equivalence. The -category being stable, the functor is an equivalence . The functors and admit as left adjoint the functors and respectively given by the bar construction. By virtue of the assumption that preserves geometric realizations of simplicial objects, and that the bar construction is given by the geometric realization of a simplicial object (see [Lurd, Rem 5.2.2.8]), there exists a natural equivalence . Let be the counit map. The composition of with the composite functor is equivalent to the natural transformation
[TABLE]
which is an equivalence because preserves finite limits. Because is an equivalence and is conservative, this implies that is an equivalence. We can prove similarly that the unit map is an equivalence by composing with the functor . The assertion for follows by passing to the limit over and form Remark 2.13. ∎
Proof of Proposition 2.15.
Let be a deformation theory for the deformation context . By definition, we have at our disposal for every , a functor which preserves small sifted colimits and such that the family is jointly conservative (see [Lurb, Cor 1.3.8, Def 1.3.9]). This implies that the functor
[TABLE]
where for every , is conservative and preserves small sifted colimits. By construction, the functor preserves finite limits, so that has the same property. Denote by the inverse of the equivalence (see [Lurb, Thm 1.3.12]). Then the composite functor satisfies all the assumptions of Lemma 2.16, by which we deduce the claim. ∎
Notation 2.17**.**
In the situation of Proposition 2.15, we denote by the inverse of the equivalence .
Notation 2.18**.**
Let be a deformation context such that is a pointed -category. Consider the restriction of the iterated loop functor . By Lemma 2.11, the functor factorizes as a functor .
Corollary 2.19**.**
Let be a deformation context admitting a deformation theory and such that is a pointed -category. Let and let denote the restriction of the functor to the subcategory of -proximate formal moduli problems. Then there exists an equivalence of functors . In particular, the functor preserves all small limits. As a consequence, for every -proximate formal moduli problem , the functor is given by the formula
[TABLE]
Proof.
By Lemma 2.11, there is an equivalence of functors , which after composition with provides by Proposition 2.15 the expected natural equivalence . Because is given by a limit and that is an equivalence, we have that preserves all small limits. Let be an -proximate formal moduli problem. Then for we have natural homotopy equivalences
[TABLE]
∎
3 Reminders on deformations of categories
We start with notations and terminology related to presentable, compactly generated and linear -categories, as well as the relation with differential graded categories. We follow the terminology of [Lur09] and [Lurd]. We also point out the reference [Rob14] where we can find an overview on this subject.
3.1 Linear -categories
We denote by the -category of not necessarily small presentable -categories together with functors which are left adjoints (see [Lur09, Def 5.5.3.1] for a construction of ).
An -category is said to be compactly generated if it is presentable and if in addition there exists a small -category which admits finite colimits and an equivalence . In this situation, by [Lur09, Thm 5.4.2.2] the small -category which generates can be choosen to be the small idempotent complete full subcategory consisting of compact objects of .
By [Lurd, Prop 4.8.1.14], the -category admits a symmetric monoidal structure in the sense of [Lurd, Def 2.1.2.13] and we denote the induced symmetric monoidal product by . If are presentable -categories of the form and with , small -categories, then we have where is the cartesian symmetric monoidal structure on . The unit of is the -category of small spaces. Moreover this symmetric monoidal structure is closed and for two presentable -categories , , the -category of colimit preserving -functors is presentable. By [Lurd, Lem 5.3.2.11], if and are compactly generated -linear -categories, the tensor product is again compactly generated and the subcategory is generated under finite colimits by .
Let be an -ring. The -category of -module spectra admits a symmetric monoidal structure given by the smash product over . Considering as an -algebra object in the symmetric monoidal -category , we define the -category of -linear -categories to be
[TABLE]
Morphisms in are called -linear functors. Because is a stable -category in the sense of [Lurd, Def 1.1.1.9], the underlying -category of any -linear -category is stable. For example if the sphere spectrum, then is the -category of presentable stable -categories with exact functors between them. The symmetric monoidal structure on induces a symmetric monoidal structure on . We denote the corresponding tensor product by . Moreover the symmetric monoidal structure on is closed. For -linear -categories , and there exists a -linear -category of -linear colimit preserving -functors, such that there exists an equivalence
[TABLE]
In the case , we denote by the -linear -category of -linear endofunctors of . Any -linear -category is enriched over . Indeed if are objects, the -functor given on objects by commutes with small colimits and by presentability of is therefore representable by an object satisfying the universal property
[TABLE]
We set the notation for the Ext groups in .
