Januray 31: The first lecture was an informal introduction to the subject. Namely, we introduced polynomial mappings between R^n and R^m, or C^n and C^m, or k^n and k^m for abitrary field k. For a polynomials map f: R^n -> R^m the set V(f)={x \in R^n | f(x) =0} is called an affine variety. They are more complicated subsets then linear subspaces of R^n. The empty set is an example of an affine variety. This lecture is based on the Introduction in the textbook.

February 2. An affine variety in R^n is the same as the zero set of some finite system of multivariate polynomials.

We showed by an example that the image of an affine variety V(\Phi) \subset R^a under a polynomial map R^a -> R^b may not be an affine variety in R^b (Here \Phi is some polynomial map from R^a to some affine space R^c). On the other hand, the preimage of an affine variety is always an affine variety. The later statement essentially boils down to the statement that the composite of two polynomial maps is again a polynomial map (substituting polynomials for the variables in a multivariate polynomial produces a polynomial).

We introduced the notion of a  rational map R^n -> R^m. This is not a map in the convential sense in that it is not defined everywhere. More precisely, if we are given polynomials F_1,...,F_m,G_1,...,G_m \in R[X_1,...,X_n] then the assignment

\xi  -> (F_1(\xi ) / G_1(\xi) ,..., F_m(\xi) / G _m(\xi))

defines an usual map from  (R^n minus the solution set to the system G_1=0,...,G_m=0) to R^m. Such maps are called "rational maps from R^n to R^m". Introduction in the textbook.

February 4. We constructed a rational one-to-one map f : R -> R^2 whose image is all of the unit circle except one point. The contruction yields a parametrization of the unit circle and also a method of finding all rational points on the uinit circle. This is the same as fining all integer solutions to the equation X^2+Y^2=Z^2. We said that a parametrization also exists for the curve in R^2 whose equation is 2X^2+Y^2=5. But, contrary to the previous case, the latter curve contains NO rational point. Then we mentioned briefly the standard classification of quadrics in R^2 - later we will retrurn to this topic in more detail. Pages 9 - 11.

February 7. We introduced the real projective plane P^2_R. It can be defined as either the set of all lines in R^3 containing the origin 0 or the equivalence classes of the triples (x,y,z) of real numbers not all of them simultaneously equal to 0, where we declare two triples (x,y,z) and (x',y',z') equivalent if there is a nonzero real number k such that (x,y,z)=(kx',ky',kz'). The equivalence class of a triple (x,y,z), i.e. the element of P^2_R that contains this triple, is denoted by (x : y : z). There is a natural embedding of the real affine plane R^2 into the real projective plane P^2_R. Namely, we mean the map (x,y) ->(x : y : 1) . In particular, one can think of the real projectuve plane as an extension of the real affine plane. Then we discussed the group of affine transformatins of R^2. A map R^2 -> R^2 is an affine transformation if it is one-to-one and can be obtained by composing a linear automorphism (i.e. non-degenerate linear map) R^2 -> R^2 with a subsequent parallel translation R^2 -> R^2. Pages 11 - 12.

February 9. We introduced projective transformations (also called "projectivities") of P^2_R. A projectivity is a special transformation of the real projective plane, associated in a natural way to a linear transformation of the underlying R^3. The restriction of a such projectivity to R^2 - the distinguished affine real plane in  P^2_R - is a rational map not defined on certain line (called the "horizon" in painting). Page 13.

