Fact-checked by Grok 2 weeks ago

Derived algebraic geometry

Derived algebraic geometry is a framework in mathematics that extends classical algebraic geometry by integrating homotopy theory and higher category theory, enabling the study of geometric objects like derived schemes and derived stacks that incorporate homotopical data to handle phenomena such as non-transverse intersections and non-free group actions.^[1] It replaces ordinary commutative rings with simplicial commutative rings or E_∞-ring spectra as structure sheaves, allowing for a more refined treatment of singularities, deformations, and moduli spaces through tools like the cotangent complex.^[2] The development of derived algebraic geometry traces its roots to mid-20th-century advances in homological algebra, including Serre's intersection formula from the 1950s and Grothendieck and Illusie's introduction of the cotangent complex in the 1960s and 1970s, which highlighted the need for homotopical methods to resolve limitations in classical intersection theory.^[1] Formal foundations emerged in the early 2000s through the work of Jacob Lurie, whose thesis and subsequent papers established derived algebraic geometry using simplicial commutative rings and ∞-categories, building on Quillen's homotopical algebra and model category theory.^[2] Parallel contributions by Bertrand Toën and Gabriele Vezzosi developed derived stacks and representability theorems in the context of ring spectra, providing a dual perspective that emphasizes analytic and synthetic approaches.^[1] At its core, derived algebraic geometry defines a derived scheme as a ringed ∞-topos (X, O_X) locally equivalent to the spectrum of a simplicial commutative ring, where the structure sheaf O_*X captures higher homotopy groups beyond the classical zeroth homology.^[2] Derived stacks extend this to functors from simplicial commutative rings to ∞-groupoids satisfying étale descent conditions, allowing the geometric realization of quotients and intersections that are ill-behaved in the classical setting, such as those involving nilpotent thickenings.^[1] The cotangent complex L_X/k, a derived sheaf encoding infinitesimal deformations, plays a pivotal role in obstruction theory and the study of shifted symplectic structures, where closed 2-forms of nonzero degree generalize classical symplectic geometry to singular or derived contexts.^[3] This framework has profound applications across algebraic geometry and beyond, including the geometric Langlands program, where derived moduli stacks of bundles carry canonical shifted symplectic forms that facilitate quantization.^[1] It also advances p-adic Hodge theory, topological modular forms, and deformation quantization of Poisson structures, providing a unified language for quantum field theories and virtual fundamental classes in enumerative geometry.^[1] By revealing "hidden smoothness" in singular varieties through homotopical approximations, derived algebraic geometry bridges algebraic topology and geometry, offering new insights into longstanding problems in moduli theory and cohomology.^[3]

Overview

Introduction

Derived algebraic geometry is an extension of classical algebraic geometry that incorporates homotopy-theoretic methods to handle phenomena such as non-transverse intersections and to properly define derived intersections.^[4]^[5] This approach addresses limitations in traditional algebraic geometry by allowing geometric objects to capture higher-order homotopical information, enabling a more refined treatment of singularities and intersections.^[2] The core goal of derived algebraic geometry is to construct a framework where classical objects like schemes and stacks are replaced by derived versions that encode essential homotopical data, facilitating the study of geometric structures in a homotopy-invariant manner.^[5] It builds upon foundational tools such as derived categories to integrate homological algebra into geometric constructions.^[6] Key innovators in the field include Jacob Lurie, whose 2004 manuscript established foundations using simplicial commutative rings and whose 2009 book Higher Topos Theory provided the homotopy-theoretic underpinnings, and Bertrand Toën, whose 2014 survey outlined general notions and recent developments.^[6]^[5] Derived algebraic geometry holds significant importance in modern mathematics, as it bridges algebraic geometry, homotopy theory, and topology through their interrelations.^[7]

Historical Development

The roots of derived algebraic geometry trace back to the development of derived categories in homological algebra during the 1960s, pioneered by Alexander Grothendieck and his student Jean-Louis Verdier as a framework for handling sheaf cohomology and duality theorems in algebraic geometry.^[8] Verdier's thesis, supervised by Grothendieck, introduced the derived category of an abelian category to formalize hyperhomology and resolve issues in classical homological methods. This innovation provided the foundational language for tracking higher homotopical information in chain complexes, setting the stage for later extensions into geometric contexts.^[9] Parallel advancements in algebraic topology during the 1960s and 1970s, particularly Daniel Quillen's introduction of simplicial methods and model categories, further influenced the field by enabling homotopical algebra on categories beyond topological spaces. Quillen's 1967 monograph Homotopical Algebra established model category structures, which allowed for fibrant and cofibrant resolutions using simplicial sets, bridging combinatorial topology with algebraic structures.^[10] These tools proved essential for modeling homotopy types in a categorical setting, providing the machinery that would later underpin derived enhancements to algebraic geometry.^[11] Derived algebraic geometry emerged as a distinct field in the early 2000s, building on these foundations through the works of Bertrand Toën and Gabriele Vezzosi, who developed homotopical algebraic geometry using dg-categories and stacks to address intersections and moduli problems beyond classical schemes.^[12] Their series, starting with "Homotopical Algebraic Geometry I: Topos Theory" (arXiv 2002, published 2005), and continuing with Part II in 2008, formalized derived stacks and higher categorical geometry in a way compatible with étale topology.^[13] Concurrently, Jacob Lurie's 2009 book Higher Topos Theory established ∞-categories as a rigorous framework for homotopy-coherent structures, enabling the integration of stable homotopy theory into algebraic geometry via E_∞-ring spectra. This work drew influence from 1990s developments in structured ring spectra by Michael A. Mandell and J. Peter May, who advanced equivariant and monoidal categories of spectra for stable homotopy computations. Key milestones include Lurie's Derived Algebraic Geometry notes, circulated around 2007–2010, which synthesized simplicial commutative rings as the affine building blocks for derived schemes and stacks, emphasizing spectral methods over differential graded algebras.^[2] Initially focused on characteristic zero settings where resolutions behave well, the field transitioned in the mid-2010s to handle positive and mixed characteristics more robustly, incorporating prismatic and crystalline techniques to overcome limitations in simplicial models.^[14] This evolution expanded derived algebraic geometry's applicability, aligning it with broader arithmetic geometry while preserving homotopical invariance.^[15]

Motivations and Prerequisites

Classical Algebraic Geometry Limitations

Classical algebraic geometry, while powerful for transverse intersections, encounters significant limitations when varieties intersect non-transversally. For instance, consider two coincident lines in the affine plane \mathbb{A}^2 both defined by the ideal (x); their classical scheme-theoretic intersection is the entire line rather than the expected point (as for generic transverse lines), with dimension 1 instead of 0. This excess dimension arises because the sum of ideals (x) + (x) = (x) does not reduce properly. Even when the dimension is as expected, the scheme-theoretic intersection may fail to fully capture higher-order infinitesimal data, as the ordinary tensor product \mathcal{O}_Y \otimes_{\mathcal{O}_X} \mathcal{O}_Z truncates to \Tor_0 and ignores higher \Tor_i terms (i > 0) that measure deviations from flatness. This issue manifests in Bézout's theorem, where non-transverse curves in \mathbb{P}^2 yield incorrect counts without incorporating the Euler characteristic \sum (-1)^i \dim \Tor_i(\mathcal{O}_Y, \mathcal{O}_Z), leading to an underestimation of multiplicity in generic cases.^[2] The loss of homotopical information in classical constructions further exacerbates these problems, particularly with naive fiber products that do not account for homotopy coherences, resulting in inaccurate computations of derived functors such as Ext and Tor. In non-flat situations, the classical pullback Y \times_X Z truncates higher homotopy sheaves, omitting terms like \pi_n(\mathcal{O}_{Y \times_X Z}) \cong \Tor^n_{\mathcal{O}_X}(\mathcal{O}_Y, \mathcal{O}_Z), which are crucial for understanding excess intersections or virtual dimensions. This rigidity prevents a geometric interpretation of obstruction spaces in deformation theory, where classical methods rely on ad hoc cohomological corrections rather than intrinsic structures.^[1] Concrete examples illustrate these inadequacies, such as the self-intersection of a divisor on a surface, where the classical diagonal embedding D \to S yields a scheme whose structure sheaf lacks the higher Tor terms needed to resolve singularities or compute virtual normal bundles accurately. Similarly, the naive cotangent complex in classical settings, defined via Kähler differentials, fails for non-smooth morphisms, as it does not extend well to Tor-independent base changes or capture obstructions in H^2. Derived enhancements address this by introducing a derived cotangent complex L_{B/A}, which rectifies these issues through homotopical algebra, providing a universal derivation that incorporates all higher-order information and restores geometric intuition even in singular or non-transverse scenarios.^[2]^[16]

