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Jacob Lurie's under-construction book Spectral Algebraic Geometry studies a generalization that he calls a spectral Deligne–Mumford stack. By definition, it is a ringed ∞-topos that is étale-locally the étale spectrum of an E ∞-ring (this notion subsumes that of a derived scheme, at least in characteristic zero.)
In the former case, equivalence of two definitions means that a mathematical object (for example, geometric body) satisfies one definition if and only if it satisfies the other definition. In the latter case, the meaning of equivalence (between two definitions of a structure) is more complicated, since a structure is more abstract than an object.
This is a list of notable theorems.Lists of theorems and similar statements include: List of algebras; List of algorithms; List of axioms; List of conjectures
Italian school of algebraic geometry. Most gaps in proofs are caused either by a subtle technical oversight, or before the 20th century by a lack of precise definitions. A major exception to this is the Italian school of algebraic geometry in the first half of the 20th century, where lower standards of rigor gradually became acceptable.
The hinge theorem holds in Euclidean spaces and more generally in simply connected non-positively curved space forms.. It can be also extended from plane Euclidean geometry to higher dimension Euclidean spaces (e.g., to tetrahedra and more generally to simplices), as has been done for orthocentric tetrahedra (i.e., tetrahedra in which altitudes are concurrent) [2] and more generally for ...
Absolute geometry is a geometry based on an axiom system consisting of all the axioms giving Euclidean geometry except for the parallel postulate or any of its alternatives. [69] The term was introduced by János Bolyai in 1832. [70] It is sometimes referred to as neutral geometry, [71] as it is neutral with respect to the parallel postulate.
In algebraic geometry, given irreducible subvarieties V, W of a projective space P n, the ruled join of V and W is the union of all lines from V to W in P 2n+1, where V, W are embedded into P 2n+1 so that the last (resp. first) n + 1 coordinates on V (resp. W) vanish. [1]
The book is written at a level appropriate for undergraduate students in mathematics, and assumes a background knowledge in real analysis and undergraduate-level geometry. [6] One goal of the book is to expose students at this level to research-level problems in mathematics whose statement is readily accessible. [2]
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