February 11. We introduced homogeneous polynomials and described the homogenization and dehomogenization processes: one derives homogeneous polynomials f^hom(X,Y,Z), not divisible by Z, out of polynomials f(X,Y) in two variables; the dehomgenization process does the opposite - it derives a polynomial in two variables f^dehom(X,Y) out of a homogeneous polynomial f(X,Y,Z), not divisible by Z. The geometry, going on behind this algebraic processes, is as follows: to a system of polynomials f_1,...,f_k\in R[X,Y] we associate the zero set V(f_1,...,f_k)\subset R^2 and to a system of homogeneous polynomials g_1,...,g_k\in R[X,Y,Z] we associate a subset of the real projective plane P^2_R. The latter is simply the set of those lines through the origin whose union is the solution set
in R^3 of the system g_1=...=g_k=0 in R^3. The mentioned subset of P^2_R is denoted by \bar V(g_1,...,g_k) and is called the zero set of the polynomials g_1,..,g_k in P^2_R. Now if start from polynomials f_1,...,f_k\in R[X,Y] then V(f_1,...,f_k) defines a set of lines in R^3 in the usual way: to a point (x,y) \in V(f_1,...,f_k) one associates the line (x:y:1). The set of lines we get is the zero set of f^hom_1,...,f^hom_k in P^2_R (actually one may mis a few points, but was is said is essentially correct). On the other hand, if we start from a system of homogeneous polynomials g_1,...,g_k \in R[X,Y,Z], not divisible by Z then \bar V(g_1,...,g_k) \subset P^2_R is a family of lines in R^3. We intersect every element of this family with the plane {(x,y,1) | x,y \in R}. What we get is the set {(x,y,1) | (x,y) \in V(g^dehom_1,...,g^dehom_k).

February 14. We said that any quadratic form in two variables can be reduced by a linear change of variables to one of the 5 quadratic forms, listed on p. 15. A quadratic form is called non-degenerate if it reduces to X^2+Y^2-Z^2. The latter can be reduced to XZ=Y^2 by a further linear change of variables and the zero set in P^2_R of the latter polynomial, i.e. \bar (XZ=Y^2)\subset P^2_R, is exactly the subset {(u^2 : uv : v^2) | u,v \in R not simultaneously =0}\subset P^2_R. Pages13 - 16.

February 16. Let F be a non-degenerate quadratic form in three variables X, Y  and Z. All the previous results on the quadric \bar(F) in the real projective plane P^2_R can be summarized by the following commutative diagram of maps:

V(F^dehom)      \subset        R^2
/
/           |                                         |
/             |                                         |  (x,z) --> (x : 1 : z)
/               |                                         |
/                 |                                         |
/                  V                                       V
f
\bar V(F)             \subset         P^2_R
/            /
/          /
/        /                  | bijection                           | projectivity
/      /                      |                                         |
/    /                         V                                       V
/  /
R--h--> P^1_R -g--> \bar (V(XZ-Y^2) \ subset  P^2_R

where   h(t) = (t : 1),  g(u : v) = (u^2 : uv : v^2) (notice g is bijective), and  f :  R --> R^2 is a rational map  - it is in general a partial map, not defined on all of R. This gives a rational parametrization of the affine variaties in R^2 of the form V(G), where G is the dehomogenization  of a non-degenerate quadratic form in three variables.

In classical analytic geometry you may heard of the following fact, which is equivalent to what is said above: all nondegenerate conic sections in R^3 (which are just ellipses, hyperbolas, parabolas) can be parametrized by rational functions of one real variable.

February 18. We proved that for a non-denerate quadratic form F(X,Y,Z) and a homogeneous form G_d(X,Y,Z) of degree d, such that neither of \bar V(F) \subset P^2_R and \bar V(G_d) \subset P_2_R is a subset of the other, the number of points in the intersection of  \bar V(F) and \bar V(G_d) is at most 2d. This is a special case of the s.c. Bezout Theorem, which will be discussed in the next lecture. The mentioned fact implies that for any 5 pairwise distinct points in P^2_R there is at most one nondegenerate quadratic form F(X,Y,Z) such that all these 5 points are in \bar V(F). On the other hand, we observed that one can arrange 4 pairwise different points in P^2_R and inifinnitely many nondegenerate quadratic forms in 3 variables the zero set of each of which contains these 4 points. Pages 16 - 18.

