What is a real algebraic variety?
A real algebraic variety is said to be non-singular if X is non-singular. In such a case A is a smooth variety, and its dimension dimA is equal to the dimension of the complex variety CA=X(C); the latter is known as the complexification of the variety A.
Is Z an affine variety?
We call an affine variety a Zariski closed set. The complement of a Zariski closed set is a Zariski open set. The Zariski topology on An is the topology whose closed sets are the affine varieties in An. The Zariski closure of a subset Z ⊂ An is the smallest variety containing Z, which is V(I(Z)), by Lemma 1.2.
What is a variety in algebraic geometry?
A variety is the set of common zeros to a collection of polynomials. In classical algebraic geometry, the polynomials have complex numbers for coefficients. Because of the fundamental theorem of algebra, such polynomials always have zeros.
Are varieties irreducible?
Definition An affine variety is reducible if it is the union of proper subvarieties . Otherwise, is irreducible. That is, an affine variety is irreducible if whenever is written in the form , where and are affine varieties, then either or .
What is the difference between Boolean algebra and real algebra?
Difference between Boolean Algebra and ordinary algebra In Boolean algebra, they take two values, i.e. 0 and 1. 2. The values assigned to a variable have a numerical significance in ordinary algebra, whereas in Boolean algebra they have a logical significance.
Is every manifold an algebraic variety?
A quick Google search turns up a result that seems to say every smooth compact manifold is algebraic. If you are French, a variety is a manifold. If not, the connection is a bit more subtle. The correct analogue of a manifold is a scheme of finite type over a field.
What are coordinate rings?
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What do you mean by polynomial rings?
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, often a field.
Are varieties schemes?
A variety is a scheme X over k such that X is integral and the structure morphism X \to \mathop{\mathrm{Spec}}(k) is separated and of finite type. This definition has the following drawback. Suppose that k \subset k’ is an extension of fields.
Is an affine variety irreducible?
Affine variety It is irreducible, as it cannot be written as the union of two proper algebraic subsets.
What is a zero locus?
The zero locus or vanishing locus of a function is the set of points where it is vanishes, in that it takes the value zero.
What is difference between binary and Boolean?
Key Difference: In the field of computers and electronics, Boolean refers to a data type that has two possible values representing true and false. Binary in mathematics and computers, refers to a base 2 numerical notation. It consists of two values 0 and 1.
What is an affine variety over a closed field?
In algebraic geometry, an affine variety, or affine algebraic variety, over a algebraically closed field k is the zero-locus in of some finite family of polynomials of n variables with coefficients in k that generate a prime ideal. If the condition of generating a prime ideal is removed,…
What do you call a quasi affine variety?
A Zariski open subvariety of an affine variety is called a quasi-affine variety . Some texts do not require a prime ideal, and call irreducible an algebraic variety defined by a prime ideal.
Which is the best antonym for the word complex?
Antonyms for complex. simple. single. uniform. unmixed. apparent. clear. direct. discernible.
Is the affine variety defined by a prime ideal?
If the condition of generating a prime ideal is removed, such a set is called an (affine) algebraic set. A Zariski open subvariety of an affine variety is called a quasi-affine variety . Some texts do not require a prime ideal, and call irreducible an algebraic variety defined by a prime ideal.