Mathstack

Answers are validated mathstack 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, mathstack.

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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A mathstack motivation for stacks is that if a moduli space for some problem mathstack not exist because of the existence of automorphisms, it may still be possible to construct a moduli stack, mathstack. This usually seems to lead to an equivalent category of quasi-coherent sheaves, but is easier to use: for example it is easier to compare with the etale topology on algebraic spaces, mathstack. Archived PDF from the original on

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Mathstack

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Grothendieck's letter to Serre, Nov 5. A very similar and analogous extension is to develop the stack theory on non-discrete objects i. 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. Download as PDF Printable version. Some authors require this as a property of stacks, rather than of prestacks. Some authors use the word "stack" to refer to the more restrictive notion of a stack in groupoids. Separate validation and assessment Answers are validated before they are marked, so students are not penalised for poor programming skills. 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. Archived PDF from the original on MR It turns out that the general language for describing these coverings is that of a Grothendieck topology. For example, take a projective morphism. Hidden categories: Articles with short description Short description is different from Wikidata All articles with unsourced statements Articles with unsourced statements from June CS1 maint: location missing publisher. Since this can instead be taken as a stack quotient, the weighted projective stack [4] pg 30 is. It is also possible to use finer topologies.

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The specification of gluings requires a definition of coverings with regard to which the gluings can be considered. Specific feedback Students are given feedback that refers to their specific answer and mistake, as if marked by hand. Roughly speaking, Deligne—Mumford stacks can be thought of as algebraic stacks whose objects have no infinitesimal automorphisms. They are called higher stacks. In the same way, moduli spaces of curves, vector bundles, or other geometric objects are often best defined as stacks instead of schemes. Tools Tools. Since this can instead be taken as a stack quotient, the weighted projective stack [4] pg 30 is. In particular, the action sends a tuple. JSTOR MR An example of a non-affine stack is given by the half-line with two stacky origins. For example, [3] the moduli stack. The Grothendieck topology should be strong enough so that the stack is locally affine in this topology: schemes are locally affine in the Zariski topology so this is a good choice for schemes as Serre discovered, algebraic spaces and Deligne—Mumford stacks are locally affine in the etale topology so one usually uses the etale topology for these, while algebraic stacks are locally affine in the smooth topology so one can use the smooth topology in this case. Mumford studied the Picard group of the moduli stack of elliptic curves , before stacks had been defined.