Homotopy Theory and Derived Categories

Derived algebraic geometry relies on foundational tools from homotopy theory and category theory to handle homotopical and higher-categorical aspects of algebraic structures. Central to this framework is the notion of derived categories, which provide a triangulated category D(A) associated to an abelian category A, equipped with a shift functor

{{grok:render&&&type=render_inline_citation&&&citation_id=1&&&citation_type=wikipedia}}

and cone constructions that formalize exact sequences as distinguished triangles. Verdier's construction of D(A) proceeds in two steps: first, form the homotopy category K(A) of complexes in A, then localize at quasi-isomorphisms. The homotopy category K(A) (also denoted Ho(Ch(A))) is obtained from the category of chain complexes Ch(A) by taking morphisms to be homotopy classes of chain maps (quotienting by the homotopy relation), yielding a triangulated category. The derived category D(A) is then the Verdier localization of K(A) at the quasi-isomorphisms, ensuring that D(A) inverts all quasi-isomorphisms while preserving the triangulated structure, with the shift functor defined by shifting degrees: for a complex X,

(X{{grok:render&&&type=render_inline_citation&&&citation_id=1&&&citation_type=wikipedia}})^n = X^{n+1}

and the differential

d_{X{{grok:render&&&type=render_inline_citation&&&citation_id=1&&&citation_type=wikipedia}}}^n = -d_X^{n+1}

. Distinguished triangles in D(A) correspond to exact sequences of complexes up to homotopy, enabling the encoding of long exact sequences in cohomology.^[17] ∞-categories, as developed by Lurie, model homotopy coherent categories and serve as a higher-categorical enhancement for derived geometry, with prominent examples including the ∞-category of spaces (modeling homotopy types) and the ∞-category of spectra (capturing stable homotopy theory). Key concepts include homotopy limits and colimits, which generalize categorical limits to account for homotopical coherence, and derived functors such as the derived Hom functor RHom and the derived tensor product L\otimes, which rectify classical functors to work in the derived setting. Quillen's model categories provide a combinatorial framework for homotopy theory, presenting categories with weak equivalences, fibrations, and cofibrations to compute homotopy categories and limits via fibrant/cofibrant replacements.

Models for Derived Structures

Simplicial Commutative Rings

Simplicial commutative rings provide a combinatorial model for the algebraic foundations of derived algebraic geometry, generalizing ordinary commutative rings by incorporating simplicial structures to encode homotopical information. A simplicial commutative ring R is a simplicial object in the category of commutative rings, consisting of commutative rings R_n for n \geq 0 together with face maps d_i: R_n \to R_{n-1} and degeneracy maps s_j: R_n \to R_{n+1} that are ring homomorphisms satisfying the simplicial identities.^[18] The category of simplicial commutative rings, denoted \mathrm{sComm} or \mathrm{SCR}, admits a model structure where weak equivalences are maps inducing isomorphisms on all homotopy groups, enabling the passage to an \infty-category that localizes at these equivalences.^[1] The homotopy groups \pi_*(R) of a simplicial commutative ring R form a graded-commutative ring, with \pi_0(R) recovering the underlying ordinary commutative ring (up to isomorphism) and \pi_i(R) for i > 0 being \pi_0(R)-modules that capture higher homotopical data, such as derived deformations or nilpotent extensions.^[19] Specifically, \pi_0(R) = R_0 / (d_1 - d_0)(R_1), and the higher groups arise from the homology of the normalized chain complex associated to R. This graded structure allows simplicial commutative rings to model "higher nilpotents" in a way that resolves singularities in intersections and other geometric operations.^[18] A key operation in this framework is the derived tensor product R \otimes^L_S T over a simplicial commutative ring S, defined as the simplicial homotopy colimit of the two-sided bar construction B(T, S, R), which realizes the classical Tor functor on homotopy groups. In particular, \pi_0(R \otimes^L_S T) \cong \Tor_*^{\pi_0(S)}(\pi_0(R), \pi_0(T)), and there is a spectral sequence

\Tor^{\pi_*(S)}_{p,q}(\pi_*(R), \pi_*(T)) \Rightarrow \pi_{p+q}(R \otimes^L_S T)

converging to the higher homotopy groups, providing a homotopical enhancement of homological algebra.^[19] This construction is central to defining derived intersections and base change in derived geometry. Affine derived schemes arise naturally from simplicial commutative rings via the functor \Spec(R), which assigns to R the representable presheaf \Hom_{\mathrm{SCR}}(R, -) on the category of simplicial commutative rings, equipped with a structure sheaf \mathcal{O}_{\Spec(R)} that extends the classical structure sheaf to incorporate higher homotopy. The underlying classical scheme is \Spec(\pi_0(R)), with higher structure sheaves \pi_i(\mathcal{O}_{\Spec(R)}) encoding the derived data as quasi-coherent sheaves.^[1] This simplicial model offers several advantages, including compatibility with arbitrary base rings (independent of characteristic) and explicit computability through cosimplicial resolutions, such as the bar construction B(T, R, S^\bullet) that provides cofibrant replacements for derived operations.^[19] Unlike analytic models, it avoids point-set topology issues and leverages the combinatorial nature of simplicial sets for rigorous foundations. The geometric realization |R| of R yields a topological commutative ring via the fat realization functor, while the normalization N(R) is the normalized Moore complex, a non-positively graded chain complex linked to R by the Dold-Kan equivalence, facilitating connections to homological algebra.^[18] The relative cotangent complex L_{R/S} for a map S \to R of simplicial commutative rings measures infinitesimal extensions. Its zeroth homology is H_0(L_{R/S}) \cong \Omega_{\pi_0(R)/\pi_0(S)}, the module of Kähler differentials, while higher homology groups are computed via André-Quillen homology groups D_n(\pi_*(R)/\pi_*(S), \pi_*(R)), incorporating the higher homotopy groups \pi_i(R) through derived constructions. This complex is connective and governs derived deformations, with finiteness properties holding when \pi_0(R) is noetherian.^[1]

