Math 418 Algebraic Geometry Notes


 Denis Perry
 2 years ago
 Views:
Transcription
1 Math 418 Algebraic Geometry Notes 1 Affine Schemes Let R be a commutative ring with 1. Definition 1.1. The prime spectrum of R, denoted Spec(R), is the set of prime ideals of the ring R. Spec(R) = {P R : P is prime}. Example 1.2. Let R = Z. Then Spec(Z) = {(p) : p Z is prime} {(0)}. Notice that the ideals (p) are maximal as well as prime. The ideal (0) is not maximal, and is contained in every other element (p) of Spec(Z). Example 1.3. Let R = F a field. Then Spec(F ) = {(0)}. Example 1.4. Let R = F [x], where F is a field. This ring is a principal ideal domain, and so its prime ideals are of the form (f(x)) where f(x) F [x] is irreducible. That is, Spec(F [x]) = {(f(x)) : f(x) F [x] is irreducible} {(0)}. The structure of Spec(F [x]) will depend on F. Example 1.5. Let R = Q[x]. Then we saw that Spec(Q[x]) = {(f(x)) : f(x) Q[x] is irreducible} {(0)}. For any q Q, the linear polynomial x q is irreducible, and so (x q) Spec(Q[x]). The polynomial x 2 +bx+c is irreducible whenever b 2 4c Q. In this case (x 2 +bx+c) Spec(Q[x]). There are polynomials of arbitrary degree in Spec(Q[x]). For example, (x n 2) Spec(Q[x]) for all n N. Example 1.6. Let R = C[x] (or F [x] where F is algebraically closed). In this case, the irreducible polynomials are exactly the linear polynomials. So Spec(C[x]) = {(x z) : z C} {(0)}. Note that the maximal ideals of C[x] are in onetoone correspondence with the elements of C. Example 1.7. Let R = C[x, y]. This ring is not a principal ideal domain, however it is Noetherian, and so its ideals are finitely generated. The maximal ideals of R are of the form (x z, y w) where z, w C, and are in bijection with the points of C 2. Other prime ideals are generated by sets of irreducible polynomials, for example (x y) and (x y 2 ) are prime. Of course, (0) is always a prime ideal in an integral domain. We would like to view X = Spec(R) as a geometric space. The points in our space will be the elements of Spec(R). We will distinguish the points corresponding to maximal ideals, and call them the closed points of X. In order to view X as a geometric object, we will need what is called a topology on X. In other words, we will need to have some concept of an open neighborhood of a point in X. 1
2 Definition 1.8. For any subset I R, the vanishing set of I is the set of prime ideals of R containing I, that is V (I) := {P Spec(R) : I P }. We consider these sets to be closed subsets of X. For any element f R, the complement X f := V ({f}) c Spec(R) is called the basic open set of f. A subset U X is said to be open if it is a union of basic open sets. The open and closed sets described above are called Zariski open and Zariski closed. structure they give on X is called the Zariski topology on X. The map V takes subsets of R to subsets of X. It is not, however, injective. Lemma 1.9. If S is a subset of R, and I(S) is the ideal generated by S in R, then V (S) = V (I(S)). Proof. This is homework problem AM 15i. Based on this result, if we want to understand open and closed subsets of X, we can restrict our attention to ideals of R. The following lists some handy properties of V. Lemma Let I, J be ideals of R. Then 1. V (I J) = V (I) V (J). 2. V (I J) = V (I) V (J). 3. If I J, then V (J) V (I). Proof. Suppose P is prime and contains I and J. If P contains neither I nor J, then there exist i I and j J, neither of which are in P. This contradicts that ij I J P. Therefore V (I J) V (I) V (J). If P contains I or J then it certainly contains I J, so V (I) V (J) V (I J). Suppose now P contains I J. Then P contains both I and J, so P lies in V (I) V (J) On the other hand, if P lies in V (I) V (J), it contains both I and J, so it lies in V (I J). The third property is left as an exercise. Note that although the second property above holds for general subsets of R, the first does not. For example, if we take R to be the integers, I to be the natural numbers and 0, and J to be the negative numbers and 0, then all prime ideals contain I J = {0}. But no proper ideals contain I or J. In fact, even if we restrict to ideals of R, the map V will not be injective. For example, if we consider R = C[x], then V ((x)) = V ((x 2 )). This is because any prime ideal containing x 2 contains the product x x, and hence contains x itself. In fact, we could show by induction that V ((x)) = V ((x n )) for any positive integer n. In order to remedy this problem, we will need the following notion. Definition Let I be an ideal of R. The radical of I is r(i) := {r R : r m I for some k N}. As a special case, the nilradical of R is r((0)), in other words, it is the set of nilpotent elements of R. The 2
