Concept

Derived scheme

In algebraic geometry, a derived scheme is a pair consisting of a topological space X and a sheaf either of simplicial commutative rings or of commutative ring spectra on X such that (1) the pair is a scheme and (2) is a quasi-coherent -module. The notion gives a homotopy-theoretic generalization of a scheme. A derived stack is a stacky generalization of a derived scheme. Over a field of characteristic zero, the theory is closely related to that of a differential graded scheme. By definition, a differential graded scheme is obtained by gluing affine differential graded schemes, with respect to étale topology. It was introduced by Maxim Kontsevich "as the first approach to derived algebraic geometry." and was developed further by Mikhail Kapranov and Ionut Ciocan-Fontanine. Just as affine algebraic geometry is equivalent (in ) to the theory of commutative rings (commonly called commutative algebra), affine derived algebraic geometry over characteristic zero is equivalent to the theory of commutative differential graded rings. One of the main example of derived schemes comes from the derived intersection of subschemes of a scheme, giving the Koszul complex. For example, let , then we can get a derived scheme where is the étale spectrum. Since we can construct a resolution the derived ring is the koszul complex . The truncation of this derived scheme to amplitude provides a classical model motivating derived algebraic geometry. Notice that if we have a projective scheme where we can construct the derived scheme where with amplitude Let be a fixed differential graded algebra defined over a field of characteristic . Then a -differential graded algebra is called semi-free if the following conditions hold: The underlying graded algebra is a polynomial algebra over , meaning it is isomorphic to There exists a filtration on the indexing set where and for any . It turns out that every differential graded algebra admits a surjective quasi-isomorphism from a semi-free differential graded algebra, called a semi-free resolution.

À propos de ce résultat
Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.
Cours associés (5)
MATH-510: Algebraic geometry II - schemes and sheaves
The aim of this course is to learn the basics of the modern scheme theoretic language of algebraic geometry.
MATH-657: Deformation Theory
We will study classical and modern deformation theory of schemes and coherent sheaves. Participants should have a solid background in scheme-theory, for example being familiar with the first 3 chapter
MATH-658: Vanishing cycles and perverse sheaves
This course will explain the theory of vanishing cycles and perverse sheaves. We will see how the Hard Lefschetz theorem can be proved using perverse sheaves. If we have more time we will try to see t
Afficher plus
Séances de cours associées (13)
Schémas équivalents: AimantsMOOC: Conversion electromécanique I
Discute de la relation entre l'intensité du champ magnétique et la densité de flux magnétique dans des schémas équivalents pour les aimants.
Analyse de publication : Comprendre les articles de recherche
Présente l'analyse de la publication, en mettant l'accent sur l'évaluation critique des articles de recherche et des stratégies de lecture efficaces.
Structure de l'anneau gradué sur la cohomologie
Explore les propriétés associatives et commutatives du produit en cohomologie, en mettant l'accent sur les structures graduées.
Afficher plus
Publications associées (43)

Linear and nonlinear excitation of TAE modes by external electromagnetic perturbations using ORB5

Laurent Villard, Emmanuel Lanti, Alberto Bottino, Mohsen Sadr

The excitation of toroidicity-induced Alfven eigenmodes (TAEs) using prescribed external electromagnetic perturbations (hereafter 'antenna') acting on a confined toroidal plasma, as well as its nonlinear couplings to other modes in the system, is studied. ...
IOP Publishing Ltd2022

Covers Of Rational Double Points In Mixed Characteristic

Javier Alonso Carvajal Rojas

We further the classification of rational surface singularities. Suppose (S, n, k) is a 3-dimensional strictly Henselian regular local ring of mixed characteristic (0, p > 5). We classify functions f for which S/(f) has an isolated rational singularity at ...
2021

Towards Efficient LPN-Based Symmetric Encryption

Serge Vaudenay, Sonia Mihaela Bogos, Dario Korolija, Thomas Locher

Due to the rapidly growing number of devices that need to communicate securely, there is still significant interest in the development of efficient encryption schemes. It is important to maintain a portfolio of different constructions in order to enable a ...
Springer International Publishing2021
Afficher plus
Concepts associés (1)
Derived algebraic geometry
Derived algebraic geometry is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts, are replaced by either differential graded algebras (over ), simplicial commutative rings or -ring spectra from algebraic topology, whose higher homotopy groups account for the non-discreteness (e.g., Tor) of the structure sheaf. Grothendieck's scheme theory allows the structure sheaf to carry nilpotent elements.

Graph Chatbot

Chattez avec Graph Search

Posez n’importe quelle question sur les cours, conférences, exercices, recherches, actualités, etc. de l’EPFL ou essayez les exemples de questions ci-dessous.

AVERTISSEMENT : Le chatbot Graph n'est pas programmé pour fournir des réponses explicites ou catégoriques à vos questions. Il transforme plutôt vos questions en demandes API qui sont distribuées aux différents services informatiques officiellement administrés par l'EPFL. Son but est uniquement de collecter et de recommander des références pertinentes à des contenus que vous pouvez explorer pour vous aider à répondre à vos questions.