Differential Graded Algebras

Differential graded algebras provide a model for derived algebraic geometry particularly suited to settings over fields of characteristic zero, where they serve as affine building blocks for derived schemes and stacks. A differential graded algebra (DGA), or more specifically a commutative differential graded algebra (cdga), over a field k of characteristic zero is a \mathbb{Z}-graded k-vector space A = \bigoplus_{n \in \mathbb{Z}} A_n equipped with a differential d: A \to A of degree -1 satisfying d^2 = 0 and a graded-commutative multiplication A \otimes_k A \to A that is associative up to homotopy and compatible with the differential, meaning d(ab) = da \cdot b + (-1)^{|a|} a \cdot db for homogeneous elements a, b \in A.^[20] These structures generalize commutative rings by incorporating homological information, allowing the resolution of singularities and intersections in a derived sense.^[20] In homological algebra, the derived category D(A) of a DG-category A captures the homotopy theory of DG-modules over A, where objects are complexes of A-modules and morphisms are derived up to quasi-isomorphisms. For quasi-free DGAs, bar-cobar resolutions provide explicit cofibrant replacements: the bar construction B(A) yields a DG-coalgebra resolving the tensor product, while the cobar construction \Omega(B(A)) recovers A up to quasi-isomorphism, facilitating computations in the derived category.^[21] This duality is central to deriving functors like tensor products, where the derived tensor M \otimes^L_A N is computed as the total complex of a Tor spectral sequence, with H_*(M \otimes^L_A N) \cong \Tor_*^A(H_*(M), H_*(N)) under suitable flatness conditions.^[20] Derived stacks arise from DG-Artin stacks, which generalize classical Artin stacks by allowing presentations via atlases of derived schemes affine over cdgas. These stacks encode higher homotopy data, with the structure sheaf taking values in cdgas rather than rings. Deformation theory of such objects is governed by André-Quillen homology: for a map of cdgas R \to A, the André-Quillen homology groups D_*(A/R, M) measure obstructions to lifting deformations, with D_1(A/R, A) parametrizing infinitesimal deformations and higher groups controlling higher-order terms.^[22] In characteristic zero, this aligns with Hochschild cohomology HH^*(A), which computes deformations of commutative structures via the isomorphism HH^n(A) \cong \bigoplus_{p+q=n} \Gamma^p(\Omega^q_A) for smooth A, where \Omega^q_A denotes Kähler differentials.^[23] Examples of DG-enhancements include the de Rham algebra \Omega_{X/k}^\bullet of a smooth variety X, which enhances the structure sheaf \mathcal{O}_X by adjoining differential forms with zero differential in positive degrees, resolving derived intersections on X. Another is the Koszul complex K(f_1, \dots, f_n) for regular sequence elements f_i \in A, providing a free cdga resolution of A/(f_1, \dots, f_n) that captures derived zeros.^[20] Despite these strengths, DG-algebras are primarily effective in characteristic zero due to issues with Koszul duality in positive characteristic, where symmetric powers fail to preserve quasi-isomorphisms, leading to ill-behaved derived schemes.^[2] In contrast, simplicial commutative rings offer a characteristic-independent alternative for broader applications.^[20]

E∞-Ring Spectra

In derived algebraic geometry, the spectral model employs E∞-ring spectra to provide a stable homotopical framework that extends commutative algebra to incorporate higher homotopy coherences. These objects arise in the ∞-category of spectra, which is symmetric monoidal under the smash product, and capture the full range of homotopical phenomena without restricting to discrete or chain complex settings.^[24] An E∞-ring spectrum E is defined as a commutative monoid in the ∞-category \mathrm{Sp} of spectra, equipped with a unit map S \to E and multiplication E \wedge E \to E satisfying the axioms of an E∞-operad up to homotopy.^[24] The homotopy groups \pi_*(E) of such an E form a graded E∞-ring, where the grading reflects the stable homotopy groups and the E∞-structure encodes coherent operations beyond ordinary commutativity.^[24] This structure ensures that modules over E, denoted \mathrm{Mod}_E(\mathrm{Sp}), inherit a closed symmetric monoidal structure via the derived smash product M \otimes^L_E N, which resolves the classical tensor product by accounting for higher Tor terms in a homotopically invariant manner.^[25] In spectral algebraic geometry, an affine spectral scheme is constructed as \mathrm{Spec}(E) for a connective E∞-ring spectrum E, representing the functor on the ∞-category of connective E∞-rings that sends A to the space of E-module maps \mathrm{Map}_{\mathrm{Mod}_E(\mathrm{Sp})}(E, A).^[26] This functoriality allows the derived smash product \otimes^L_E to define relative tensor products over spectral affines, enabling the gluing of modules and sheaves in a way that unifies classical and homotopical constructions.^[25] One key advantage of this model is its ability to handle modules and morphisms uniformly across all characteristics, as the smash product does not depend on inverting primes, unlike derived categories over fields of positive characteristic.^[26] Furthermore, E∞-ring spectra connect naturally to chromatic homotopy theory through examples like the spectrum of topological modular forms \mathrm{TMF}, which encodes elliptic cohomology, and to motivic homotopy via motivic spectra that extend Voevodsky's triangulated categories to stable ∞-categories.^[25] A foundational result is Lurie's representability theorem, which asserts that certain ∞-functors from the ∞-category of connective E∞-rings to spaces are representable by spectral schemes if they satisfy prorepresentability, smoothness, and properness conditions in the homotopical sense.^[27] This theorem generalizes classical representability criteria to the spectral setting, allowing the construction of moduli spaces of E∞-algebras as geometric objects. In parallel, the cotangent complex in spectral algebraic geometry is defined for a map of connective E∞-rings A \to B as the spectrum L_{B/A} in \mathrm{Mod}_B(\mathrm{Sp}) that corepresents derivations \mathrm{Der}_A(B, M) for B-modules M, capturing infinitesimal deformations and extending Quillen’s algebraic cotangent complex to a stable setting.^[25] Adams operations on an E∞-ring spectrum E are natural transformations \psi^k: E \to E for integers k \geq 1, compatible with the E∞-multiplication and lifting the classical Adams operations on \pi_*(E) via power maps on homotopy groups; these are induced by endomorphisms of the sphere spectrum and satisfy \psi^k \circ \psi^l = \psi^{kl}.^[24]

\psi^k(E) \simeq E \wedge \Sigma^\infty_+ (B \Sigma_k)_+,

where \Sigma_k is the symmetric group and the smash accounts for the symmetric monoidal structure. Another central construction is topological Hochschild homology, defined for an E∞-ring spectrum E as \mathrm{THH}(E) = E \otimes_{E \wedge_{S^1} E} E, the derived tensor product relative to the free circle action on E, or equivalently the geometric realization of the cyclic bar construction; this spectrum carries a natural circle action and computes a homotopical analog of Hochschild homology, with applications to trace methods and cyclotomic structures.^[24]