We now recall the relation of the above with the homotopy theory of -linear differential graded categories. Let be a discrete commutative ring. We consider two -categories of -linear dg-categories (we use the terminology of [Rob14, §6.1.1]):
- •
: the -category encoding the homotopy theory of presentable -linear dg-categories up to quasi-equivalence. Morphisms are dg-functors which commute with small sums. The dg-nerve construction provides an -functor
[TABLE]
- •
: the -category encoding the homotopy theory of presentable compactly generated -linear dg-categories up to quasi-equivalence. Morphisms are dg-functors which commute with small sums and preserve compact objects. The dg-nerve restricts to an -functor
[TABLE]
By the work of Cohn [Coh13] there exists a commutative diagram of -categories
[TABLE]
where is the -category of -linear compactly generated -categories together with -linear functors which preserve small colimits. Moreover the bottom horizontal arrow is an equivalence of -categories.
Notation 3.1**.**
Let be an -ring and a -linear -category. If is a collection of objects of , we denote by the full subcategory of consisting of objects such that for every .
Definition 3.2**.**
Let be an -ring. Let be a -linear -category.
A collection of objects of generates if consists of zero objects of . 2. 2.
A collection of objects of is a family of compact generators of if each object is compact and if the collection generates . 3. 3.
A compact generator of is a compact object of such that the family generates .
Definition 3.3**.**
Let be an -ring. A -linear -category is said to be tamely compactly generated if is compactly generated and if for any compact objects of we have for .
Remark 3.4**.**
Let be an -ring and a -linear -category. Suppose that admits a single compact generator . Then is tamely compactly generated if and only if for .
3.2 Deformations of objects
We recall from [Lurb, §5.2] the fundamental results about the deformation functor of an object in a linear -category over a field.
Let be a field, let be a -linear -category and let be an object. Recall from [Lurb, Const 5.2.1] that we can define a functor
[TABLE]
from artinian -algebras over to spaces which satisfies the following properties.
- •
For every artinian -algebra over , the space classifies deformations of over in , or in other words it is the classifying space of pairs with a right -module object in and an equivalence in .
- •
In general, the functor does not satisfy the derived Schlessinger conditions and is not a formal -moduli problem. Let denote the associated formal -moduli problem.
- •
By [Lurb, Cor 5.2.5], the functor is a -proximate formal -moduli problem, or equivalently the unit map is -truncated.
- •
Let be the -algebra of endomorphisms of in . By [Lurb, Thm 5.2.8] the augmented -algebra which controls the formal -moduli problem is given by . In other words there exists for every a natural equivalence
[TABLE]
In particular the tangent spectrum of is given by .
- •
Koszul duality for modules implies that for every artinian -algebra we have a natural homotopy equivalence
[TABLE]
with the space of deformations of as an Ind-coherent right -modules in , where is the -category of Ind-coherent right -modules in (see [Lurb, Rem 5.2.16]).
- •
By [Lurb, Prop 5.2.14], under the assumption that admits a left complete -structure and that is connective, then is a formal -moduli problem.
Notation 3.5**.**
Let be a field, let be a -linear -category and let be an object. We will denote by the restriction of the functor to artinian -algebras along the forgetful functor .
3.3 Deformations of categories
We first recall from [Lurb, §5.3] the fundamental results about the deformation functor of a linear -category over a field. Then we recall the construction of this functor.
Let be field and a -linear -category. In [Lurb, §5.3] is constructed a functor
[TABLE]
from artinian -algebras over to not necessarily small spaces which satisfies the following properties.
- •
For any artinian -algebra over , the space classifies deformations of over , or in other words pairs with a right -linear -category and an equivalence in (see below for the definition of linear -category over an -algebra). We observe that the space is contractible.
- •
In general, the functor does not satisfy the derived Schlessinger conditions, and is not an -formal moduli problem. Let denote the associated -formal moduli problem and
[TABLE]
denote the unit map. In other words, the map is in general not an equivalence. We will see a simple example below.
- •
However, the map is rather close to be an equivalence as shown by Lurie. Namely, by [Lurb, Cor 5.3.8] the functor is a -proximate -formal moduli problem in the sense of [Lurb, Def 5.1.5] (with values in essentially small spaces). By [Lurb, Thm 5.1.9], it means that the map has [math]-truncated fibers, or in other words, that the map induces an isomorphism on for and is injective on (all homotopy groups are based at the trivial deformation of ).
- •
Let denote the -algebra over given by the center of the -category in the sense of [Lurb, Def 5.3.10]. The underlying -module spectrum of is given by so that the homotopy groups of are isomorphic to the Hochschild cohomology groups of . By [Lurb, Thm 3.5.1], the augmented -algebra which controls the -formal moduli problem is given by . In other words, for any artinian -algebra over there exists a natural equivalence
[TABLE]
where is the -Kozsul duality functor. In particular the tangent spectrum of is given by .
We have stronger properties if we restrict to compactly generated deformations as observed by Lurie (see [Lurb, Variant 5.3.4]), a condition which is reasonable in most examples of interest. Let denote the subfunctor of spanned by pairs where is compactly generated. By [Lurb, Variant 5.3.6], the functor has values in essentially small spaces, and can therefore be viewed as a functor
[TABLE]
which satisfies the following properties.
