Cotangent Complex Research Articles

Deformations of commutative Artinian algebras

The paper is devoted to the study of deformations of Artinian algebras and zero-dimensional germs of varieties. In particular, an approach is developed to solving the open problem about the nonexistence of rigid Artinian algebras; it is based essentially on the use of the canonical duality in the cotangent complex. Thus, it is shown that there are no rigid Gorenstein Artinian algebras and rigid almost complete intersections. The proof of the latter statement is based on the properties of the torsion functors. More precisely, the tensor product of the conormal and canonical modules of the corresponding Artinian algebra is calculated. In this case, the homology and cohomology groups of higher degrees are also found. Among other things, some estimates are obtained for the dimension of the spaces of the first lower and upper cotangent functors of Artinian algebras, and the relationship between them is described. In conclusion, several examples of nonsmoothable Artinian noncomplete intersections are examined, and some unusual properties of such algebras are discussed.

Graded quotients of ramification groups of local fields with imperfect residue fields

abstract: We prove that the graded quotients of the filtration by ramification groups of any henselian discrete valuation field of residue characteristic $p>0$ are ${\bf F}_p$-vector spaces. We define an injection of the character group of each graded quotient to a twisted cotangent space defined using a cotangent complex.

Integral period relations and congruences

Under relatively mild and natural conditions, we establish an integral period relations for the (real or imaginary) quadratic base change of an elliptic cusp form. This answers a conjecture of Hida regarding the {\it congruence number} controlling the congruences between this base change and other eigenforms which are not base change. As a corollary, we establish the Bloch-Kato conjecture for adjoint modular Galois representations twisted by an even quadratic character. In the odd case, we formulate a conjecture linking the degree two topological period attached to the base change Bianchi modular form, the cotangent complex of the corresponding Hecke algebra and the archimedean regulator attached to some Beilinson-Flach element.

Analytification, localization and homotopy epimorphisms

We study the interaction between various analytification functors, and a class of morphisms of rings, called homotopy epimorphisms. An analytification functor assigns to a simplicial commutative algebra over a ring R, along with a choice of Banach structure on R, a commutative monoid in the symmetric monoidal model category of simplicial ind-Banach R-modules. We show that several analytifications relevant to analytic geometry - such as Tate, overconvergent, Stein analytification, and formal completion - are homotopy epimorphisms. Another class of examples of homotopy epimorphisms arises from Weierstrass, Laurent and rational localizations in derived analytic geometry. As applications of this result, we prove that Hochschild homology and the cotangent complex are computable for analytic rings, and the computation relies only on known computations of Hochschild homology for polynomial rings. We show that in various senses, Hochschild homology as we define it commutes with localizations, analytifications and completions.

Homological characterization of regularity in logarithmic algebraic geometry

We characterize K. Kato’s log regularity in terms of vanishing of (co)homology of the logarithmic cotangent complex.

A non-commutative differential module approach to Alexander modules

Abstract The theory of R. Crowell on derived modules is approached within the theory of non-commutative differential modules. We also seek analogies to the theory of cotangent complex from differentials in the commutative ring setting. Finally, we give examples motivated from the theory of Galois coverings of curves.

On the relationship between logarithmic TAQ and logarithmic THH

We provide a new description of logarithmic topological André-Quillen homology in terms of the indecomposables of an augmented ring spectrum. The new description allows us to interpret logarithmic TAQ as an abstract cotangent complex, and leads to a base-change formula for logarithmic topological Hochschild homology. The latter is analogous to results of Weibel-Geller for Hochschild homology of discrete rings, and of McCarthy-Minasian and Mathew for topological Hochschild homology. For example, our results imply that logarithmic THH satisfies base-change for tamely ramified extensions of discrete valuation rings.