3 Note that r(r(i)) = r(i) for every ideal I of R, so we may call an ideal whose radical is itself a radical ideal. Lemma For any ideal I of R, V (r(i)) = V (I). Proof. This proof is homework problem AM 15i. There is another characterization of radical ideals which will be useful to us. Proposition If I is an ideal in R, then r(i) is the intersection of all prime ideals containing I. In particular, the nilradical of R is the intersection of all prime ideals of R. Proof. First, if x r(i) then x m I for some positive integer m. Hence x lies in every prime ideal containing I. This shows one containment. Now we must show that the intersection of all prime ideals containing I lies in r(i). Suppose first the I = (0), and r(i) is the nilradical of R. We wish to show that the intersection of all prime ideals in R contains only nilpotent elements. Suppose then that f is not nilpotent. Consider the set of ideals Σ which contain no power of f. Since f is not nilpotent, the 0 ideal is in Σ. Since Σ is not empty, by Zorn s Lemma it contains a maximal element under inclusion, call it M. If x, y M then the ideal (x) + M contains M, so f m (x) + M for some positive integer m. Similarly, f n (y) + M for some positive integer n. Therefore f n+m (xy) + M, and xy M. This shows M is prime, so f is not in the nilradical of R. Now for a general ideal I, we can apply the previous argument to the ring R/I to get the result. Corollary If I and J are ideals of R such that V (I) = V (J), then r(i) = r(j). In particular, the map V is an injective map from the set of radical ideals of R to subsets of X. Definition The space X = Spec(R) together with the Zariski topology is called an affine scheme. We may now wish to consider maps between different affine schemes. There is a close relationship between such morphisms and ring homomorphisms. Definition Let X = Spec(R) and Y = Spec(S). A morphism of affine schemes f : X Y is a map of sets given by a ring homomorphism f : S R as follows: For any prime ideal P X, f(p ) := (f ) 1 (P ). The map f is called the pullback map of f. Note that this map is well defined, since if xy (f ) 1 (P ), then f (xy) P, hence at least one of (f ) 1 (x), (f ) 1 (y) lies in (f ) 1 (P ). It is important to note, however, that if Q Y is prime, f (Q) need not be prime. Example Suppose we start with the quotient map f : C[x, y] C[x, y]/(xy). We can write any ideal of C[x, y]/(xy) as I + (xy) for an ideal I of C[x, y] containing (xy). Therefore, the map f is simply the inclusion map of prime ideals of C[x, y] containing (xy) back into C[x, y]. Note that the maximal ideals containing (xy) are those of the form (x, y a) and (x b, y) for some a, b C. This example shows that in a sense, we can think of the scheme Spec(C[x, y]/(xy)) as a subscheme of Spec(C)[x, y]. We will now make this precise. 3
4 Definition A subscheme of an affine scheme X = Spec(R) is a scheme Y = Spec(S) such that there is a ring homomorphism i : R S which induces the inclusion i: Y X. Lemma Let I be an ideal of R. i: Spec(R/I) Spec(R). The quotient map i : R R/I induces an injection This shows that there is a correspondence between ideals of R and subschemes of Spec(R). However, this correspondence is not onetoone. As we ve seen, the prime ideals containing one ideal may be the same as the prime ideals containing another. The following proposition gives the desired onetoone correspondence. Proposition The subschemes of X = Spec(R) are in one to one correspondence with the radical ideals of R. 2 Polynomial Rings Let k be an algebraically closed field, such as C. We will now consider the special case that R = k[x 1,..., x n ]. In this case, Spec(R) is sometimes referred to as affine nspace, and is often denoted A n k. This case has many advantages. One is that R is a finitely generated kalgebra. In particular, it is Noetherian. Therefore the ideals we wish to study will always be finitely generated. Further, techniques exist to compute the radical of an ideal in