Core Objects in Derived Algebraic Geometry

Derived Schemes

Derived schemes are the fundamental geometric objects in derived algebraic geometry, generalizing classical schemes by incorporating homotopical data through structure sheaves valued in simplicial or differential graded commutative rings. Formally, a derived scheme is a ringed ∞-topos (X, \mathcal{O}_X) admitting an étale atlas by affine derived schemes \Spec A_i for simplicial commutative rings A_i, with the structure sheaf \mathcal{O}_X taking values in the \infty-category of simplicial commutative rings (or equivalently, connective commutative differential graded algebras).^[2]^[28] This definition ensures that derived schemes capture higher homotopy groups in the structure sheaf, allowing for a rigorous treatment of infinitesimal thickenings and intersections that fail transversality conditions in the classical setting.^[29] The construction of derived schemes proceeds from simplicial commutative ringed spaces equipped with the Zariski topology, where the relative spectrum \Spec_R A over a base ring R is formed by gluing affines \Spec A via Zariski open immersions. Specifically, for an affine derived scheme X = \Spec A, the structure sheaf satisfies \mathcal{O}_X(X) = A, and on a basic open D(f) \subseteq X (for f \in \pi_0 A), \mathcal{O}_X(D(f)) = A[1/f], where [1/f] denotes the derived localization at f.^[30] General derived schemes are then obtained by gluing such affines via Zariski open immersions, ensuring the structure is locally affine. Key properties of derived schemes include the étale topology, which equips them with a finer site for sheafification beyond the Zariski topology, enabling the study of descent and cohomology in the derived setting. A derived analogue of the Artin-Rees lemma holds in the context of Noetherian simplicial commutative rings and their modules, ensuring compatibility of completions with the homotopical structure.^[31] Similarly, derived completion theorems assert that, under suitable hypotheses (e.g., X proper over a Noetherian base), the formal completion X^\wedge_I along the inverse image of an ideal I induces an equivalence between the \infty-categories of perfect quasi-coherent sheaves on X and on X^\wedge_I.^[31] Higher direct images under a morphism f: X \to Y of derived schemes are computed as R^i f_* \mathcal{F} = \pi_i (f_* \mathcal{F}) for a quasi-coherent sheaf \mathcal{F} on X, preserving the derived structure.^[2] In relation to classical algebraic geometry, a derived scheme X reduces to an underived scheme when the homotopy groups \pi_{\geq 1} \mathcal{O}_X = 0, recovering the classical moduli problems via truncation \tau_{\leq 0} X.^[2] However, derived schemes provide enhancements for non-reduced intersections, where the higher structure resolves singularities and virtual dimensions through the cotangent complex, enabling precise counts in enumerative geometry that classical schemes cannot handle.^[30]

Derived Stacks

Derived stacks generalize the notion of algebraic stacks to incorporate homotopy-theoretic structures, allowing for the treatment of derived rings and higher categorical data. They are defined as fibered categories over the site of simplicial commutative rings (or derived rings) equipped with the étale topology, satisfying descent conditions with respect to representable presheaves on derived affine schemes. Equivalently, a derived stack can be viewed as an S-valued functor on the category of simplicial commutative rings that is a sheaf for the étale topology, where S denotes the ∞-category of spaces; these functors are often defined inductively as n-stacks, admitting a smooth surjection from a disjoint union of affine derived schemes whose fibers over derived rings are (n-1)-stacks.^[2] This framework extends classical stacks by ensuring compatibility with hypercovers and smooth descent, enabling the gluing of derived geometric objects along homotopy-coherent equivalences.^[2] Higher stacks in the derived context, often referred to as ∞-stacks, arise as sheaves of ∞-groupoids on derived sites, capturing the full homotopy theory of algebraic structures. These are ∞-categorical analogs of Artin stacks, where the descent data involves higher coherences modeled by simplicial sets or ∞-categories, as developed in the theory of higher topoi. In derived algebraic geometry, ∞-stacks provide a setting for Artin stacks with derived enhancements, allowing for the resolution of singularities and intersections via homotopy limits rather than classical quotients.^[2] Key constructions of derived stacks include quotient stacks in the derived sense, such as [X/G] where X is a derived scheme and G is a group stack acting on it homotopy-theoretically. This quotient is formed by taking the homotopy colimit of the action groupoid, formally adding paths between points related by the group action to account for higher homotopies. Derived moduli stacks, which parametrize families of objects up to homotopy (e.g., derived moduli of curves or vector bundles), are representable as derived stacks under suitable finiteness conditions, often via the cotangent complex ensuring local finite presentation.^[1]^[32] Properties of derived stacks are defined relative to a base, with smoothness characterized by the cotangent complex L_{X/Y} being connective (i.e., concentrated in non-positive degrees) and perfect, implying that the stack is locally of finite presentation and admits a good obstruction theory for deformations. Properness is adapted to mean that the stack is quasi-compact and universally closed in a derived sense, often verified via the existence of a proper atlas. Obstruction theories for lifting maps or deformations into derived stacks are governed by the cohomology of the cotangent complex, where obstructions lie in H^2(X, L_{X/Y}) and infinitesimal automorphisms in H^1(X, L_{X/Y}), providing a homotopy-coherent replacement for the classical tangent-obstruction complex. Derived schemes serve as the base case, where representable functors yield 0-stacks.^[2]^[1] A fundamental construction is the inertia stack I_Q of a derived stack Q, given by the formula

I_Q = Q \times_{Q \times Q} Q,

which parametrizes pairs of parallel arrows in Q, encoding the stack's automorphisms and loop spaces in a derived setting. This 2-stack structure is crucial for derived loop stacks and higher cohomology computations.^[32]

Spectral Schemes

Spectral schemes provide a stable homotopy-theoretic enhancement of classical schemes, incorporating the full structure of E∞-ring spectra to capture exotic phenomena in algebraic geometry. An affine spectral scheme is defined as \operatorname{Spec}^h(E) for an E∞-ring spectrum E, regarded as a spectrally ringed ∞-topos with underlying ∞-topos \operatorname{Shv}(\operatorname{Spec} \mathbb{Z}(\pi_0 E)) and structure sheaf \mathcal{O}(U_f) = E[1/f].^[26] More generally, spectral schemes are spectrally ringed ∞-topoi (X, \mathcal{O}_X) locally isomorphic to such affine objects, equipped with the big Zariski site on the category of E∞-rings to encode the appropriate topology and coverings.^[26] The geometry of spectral schemes revolves around coherent sheaves on the structure sheaf \mathcal{O}_X, whose homotopy groups \pi_i \mathcal{O}_X form quasi-coherent sheaves of \mathcal{O}_X-modules for all integers i, distinguishing even-degree (nonnegative homotopy) and odd-degree components that encode higher homotopical data beyond classical coherent sheaves.^[26] This framework interfaces with synthetic spectra in the condensed mathematics of Clausen and Scholze, where solid and liquid modules over E∞-ring spectra provide algebraic models for analytic and p-adic structures, enabling synthetic descriptions of spectra within condensed abelian groups.^[33] Key properties include the use of Postnikov towers to approximate the structure sheaf: \mathcal{O}_X is hypercomplete, constructed as the inverse limit of a convergent tower of truncations \tau_{\leq n} \mathcal{O}_X, yielding successive E∞-rings A_{\leq n} that recover classical approximations.^[26] Spectral schemes also relate to motivic spectra through realizations that preserve Galois actions, facilitating uniform treatment of arithmetic and topological invariants, as seen in connections to Bhatt-Morrow-Scholze filtrations on topological Hochschild homology.^[34] A primary advantage of spectral schemes over unstable derived schemes lies in their ability to handle p-adic and chromatic phenomena uniformly via nonconnective E∞-rings, such as Morava E-theory spectra, which classical approaches cannot accommodate without ad hoc extensions.^[26] Fundamental operations include the smash product on spectra, serving as the derived tensor product E \wedge_R F over an E∞-ring R, which underlies monoidal structures on quasi-coherent sheaves.^[26] Moreover, the global sections functor satisfies \Gamma(\operatorname{Spec}(E), \mathcal{O}) = E, linking the homotopical algebra directly to the spectrum's sections.^[26]

