Search results

..., the base change <math>X \times_Y Y' \to Y'</math> is a [[homeomorphism]] of topological spaces. ...f>EGA IV₄, 18.12.11.</ref> In particular, a morphism of locally of finite type is a universal homeomorphism if and only if it is [[finite morp ...

...definitions of an '''fpqc morphism''', both variations of faithfully flat morphisms. ...er it is more common to define an fpqc morphism <math>f: X \to Y</math> of schemes to be a [[faithfully flat morphism]] that satisfies the following equivalen ...

In [[algebraic geometry]], a morphism <math>f: X \to S</math> of [[scheme (mathematics)|scheme]]s is said to be '''locally acyclic''' if, ro [[Category:Morphisms of schemes]] ...

...er field, or the pullback of a family of varieties, or a fiber of a family of varieties. '''Base change''' is a closely related notion. ..., rather than simply studying [[algebraic curve]]s, one can study families of curves over any base scheme ''Y''. Indeed, the two approaches enrich each o ...

...)|module]]. An <math>R</math>-algebra can be thought as a [[homomorphism]] of [[ring (mathematics)|rings]] <math>f\colon R \to A</math>, in this case <ma ...itely generated algebra#finite algebras vs algebras of finite type|algebra of finite type]]. ...

...opology]]. The precise definition depends on one's formalism. But the idea of the definition itself is simple. The usual prime spectrum Spec enjoys the r ...Spét to be the right adjoint to the global section functor on the category of "spaces" with étale topology.<ref>{{harvnb|Lurie|loc=Remark 1.2.3.6.}}</ref ...

{{short description|Type of space in mathematics}} ...formal functions]], which is used to deduce theorems of interest for usual schemes. ...

...X</math> in some category <math>\mathcal{C}</math>. There is a dual notion of undercategory, which is defined similarly. Let <math>\mathcal{C}</math> be a category and <math>X</math> a fixed object of <math>\mathcal{C}</math><ref>{{cite arXiv|last=Leinster|first=Tom|date=2016 ...

...y sheaves]] <math>\pi_* F \to \pi_* G</math> is an isomorphism. A '''sheaf of spectra''' is then a fibrant/cofibrant object in that category. ....ku.dk/~jg/homotopical2008/goerss.lec2.pdf |first=Paul |last=Goerss |title=Schemes |work=TAG Lecture 2 |date=16 June 2008 }} ...

...aracteristic (algebra)|characteristic]] zero which have [[residue field]]s of characteristic [[prime number|prime]] ''p''. ...ive]] on ''K''°/''p'' where ''K''° denotes the [[ring (mathematics)|ring]] of power-bounded elements. ...

...poid]] which is built on richer structures than sets, and a generalization of a [[group object]]s when the multiplication is only [[partial function|part ...ng finite [[pullback (category theory)|fiber product]]s consists of a pair of [[Object (category theory)|objects]] <math>R, U</math> together with five [ ...

...>-algebra unless <math>A</math> = 0 or <math>n</math> = 0. Another example of a finite-type homomorphism that is not finite is <math>\mathbb{C}[t] \to \m {{expert needed|Mathematics |talk=distinction between finite type and locally of finite type|date=August 2023}} ...

In [[topology]], '''constructible sets''' are a class of subsets of a [[topological space]] that have a relatively "simple" structure. ...t the image of a constructible set is constructible for an important class of [[map (mathematics)|mappings]] ...

...dsky]] to study the [[homology theory|homology]] of [[scheme (mathematics)|schemes]].<ref>{{citation | title = Homology of schemes ...

...s generated by a [[regular sequence]] of length ''r''. A regular embedding of codimension one is precisely an [[effective Cartier divisor]]. ...-morphism, then ''i'' is a regular embedding. In particular, every section of a smooth morphism is a regular embedding.<ref>{{harvnb|Sernesi|2006|loc=D. ...

{{Short description|Type of ringed space}} ...af of commutative rings]] on ''X'' that is also a [[sheaf of modules|sheaf of <math>\mathcal{O}_X</math>-modules]]. It is [[quasi-coherent sheaf|quasi-co ...

...e.g. an [[algebraic variety]]) can be obtained by '''gluing''' existing '''schemes''' through gluing maps. Suppose there is a (possibly infinite) family of schemes <math>\{ X_i \}_{i \in I}</math> and for pairs <math>i, j</math>, there are ...

...open cover <math>\mathcal U_1</math>, and to let the triple intersections of this cover to be covered by yet another open cover <math>\mathcal U_2</math ...ath>. A '''hypercover''' is a semisimplicial object <math>U_\bullet</math> of this category such that <math>U_0 \to X</math> is an étale cover and such t ...

...the latter subject, a [[Bohr topos]] is a ringed topos that plays the role of a quantum [[phase space]].<ref>{{Cite web|url=https://golem.ph.utexas.edu/c ...posé IV, Exercise 13.9), which is equivalent to saying that all the stalks of the structure ring object are local rings when there are [[enough points]]. ...

...heory|Hodge-theoretic]] concepts. This idea has applications in the theory of [[moduli spaces]], in [[deformation theory]] and Fontaine's [[p-adic Hodge ...s could provide one which vanishes), and we only get a sheaf of submonoids of <math> \mathcal{O}_X </math>, multiplicatively. Remembering this additional ...