Motivic infinite loop spaces

We prove a recognition principle for motivic infinite P1-loop spaces over a perfect field. This is achieved by developing a theory of framed motivic spaces, which is a motivic analogue of the theory of E-infinity-spaces. A framed motivic space is a motivic space equipped with transfers along finite syntomic morphisms with trivialized cotangent complex in K-theory. Our main result is that grouplike framed motivic spaces are equivalent to the full subcategory of motivic spectra generated under colimits by suspension spectra. As a consequence, we deduce some representability results for suspension spectra of smooth varieties, and in particular for the motivic sphere spectrum, in terms of Hilbert schemes of points in affine spaces.

Shifted Poisson Structures on Differentiable Stacks

Abstract The purpose of this paper is to investigate $(+1)$-shifted Poisson structures in the context of differential geometry. The relevant notion is that of $(+1)$-shifted Poisson structures on differentiable stacks. More precisely, we develop the notion of the Morita equivalence of quasi-Poisson groupoids. Thus, isomorphism classes of $(+1)$-shifted Poisson stacks correspond to Morita equivalence classes of quasi-Poisson groupoids. In the process, we carry out the following program, which is of independent interest: (1) We introduce a ${\mathbb{Z}}$-graded Lie 2-algebra of polyvector fields on a given Lie groupoid and prove that its homotopy equivalence class is invariant under the Morita equivalence of Lie groupoids, and thus they can be considered to be polyvector fields on the corresponding differentiable stack ${\mathfrak{X}}$. It turns out that $(+1)$-shifted Poisson structures on ${\mathfrak{X}}$ correspond exactly to elements of the Maurer–Cartan moduli set of the corresponding dgla. (2) We introduce the notion of the tangent complex $T_{\mathfrak{X}}$ and the cotangent complex $L_{\mathfrak{X}}$ of a differentiable stack ${\mathfrak{X}}$ in terms of any Lie groupoid $\Gamma{\rightrightarrows } M$ representing ${\mathfrak{X}}$. They correspond to a homotopy class of 2-term homotopy $\Gamma$-modules $A[1]\rightarrow TM$ and $T^{\vee } M\rightarrow A^{\vee }[-1]$, respectively. Relying on the tools of theory of VB-groupoids including homotopy and Morita equivalence of VB-groupoids, we prove that a $(+1)$-shifted Poisson structure on a differentiable stack ${\mathfrak{X}}$ defines a morphism ${L_{\mathfrak{X}}}[1]\to{T_{\mathfrak{X}}}$.

The cotangent complex and Thom spectra

The cotangent complex of a map of commutative rings is a central object in deformation theory. Since the 1990s, it has been generalized to the homotopical setting of E_infty -ring spectra in various ways. In this work we first establish, in the context of infty -categories and using Goodwillie’s calculus of functors, that various definitions of the cotangent complex of a map of E_infty -ring spectra that exist in the literature are equivalent. We then turn our attention to a specific example. Let R be an E_infty -ring spectrum and mathrm {Pic}(R) denote its Picard E_infty -group. Let Mf denote the Thom E_infty -R-algebra of a map of E_infty -groups f:Grightarrow mathrm {Pic}(R); examples of Mf are given by various flavors of cobordism spectra. We prove that the cotangent complex of Rrightarrow Mf is equivalent to the smash product of Mf and the connective spectrum associated to G.

Representability theorem in derived analytic geometry

We prove the representability theorem in derived analytic geometry. The theorem asserts that an analytic moduli functor is a derived analytic stack if and only if it is compatible with Postnikov towers, has a global analytic cotangent complex, and its truncation is an analytic stack. Our result applies to both derived complex analytic geometry and derived nonarchimedean analytic geometry (rigid analytic geometry). The representability theorem is of both philosophical and practical importance in derived geometry. The conditions of representability are natural expectations for a moduli functor. So the theorem confirms that the notion of derived analytic space is natural and sufficiently general. On the other hand, the conditions are easy to verify in practice. So the theorem enables us to enhance various classical moduli spaces with derived structures, thus providing plenty of down-to-earth examples of derived analytic spaces. For the purpose of proof, we study analytification, square-zero extensions, analytic modules and cotangent complexes in the context of derived analytic geometry. We will explore applications of the representability theorem in our subsequent works. In particular, we will establish the existence of derived mapping stacks via the representability theorem.