Examples and Properties

Basic Examples of Derived Objects

One basic example of a derived object is the derived point, which enhances the classical point \operatorname{Spec} k with higher homotopy structure captured by the Koszul complex. In the framework of derived schemes, the derived point arises as the spectrum of the Koszul complex associated to the zero section of the affine line, providing a resolution that accounts for infinitesimal thickenings beyond the dual numbers k[\epsilon]/\epsilon^2. Specifically, for a field k, the Koszul complex K^\bullet on the structure sheaf of \mathbb{A}^1_k yields a simplicial commutative ring whose homotopy groups encode the derived structure, with H_0 = k and higher terms vanishing in low degrees but enabling non-trivial Tor contributions in intersections.^[2]^[3] A prominent illustration of derived intersections is the derived loop space, realized as the derived fiber product of a point with itself over another point. For a derived scheme X = \operatorname{Spec} A, the loop space LX is given by the homotopy pullback X \times_{X \times X} X, which computes to \operatorname{Spec}(A \otimes_{A \otimes A} A) and corresponds to the Hochschild chains of A, enriching the classical loop space with algebraic data from flat connections. This construction, valid for schemes with affine diagonals, equates LX to the shifted tangent bundle TX[-1], where constant loops form the zero section, and the structure sheaf involves the negative de Rham complex \Omega^{-\bullet}_X.^[35]^[1] The derived projective space \mathbb{P}^n extends the classical projective space via a simplicial resolution of its homogeneous coordinate ring. In derived algebraic geometry, \mathbb{P}^n_k is constructed as the quotient stack [U / \mathbb{G}_m] where U covers the affine opens defined by homogeneous coordinates, resolved simplicially to incorporate higher syzygies in the Koszul complex of the irrelevant ideal. This resolution ensures that the derived Proj functor applied to the polynomial ring k[x_0, \dots, x_n] yields a derived scheme whose quasi-coherent sheaves include twisted line bundles \mathcal{O}(m) with non-trivial higher cohomology, distinguishing it from the classical truncation.^[2]^[36] Derived moduli spaces of curves provide another concrete example, enhanced by obstruction spaces arising from differential graded algebras (DGAs). The moduli stack \mathcal{M}_g of genus g curves is represented as a derived 0-stack over \operatorname{Spec} \mathbb{Z}, smooth and representable, where deformations are governed by the cotangent complex whose higher Ext groups serve as obstruction spaces for lifting infinitesimal deformations. For instance, in the DGA model, the tangent cohomology D^i(F, M) at a point, computed as \pi_j T_x(F, M[-n]), measures obstructions in higher degrees, allowing the inclusion of singular or stable curves with derived structure that classical moduli cannot capture.^[2]^[3] In the spectral setting, KU-module spectra serve as affine schemes over connective K-theory, illustrating derived objects in E∞-ring contexts. The periodic complex K-theory spectrum KU, as an E∞-ring, gives rise to affine spectral schemes \operatorname{Spf}_{\mathrm{et}} KU, whose points correspond to KU-algebras and whose quasi-coherent sheaves are modules over KU, with the connective cover ku providing a truncation that aligns with classical affine space but incorporates homotopy-invariant structures like Adams operations. This construction unifies algebraic and topological K-theory, where KU-modules parametrize vector bundles with higher Chern classes encoded in the spectral geometry.^[37]^[2]

Key Properties and Constructions

In derived algebraic geometry, derived étale cohomology extends classical étale cohomology by computing the hypercohomology of derived sheaves on the étale site of a derived scheme or stack. This construction preserves the sheaf-theoretic properties of the étale topology while incorporating higher homotopical information from the derived category, enabling the study of invariants such as Brauer groups of Azumaya algebras over derived rings.^[38] For a derived scheme X, the derived étale cohomology groups H^i_{\ét}(X, \mathcal{F}) for a derived sheaf \mathcal{F} are the cohomology of the total complex associated to a Godement resolution or Čech complex on the étale site, yielding a richer theory that detects obstructions and extensions in the derived setting.^[38] Deformation theory in derived algebraic geometry formalizes infinitesimal extensions using the cotangent complex L_{A/B} of an E_\infty-ring map B \to A, which controls square-zero extensions via derivations

\eta: L_{A/B} \to M{{grok:render&&&type=render_inline_citation&&&citation_id=1&&&citation_type=wikipedia}}

for a connective module M. The Maurer-Cartan equation in this context governs the compatibility of such extensions, stating that a Maurer-Cartan element \pi \in \pi_0(\MC(G)) in the Maurer-Cartan space of a dg-Lie algebra G associated to the cotangent complex corresponds to a flat deformation, with higher homotopy groups encoding obstruction classes.^[39] Lurie proves that for connective E_\infty-rings, the space of square-zero extensions is equivalent to the derived moduli of derivations, providing a universal framework for lifting structures along nilpotent thickenings.^[39] This derived perspective resolves singularities in classical deformation functors by incorporating Tor-amplitude and homotopy coherence. Key constructions include the derived Hilbert scheme, which represents the moduli functor \Hilb_{X/S} associating to each test object S' the \infty-groupoid of proper, flat derived subschemes of X_{S'} = X \times_S S' that are almost of finite presentation, realized as a relative derived algebraic space locally almost of finite presentation over S.^[2] Unlike its classical counterpart, it is not always locally of finite presentation but captures non-transverse intersections via derived structure, with its cotangent complex corepresented by almost perfect modules \Hom_{\QCoh_{Y/X'}}(L_{Y/X'}, q^* M). Perfect complexes on derived schemes are defined as pseudo-coherent objects in the derived category of quasi-coherent sheaves with finite Tor-dimension over the structure sheaf, locally quasi-isomorphic to bounded complexes of finite-rank locally free sheaves; they form a thick triangulated subcategory stable under base change and form the building blocks for derived coherent sheaves.^[40] Fundamental theorems include Thomason's classification, which asserts that in the derived category of quasi-coherent sheaves on a scheme X, a complex E^\bullet is perfect if and only if it is pseudo-coherent and has locally finite Tor-amplitude, with supports providing a geometric invariant for thick triangulated subcategories via the map to the spectrum of supports.^[41] Lurie's representability theorem establishes that a homotopy-preserving functor F: \sCRing_R \to \cS from simplicial commutative R-algebras to spaces is an almost finitely presented derived geometric n-stack if and only if it commutes with filtered colimits on k-truncated objects for k \geq 0, is n-truncated on discrete rings, is a hypersheaf for the étale topology, is cohesive and nilcomplete, and has finitely generated tangent modules D_{n-i}^x(F, C) for finitely generated domains C.^[42] For a proper morphism f: X \to Y of derived schemes, the derived pullback functor satisfies f^* \simeq Rf^! on perfect complexes in the six-functor formalism, reflecting the compatibility of exceptional pullback with proper pushforward and enabling Grothendieck's existence theorem for coherent cohomology.^[31] In derived intersections, the virtual fundamental class [X]^{\vir} is constructed via the intrinsic normal cone C_{X/Y} of a morphism X \to Y, defined as the spectrum of the symmetric algebra on the shifted conormal complex, yielding a class in A_*(X) that refines intersection products for obstructed situations and satisfies a deformation invariance property. For instance, in the derived Hilbert scheme, this class computes virtual counts of points in non-transverse loci.