- •
The natural embedding has [math]-truncated fibers, which implies that is a -proximate -formal moduli problem and that the induced map is an equivalence. We denote by
[TABLE]
the unit map.
- •
Suppose that is tamely compactly generated (see Definition 3.3). Then the functor is a -proximate -formal moduli problem by [Lurb, Prop 5.3.21]. In other words, the map induces an isomorphism on for and an injection on . Our main result gives additional conditions under which the surjectivity on holds for formal deformations.
We recall now the construction of the functor from [Lurb, Const 5.3.2]. By [Lurd, Thm 4.8.5.16] there exists a functor
[TABLE]
which extends to a symmetric monoidal functor. Passing to algebra objects we obtain a functor
[TABLE]
which to an -algebra associates the monoidal -category of left -modules.
Let be an -algebra over . A right -linear -category (resp. left -linear -category) is by definition a right -module in (resp. a left -module in ). If is a morphism of -algebras over , and a right -linear -category, we denote by the extension of scalars .
Consider the cocartesian fibration with fiber over a -linear monoidal -category given by the -category of -categories right tensored over . Set and consider the induced cocartesian fibration with fiber over an -algebra given by the -category of right -linear -categories. Consider the subcategory consisting of -cocartesian morphisms so that we obtain a left fibration with fiber over the space of right -linear -categories.
Let be a -linear -category and consider the pair as an object of . The -category of deformations of is the -category . The induced left fibration is classified by a functor whose restriction to artinian -algebras is denoted by .
4 Generators and curvature in deformations of categories
In this section we first set up some preliminary results concerning simultaneous deformations and deformations of -algebras. Finally, in the last subsection, we use them to prove our main results concerning formal deformations.
4.1 Simultaneous deformations
Let be a field. We now study the deformation functor of a pair with a -linear -category and an object.
We first give a construction of this functor which consists in mimicking the construction of in [Lurb, Const 5.3.2] but replacing the -category by the slice -category of pairs with a -linear -category and an object of .
Construction 4.1**.**
[Lurd, Thm 4.8.5.16] provides a symmetric monoidal functor
[TABLE]
where the -category is endowed with the natural symmetric monoidal structure from [Lurd, Rem 2.2.2.25] coming from the fact that is the unit in . Applying algebra objects we obtain a functor
[TABLE]
Consider the cocartesian fibration
[TABLE]
with fiber over a monoidal -category the -category of pairs where is a -linear -category right tensored over and an object. Set and consider the induced cocartesian fibration with fiber over an -algebra the -category of right -linear -categories together with a distinguished object. Let be the subcategory of -cocartesian morphisms so that we have an induced left fibration . Let be a -linear -category and an object of . The -category of deformations of is the -category
[TABLE]
The induced left fibration is classified by a functor whose restriction to artinian -algebras is denoted by
[TABLE]
For any artinian -algebra over , the space classifies -uples where is a right -linear -category, is an equivalence, is an object and is an equivalence in . We have that is a contractible space.
The deformation functor is closely related to the deformation functors and . We construct the evident maps between them.
Construction 4.2**.**
Let be a field, a -linear -category and an object of . The forgetful functor induces a functor which in turn induces a functor commuting with the projection to . We obtain natural transformations
[TABLE]
[TABLE]
given on objects by forgetting the distinguished object deformation.
The homotopy fiber of the forgetful functor at is equivalent to the Kan complex . The induced functor is given on object by sending an object of to the pair . This functor induces a functor commuting with the projection to . We therefore obtain a natural transformation
[TABLE]
where is the restriction of the functor to artinian -algebras over along the forgetful functor . The functor is given on object by sending a deformation of over an -algebra to the pair . Moreover, whenever is compactly generated, we have that the map factors as a map .
Proposition 4.3**.**
Let be a field, a -linear -category and an object. The sequence of natural transformations
[TABLE]
from Construction 4.2 is a fiber sequence in . Consequently, the functor is a -proximate formal -moduli problem (in a larger universe). Moreover it induces a homotopy commutative diagram
[TABLE]
where the bottom line is a fiber sequence in .
Proof.
The first assertion follows from the fact that the homotopy fiber of the functor at is given by the Kan complex . The second assertion follows directly from the existence of the fiber sequence and the fact that and are -proximate and -proximate respectively. The last assertion follows from the fact that the three functors are -proximate formal moduli problems for some and from the fact (after suitable enlargement of the universe) that the functor preserves small limits by Corollary 2.19. ∎
Remark 4.4**.**
We can already deduce from Proposition 4.3 the tangent spectrum of the formal -moduli problem . However, to avoid dealing with the -algebra associated to the formal -moduli problem , we give a direct construction of the -algebra associated to , which will be useful in the sequel.
