Mathstack

Answers are validated before they are marked, mathstack, so students are not penalised for poor programming skills.

In mathematics a stack or 2-sheaf is, roughly speaking, a sheaf that takes values in categories rather than sets. Stacks are used to formalise some of the main constructions of descent theory , and to construct fine moduli stacks when fine moduli spaces do not exist. Descent theory is concerned with generalisations of situations where isomorphic , compatible geometrical objects such as vector bundles on topological spaces can be "glued together" within a restriction of the topological basis. In a more general set-up the restrictions are replaced with pullbacks ; fibred categories then make a good framework to discuss the possibility of such gluing. The intuitive meaning of a stack is that it is a fibred category such that "all possible gluings work". The specification of gluings requires a definition of coverings with regard to which the gluings can be considered.

Mathstack

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Descent theory is concerned with generalisations of situations where isomorphicmathstack, compatible geometrical objects such as vector bundles on topological spaces can be "glued together" mathstack a restriction of the mathstack basis. The concept of stacks has its origin in the definition of effective descent data in Grothendieck

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Mathstack

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Randomisation STACK can generate random questions so students are shown different variants of questions, and can repeat quizzes with new variants. Generalisation of a sheaf; a fibered category that admits effective descent. A major motivation for stacks is that if a moduli space for some problem does not exist because of the existence of automorphisms, it may still be possible to construct a moduli stack. The fiber product of stacks is defined using the usual universal property , and changing the requirement that diagrams commute to the requirement that they 2-commute. If A is a quasi-coherent sheaf of graded algebras in an algebraic stack X over a scheme S , then there is a stack Proj A generalizing the construction of the projective scheme Proj A of a graded ring A. Separate validation and assessment Answers are validated before they are marked, so students are not penalised for poor programming skills. Students are given feedback that refers to their specific answer and mistake, as if marked by hand. For example, take a projective morphism. Main article: Quasi-coherent sheaf on an algebraic stack. For example, [3] the moduli stack. It is also possible to use finer topologies. Instead of using the smooth topology for algebraic stacks one often uses a modification of it called the Lis-Et topology short for Lisse-Etale: lisse is the French term for smooth , which has the same open sets as the smooth topology but the open covers are given by etale rather than smooth maps. Roughly speaking, Deligne—Mumford stacks can be thought of as algebraic stacks whose objects have no infinitesimal automorphisms. Stacks are used to formalise some of the main constructions of descent theory , and to construct fine moduli stacks when fine moduli spaces do not exist.

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Article Talk. There are some minor set theoretical problems with the usual foundation of the theory of stacks, because stacks are often defined as certain functors to the category of sets and are therefore not sets. When defining quotients of schemes by group actions, it is often impossible for the quotient to be a scheme and still satisfy desirable properties for a quotient. For example, take a projective morphism. Instead of using the smooth topology for algebraic stacks one often uses a modification of it called the Lis-Et topology short for Lisse-Etale: lisse is the French term for smooth , which has the same open sets as the smooth topology but the open covers are given by etale rather than smooth maps. They are called higher stacks. ISSN X. Keys Action? A very similar and analogous extension is to develop the stack theory on non-discrete objects i. An algebraic stack or Artin stack is a stack in groupoids X over the fppf site such that the diagonal map of X is representable and there exists a smooth surjection from the stack associated to a scheme to X. Roughly speaking, Deligne—Mumford stacks can be thought of as algebraic stacks whose objects have no infinitesimal automorphisms. Tools Tools. Students are given feedback that refers to their specific answer and mistake, as if marked by hand. In particular, the action sends a tuple. MR