Applications

In Algebraic Geometry and Topology

Derived algebraic geometry enhances classical intersection theory by providing a framework for defining virtual classes on singular schemes, allowing for the computation of intersection numbers in cases where classical transversality fails. In particular, the derived intersection of two subschemes captures higher homotopy information through the Tor spectral sequence, enabling the construction of virtual fundamental classes that refine enumerative invariants. For example, in enumerative geometry, these derived virtual classes facilitate the study of node-smoothing problems on singular curves, where the virtual degree aligns with expected counts from deformation theory. This approach, building on obstruction theories, resolves singularities by embedding them into derived stacks that encode the full homotopical structure of intersections.^[43] In motivic homotopy theory, derived algebraic geometry underpins Voevodsky's construction of derived motives, which are triangulated categories of mixed motives equipped with an A¹-homotopy structure and localized with respect to the Nisnevich topology. This setup allows for the definition of motivic cohomology as the homotopy groups of motivic spectra, bridging algebraic cycles and homotopy invariants. Derived motives thus provide a universal cohomology theory for varieties, where the A¹-invariance ensures that affine line bundles act trivially, and the Nisnevich topology handles étale-local phenomena like residue fields. Voevodsky's framework, realized through derived categories of presheaves on smooth schemes, enables the proof of key results such as the Beilinson-Lichtenbaum conjectures on étale cohomology.^[44]^[45] Topological realizations in derived algebraic geometry are facilitated by the Betti realization functor, which maps derived stacks over the complex numbers to topological spaces by taking classical truncations and applying singular cohomology. This functor preserves homotopy limits and colimits, providing a bridge between algebraic and topological invariants; for instance, it realizes the étale homotopy type of a derived stack as a pro-space. In the context of derived stacks, the Betti realization extracts the topological data from algebraic objects, such as the fundamental group of moduli stacks of curves.^[46]^[47] Derived algebraic K-theory extends Quillen's classical definition by incorporating spectral methods, where the plus construction is lifted to E∞-ring spectra to produce connective K-theory spectra from ring spectra. This spectral plus construction resolves the perfect group issue in the classical case by applying Bousfield localization, yielding deloopings that compute higher K-groups as homotopy groups. In derived terms, it applies to simplicial commutative rings, enhancing computations for singular rings via derived schemes.^[48]^[49] Significant results include the resolution of aspects of Beilinson's conjectures through derived categories, where regulators map K-groups to étale cohomology, verified using perfect complexes on varieties. Additionally, derived enhancements of Deligne-Mumford stacks for curves, such as the moduli stack of stable maps, incorporate perfect obstruction theories to define virtual classes for enumerative counts of curve intersections. These derived stacks refine the classical coarse moduli spaces by tracking homotopical obstructions in deformation spaces. A landmark achievement is the 2024 proof of the categorical unramified geometric Langlands conjecture by Gaitsgory and collaborators, which establishes a correspondence between automorphic sheaves on moduli stacks of bundles and representations of the Langlands dual group, relying on derived algebraic geometry to construct and analyze these derived moduli stacks equipped with shifted symplectic structures. This proof, spanning over 800 pages across five papers, resolves a central problem in modern mathematics and highlights the power of derived methods in bridging number theory, representation theory, and geometry.^[50]^[51]^[2]^[52]

In Homotopy Theory and Higher Categories

Derived algebraic geometry provides a framework for constructing ∞-topoi from derived geometric sites, particularly through the ∞-topos of sheaves on derived stacks as developed by Jacob Lurie. In this setting, a derived stack is equipped with a topology, such as the étale or Nisnevich topology, allowing the formation of the ∞-category of sheaves on that site, which inherits the structure of an ∞-topos. This construction generalizes classical sheaf theory to the derived context, where sheaves of E_\infty-rings on the ∞-topos enable the study of quasicoherent sheaves and modules over derived stacks. Lurie's work in Higher Topos Theory establishes the foundational theory of ∞-topoi, while Derived Algebraic Geometry VIII: Quasicoherent Sheaves details the sheafification process for modules over these structures.^[53] A significant application arises in synthetic homotopy theory through the lens of condensed mathematics, as introduced by Dustin Clausen and Peter Scholze, which interfaces with spectral schemes in derived algebraic geometry. Condensed mathematics reformulates topological structures using solid and liquid modules over extremally disconnected sets, providing a synthetic approach to homotopy that avoids classical analytic pathologies. Spectral schemes, as ∞-topoi of sheaves on spectral Deligne-Mumford stacks, serve as the geometric foundation for this theory, enabling the definition of condensed spectra and their homotopy invariants. Clausen and Scholze's lectures demonstrate how this yields a cohesive framework for synthetic spectra, bridging derived geometry with analytic homotopy.^[54]^[55] In higher algebra, derived algebraic geometry incorporates E_n-algebras as central objects, generalizing operadic structures to the derived setting. An E_n-algebra in a stable ∞-category arises from an algebra over the E_n operad, capturing n-fold homotopy coherent operations, and extends to modules and bimodules in the derived context. This allows the study of derived stacks classified by E_n-algebras, where deformation theory and obstruction spaces are governed by operadic cotangent complexes. John Francis's thesis on Derived Algebraic Geometry Over E_n-Rings elucidates these constructions, showing how E_n-algebras underpin the homotopy theory of derived schemes and stacks. Lurie's Higher Algebra provides the ∞-categorical foundations for operads and their algebras in this framework.^[56]^[18] Key results include recognition principles for algebraic ∞-categories, which characterize presentable ∞-categories with finite colimits as modules over an E_\infty-ring spectrum, and derived Morita theory, which equates derived categories of modules over dg-categories via perfect complexes. Lurie's recognition principle in Higher Algebra (Theorem 4.8.5.18) identifies algebraic ∞-categories as those equivalent to \mathrm{Mod}_R(R\text{-}\mathrm{mod}^\hearts) for some E_\infty-ring R, facilitating the passage from homotopy to algebraic structures. Derived Morita theory, extended to derived stacks by Bertrand Toën, establishes that two derived stacks are Morita equivalent if their categories of quasicoherent sheaves are derived equivalent, preserving homotopy invariants.^[18]^[57] Connections to Goodwillie calculus emerge through derived functors on spectra, where the Taylor tower of a functor between ∞-categories of E_\infty-ring spectra approximates deformation functors. In this derived setting, the n-th derivative of a functor corresponds to a spectrum-valued multilinear map, enabling the analysis of higher-order obstructions in algebraic homotopy. Lurie's Derived Algebraic Geometry IV: Deformation Theory employs Goodwillie calculus to study the deformation theory of E_\infty-ring maps, linking excisive approximations to derived stacks.^[39]

In Physics and Quantum Field Theory

Derived algebraic geometry provides foundational tools for modeling observables in quantum field theory (QFT) through the framework of derived stacks and factorization algebras. In particular, Kevin Costello and Owen Gwilliam developed factorization algebras as local-to-global structures that encode the observables of QFTs, reworking the Batalin-Vilkovisky formalism using derived geometry to handle renormalization and interactions in a rigorous manner.^[58] These algebras arise naturally from classical and quantum field theories on derived stacks, capturing homotopy-theoretic aspects of field configurations that classical geometry overlooks. In topological field theories (TFTs), derived algebraic geometry underpins the classification via ∞-categories derived from stacks. Jacob Lurie's proof of the Baez-Dolan cobordism hypothesis demonstrates that fully extended n-dimensional TFTs are determined by the data of a fully dualizable object in a symmetric monoidal ∞-category, with derived stacks providing the geometric realization of these categories and enabling the extension to higher dimensions.^[59] This framework integrates cobordisms as morphisms in derived stack quotients, facilitating the computation of invariants in TFTs that align with physical models of topological phases.^[60] Applications extend to string theory, where derived symplectic geometry addresses moduli spaces of curves central to mirror symmetry. Derived symplectic structures on stacks of stable curves resolve singularities and obstructions in the classical moduli, allowing for a homotopical enhancement that matches symplectic invariants on the mirror side through homological mirror symmetry. This approach refines the computation of Gromov-Witten invariants and their mirrors, providing a derived enhancement to the A-model and B-model correspondences.^[61] Seminal contributions include Mikhail Kapranov's derived non-commutative geometry, which formalizes vertex algebras via ind-schemes modeling formal loop spaces, linking algebraic structures in conformal field theory to derived enhancements of loop group representations. Similarly, Bertrand Toën's derived Hall algebras, constructed from dg-categories, generalize classical Hall algebras to homotopy settings, offering tools for quantizing structures relevant to vertex operator algebras in QFT.^[62] Representative examples include derived loop spaces in conformal field theory, where Kapranov and Éric Vasserot's construction equips the formal loop space of an algebraic variety with a vertex algebra structure, capturing chiral symmetries and operator product expansions in two-dimensional CFTs.^[63] Additionally, spectral sequences in derived algebraic geometry compute partition functions in TFTs by filtering derived pushforwards along stacky fibrations, converging to homotopy-invariant traces that encode vacuum expectations in low-dimensional QFTs. Derived moduli stacks serve briefly as models for physical configurations, such as Higgs bundles in gauge theories.^[42]

