Derived equivalences of upper-triangular ring spectra via lax limits

We extend a theorem of Ladkani concerning derived equivalences between upper-triangular matrix rings from ordinary rings to ring spectra. Our result also extends an analogous theorem of Maycock for differential graded algebras.

The purpose of this short article is to extend the following theorem of Ladkani [Lad11] from ordinary rings to ring spectra in the sense of stable homotopy theory; we note that this theorem was extended to differential graded algebras by Maycock [May11].Recall that to rings R and S and an S-R-bimodule M one associates the upper-triangular matrix ring ( S M 0 R ) = {( s m 0 r ) | r ∈ R, s ∈ S, m ∈ M } with sum and product operations given the corresponding matrix operations.We denote the (triangulated) derived category of right modules over a ring R by D(Mod(R)) and recall than an object X ∈ D(Mod(R)) is compact if the functor Hom R (X, −) : D(Mod(R)) −→ Ab preserves small coproducts.
Theorem (Ladkani).Let R and S be rings.Suppose given an S-R bimodule M such that M R is compact as an object of D(Mod(R)) and an R-module T such that the functor is an equivalence of triangulated categories, where E = Hom R (T, T ) is the ring of endomorphisms of T .Suppose, moreover, that Ext >0 R (M, T ) = 0.Then, there is an equivalence of triangulated categories D(Mod( S M 0 R )) ≃ D Mod E HomR(M,T ) 0 S .
As Ladkani explains in loc.cit., interesting equivalences of derived categories are obtained from appropriate choices of R, S, M and T .The main focus of this article is to illustrate how formal properties of a higher-categorical upper-triangular gluing construction yield a simple and conceptual proof of (a vast generalisation of) the above theorem.
We use freely the theory of ∞-categories developed by Joyal, Lurie and others; our main references are [Lur09,Lur17,Lur18].Here we only recall that an ∞-category C is stable if it is pointed, admits finite colimits and the suspension functor Σ : The homotopy category of a stable ∞-category is additive (in the usual sense) and is canonically triangulated in the sense of Verdier [Lur17, Theorem 1.1.2.14].Working with ∞-categories rather than with triangulated categories permits us to construct the (homotopy) limit of a diagram of exact functors between stable ∞-categories, a construction that is not available in the realm of triangulated categories.We also mention that the gluing construction that we utilise below is used by Ladkani in [Lad11] to glue (abelian) module categories; notwithstanding, our proof of the main theorem is different in the case of ordinary rings and of differential graded algebras in that it does not rely on explicit computations.
Let k be an E ∞ -ring spectrum, for example the sphere spectrum S or the Eilenberg-Mac Lane spectrum of an ordinary commutative ring [Lur17 Let C and D be k-linear presentable stable ∞-categories and F : C → D a k-linear colimit-preserving functor.Define L * (F ) via the the pullback square The above pullback is well defined since the ∞-category Fun ∆ 1 , D is presentable [Lur09, Proposition 5.5.3.6] and stable [Lur17, 1.1.3.1] and inherits a k-linear structure from D via the equivalence of ∞-categories Above, S denotes the ∞-category of spaces, LFun (−, −) (resp.RFun (−, −)) denotes the ∞-category of functors that admit a right adjoint (resp.a left adjoint), and the symbol ⊗ denotes Lurie's tensor product of presentable ∞-categories [Lur17, Propositions 4.8.1.15and 4.8.1.17](see also [Lur09, Theorem 5.1.5.6 and Proposition 5.2.6.2]).Similarly, the restriction functor has a canonical k-linear structure.When the right adjoint G : D → C of F , which exists by [Lur09, Corollary 5.5.2.9], is also colimit-preserving we may also form the pullback square We also remind the reader of the equivalence of k-linear presentable stable ∞-categories induced by the passage from a morphism to its cofibre, that we regard as a very general version of the Bernšteȋn-Gel ′ fand-Ponomarev reflection functors [BGP73].The gluing operation F → L * (F ) is an example of a lax limit [GHN17] and is also considered in the setting of differential graded categories, see for example [KL15].
For a given k-algebra spectrum R, that is an E 1 -algebra object of the symmetric monoidal ∞-category D(k), we denote the k-linear stable ∞-category of (right) R-module spectra by D(R), see also [ Given a bimodule spectrum M ∈ D(S op ⊗ k R), we denote the right adjoint to the tensor product functor − ⊗ S M by Map R (M, −).We also introduce the k-linear presentable stable ∞-category Upper-triangular ring spectra are considered for example in [Sos22].
We are ready to state and prove the main result in this article.
Theorem.Let R, S and E be k-algebra spectra.Suppose given a bimodule spectrum M ∈ D(S op ⊗ k R) such that the R-module spectrum M R = S ⊗ S M is compact and a bimodule spectrum is an equivalence.Then, there is an equivalence of k-linear presentable stable ∞-categories Proof.The commutative square in which the left vertical functor is an equivalence by assumption, induces an equivalence of k-linear presentable stable ∞-categories are equivalent.Consequently, there is an equivalence of k-linear presentable stable ∞-categories We conclude the proof by considering the following composite of equivalences of k-linear presentable stable ∞-categories (recall that N = Map R (M, T )): Remark.When k is the Eilenberg-Mac Lane spectrum of the ordinary ring of integer numbers, Ladkani's theorem is recovered from the previous theorem by considering the case where the underlying spectra of R, S, M and T are discrete, that is their stable homotopy groups vanish in non-zero degrees.The assumptions in Ladkani's theorem are sufficient to guarantee that the upper-triangular ring spectra in the statement in the previous theorem are both discrete.Ladkani's theorem then follows from the fact that the ∞-category of module spectra over a discrete ring spectrum A is equivalent to the derived ∞category of modules over the ordinary ring π 0 (A), see Example.Let R = S = E be arbitrary k-algebra spectra and M = T = R with its canonical R-bimodule structure.The functors − ⊗ R R and − ⊗ R Map R (R, R) are both equivalent to the identity functor of D(R) and the equivalence in the main theorem reduces to the (non-trivial) equivalence of k-linear presentable stable ∞-categories given by the passage from a morphism in D(R) to its cofibre.
We conclude this article by describing certain canonical equivalences attached to an algebra spectrum (or, more generally, a morphism between such) that satisfies suitable finiteness/dualisability conditions.The bimodule spectra that arise play a central role in the study of right/left Calabi-Yau structures [Gin06,KS06] and their relative variants [Toë14 for the k-linear dual of A. Setting R = E = k, S = A e , M = A and T = k, the main theorem affords an equivalence of k-linear presentable stable ∞-categories between the derived ∞-category of the 'one-point extension' of A e by the diagonal A-bimodule spectrum and that of the 'one-point coextension' of A e by DA (this terminology originates in representation theory of algebras [Rin84]).Suppose that A is smooth and that f * (A) is compact as a B-module spectrum, so that the source and target of the morphism ε are compact A-bimodule spectra and, consequently, so is its cofibre.that specialises to the equivalence in ii when B = 0.
stemming from the fact that both ∞-categories L * (F ) and L * (G) are equivalent to the ∞-category of sections of the biCartesian fibration over ∆ 1 classified by the adjunction F ⊣ G, see [Lur09, Lemma 5.4.7.15].