Notation 4.5**.**
Let be a field, a -linear -category and an object. Consider the object of the slice -category and its endomorphism monoidal -linear -category . Denote by the -algebra over given by endomorphisms of the unit. By adjunction, this -algebra satisfies the following universal property: for any -algebra over , there exits a homotopy equivalence
[TABLE]
where the right hand side is the space of left -linear structures on the pair , or in other words, the space of left -linear structures on acting trivially on up to equivalence.
Remark 4.6**.**
Let be a field, a -linear -category and an object. By definition of the slice -category , the -linear -category is the homotopy fiber of the -linear functor given by evaluation at . We deduce that the induced sequence
[TABLE]
is a fiber sequence of -module spectra. We observe that the first map in the sequence is a map of -algebras over .
Proposition 4.7**.**
Let be a field, a -linear -category and an object. Consider the augmented -algebra given by and its associated -formal moduli problem . Then there exists a natural transformation which is [math]-truncated. Therefore the formal -moduli problem is given by the formula
[TABLE]
Proof.
We mimic the proof of [Lurb, Thm 5.3.16], replacing the center by . Following the same lines as in [Lurb, Const 5.3.18], we see that there exits a homotopy commutative diagram
[TABLE]
where the upper functor sends a deformation over an augmented -algebra to seen as left tensored over via the equivalence together with the distinguished object . This functor induces a natural transformation
[TABLE]
The same proof as [Lurb, Prop 5.3.17] but replacing the center by the -algebra shows that this map is [math]-truncated. Finally [Lurb, Thm 5.1.9] implies that the induced map is an equivalence. ∎
Remark 4.8**.**
In the situation of Proposition 4.7, the universal property of the -algebra (see Notation 4.5) implies that we have for each artinian -algebra over a natural homotopy equivalence
[TABLE]
with the space of left -linear structures on the -category such that acts trivially on the object .
Remark 4.9**.**
Let be a field, a -linear -category and an object. Recall from Proposition 4.3 the existence of a fiber sequence of formal -moduli problems
[TABLE]
It induces a fiber sequence
[TABLE]
which after taking the tangent spectrum gives a fiber sequence of -module spectra
[TABLE]
It can be seen that for each artinian -algebra over , the map sends an -linear autoequivalence of the trivial deformation to the image of the trivial deformation of in . From this we can deduce that the fiber sequence of tangent spectra above is equivalent to the fiber sequence given in Remark 4.6, although we will not use this fact. We denote by the map of -modules induced by evaluation at and call it the obstruction map of .
Remark 4.10**.**
Let be a field, a -linear -category and an object. Consider the homotopy commutative diagram
[TABLE]
where the right square is induced by Proposition 4.3. The bottom left map is an equivalence because is a formal moduli problem. By [Lurb, Thm 3.5.1], there exists an isomorphism . Moreover by [Lurb, Thm 5.2.8] there is an isomorphism . From Proposition 4.3 and taking , we have an induced commutative diagram
[TABLE]
where the bottom line is exact in the middle. Let be a deformation of over and let . Suppose deforms or in other words that in . In this case we have and thus .
Example 4.11**.**
We can now explain a simple example in which fails to be a formal moduli problem. It is taken from [KL09, Ex 3.14]. Let be a field of characteristic zero and the free graded commutative -algebra generated by a degree variable and its inverse over . We regard as an -algebra and let be the -linear -category of -module spectra. A computation gives generated by the multiplication by whose corresponding cocycle is denoted by . Then for every non zero -module , we have that in because multiplication by on is an equivalence. Suppose now that is the image by the map of a deformation of over . Then any in satisfies in , and by Remark 4.10 we have , which is a contradiction. We deduce that is not in the image of the map and that is not a formal -moduli problem.
Notation 4.12**.**
Let be a field, a compactly generated -linear -category and an object. For each artinian -algebra over we denote by the summand of consisting of -uplets with being compactly generated. We obtain a functor
[TABLE]
Proposition 4.13**.**
Let be a field, a compactly generated -linear -category and an object. The functor is a -proximate -moduli problem and the inclusion induces an equivalence of formal -moduli problems . If moreover is supposed to be tamely compactly generated, then is a -proximate formal -moduli problem.
Proof.
A variant of Proposition 4.3 implies the existence of a homotopy commutative diagram
[TABLE]
where the two horizontal top maps are constructed in Construction 4.2. This diagram implies the first two assertions. The last one follows from this diagram and the fact that if is tamely compactly generated then by [Lurb, Prop 5.3.21] the functor is -proximate. ∎
Remark 4.14**.**
Let be a field, a tamely compactly generated -linear -category and an object. Let be an integer and consider the following commutative diagram induced by the commutative diagram of the proof of Proposition 4.13, similarly to Remark 4.10.
[TABLE]
The bottom line is exact in the middle. All vertical maps are injective because when is tamely compactly generated, the functors and are -proximate and moreover is -proximate. This implies that the upper line is also exact in the middle, as a sequence of pointed sets. We deduce from this that given a compactly generated deformation of over , there exists an object deforming if and only if .