February 21. Bezout Theorem says that two differents curves in P^2_C, i. e. the solution sets \bar V(f) and \bar V(g) for some homogeneous polynomials f,g \in C[X,Y,Z], intersect in exactly deg(f) * deg(g) points where the intersectrion points are computed with multilicities. We explained by examples why one needs to work in the projective plane instead of the affine plane, why one needs to count the points of intersection with multiplicites and, finally, why it is necessary to work over C and not over R. The Bezout theorem is only mentioned in the first paragraph of Section 1.9 (page 17) and the text descusses in detail only some easy cases of the general statement. (One of these cases, when deg(f)=2, f is a non-degenrate quadratic form, and the equality is changed to the "< =" inequality, was discussed in the previous lecture.)

February 23. We stated the theorem on the impossibility of parametrizing by rational functions the set V(y^2-x(x-1)(x-\lambda x)) \subset R^2 where \lambda is not 1 or 0. We also recalled the concept of a UFD and that k[t] is a UFD for a field k. As a corollary, if f and g are coprime polynomials in k[t] and g divides the product f*h for some h \in k[t] then g divides h. As another corollary, if a field k is closed under taking square roots and f,g \in k[t] are two coprime polynomials then the product f*g is a square in k[t] if and only if both f and g are squares in k[t]. That's all you need to understand the proof of the theorem and the related lemma on p. 28 - 29. This material is covered in teh course Math 335.

February 28. We introduced a group structure on an irreducible smooth cubic in P^2_R. In other words, there is a group structure, considered by Henri Poincare more than a centrury ago, on \bar V(F) \subset P^2_R where F(X,Y,Z) is a homogeneous irreducible cubic polynomial such that for any point p\in \bar (F) there is a tangent line L=\bar V(aX+bY+cZ) \subset P^2_R to \bar V(F) at p. Any such a cubic can be projectively transformed to a cubic whose dehomogenization w.r.t. Z is Y^2 - X^3 - aX - b. After such a linear change of coordinates the set \bar V(F) becomes the curve V(
Y^2 - X^3 - aX - b) \subset R^2 and exactly one point at infinity, namely the point O \in P^2_R \ R^2 which determined by the condition that all lines through it are represented by vertical straight lines in R^2. Then we have the following nice description of the aforementioned group structure:
(1) O (not to be confused with (0,0) \in R^2) is the zero element,
(2) for any point (x,y) on
V(Y^2 - X^3 - aX - b) \subset R^2 its opposite (w.r.t. to the group structure) is the point (x,-y),
(3) A + B + C = O for A, B, C \in
V(Y^2 - X^3 - aX - b) if and only if these 3 points lie on one line.
Section 2.8, page 33; Section 2.13. pages 39 - 40.

March 2. First we recalled the Fundamental Theorem on the structure of finitely generated abelian groups, and the notions of the rank and the torsion part of such group. We then stated Mordell's 1929 theorem that the group of rational points on a smooth irreduicible cubic in the projective plane is a finitely generated abelian group. This had been conjectured by Poincare at the beginning of the 20th century. We also stated B. Mazur's theorem (the 1970s) that the torsion part of the mentioned group of rational points has order at most 12. On the other hand it is still a big open question whether the rank of the mentioned group can be arbitrarily large. A part of this information is depicted on page 46. We will elaborate on this diagram/picture on Friday, March 4.

March 4. We elaborated on the information on page 46: namely, we descussed the relationship between the genus of all complex solutions
\bar V(F) \subset P^2_C and the rational points \bar V(F)(Q), where F is an irreducible smooth cubic with rational coefficients. This includes such milestone results as Faltings' and Wiles' theorems, saying respectively that if def(F)>3 then there are only finitely many rational solutions and, in the special case F=X^n+Y^n-Z^n, there are no nontrivial integer solutions (Fermat's Last Theorem). Such an achievement was made possible after Alexandre Grothendieck's revolutionary advancement of Algebraic Geometry (in parallel with Jean-Pierre Serre's contribution) in the 50s and 60s of the last century.