this setting. In particular, this is where the study of Gröbner bases intersects algebraic geometry. In order to study this specific case, we first want to understand the closed points of A n k. This result is foundational to the field. Theorem 2.1 (Hilbert s Nullstellensatz). The maximal ideals of R are the ideals (x 1 a 1,..., x n a n ) where a 1,..., a n k. Proof. This proof will assume that k = C, but the result holds for any algebraically closed field k. Note that it is clear that ideals of the form I = (x 1 a 1,..., x n a n ) are maximal, since the quotient R/I = C. Suppose then that I is a maximal ideal. The ring R is Noetherian, so we may write I = (f 1,..., f m ) as being generated by a finite number of polynomials f 1,..., f m R. We can define a new field K to be the extension of Q obtained by adjoining all coefficiants of the polynomials f 1,..., f m to Q. This is a subfield of C. Define I 0 to be the maximal ideal I K[x 1,..., x n ] in K[x 1,..., x n ]. The quotient K[x 1,..., x n ]/I 0 is a field. In fact, it is a field which embeds in C. If we define a i to be the image of the class x i + I 0 in this embedding, then f i (a 1,..., a n ) = 0 for each i = 1,..., m. Hence I 0 (x 1 a 1,..., x n a n ). Corollary 2.2. If I is a proper ideal of k[x 1,..., x n ], then V (I) is nonempty. Proof. This follows from Hilbert s Nullstellensatz, and the fact that every proper ideal is contained in a maximal ideal. Now we have a onetoone correspondence between the points in A n k and the maximal ideals of k[x 1,..., x n ]. We know further that there is a bijection between radical ideals of k[x 1,..., x n ] and subschemes of A n k, given in one direction by the map V. We can describe the inverse of V nicely in this case. 4
5 Definition 2.3. If X is a subset of points in A n k, we define the ideal of X to be I(X) := {f k[x 1,..., x n ] : f(p ) = 0 for all P X}. If X is a subset of points in A n k, then V (I(X)) = X. However, it is not always true that for an ideal J of R, I(V (J)) = J. For example, in A 2 k, I(V ((x2 ))) = (x). However, if we restrict our attention to radical ideals, I and V are inverse. Corollary 2.4. If J is a radical ideal of R. then I(V (J)) = J. Proof. Clearly J I(V (J)). Our ring is Noetherian, so we can write J = (f 1,..., f m ) for some polynomials f i k[x 1,..., x n ]. Suppose now that f I(V (J)). We can then include f into the ring k[x 1,..., x n, t], then consider the ideal A = J + (ft 1) in this ring. A point in A n+1 k is in V (A) if and only if it is both in V (J) in A n k and it is in V (ft 1). But f I(V (J)) so both conditions cannot be met at the same time. Therefore V (A) =. By the previous corollary of Hilbert s Nullstellensatz, this implies that A = k[x 1,..., x n, t]. Therefore there are g 0,..., g m k[x 1,..., x n, t] such that g 0 (ft 1) + g 1 f g m f m = 1. Let N be the highest power of t appearing on the left hand side. We can multiply both sides of the equation by f N to get f N g 0 (ft 1) + f N g 1 f f N g m f m = f N. Because f k[x 1,..., x n ], we can replace each g i (x 1,..., x n, t) with G i (x 1,..., x n, ft). Then G 0 (x 1,..., x n, ft)(ft 1) + G 1 (x 1,..., x n, ft)f G m (x 1,..., x n, ft)f m = f N. If we now consider this polynomial in the quotient k[x 1,..., x n, t]/(ft 1), we have G 1 (x 1,..., x n, 1)f G m (x 1,..., x n, 1)f m = f N. Since k[x 1,..., x n ] can be included into this quotient ring, hence the equation holds there as well. It shows that f N J. Since r(j) = J, f J. The ideals in polynomial rings are generated by a finite number of polynomials. The previous results show that we can identify closed sets in A n k as the solution sets to a finite list of polynomials in k[x 1,..., x n ]. Definition 2.5. A set X A n k is an algebraic set if it is a solution set to a finite number of polynomial equations in k[x 1,..., x n ]. Note that given an algebraic set X A n k, we can view X as an affine subscheme of An k, as it is X = Spec(k[x 1,..., x n ]/I(X)), and with the quotient map by the ideal I(X) inducing the inclusion map of X into A n k. Example 2.6. In A 2 C, the solutions set to one polynomial is called a plane curve. This name can be a bit misleading, since a plane curves is in fact two dimensional over R. For example, an elliptic curve, an important object of study not only in algebraic geometry but also in number theory and cryptography, is the solution set to a degree 3 equation. In fact, such a curve is a torus (doughnut)! 5