4.2 Deformations of associative algebras
We recall some known facts about the deformation functor of an associative algebra and construct maps to the formal -moduli problem of its -category of left modules and from the formal -moduli problem of simultaneous deformations of pairs given by taking endomorphisms.
Construction 4.15**.**
Consider the functor given on objects by (see [Lurb, Not 5.3.1] or §3.3). For an -algebra over , we denote by the presentable -category of -algebras. Composing the functor with the functor given by taking algebra objects, we obtain a functor sending an -algebra to the -category of -algebras. This functor is classified by a cocartesian fibration with fiber over an -algebra given by the -category . Consider the subcategory of -cocartesian morphisms and the induced left fibration .
Let be an -algebra over . The -category of deformation of is the -category . The induced left fibration is classified by a functor whose restriction to artinian -algebras is denoted by
[TABLE]
For any artinian -algebra , the space classifies pairs with an -algebra and an equivalence of -algebras over .
Proposition 4.16**.**
Let be a field and an -algebra over . For every pullback diagram
[TABLE]
in , the induced map
[TABLE]
has -truncated homotopy fibers (or equivalently induces a homotopy equivalence onto its essential image).
Proof.
By [Lura, Thm 7.4], for every such pullback square in , the induced functor
[TABLE]
is fully faithful. Because the functor given by applying algebra objects preserves small limits and preserves fully faithful functors, the induced functor
[TABLE]
is fully faithful, which implies our assertion. ∎
Corollary 4.17**.**
Let be a field and an -algebra over . For every artinian -algebra over , the space is essentially small. Therefore can regarded as a functor with values in the -category of small spaces.
Proof.
Let be an integer. By Proposition 4.16 the map induces a homotopy equivalence onto its essential image. Moreover the target of this map is homotopy equivalent to the homotopy fiber of the restriction map at and is therefore essentially small. This implies that is essentially small for any . Let now be an artinian -algebra and let be a finite sequence of maps such that for every there exists an integer and a pullback diagram
[TABLE]
By Proposition 4.16 the induced map
[TABLE]
is a homotopy equivalence onto its essential image. We then deduce by descending induction on , using the first part of the proof that is essentially small. ∎
Corollary 4.18**.**
Let be a field and an -algebra over . The the functor is a -proximate formal -moduli problem.
Proof.
It follows by combining Proposition 4.16 and Corollary 4.17. ∎
Proposition 4.19**.**
Let be a field and let be an -connective -algebra over for some integer (i.e. the underlying -module of is -connective). Then the functor is a formal -moduli problem.
Proof.
For an -algebra , let denote the full subcategory of spanned by -connective left -modules. Moreover let denote the full subcategory of spanned by the -connective -algebras. By a variant of [Lura, Prop 7.6], for every pullback diagram
[TABLE]
in such that the maps are surjective, the induced functor
[TABLE]
is an equivalence, where is the -category of -connective left -modules. Because for every -algebra the forgetful functor is conservative, the proof of [Lura, Prop 7.6] also shows that the induced functor
[TABLE]
is an equivalence. Remark that if is an -connective -module and if is a deformation of over an artinian -algebra , then the -module is -connective (see the proof of [Lurb, Prop 5.2.14]). By the above, this implies that the induced map
[TABLE]
is a homotopy equivalence and therefore that is a formal -moduli problem. ∎
Notation 4.20**.**
Let be a field and an -algebra over . Consider the cocartesian fibration from Construction 4.15 and the cocartesian fibration from §3.3. By [Lurd, Not 4.8.5.10] there exists a functor which to a pair with an -algebra over and an -algebra, assigns the pair where is seen as right tensored over . By [Lurd, Prop 4.8.5.1] the functor sends -cocartesian morphisms to -cocartesian morphisms and therefore induces a natural transformation which factors as a natural transformation
[TABLE]
given on objects by applying the -category of left modules.
Notation 4.21**.**
Let be a field, let be a compactly generated -linear -category and let be an object. Consider the cocartesian fibration from Construction 4.15 and the cocartesian fibration from Construction 4.1. By [Lurd, Thm 4.8.5.11], there exists a functor which to a triple with an -algebra over , a right -linear -category and an object, assigns the pair where is seen as an -algebra. It is easy to see that the functor sends -cocartesian morphisms to -cocartesian morphisms and therefore induces a natural transformation
[TABLE]
given on objects by taking endomorphisms.
Remark 4.22**.**
Let be a field, let be a tamely compactly generated -linear -category and let be a compact generator of . Let be an artinian -algebra over , and suppose given a deformation of over and a deformation of . Then by [Lurb, Lem 5.3.31] we have that is tamely compactly generated, and [Lurb, Lem 5.3.37] implies that the object is a compact generator of . Therefore by [Lurd, Thm 7.1.2.1] (Schwede–Shipley result), there exists a natural -linear equivalence
[TABLE]
This fact implies that the diagram
[TABLE]
commutes up to natural homotopy, where is the natural transformation from Construction 4.2 given by forgetting the distinguished object. Indeed, unwinding the construction of each of these functors, we see that it suffices to produce for each artinian -algebra over an equivalence as above.