This completes the first part of the course.

March 7. We recalled the basics from Math 335 - a prerequisite course for MAth 850 - on rings theory: the definition of a (commutative) ring, an integral domain, a field, as well as the efinition of a ring homomorphism.We also recalled several elementary facts: a field is a domain, a finite domain is a field. Then we discussed finite rings of type Z/mZ - it is a domain iff it is a field iff m is prime. This material is given in any text-book used in any course similar to Math 335 here. The books used at SFSU during 2004 srping and fall were S. Lang's "Undergraduate Algebra" and J. Gallian's "Contemporary Abstract Algebra". Finally, we introduced a new terminology: an algebra R over a field k. This simply means a ring R together with an injective ring homorphism from a field k. Such a homomorphism can be also called a k-algebra structure on R. In this situation R is called an k-algebra. A k-algebra homomorphism of two k-algebras R and S is a ring homomorphism \phi : R ---> S which respects the given embeddings of k into R and  S. In other words, the composite k ---> R --\phi--> S, where k ---> R is the k-algebra structure on R, is the k-algebra structure on S.

March 9. We introduced the notion of a generating set of a k-algebra R where k a field. If a k-algebra R has a finite generating set then R is said to be a finitely generated k-algebra. Theorem: a k-algebra R is finitely generated if and only if there is a surjective k-algebra homomorphism k[X_1,...,X_n] --> R for some natural n. Examples: the field of complex numbers is a finitely generated algebra over the reals, but the field of real numbers is NOT a finitely generated Q-algebra. (Again, all of this is found in almost all of modern algebra text-books.)

March 11. We recalled from Math 335 the notion of (i) an ideal I \subset R, (ii) an ideal I \subset R, generated by a subset a_1,...,a_n \in R, (ii) the set of cosets of an ideal I \subset R. The lattrer is denoted by R/I. A coset is a subset of the form \bar r = r + I \subset R. Two cosets either don't intersect or are disjoint. For a ring homomorphism f : R --> S its kernel Ker(f) is an ideal in R. Moreover, f is injective if and only of Ker(f)=0. Any text book for Math 335.

March 14. For a proper ideal I \subset R the set of cosets R/I carries a ring structure, defined as follows: \bar r + \bar s = \bar (r + s) and \bar r * \bar s = \bar ( r * s). The canonical map R --> R/I, given by r --> \bar r is a ring homomorphism whose kernel is the ideal I. Therefore, any proper ideal in a ring  is just the kernal of some ring homomorphism from this ring. The Isomorphism Theorem says that for a ring homomorphism f : R --> S we have the natural isomorphism R/Ker(f) --> Im(f), given by \bar r --> f(r). This gives us a possibility to visualize a quotient ring of the form R/I, defined in the abstract terms of cosets, as the image of some ring homomorphism f : R --> S whose kernel is I. An example (worked out in class): let k be a field and a \ in k; then k[X] / (X-a) and k are isomorphic as k-algebras - this isomorphism is given by \bar f(X) --> f(a) for f(X) \in k[X]. Any textbook for Math 335.

March 16. We introduced noetherian rings: they are rings in which all ideals are finitely generated. Hilbert's Theorem on Basis says that if R is a noetherian ring then R[X] is also noetherian. Since any field k and the ring of integers Z are noetherian (even more is true: their ideals are generated by one element), Hilbert's theorem implies that the multivariate polynomial rings k[X_1,...,X_n] and Z[X_1,...,X_n] are also noetherian. This follows from the general fact that for any ring R the polynomial ring  R[X_1,...,X_n] can be iteratively defined as R[X_1][X_2]...[X_n] and, therefore, at each steps hilbert's theorem applies. The image of a noetherian ring under a ring homomorphism is again noetherian. Therefore, for any field k any finitely generated k-algebra is noetherian. In fact, in one of the previous lectures we stated that a k-algebra is finitely generated if and onlyt if it is a surjective image of a polynomial ring k[X_1,....,X_n].  However, a k-algebra need not be finitely generated to be noetherian - so the converse result does not hold. Example: the field of reals R viewed as Q-algebra. Pages 48, 49. No proofs of these statements (given on these pages) were discussed. This will be done in the next lecture.

