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Все для школьников, студентов, учащихся, преподавателей и родителей. Jump to navigation Jump to search This article is about algebraic varieties. The twisted cubic is a projective algebraic variety. Algebraic varieties are the central objects of study in algebraic geometry. Conventions regarding the definition of an algebraic variety differ slightly. For example, some definitions require an algebraic variety to be irreducible, which means that it is not the union of two smaller sets that are closed in the Zariski topology. Under this definition, non-irreducible algebraic varieties are called algebraic sets.

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Other conventions do not require irreducibility. The concept of an algebraic variety is similar to that of an analytic manifold. An important difference is that an algebraic variety may have singular points, while a manifold cannot. An affine variety over an algebraically closed field is conceptually the easiest type of variety to define, which will be done in this section. Next, one can define projective and quasi-projective varieties in a similar way.

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The most general definition of a variety is obtained by patching together smaller quasi-projective varieties. A nonempty affine algebraic set V is called irreducible if it cannot be written as the union of two proper algebraic subsets. An irreducible affine algebraic set is also called an affine variety. Affine varieties can be given a natural topology by declaring the closed sets to be precisely the affine algebraic sets.

This topology is called the Zariski topology. For any affine algebraic set V, the coordinate ring or structure ring of V is the quotient of the polynomial ring by this ideal. An irreducible projective algebraic set is called a projective variety. Projective varieties are also equipped with the Zariski topology by declaring all algebraic sets to be closed. For any projective algebraic set V, the coordinate ring of V is the quotient of the polynomial ring by this ideal. A quasi-projective variety is a Zariski open subset of a projective variety.

Notice that every affine variety is quasi-projective. In classical algebraic geometry, all varieties were by definition quasiprojective varieties, meaning that they were open subvarieties of closed subvarieties of projective space. So classically the definition of an algebraic variety required an embedding into projective space, and this embedding was used to define the topology on the variety and the regular functions on the variety. The disadvantage of such a definition is that not all varieties come with natural embeddings into projective space. However, any variety that admits one embedding into projective space admits many others by composing the embedding with the Veronese embedding. The earliest successful attempt to define an algebraic variety abstractly, without an embedding, was made by André Weil. In his Foundations of Algebraic Geometry, Weil defined an abstract algebraic variety using valuations.

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One of the earliest examples of a non-quasiprojective algebraic variety were given by Nagata. For example, every open subset of a variety is a variety. Hilbert’s Nullstellensatz says that closed subvarieties of an affine or projective variety are in one-to-one correspondence with the prime ideals or homogeneous prime ideals of the coordinate ring of the variety. A2 be the two-dimensional affine space over C. Polynomials in the ring C can be viewed as complex valued functions on A2 by evaluating at the points in A2. This is called a line in the affine plane. In the classical topology coming from the topology on the complex numbers, a complex line is a real manifold of dimension two.

The set V is not empty. It is irreducible, as it cannot be written as the union of two proper algebraic subsets. Thus it is an affine algebraic variety. The following example is neither a hypersurface, nor a linear space, nor a single point. Let A3 be the three-dimensional affine space over C.

C is an algebraic variety, and more precisely an algebraic curve that is not contained in any plane. The irreducibility of this algebraic set needs a proof. A projective variety is a closed subvariety of a projective space. That is, it is the zero locus of a set of homogeneous polynomials that generate a prime ideal. The corresponding projective curve is called an elliptic curve. A plane projective curve is the zero locus of an irreducible homogeneous polynomial in three indeterminates.

Let V be a finite-dimensional vector space. V and the bracket means the line spanned by the nonzero vector w. An algebraic variety can be neither affine nor projective. It is an algebraic variety since it is a product of varieties. V is a variety if and only if its coordinate ring is an integral domain. The dimension of a variety may be defined in various equivalent ways.

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See Dimension of an algebraic variety for details. This section includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. The basic definitions and facts above enable one to do classical algebraic geometry. To be able to do more — for example, to deal with varieties over fields that are not algebraically closed — some foundational changes are required.