(
ii) Let A be a smooth k-algebra spectrum, that is A ∈ D(A e ) is a compact object [Lur18, Definition 11.3.2.1]; equivalently, A is a left dualisable object of the ∞-category of A e -k-bimodule spectra, see [Lur17, Definition 4.6.4.13] and [Lur18, Remark 11.3.2.2].The A-bimodule spectrum Ω A = Map A e (A, A e ) is called the inverse dualising A-bimodule (not to be confused with the based-loops functor on D(A)).Setting R = E = A e , S = k, M = A and T = A e , the main theorem yields an equivalence of k-linear presentable stable ∞-categories D k A 0 A e ∼ −→ D A e ΩA 0 k .(iii) Let A be a smooth and proper k-algebra spectrum.In this case there are mutually-inverse equivalences of k-linear presentable stable ∞-categories − ⊗ A Ω A : D(A) ∼ ←→ D(A) : − ⊗ A DA, see [Lur17, Proposition 4.6.4.20]where DA is called the Serre A-bimodule [Lur17, Definition 4.6.4.5] and Ω A is called the dual Serre A-bimodule [Lur17, Definition 4.6.4.16] (the fact that DA and Ω A are the right and left duals of A in the ∞-category of A e -k-bimodule spectra in the sense of [Lur17, Definition 4.6.2.3] follows from [Lur17, Proposition 4.6.2.1 and Remark 4.6.2.2]).Setting R = E = S = A, M = A and T = DA or T = Ω A , the main theorem provides equivalences of k-linear presentable stable ∞-categories D( that Map A (A, DA) ≃ DA and Map A (A, Ω A ) ≃ Ω A as A-bimodule spectra.(iv) Let f : B → A be a morphism of k-algebra spectra that is not necessarily unital.By the Eilenberg-Watts Theorem, the counit of the induced adjunction − ⊗ B A ≃ f !: D(B) ←→ D(A) : f * can be interpreted as a morphism of A-bimodule spectra ε : A ⊗ B A −→ A.
[Kel11,Yeu16,BD21,BCS,KW23,Wu23b,Wu23a]u23b,Wu23a].Given a k-algebra spectrum A, we write A e = A ⊗ k A op and recall that A can be viewed either as a right or as a left A e -module spectrum [Lur17, Construction 4.6.3.7 and Remark 4.6.3.8].We also make implicit use of the canonical equivalences between the k-linear ∞-category of A-bimodule spectra and those of A e -k-bimodule spectra and of k-A e -bimodule spectra, see [Lur17, Proposition 4.6.3.15] and the discussing succeeding it.(i)Let A be a proper k-algebra spectrum, that is the underlying k-module spectrum of A is compact; equivalently, A is a right dualisable object of the ∞-category of A e -k-bimodule spectra, see [Lur17, Definition 4.6.4.2] and [Lur18, Example D.7.4.2 and Remark D.7.4.3].We write