4.3 Formal deformations
Let be a field and let be a -linear -category. Recall that under the assumption that is tamely compactly generated (see Definition 3.3), we have by [Lurb, Prop 5.3.21] that the natural transformation
[TABLE]
is -truncated. Under some additional assumptions on , we now prove that the map induces a surjection on for formal deformations, and therefore is an equivalence on formal deformations (that is, over the adic ring of formal power series). To prove this, we prove a stronger statement which implies the existence of a compact generator for formal deformations.
We first set up some conventions about formal deformations.
Let be an integer, let be a field and let be a functor. Consider the discrete commutative algebra of formal power series in one variable as an augmented -algebra over via the forgetful functor . The space of formal deformations in is by definition the space
[TABLE]
where is the formal spectrum (see Notation 2.3).
Lemma 4.23**.**
Let be a field and let
[TABLE]
be an inverse system of discrete local artinian commutative -algebras having same residue field . Let be the limit. Let and consider for each the algebra as an artinian -algebra and as an augmented -algebra via the forgetful functor . Then the natural map is an equivalence of formal -moduli problems. Equivalently, the natural map is an equivalence of augmented -algebras over .
Proof.
The statement for is given by [Lurc, Lem 6.3.3]. More precisely, let as an artinian -algebra for some integer and consider the homotopy commutative diagram
[TABLE]
where the vertical maps are given by the forgetful functor . By [Lurc, Lem 6.3.3] the horizontal top map is a homotopy equivalence. Denote by and the subcategories of discrete augmented -algebras over and of discrete augmented -algebras over respectively. As the forgetful functor induces an equivalence of categories . The -algebras and being discrete, this implies that the vertical maps in the diagram are homotopy equivalences. Therefore the horizontal bottom map is a homotopy equivalence as well. We have proven that the map induces an equivalence on tangent spectra, which implies that it is an equivalence of formal -moduli problems. The last statement follows from Lemma 2.5 by applying the equivalence . ∎
Remark 4.24**.**
Let be an integer, let be a field and let be a functor. Applying Lemma 4.23 to the ring of formal power series in one variable over , we obtain a natural homotopy equivalence:
[TABLE]
Therefore a point in is essentially given by a formal family of points together with for every , a path in , where is the map induced by the projection , plus higher coherences.
Remark 4.25**.**
Let be a field and let be a compactly generated -linear -category. Remark 4.24 applied to the functor implies that a point in is essentially given by a formal family with a compactly generated -linear -category together with for each a -linear equivalence .
Remark 4.26**.**
Let be a field and let be a -linear -category. Consider the space of formal deformations . Applying the equivalence , by Lemma 2.5 and [Lurb, Thm 3.5.1] there exists natural homotopy equivalences
[TABLE]
where is the -algebra given by the center of .
The usual computation using the Koszul–Tate resolution gives where is the augmented -algebra over associated to the free augmented graded commutative -algebra on one generator in cohomological degree via the forgetful functor. We deduce from this and from the universal property of the center ([Lurb, Rem 5.3.11]) that there exists a natural homotopy equivalence
[TABLE]
with the space of left -linear structures on the -category .
Here is our first main result.
Theorem 4.27**.**
Let be a field and let be a tamely compactly generated -linear -category which has a compact generator. Then the map
[TABLE]
induced by is a homotopy equivalence.
We will deduce Theorem 4.27 from the following intermediate statement.
Proposition 4.28**.**
Let be a field and let be tamely compactly generated -linear -category which has a compact generator . Let be any point. Then there exists another compact generator of and a formal deformation of its endomorphism algebra such that the connected component of is sent to the connected component of by the composite map
[TABLE]
In particular, the map is surjective on .
Proof of Theorem 4.27.
By [Lurb, Prop 5.3.21] the map is -truncated. This implies that the induced map
[TABLE]
is -truncated as well. By Proposition 4.28 we have that the map is surjective on , which proves that it is a homotopy equivalence. ∎
Here is another consequence of Proposition 4.28.
Theorem 4.29**.**
Let be a field and let be tamely compactly generated -linear -category which has a compact generator . Let be a compactly generated formal deformation of . Then there exists another compact generator of and a formal deformation of its endomorphism algebra such that for every there exists a -linear equivalence . In particular, each is tamely compactly generated and admits a compact generator.
Proof.
Let be a compactly generated formal deformation of . Consider the vertex . Let be the compact generator provided by Proposition 4.28 so that there exists a formal deformation of such that the connected component of is sent to the connected component of by the map . But is injective on by Theorem 4.27, so that for every there exists a -linear equivalence . As is tamely compactly generated, the -algebra is -connective for some integer . By Proposition 4.19 this implies that for every the -algebra is -connective, and therefore that is tamely compactly generated. ∎
Remark 4.30**.**
In the situation of Theorem 4.29, the fact that each is tamely compactly generated for every compactly generated formal deformation of is already a consequence of [Lurb, Lem 5.3.31] and does not require the assumption that has a compact generator.