References

[1]
[PDF] Derived algebraic geometry
Derived algebraic geometry is an extension of algebraic geometry whose main purpose is to propose a setting to treat geometrically special situations ...
[2]
[PDF] DERIVED ALGEBRAIC GEOMETRY 1. Introduction 1.1. Bezout's ...
To obtain derived algebraic geometry, we need a formalism of “generalized rings” in which imposing the equation x = 0 twice is not equivalent to imposing the ...
[3]
[PDF] An introduction to derived (algebraic) geometry
Aug 26, 2025 · Focusing on the characteristic 0 theory, Section 1 introduces dg-algebras as the affine building blocks for derived geometry. It then gives a ...
[4]
[PDF] Derived algebraic geometry - HAL
Jan 12, 2016 · Introduction. Derived algebraic geometry is an extension of algebraic geometry whose main purpose is to propose a setting to treat ...
[5]
[1401.1044] Derived Algebraic Geometry - arXiv
Jan 6, 2014 · This text is a survey of derived algebraic geometry. It covers a variety of general notions and results from the subject with a view on the recent developments.
[6]
[PDF] Derived Algebraic Geometry - DSpace@MIT
May 10, 2004 · The purpose of this document is to establish the foundations for a theory of derived algebraic geometry based upon simplicial commutative rings.
[7]
[PDF] arXiv:1401.1044v2 [math.AG] 12 Sep 2014
Sep 12, 2014 · The interrelations between ideas from algebraic geometry and algebraic topology is one of the feature of derived algebraic geometry, and the.
[8]
[1501.06731] Introduction to Derived Categories - arXiv
Jan 27, 2015 · Derived categories were invented by Grothendieck and Verdier around 1960, not very long after the "old" homological algebra (of derived functors ...
[9]
[PDF] Notes on Derived Functors and Grothendieck Duality. - Purdue Math
Grothendieck also applied his ideas in the context of étale cohomology. The fundamental technique of derived categories was developed by Verdier, who used ...
[10]
[math/0609537] Model Categories and Simplicial Methods - arXiv
Sep 19, 2006 · Here we'd like to treat model categories as a way to build and control resolutions. This an historical approach, as in his original and ...
[11]
[PDF] Model Categories and Simplicial Methods
This an historical approach, as in his original and spectacular applications of model categories, Quillen used this technology as a way to construct resolutions ...<|control11|><|separator|>
[12]
[math/0207028] Homotopical Algebraic Geometry I: Topos theory
Jul 2, 2002 · This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts.
[13]
AMS eBooks: Memoirs of the American Mathematical Society
Homotopical algebraic geometry. II. Geometric stacks and applications. About this Title. Bertrand Toën and Gabriele Vezzosi ... Publication Year: 2008; Volume ...
[14]
[1412.2203] Positive characteristic algebraic geometry - arXiv
Dec 6, 2014 · These are notes for the Bootcamp volume for the 2015 AMS Summer Institute in Algebraic Geometry. They are based on earlier notes for the Positive ...
[15]
higher category theory - Spectral algebraic geometry vs derived ...
Jun 17, 2018 · This is already one example of how the spectral and derived worlds diverge in positive characteristic. Another example comes from considering ...
[16]
Complexe Cotangent et Deformations I - SpringerLink
Book Title: Complexe Cotangent et Deformations I · Authors: Luc Illusie · Series Title: Lecture Notes in Mathematics · Publisher: Springer Berlin, Heidelberg.
[17]
[PDF] Higher Algebra - Harvard Mathematics Department
Jun 2, 2015 · ... Simplicial Models for Algebras and Modules ... Recall that ordinary commutative ring R can be viewed as a commutative algebra ...
[18]
[PDF] Simplicial Commutative Rings - The University of Chicago
May 22, 2012 · Simplicial commutative rings are used to make categories more "homotopyish" and are a "homotopical" replacement for topological commutative ...
[19]
[2109.14594] An introduction to derived (algebraic) geometry - arXiv
Sep 29, 2021 · These notes introduce derived geometry, mainly derived algebraic geometry in characteristic 0, for those with geometry and homological algebra ...
[20]
[PDF] Differential graded categories
Introduction. 1.1. Triangulated categories and dg categories. Derived categories were invented by. Grothendieck-Verdier in the early sixties in order to ...
[21]
[PDF] Andre-Quillen homology of commutative algebras - MIT Mathematics
ABSTRACT. These notes are an introduction to basic properties of Andre-. Quillen homology for commutative algebras. They are an expanded version of.
[22]
hochschild (co)homology in commutative algebra. a survey
Feb 10, 2015 · The reciprocal has a diﬃcult proof. 3.11 Theorem [29]: Let K be a ﬁeld of characteristic zero and A a K-algebra.<|control11|><|separator|>
[23]
[PDF] Higher Algebra
Sep 18, 2017 · ... Simplicial ... The E∞-rings play a role in stable homotopy theory analogous to the role played by commutative rings in ordinary algebra.
[24]
[PDF] Spectral Algebraic Geometry (Under Construction!)
Apr 1, 2011 · Spectral Algebraic Geometry (Under Construction!) if the right hand side is again interpreted as counting the number of points in the set- ...
[25]
[PDF] Derived Algebraic Geometry VII: Spectral Schemes
Nov 5, 2011 · Our goal in this paper is to introduce a variant of algebraic geometry, which we will refer to as spectral algebraic geometry.
[26]
[PDF] Derived Algebraic Geometry XIV: Representability Theorems
Mar 14, 2012 · Introduction to commutative algebra. Addison-Wesley Publishing Co ... Lurie. Vanishing Theorems for Higher Tate Cohomology. In ...
[27]
[PDF] Homotopical algebraic geometry I: topos theory
Abstract. This is the first of a series of papers devoted to lay the foundations of Algebraic Geometry in homotopical and higher categorical contexts.
[28]
[PDF] A brief introduction to derived schemes - arXiv
Nov 30, 2018 · Throughout this paper, we have proposed a point of view of derived schemes as a means ... [2] Jacob Lurie, Derived algebraic geometry, Ph.D.
[29]
[PDF] Introductory topics in derived algebraic geometry - (DIMAI) | UniFI
We discuss affine derived schemes, derived algebraic stacks, and the Artin-Lurie representability theorem. Through the example of deformations of smooth and ...
[30]
[PDF] Derived Algebraic Geometry XII: Proper Morphisms, Completions ...
Nov 8, 2011 · This paper aims to prove a version of Grothendieck's existence theorem in spectral algebraic geometry, developing a theory of proper morphisms.<|control11|><|separator|>
[31]
[PDF] Higher and derived stacks: a global overview
Considering F as a derived stack i(F) we get a derived inertia stack. RIF := Ii(F) := i(F) ×i(F)×i(F) i(F) ∈ dSt(k). The derived stack RIF is of course a ...
[32]
[PDF] ANALYTIC STACKS Lectures by Dustin Clausen and Peter Scholze ...
