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The sheaf of rational functions K X of a scheme X is the generalization to scheme theory of the notion of function field of an algebraic variety in classical algebraic geometry. In the case of algebraic varieties , such a sheaf associates to each open set U the ring of all rational functions on that open set; in other words, K X ( U ) is the ...
A subset V of A n is called an affine algebraic set if V = Z(S) for some S. [1]: 2 A nonempty affine algebraic set V is called irreducible if it cannot be written as the union of two proper algebraic subsets. [1]: 3 An irreducible affine algebraic set is also called an affine variety.
In algebraic geometry, the function field of an algebraic variety V consists of objects that are interpreted as rational functions on V.In classical algebraic geometry they are ratios of polynomials; in complex geometry these are meromorphic functions and their higher-dimensional analogues; in modern algebraic geometry they are elements of some quotient ring's field of fractions.
In universal algebra, a subalgebra of an algebra A is a subset S of A that also has the structure of an algebra of the same type when the algebraic operations are restricted to S. If the axioms of a kind of algebraic structure is described by equational laws , as is typically the case in universal algebra, then the only thing that needs to be ...
These are two examples in which both the subset and the whole set are infinite, and the subset has the same cardinality (the concept that corresponds to size, that is, the number of elements, of a finite set) as the whole; such cases can run counter to one's initial intuition. The set of rational numbers is a proper subset of the set of real ...
Suppose that X, Y, and Z as above are sets, and that f : Z → X and g : Z → Y are set functions. The pushout of f and g is the disjoint union of X and Y , where elements sharing a common preimage (in Z ) are identified, together with the morphisms i 1 , i 2 from X and Y , i.e. P = ( X ⊔ Y ) / ∼ {\displaystyle P=(X\sqcup Y)/\!\sim } where ...
In algebraic geometry, an affine variety (or, more generally, an affine algebraic set) is defined as the subset of an affine space that is the set of the common zeros of a set of so-called polynomial functions over the affine space. For defining a polynomial function over the affine space, one has to choose an affine frame.
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometrical problems.Classically, it studies zeros of multivariate polynomials; the modern approach generalizes this in a few different aspects.
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