March 18. We discussed the idea behind the proof of Hilbert's Theorem on Basis.
Pages 48, 49.

March 28. We discussed the correspondence V from ideals in k[X_1,...,X_n] to algebraic sets in the affine space A_k^n. Namely, V(I) is the solution set to the polynomials in an ideal I. By Hilbert Theorem on Basis, all algebraic sets can be given by finitely many polynomial equalities. We then stated the basic properties of the V-correspondence:

(a) if I \subset J then V(I) \overset V(J),
(b) V(I) \cup V(J) = V(I \cap J),
(c) V(\sum I_t)= \cap_t V(I_t),
(d) \emptyset = V(k[X_1,...,X_n]) and A_k^n=V(0).

Page 50, until the discussion on Zariski topology.

March 30. We recalled the notion of a topological space, defined in terms of open sets or in terms of closed sets. Standard examples are Euclidean spaces - R^n with the usual topology, induced by the Euclidean metric. Not so standrad examples are discrete and antidiscrete spaces. For any field k the affine space A_k^n carries the s. c. Zairiski topology. In this topology, the closed sets are exactly the solution sets of systems of polynlmials eaquations, i. e. sets of the form V(I) where I is an ideal in k[X_1,...,X_n]. In the case k=R we get two essentially different topologies on the same set R^n. For instance, when n=1, then Zariski closed subsets of R are just the empty set, R itself, and the finite subsets of R, whereas the Euclidean open sets of R are the sets containing all there adherent points (an closed interval is an example). P. 50 and the beginning of p. 51.

April 1. We considered the correspondence I from subsets of A^n_k to those of k[X_1,...,X_n]. Namely, I(X) is the set of polynomials that vanish on all points of X. This is always an ideal in k[X_1,...,X_n]. The correspondences V and I satisfy the following conditions: X \ subset V(I(X)) for any X \subset A^n_k and S \subset I(V(S)) for any S \subset k[X_1,...,X_n]. We also "computed" I(X) in a few special cases: I(\emptyset) = k[X_1,...,X_n],  I(A^n_k) = {0} and, also, for any point (a_1,...,a_n) \ in A^n_k we have  I{(a_,...,a_n)}=(X_1-a_1,...,X_n-a_n) - the ideal generated by X_1-a_1,...,X_n-a_n. Pages 51, 52

April 4. We introduced affine varieties - these are the algebraic sets in A^n that are not unions of two (equivalently, finitely many) smaller algebraic sets. Then we proved that X \subset A^n is an affine variety if and only if the ideal I(X) \subset k[X_1,...,X_n] is prime. Prime ideals were introduced in math 335: an ideal I in a ring R is called prime if ab \in I implies a \in I or b \in I; one proves that an ideal I \subset R is prime if and only if the quotient ring R/I is an integral domain. Pages 52,53. Only the (a)-part of the propositon there.

April 6. Hilbert Nullstellensatz can be stated as follows. For an algebraically closed field k we have the following  diagram of vertical inclusions and horizontal mutually inverse I and V correspondences (in all three levels):

V
radical ideals     =      intersection of finitely    ---->       algebraic sets in A^n,

many prime ideals        <----       same as Zariski closed sets in A^n
I
U                                              U

V
prime ideals               ---->    affine varieties in A^n,
<----      same as irreducible algebraic sets
I
U                                               U

V
ideals of the form   =      maximal ideals      ---->       points in A^n
(X_1-a_1,...,X_n-a_n)                              <----
I

where a "radical ideal" means an ideal satisfying the condition (a^m for some ring element a and natural number m) ==> (a is in the ideal), and a "maximal ideal" means a proper ideal which is not contained in a bigger and proper ideal. Pages 54, 55. Our diagram above is slighly more detailed then the one given on page 55. We will elaborate further on this diagarm, explaining its various aspects. We will prove only a small part of the statement.