So one can give some purely synthetic algebraic descriptions of ... condensed being groups, you also have solid spectra inside of all condensed spectra.
[33]
[PDF] Cyclotomic synthetic spectra
Dec 2, 2024 · We call Sev the synthetic sphere spectrum, or the even sphere spectrum. It is complete with underlying object given by S. Remark 2.8. The E1- ...
[34]
[PDF] arXiv:1002.3636v2 [math.AG] 5 Jul 2011
Jul 5, 2011 · We examine the geometry of loop spaces in derived algebraic geometry ... for the loop space as a fiber product of derived stacks along diagonal ...
[35]
[PDF] an introduction to derived algebraic geometry - adeel a. khan
Jun 19, 2023 · These lecture notes aim to provide a working knowledge of the languages of. ∞-categories and derived algebraic geometry.Missing: review | Show results with:review
[36]
[PDF] SPECTRAL ALGEBRAIC GEOMETRY 1. Introduction This chapter is ...
Spectral algebraic geometry replaces “rings” with an ∞-categorical generalization, namely commutative ring spectra, which (following Lurie) we will here call E∞ ...
[37]
Brauer groups and étale cohomology in derived algebraic geometry
Oct 1, 2012 · In this paper, we study Azumaya algebras and Brauer groups in derived algebraic geometry. We establish various fundamental facts about Brauer groups in this ...
[38]
[PDF] Derived Algebraic Geometry IV: Deformation Theory
Oct 8, 2009 · derived algebraic geometry; see [45]. Remark 0.1. The theory of ... cotangent complex LB/A and the absolute cotangent complex LB. 21 ...
[39]
Section 75.26 (0D1X): Perfect complexes—The Stacks project
We first talk about jumping loci for betti numbers of perfect complexes. First we have to define betti numbers. Let S be a scheme. Let X be an algebraic space ...
[40]
[PDF] Higher Algebraic K-Theory of Schemes and of Derived Categories
Feb 1, 2015 · In this paper we prove a localization theorem for the A-theory of com- mutative rings and of schemes, Theorem 7.4, relating the A'-groups of.
[41]
[1011.2189] Representability of derived stacks - arXiv
Nov 9, 2010 · Lurie's representability theorem gives necessary and sufficient conditions for a functor to be an almost finitely presented derived geometric stack.
[42]
[PDF] arXiv:2504.19542v1 [math.AG] 28 Apr 2025
Apr 28, 2025 · The purpose of this paper is to shed a new light on classical constructions in enumerative geometry from the view point of derived algebraic ...
[43]
[PDF] Voevodsky's Nordfjordeid Lectures: Motivic Homotopy Theory
Motivic homotopy theory is a new and in vogue blend of algebra and topology. Its primary object is to study algebraic varieties from a homotopy theoretic.
[44]
[PDF] Lecture Notes on Motivic Cohomology - Clay Mathematics Institute
Mazza, Carlo, 1974–. Lecture notes on motivic cohomology / Carlo Mazza, Vladimir Voevodsky, Charles A. ... motivic cohomology as well (such as homotopy invariance) ...
[45]
[PDF] Betti Realization, Gluing, and the Motivic Homotopy Theory
Mar 2, 2025 · The concrete goals of this dissertation are to define the motivic homotopy theory of complex analytic stacks, establish some of its properties, ...
[46]
[PDF] arXiv:2303.10146v2 [math.AG] 15 Sep 2023
Sep 15, 2023 · Betti stacks and affinization. We define an important class of derived stacks, called. Betti stacks, which correspond to spaces. Definition ...
[47]
[PDF] The plus construction, Bousfield localization, and derived completion
Jun 28, 2009 · We define a plus-construction on connective augmented algebras over operads in symmetric spectra using Quillen homology. For asso- ciative and ...
[48]
[PDF] Quillen's work in algebraic K-theory - School of Mathematics
Dec 2, 2011 · Quillen then defined the higher algebraic K-groups by setting. Kn(R) := πnB GL∞(R)+ for n > 0. The groups are abelian: K1(R) by construction, ...
[49]
[PDF] Beilinson's conjectures - Mathematics
Introduction. In his seminal paper [1], A.A. Beilinson formulated far reaching conjectures about values of motivic L-functions at integers, and produced a ...
[50]
Derived algebraic geometry, determinants of perfect complexes, and ...
Feb 6, 2011 · Authors:Timo Schürg, Bertrand Toën, Gabriele Vezzosi. View a PDF of the paper titled Derived algebraic geometry, determinants of perfect ...
[51]
[PDF] Derived Algebraic Geometry VIII: Quasicoherent Sheaves
May 18, 2011 · 2 Sheaves of Modules. 2.1 Sheaves on an ∞-Topos. Definition 2.1.1. Let X be an ∞-topos and let O ∈ ShvCAlg(X) be a sheaf of E∞-rings on X.
[52]
[PDF] Lectures on Condensed Mathematics Peter Scholze (all results joint ...
The material presented is part of joint work with Dustin Clausen. The goal of the course is to define the (derived) category of solid A-modules for any ring A,.Missing: synthetic | Show results with:synthetic
[53]
[PDF] Condensed Mathematics and Complex Geometry Dustin Clausen ...
Lecture I: Introduction. Over the last few years, we have been working on an alternative foundation for the development of a (very general) “analytic ...Missing: synthetic | Show results with:synthetic
[54]
[PDF] Derived Algebraic Geometry Over En-Rings John Francis
Derived algebraic geometry over En-rings uses En-rings, whose multiplication is parametrized by configuration spaces, instead of commutative rings, and ...
[55]
The homotopy theory of dg-categories and derived Morita theory
Aug 24, 2004 · We use these two results in order to prove a derived version of Morita theory, describing the morphisms between dg-categories of modules over ...
[56]
Factorization Algebras in Quantum Field Theory
Along with a systematic reworking of the Batalin–Vilkovisky formalism via derived geometry and factorization algebras, this book offers concrete examples ...<|separator|>
[57]
[0905.0465] On the Classification of Topological Field Theories - arXiv
May 4, 2009 · This paper provides an informal sketch of a proof of the Baez-Dolan cobordism hypothesis, which provides a classification for extended topological quantum ...
[58]
[PDF] On the Classification of Topological Field Theories
Apr 26, 2010 · This paper discusses the classification of topological field theories, using extended theories and the cobordism hypothesis, and the language ...
[59]
Lectures on Mirror Symmetry, Derived Categories, and D-branes
Aug 18, 2003 · This paper is an introduction to Homological Mirror Symmetry, derived categories, and topological D-branes aimed mainly at a mathematical audience.
[60]
[math/0501343] Derived Hall Algebras - arXiv
Jan 21, 2005 · The purpose of this work is to define a derived Hall algebra \mathcal{DH}(T), associated to any dg-category T (under some finiteness conditions).Missing: Toën | Show results with:Toën
[61]
Vertex algebras and the formal loop space
Nov 1, 2004 · We construct a certain algebro-geometric version L ( X ) of the free loop space for a complex algebraic variety X. This is an ind-scheme ...