Proof of Proposition 4.28.
Let . Recall from Remark 4.26 the existence of a homotopy equivalence with the space of left -linear structures on . We denote by the left -linear structure corresponding to . Let be a compact generator of . We exhibit another compact generator of on which acts trivially. The left -action gives a map of -modules and consider the map given by the image of . Define to be the homotopy fiber of in , it is well defined up to a contractible space of choice. We claim that the object is still a compact generator of . That it is compact follows from the fact that the subcategory spanned by compact objects is stable under finite colimits in and that is stable. To see that it is a generator, let be an object of such that and we must prove that . For this we can suppose that is compact. The fiber sequence induces a fiber sequence of -module spectra
[TABLE]
which implies that . Because is tamely compactly generated, we deduce that . But is a generator of and thus .
We adopt the notation
[TABLE]
for the space of left -linear structures on and similarly for left actions of on the pair . Consider the diagram
[TABLE]
where the notations , , and have been removed indices for simplicity. By construction, the two smaller squares in this diagram commutes up to natural homotopy. Moreover the outer square commutes up to natural homotopy by virtue of Remark 4.22. Therefore the right square is homotopy commutative as well, up to natural homotopy. The two horizontal maps in the right square are equivalences by [Lurb, Thm 5.3.16] and Proposition 4.7. Because is supposed to be tamely compactly generated and compact, the -algebra is -connective for some integer which implies that the map is a homotopy equivalence by Proposition 4.19.
We claim that the object of provides a lift of through the map . Unwinding the definitions, we must show that admits a lift through the functor induced by the monoidal functor itself induced by the augmentation map . By the right handed variant of [Lurd, Thm 4.8.4.1], there exists an equivalence of -categories . Therefore we must show that there exists a lift of through the forgetful functor . By a variant of [Lurd, Cor 4.2.4.4] this forgetful functor admits a right adjoint given by the cofree right -module functor which to an object assigns the right -module given by the exponential of by the left -module . There is a fiber sequence in , where the first map is multiplication by . This implies the existence of a fiber sequence in and thus of an equivalence in , which shows that is sent to via the forgetful functor .
We deduce that the pair provides a well defined object in the -category and therefore, by virtue of the commutative diagram above, of a well defined object in corresponding to a formal deformation of the -algebra . By commutativity of the diagram, we have the equality of connected components in .
∎
Appendix A Deformations as !-group actions
As a consequence of Corollary 2.19, we give a description of the formal -moduli problem for a -linear -category over a field in terms of -actions of the formal group on .
Notation A.1**.**
Let be a field and let be a -linear -category, we set the notations and . The natural transformation induces a natural transformation .
Remark A.2**.**
Let be a field and let be a compactly generated -linear -category. Because the embedding of compactly generated deformations into all deformations is fully faithful, the induced map is a homotopy equivalence.
Remark A.3**.**
Let be a field and let be a -linear -category. By [Lurb, Cor 5.3.8] the functor is a -proximate formal -moduli problem, which implies by Remark 2.10 that the functor is a -proximate formal -moduli problem and that the induced map is an equivalence. By Corollary 2.19, we deduce that the induced map is an equivalence.
If in addition is tamely compactly generated, then by [Lurb, Prop 5.3.21] we have that is -proximate and we deduce from Remark A.2 that is a formal -moduli problem and that the natural map is an equivalence.
Remark A.4**.**
Let be a field and let be a -linear -category. It can be shown that there exists a natural transformation which is an equivalence, where is seen as an object of the -linear -category of -linear endofunctors of . Having this, we see that for every artinian -algebra the space parametrizes pairs with an object of the -category (which is necessarily an -linear autoequivalence of ) and an equivalence in . Moreover the space is the classifying space of right Ind-coherent -modules in which deforms , or in other words of pairs with an object in and an equivalence.
Corollary A.5**.**
Let be a field and a -linear -category. Then the formal -moduli problem is given by the formula
[TABLE]
If moreover is tamely compactly generated, then the formal -moduli problem is given by the formula
[TABLE]
Proof.
The first assertion follows from the definition of and from Proposition 2.15. The second follows from the first combined with Remark A.3. ∎
Construction A.6**.**
(Ind-coherent sheaves on formal -moduli problems as a symmetric monoidal functor). Let be a field. Consider the cocartesian fibration of [Lurb, Const 3.4.11] which is classified by the functor given by and functoriality for a morphism in is given by the -pullback . The functor preserves small colimits. It follows that factors through the subcategory of presentable -categories with functors that are left adjoints. The full subcategory is stable under tensor product and therefore inherits a symmetric monoidal structure from . A variant of [Lurd, Thm 4.8.5.16] allows us to promote the functor to a symmetric monoidal functor .