Tangent categories of algebras over operads

Associated to a presentable ∞-category $$\mathcal{C}$$ and an object X ∈ $$\mathcal{C}$$ is the tangent ∞-category $$\mathcal{T}_X\mathcal{C}$$, consisting of parameterized spectrum objects over X. This gives rise to a cohomology theory, called Quillen cohomology, whose category of coefficients is $$\mathcal{T}_X\mathcal{C}$$. When $$\mathcal{C}$$ consists of algebras over a nice ∞-operad in a stable ∞-category, $$\mathcal{T}_X\mathcal{C}$$ is equivalent to the ∞-category of operadic modules, by work of Basterra–Mandell, Schwede and Lurie. In this paper we develop the model-categorical counterpart of this identification and extend it to the case of algebras over an enriched operad, taking values in a model category which is not necessarily stable. This extended comparison can be used, for example, to identify the cotangent complex of enriched categories, an application we take up in a subsequent paper.

Deformations of modules through butterflies and gerbes

Classifying obstructions to the problem of finding extensions between two fixed modules goes back at least to L. Illusie's thesis. Our approach, following in the footsteps of J. Wise, is to introduce an analogous Grothendieck Topology on the category A-mod of modules over a fixed ring A in a topos E. The problem of finding extensions becomes a banded gerbe and furnishes a cohomology class on the site A-mod. We compare our obstruction and that coming from Illusie's work, giving another construction of the exact sequence Illusie used to obtain his obstruction. Our work circumvents the cotangent complex entirely and answers a question posed by Illusie.

Vanishing theorems for representation homology and the derived cotangent complex

Let $G$ be a reductive affine algebraic group defined over a field $k$ of characteristic zero. In this paper, we study the cotangent complex of the derived $G$-representation scheme $ {\rm DRep}_G(X)$ of a pointed connected topological space $X$. We use an (algebraic version of) unstable Adams spectral sequence relating the cotangent homology of $ {\rm DRep}_G(X) $ to the representation homology $ {\rm HR}_*(X,G) := \pi_*{\mathcal O}[{\rm DRep}_G(X)] $ to prove some vanishing theorems for groups and geometrically interesting spaces. Our examples include virtually free groups, Riemann surfaces, link complements in $ {\mathbb R}^3 $ and generalized lens spaces. In particular, for any f.g. virtually free group $ \Gamma $, we show that $\, {\rm HR}_i({\rm B}\Gamma, G) = 0 \,$ for all $ i > 0 $. For a closed Riemann surface $\Sigma_g $ of genus $ g \ge 1 $, we have $\, {\rm HR}_i(\Sigma_g, G) = 0 \,$ for all $ i > \dim G $. The sharp vanishing bounds for $ \Sigma_g $ depend actually on the genus: we conjecture that if $ g = 1 $, then $\, {\rm HR}_i(\Sigma_g, G) = 0 \,$ for $ i > {\rm rank}\,G $, and if $ g \ge 2 $, then $\, {\rm HR}_i(\Sigma_g, G) = 0 \,$ for $ i > \dim\,{\mathcal Z}(G) \,$, where $ {\mathcal Z}(G) $ is the center of $G$. We prove these bounds locally on the smooth locus of the representation scheme $ {\rm Rep}_G[\pi_1(\Sigma_g)]\,$ in the case of complex connected reductive groups. One important consequence of our results is the existence of a well-defined $K$-theoretic virtual fundamental class for $ {\rm DRep}_G(X)$ in the sense of Ciocan-Fontanine and Kapranov. We give a new `Tor formula' for this class in terms of functor homology.