April 8. During the coming several lectures we are going to elaborate on various aspects of the diagram above (from the lecture on April 6). Today we proved that any algebraically closed set X \subset A^n_k is the union of finitely many affine varieties. Moreover, if there is no containment between these affine varieties allowed, then such a representation is unique (up to permutation). These affine varieties are called the irreducible components of X. Proposition (b) and its proof on pages 52,53.

April 11. Today we proved the Nullstellensatz (3.10), parts (a) and (b), based on (what is called in the text) Hard Fact. Page 54,55.

April 13. We elaborated on the `Hard Fact' on p. 54 that is used in the proof of the Hilbert N.S. Namely, we showed on examples that neirther the finite generation condition over k nor the condition of being a field can be dropped on the given k-algebra A to guarantee that ever element a \in A is algebraic over k. On the other hand, there are examples of algebraic extensions k \subset A where A are fields, but not finitely generated as k-algebras. As a classical example, we have Q \subset \bar Q where \bar Q is the set of all complex numbers which are algebraic over Q. This is a field, called the algebraic closure of Q. We then elaborated on the group of automtorphisms of \bar Q, which is called the Absolute Galois Group and whose structure is one of the big challenges in mathematics. It is a topological group in some natural way and as such is generated by two elements - this is not the same algebraic generation by two elements. The famous Inverse Galois Problem asks whether all finite groups can be surjective group homomorphic images of the absolute Galois group. The latter question reduces to the special case of finite simple groups. This question has been answered in the positive for great many examples of finite groups, but the current state of art is far from beeing complete. This was a short glimpse into Galois theory which some day may be the main topic of a Math 850 course at SFSU.

April 15. We proved that for any commutative ring R and its any proper ideal I the intersection of all prime ideals containing I is exactly the radical of I, i.e. the set of elements of R whose some powers belong to I. (The radical of I is itself an ideal in R.) In the case when R is noetherian, the radical of I can be represented as the intersection of finitely many prime ideals containing I. The geometric version of the argument presented in the lecture is given by the proof of the implication (b) ==> (c) on pp. 55 - 56.

April 18. For any ring R we let Ideals(R), Rad.Ideals(R), spec(R) and max(R) repsectively denote the set of all ideals, radical ideals, prime ideals, and maximal ideals in R. Let J \subset R be an ideal then we have the following bijective correspondences:

Ideals (R/J)    <-----> {I \in Ideals(R) | J \subset I}
Rad.Ideals (R/J)    <-----> {I \in Rad.Ideals(R) | J \subset I}
spec (R/J)    <-----> {I \in spec(R) | J \subset I}
max (R/J)    <-----> {I \in max(R) | J \subset I}

defined by

\alpha ---> \pi^{-1}(\alpha) for an ideal \alpha \subset R/J

and

I ---> \pi(I)

where \pi : R ---> R/I refers to the canonical map (that is, \pi(r) = \bar(r), the residue class of r modulo J).

Assume R is an integral domain and S \subset R is a non-empty subset, not containing 0. Then we have the following bijective correspondence

spec(R[S^{-1}]  <----> {P \in spec(R) | P does not intersect S}

defined by

p ---> the intersection of p with R, where p \subset R[S^{-1}] is a prime ideal

and

P ---> the ideal P generates in the bigger ring R[S^{-1}], where P \in spec(R) is such that P does not intersect S.

Observe, we do not state a similar claim for Ideals(R), Rad.Ideals(R), and max(R). Try to prove these claims. The proof should represent a routine check that the mentioned maps are well defined and are mututally inverse.

