By Walker R. J.

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8 Projective Varieties Kepler in 1604 and Desargues in his book in 1639 realized that the introduction of imaginary points could substantially simplify Euclidean geometry. Each line in the plane was given one new point, "at infinity," in such a way that two parallel lines would meet at their points at infinity. In this way many geometric results became simpler (for example, the statement that every two distinct lines meet in exactly one point), and a remarkable duality between points and lines was introduced.

0), (0,1,0, ... ,0), ... , (0, ... ,0,1). If M is a finitely generated free module, that is M ~ Rn for some n, then the number n is an invariant of M (in the case when R is a field this is just the dimension of M as a vector space). It is called the rank of M. 5. If A, B, and Care R-modules, and a : A -+ B, {3 : B -+ C are homomorphisms, then a pair of homomorphisms is exact if the image of a is equal to ker {3, the kernel of {3. In general, a sequence of maps between modules like is exact if each pair of consecutive maps is exact.

8, reduced finitely generated kalgebras are often called affine k-algebras, or, when it is not necessary to refer explicitly to the field k, simply affine rings. 8 we must choose a set of k-algebra generators for A(X). But it turns out that X is in a certain sense independent of this choice. First, note that for any field the ideal of polynomials in k[XI, ... ,xnl vanishing at the point p = (al, ... ,an) E An is tllp := (Xl - al,"" Xn - an). ) The Nullstellensatz shows that every maximal ideal has this form.

### Algebraic Curves by Walker R. J.

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