Recall that the Yoneda embedding can be promoted to a symmetric monoidal functor where we endow the -category with its cartesian monoidal structure. Applying the monoidal variant of [Lur09, Thm 5.1.5.6], we deduce that admits an essentially unique factorization as a composition
[TABLE]
such that the functor preserves small limits (recall that a limit in can be computed as the limit of the corresponding diagram in ). For any functor , the -category is the -linear -category of right Ind-coherent sheaves on and the symmetric monoidal structure expresses the fact that for any two functors , the natural functor is an equivalence in . Similarly we can define left Ind-coherent sheaves.
Passing to right adjoints (through the symmetric monoidal equivalence ), we obtain a symmetric monoidal functor
[TABLE]
where the functoriality is now given by the pushforward.
Consider the lax symmetric monoidal inclusion .
Lemma A.7**.**
Let be a field. The restriction of the composition
[TABLE]
to the symmetric monoidal full subcategory spanned by formal -moduli problems factors through the lax symmetric monoidal inclusion and is symmetric monoidal, providing a symmetric monoidal functor
[TABLE]
Proof.
Let be a map of -formal moduli problems over and let be the induced functor. We wish to show that admits a right adjoint, or equivalently by the adjoint functor theorem [Lur09, Cor 5.5.2.9] that it preserves small colimits. Let and be the augmented -algebras over corresponding to and respectively through the equivalence . Let be the map corresponding to . By the Koszul duality for modules [Lurb, Prop 3.5.1] there exists equivalences and and the induced diagram
[TABLE]
commutes up to homotopy, where is the left adjoint to the forgetful functor along . The functor preserves small colimits which implies that has the same property. The second assertion follows directly from the existence of the symmetric monoidal functor . ∎
Notation A.8**.**
The symmetric monoidal functor provided by Lemma A.7 will be simply denoted by . Consider the unit in as a commutative algebra object. We have an equivalence of commutative algebra objects in . Therefore induces a symmetric monoidal functor
[TABLE]
into the symmetric monoidal -category of presentable -linear -categories.
For any -formal moduli problem over , the space is contractible, which implies that the projection is an equivalence of -categories. Because and are the units of the symmetric monoidal -categories and respectively, we have natural symmetric monoidal structures and on the slice -categories (see [Lurd, Rem 2.2.2.5]). Moreover the symmetric monoidal functor induces a symmetric monoidal functor
[TABLE]
The data of an object of the -category is essentially given by a pair with a -linear -category and an object of . The functor sends a formal moduli problem to the pair where is the -linear -category of Ind-coherent sheaves on and is the skyscraper sheaf with value at the -point of . If is the augmented -algebra of , the sheaf corresponds to the left -module through the equivalence .
We have the following consequence of [Lurd, Thm 4.8.5.11].
Proposition A.9**.**
Let be a field. The symmetric monoidal functor
[TABLE]
admits a lax symmetric monoidal right adjoint which sends a pair to the formal -moduli problem of deformations of in .
Proof.
By [Lurd, Cor 7.3.2.7], it suffices to show that the underlying functor of admits a right adjoint. By Koszul duality for modules [Lurb, Prop 3.5.1], there exists a homotopy commutative diagram
[TABLE]
where is the equivalence of [Lurb, Thm 3.0.4] and is the composition of the functor constructed in [Lurd, Const 4.8.3.24] with the forgetful functor . Therefore is given on objects by . By [Lurd, Thm 4.8.5.11], has a right adjoint which sends a pair to the augmented -algebra , with the -algebra of endomorphisms of in . This implies that has a right adjoint, which by [Lurb, Thm 5.2.8] recalled in §3.2, is given on objects by . ∎
Remark A.10**.**
Let be a field. Consider the restriction functor which admits a left adjoint given by the left Kan extension along the forgetful functor . By composition we obtain from Lemma A.9 that the symmetric monoidal functor
[TABLE]
admits as right adjoint the lax symmetric monoidal functor
[TABLE]
where is the restriction of the formal -moduli problem on -algebras. Passing to algebra objects, we obtain a functor
[TABLE]
with right adjoint given by the functor sending a pair where is a monoidal -linear -category and its unit object to the formal moduli problem of deformations of in .
Corollary A.11**.**
Let be a field and let be a -linear -category. Then the formal -moduli problem is given by the formula
[TABLE]
In other words is naturally homotopy equivalent to the space
[TABLE]
of left actions of the monoidal -category on .
Proof.
By Corollary A.5, for an artinian -algebra, there is a natural homotopy equivalence
[TABLE]
Besides, there exists an equivalence of functors which induces a natural equivalence . Applying the adjunction of Remark A.10, we obtain a natural homotopy equivalence
[TABLE]
which proves our claim. ∎
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