April 20. For any ring R we considered the sets
Ideals(R), Rad.Ideals(R), spec(R) and max(R) as partially ordered sets w.r.t. to inclussion. The set Ideals(R) then has the property that for any two elements I_1, I_2 \in Ideals(R) there is the biggest lower bound (which is I_1 \cap I_2) and the smallest upper bound (which is I_1+I_2).  This is no longer true for the smaller sets spec(R) and max(R).

April 21. We introduced the (Krull) dimension of a commuttive ring - the suprerum of the lengthes of ascending chains of prime ideals. When R is noetherian, then dim R[X_1,...,X_n]=n + dim R (Krull-Serre Theorem). We also introduced the notion of the height ht(p) of a prime ideal in a commutative ring -
the suprerum of the lengthes of ascending chains of prime ideals whose last term is the given prime ideal. For a finitely generated algebra A over any field k and any prime ideal P \subset A we have the following theorem: dim(A/P) + ht P = dim A. In general rings one only has the "< or =" inequality.

April 22 - 29: Notes for Dr. Meredith three lectures (click on the link)

May 2. We gave a survey of the correspondence between k-algebras and algebraic subsets of affine spaces, including the relationship between k-algebra homomorphisms  of the coordiante rings and polynomial homomorphisms bewteen the coresponding algebraic subsets. (We will continue excploration of this correspondence - these are critical concluding topics of the course and you need to come to classes to shed much light in your understanding to what we have been up to during the semester)   Pages 66 - 70.

May 4. We discussed in more detail the correspondence between algebraic sets in affine spaces and finitely generated k-algebras, mentioned in the previous lecture. In particular, the two "worlds" - algenraic subsets of affine spaces and polynomial maps between them on one side and finitely generated k-algebras and k-algebra homomorphisms between them on the other side - are completely equivalent (we assume k is an algebraically closed field). Pages 66 - 70. (Have a look also at the notes of Dr. Meredith, posted above.)

May 6. We introduced the field of rational function k(V) on an affine variety V and the local ring
O_{V,P} of a point P in V, the same as the ring of regular functions at P. In particular, k(V) is the fraction field of the coordinate ring k[V] and the ring O_{V,P} is a subring of k(V). Moreover, O_{V,P} has only one maximal ideal, namey {f/g | f,g \in k[V], f(P)=0, g(P)\not=0}. Pages 71 - 72.

May 9. We proved that a rational function on an affine variety that is regular at every point of the variety is necessarily polynomial. That is, if f \in k(V) and dom(f) = V then f \in k[V]. Here k is assumed to be an algebraically closed field and the argument uses in an essential way the Hilbert Nullstellensatz stated for V. For not algebraically closed fields there are everywhere defined rational functions that are not polynomial. A simple example is given by 1/(X^2+1) on R. Page 72.

May 11. We proved that for an affine variety V and a polynomial function on it g \in k[V] the rational functions regular on V \ V(g) constitute the subalgebra k[V](1/g) of the coordiante ring k[V] (Page 72, 74). Since k[V](1/g) is a finitely generated k-algebra which is a domain, in view of the fundamental correspondence between algebraic sets and fin. gen. k-algebras, one can consider in a natural way the set V \ V(g) as an affine viriety on it own right.

May 13. On the open day we discussed concepts related to Zelmanov's Fong Symposium 2005 public lecture: the Burnside problems.

May 16. We introduced the notion of a morphism between two Zariski open sets of some affine varieties: it is simply a rational map (which is a partial map) between the ambient affine varieties that is defined everywhere on the source open set. This notion allows us to talk on isomorphisms between Zariski open sets in algebraic sets and affine verieties; an example - an affine line without the origin (a Zariski open subset of the line) is isomoprhic to the affine variety V(XY-1)\subset A. Page 74.

May 18 and May 20 will be review classes. We will essentially follow the notes above.