This also leads to difficulties since one can introduce somewhat pathological objects, e. Such objects are usually not considered varieties, and are eliminated by requiring the schemes underlying a variety to be separated. Some modern researchers also remove the restriction on a variety having integral domain affine charts, and when speaking of a variety only require that the affine charts have trivial nilradical. A complete variety is a variety such that any map from an open subset of a nonsingular curve into it can be extended uniquely to the whole curve. Every projective variety is complete, but not vice versa. These varieties have been called “varieties in the sense of Serre”, since Serre’s foundational paper FAC on sheaf cohomology was written for them. They remain typical objects to start studying in algebraic geometry, even if more general objects are also used in an auxiliary way.

R may not be integral domains. A more significant modification is to allow nilpotents in the sheaf of rings, that is, rings which are not reduced. Allowing nilpotent elements in rings is related to keeping track of “multiplicities” in algebraic geometry. There are further generalizations called algebraic spaces and stacks. An algebraic manifold is an algebraic variety that is also an m-dimensional manifold, and hence every sufficiently small local patch is isomorphic to km.

R, algebraic manifolds are called Nash manifolds. Algebraic manifolds can be defined as the zero set of a finite collection of analytic algebraic functions. Algebraic Geometry and Arithmetic Curves, p. On the imbedding problem of abstract varieties in projective varieties”, Memoirs of the College of Science, University of Kyoto. On the imbeddings of abstract surfaces in projective varieties”, Memoirs of the College of Science, University of Kyoto.

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Algebraic Geometry – A first course. Commutative Algebra with a View Toward Algebraic Geometry. Jump to navigation Jump to search This article is about vector spaces equipped with some kind of multiplication. The multiplication operation in an algebra may or may not be associative, leading to the notions of associative algebras and nonassociative algebras. An algebra is unital or unitary if it has an identity element with respect to the multiplication. The ring of real square matrices of order n forms a unital algebra since the identity matrix of order n is the identity element with respect to matrix multiplication.

Many authors use the term algebra to mean associative algebra, or unital associative algebra, or in some subjects such as algebraic geometry, unital associative commutative algebra. Replacing the field of scalars by a commutative ring leads to the more general notion of an algebra over a ring. The following statements are basic properties of the complex numbers. In other words, multiplying a complex number by the sum of two other complex numbers, is the same as multiplying by each number in the sum, and then adding. This shows that complex multiplication is compatible with the scalar multiplication by the real numbers.

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This example fits into the following definition by taking the field K to be the real numbers, and the vector space A to be the complex numbers. These three axioms are another way of saying that the binary operation is bilinear. An algebra over K is sometimes also called a K-algebra, and K is called the base field of A. The binary operation is often referred to as multiplication in A. Notice that when a binary operation on a vector space is commutative, as in the above example of the complex numbers, it is left distributive exactly when it is right distributive. The real numbers may be viewed as a one-dimensional vector space with a compatible multiplication, and hence a one-dimensional algebra over itself.

Likewise, as we saw above, the complex numbers form a two-dimensional vector space over the field of real numbers, and hence form a two dimensional algebra over the reals. The quaternions were soon followed by several other hypercomplex number systems, which were the early examples of algebras over a field. An example of a non-associative algebra is a three dimensional vector space equipped with the cross product. A K-algebra isomorphism is a bijective K-algebra homomorphism. For all practical purposes, isomorphic algebras differ only by notation.

A subalgebra of an algebra over a field K is a linear subspace that has the property that the product of any two of its elements is again in the subspace. In other words, a subalgebra of an algebra is a subset of elements that is closed under addition, multiplication, and scalar multiplication. In the above example of the complex numbers viewed as a two-dimensional algebra over the real numbers, the one-dimensional real line is a subalgebra. A left ideal of a K-algebra is a linear subspace that has the property that any element of the subspace multiplied on the left by any element of the algebra produces an element of the subspace. In symbols, we say that a subset L of a K-algebra A is a left ideal if for every x and y in L, z in A and c in K, we have the following three statements.