Algebra is written geometry: geometry is drawn algebra. Sophie Germain.
Commentary on Algebraic Geometry, Robin Hartstone pages 1 to 3.
A field (i.e. displays additive and multiplicative associativity, commutativity, distributivity, identity and inverses) is algebraically closed if every polynomial with coefficients in has a root in .
Note: a field of real numbers is not algebraically closed as there is a uni-variate polynomial e.g. that does not have a solution in the real numbers.
An affine -space over , is a set of all -tuples of elements of . An element of affine -space, is called a point and for , then are called the coordinates of .
Let be the polynomial ring in variables over . The elements of can be interpreted as functions from the affine -space to by defining and . If is a polynomial the set of zeros of f are . Generally, if is any irreducible subset of the zero set of is defined as the common zeros of all the elements of , and called an algebraic set (if is not irreducible the zero set is called an algebraic variety) i.e. So an affine algebraic variety is the set of common zeros of a collection of polynomials.
Ideals are subsets of rings i.e. in the integers even numbers are an ideal as it is an additive subgroup of the ring of integers that absorbs multiplication by elements of the ring, i.e. an ideal in an ring is closed under addition and multiplication with arbitrary ring elements,
An ideal of a ring is called principal if there is an element such that . The ideal is generated by . e.g the ideals of the ring of integers , are all principal as are all the ideals of . We note that a function has the same zero set as its powers, so e.g. although the ideals are, of course, different. If is and ideal of generated by , then .
A ring of integers and the polynomial ring over a field are both Noetherian. They satisfy the ascending chain condition on ideals, for there exists an such that i.e. the ascending chain of ideals terminates. is a Noetherian ring, is a finite set of generators . Thus can be expressed as the common zero’s of the finite set of polynomials .
Definition 1.1 A subset of is an algebraic set if there exits a subset such that .
Proposition 1.1 The union of two algebraic sets is an algebraic set. The intersection of any family of algebraic sets is an algebraic set. The empty set and the whole space are algebraic sets.
Definition 1.2 A set , along with a collection of subsets of it, is a topology if the subsets in obey, 1) the trivial subset and the empty set are in , 2) whenever sets and are in , the so is , 3) whenever two or more sets are in , then so is . A set for which a topology has been specified is called a topological space. The subsets of that belong to are called the open sets of . If is an open set then its complement ( is called a closed set. Note we can define a topology on a set by listing all the closed sets, and taking the open sets to be all their complements. The null set is neither open nor closed.
Example 1.1 with subsets comprise a topology and is a topological space.
Definition 1.3 We define the Zariski topology on to be the topology whose closed sets are all the algebraic sets in . Moreover, any subset of will have the topology induced by the Zariski topology on . This will be called the Zariski topology on . The Zariski topology is the standard topology in algebraic geometry.
Example 1.2 The Zariski topology of the affine line . Here the point and with . Let be the polynomial ring in one variable over . and . Every ideal in is a principle. Every algebraic set is the set of simple zeros of a single monic polynomial of the form , with all . Then . Thus the algebraic sets in are the finite subsets (with the empty set) and whole space (). The open sets are the empty sets and the complements of the finite subsets.
Definition 1.3 A nonempty subset of a topological space is irreducible if it cannot be expressed as the union of two proper subsets each of which is closed in . The empty set is not irreducible.
Definition 1.4 Space is disconnected if it is the union of closed subsets with the intersections between them being empty. Otherwise the space is called connected.
Example 1.3 is irreducible.
Definition 1.4 An affine algebraic variety is an irreducible closed subset of . An open subset of the affine algebraic variety is a quasi-affine variety. Using the above definitions we can establish the connection between geometry and algebra by studying the connections between the algebraic sets in and the ideals in . The operation takes a subset of or an ideal to an algebraic set. With the following definition we establish a two way correspondence.
Definition 1.5 For a subset we call the ideal Hence we have defined the following two way correspondence; we have an operation which maps ideals, subsets, of the polynomial ring in variables over to algebraic sets in and an operation which maps algebraic sets in to ideals in the polynomial ring in variables over .