Stabilization of the cohomology of thickenings

For a local complete intersection subvariety $X=V({\mathcal I})$ in ${\mathbb P}^n$ over a field of characteristic zero, we show that, in cohomological degrees smaller than the codimension of the singular locus of $X$, the cohomology of vector bundles on the formal completion of ${\mathbb P}^n$ along $X$ can be effectively computed as the cohomology on any sufficiently high thickening $X_t=V({\mathcal I^t})$; the main ingredient here is a positivity result for the normal bundle of $X$. Furthermore, we show that the Kodaira vanishing theorem holds for all thickenings $X_t$ in the same range of cohomological degrees; this extends the known version of Kodaira vanishing on $X$, and the main new ingredient is a version of the Kodaira-Akizuki-Nakano vanishing theorem for $X$, formulated in terms of the cotangent complex.

A simple universal property of Thom ring spectra

We give a simple universal property of the multiplicative structure on the Thom spectrum of an $n$-fold loop map, obtained as a special case of a characterization of the algebra structure on the colimit of a lax $\mathcal{O}$-monoidal functor. This allows us to relate Thom spectra to $\mathbb{E}_n$-algebras of a given characteristic in the sense of Szymik. As applications, we recover the Hopkins--Mahowald theorem realizing $H\mathbb{F}_p$ and $H\mathbb{Z}$ as Thom spectra, and compute the topological Hochschild homology and the cotangent complex of various Thom spectra.

The abstract cotangent complex and Quillen cohomology of enriched categories

In his fundamental work, Quillen developed the theory of the cotangent complex as a universal abelian derived invariant, and used it to define and study a canonical form of cohomology, encompassing many known cohomology theories. Additional cohomology theories, such as generalized cohomology of spaces and topological Andr\'e-Quillen cohomology, can be accommodated by considering a spectral version of the cotangent complex. Recent work of Lurie established a comprehensive $\infty$-categorical analogue of the cotangent complex formalism using stabilization of $\infty$-categories. In this paper we study the spectral cotangent complex while working in Quillen's model categorical setting. Our main result gives new and explicit computations of the cotangent complex and Quillen cohomology of enriched categories. For this we make essential use of previous work, which identifies the tangent categories of operadic algebras in unstable model categories. In particular, we present the cotangent complex of an $\infty$-category as a spectrum valued functor on its twisted arrow category, and consider the associated obstruction theory in some examples of interest.

On the non-existence of torus actions

In this paper we compute the higher left derived functors of the indecomposable functor, in certain degrees, for a general class of algebras. The techniques do not depend on the existence of a Projective extension sequence—a common method used to make such computations throughout the literature and as a result, we generalize all known computations of these higher derived functors. As a result of these calculations we have the following applications. First, we prove that certain torus actions on a Quasitoric manifold are restricted due to the combinatorial structure of the orbit implying the non-existence of certain torus actions. Second, we obtain a refinement of the class of equivariantly formal torus actions, splitting them into two classes. In the context of augmented simplicial algebras over a field, the results provide explicit computations of the cotangent complex. We also show very explicitly how one higher derived functor depends on another and in the case of Toric Topology, the generators of these derived functors are linked backed to the combinatorics of the orbit.

Deformation of formal schemes through local homology

We define the cotangent complex of a morphism f:X→Y of locally noetherian formal schemes as an object in the derived category D−(X) through local homology. We discuss its basic properties and establish the basics results of a deformation theory, providing a characterization of smooth and étale morphisms. This leads to simpler lifting results depending on a differential module, for a class of non-smooth morphism of usual schemes. We also give descriptions of the cotangent complex in the case of regular closed immersions and complete intersection morphisms of formal schemes.

The deformation theory of sheaves of commutative rings II

We show that the obstructions to infinitesimal deformation of sheaves of rings defined in our earlier work are equivalent to those originally defined by Illusie. We also give sheaf-theoretic proofs of several fundamental properties of the cotangent complex, including the transitivity triangle and flat base change, avoiding simplicial methods.