6 We have discussed morphisms of affine schemes. We can say exactly what these look like for algebraic sets. Proposition 2.7. If X A m k and Y An k are algebraic sets, then f : X Y is a morphism of algebraic sets if and only if we can write f(p ) = (f 1 (P ),..., f n (P )) for polynomials f 1,..., f n k[x 1,..., x m ] and P X. Proof. Suppose f : k[x 1,..., x n ]/I(Y ) k[x 1,..., x m ]/I(X) is a ring homomorphism. Then f must restrict to the identity on k, hence it is also a kalgebra homomorphism. It is completely determined by the n polynomials g i = f(x i ) k[x 1,..., x m ]. For any point P X, f(p ) = (g 1 (P ),..., g n (P )) Y. 2.1 Affine varieties We now wish to study the geometric properties of algebraic sets which can be determined using algebra. The first property we will study is irreducibility. Definition 2.8. An algebraic set X is reducible if it can be written as a union X = X 1 X 2 of nonempty algebraic sets. Otherwise, X is called irreducible. If X is reducible and X 1 and X 2 can be chosed to be disjoint, then X is disconnected. Otherwise X is connected. There is a special name for irreducible algebraic sets in A n k. Definition 2.9. An irreducible algebraic set in A n k is called an (affine) algebraic variety. We will now show that the points of Spec(k[x 1,..., x n ]) are in bijection with the irreducible algebraic varieties in A n k. In other words, the subvarieties of An k are the vanishing sets of prime ideals of k[x 1,..., x n ]. Proposition If I is an ideal of k[x 1,..., x n ], then V (I) is irreducible if and only if I is prime. Proof. First suppose V (I) is irreducible. If the product fg I, then (fg) I, so V (fg) V (fg). Then since V (fg) = V (f) V (g), X = (V (f) X) (V (g) X). Both V (f) X and V (g) X are algebraic sets, so X = V (f) X or X = V (g) X. Hence f or g is in I. This shows I is prime. Now suppose I is a prime ideal. If we can write X = X 1 X 2 for algebraic sets X 1 = V (I 1 ) and X 2 = V (I 2 ), for radical ideals I 1 and I 2, then I = I 1 I 2. Since I is prime, I = I 1 or I = I 2. Hence X is irreducible. Every algebraic set can be written as a finite union of algebraic varieties. In this way, Spec(k[x 1,..., x n ]) contains the building blocks of all the algebraic sets in A n k. 2.2 Dimension When thinking of a geometric object we can visualize, we have an intrinsic sense of dimension. We know that a line is onedimensional, a square and a triangle are twodimensional, and we are threedimensional. But in this abstract setting, we will often describe our varieties as solutions sets of several polynomials in some large dimensional space. We need a definition of dimension that fits in this setting. 6
7 Definition The dimension of an algebraic set X is the largest number n so that there is a chain of closed irreducible subsets of X X 0 X 1 X n 1 X n = X. Example The chain shows that the dimension of A n k exactly n. 0 A 1 k An 1 k A n k is at least n. It is hard to show, however, that the dimension is We can change the question of dimension into an algebraic question. By using the map I we can instead look for the longest chain of ideals I(X) I(X n 1 ) I(X 1 ) I(X 0 ) k[x 1,..., x n ]. In particular, we could require that the subsets X i are irreducible, and look for the longest chains of prime ideals of this form. Or rather, we could look at the longest chain of prime ideals in the ring k[x 1,..., x n ]/I(X). I(X n 1 )/I(X) I(X 1 )/I(X) I(X 0 ) k[x 1,..., x n ]/I(X) Definition The Krull dimension of a ring R is the largest n such that there is a chain of prime ideals P 0 P 1 P n. It is immediate that the dimension of the algebraic set X in A n k dimension of the ring k[x 1,..., x n ]/I(X). is the same as the Krull Example The ring k[x] is a principal ideal domain. Hence the longest possible chain of prime ideals is 0 (f) where f is irreducible. This shows A 1 k is onedimensional. Example Suppose P is a prime ideal in the ring k[x, y]. If P is maximal, we know P = (x a, y b) for some a, b k. Otherwise, P must be finitely generated, and k[x, y]/p must be an integral domain. In fact, we can show P = (f) for some irreducible polynomial f, and so the longest possible chain of prime ideals is 0 (f) (x a, y b), showing A 2 k is twodimensional. Example Suppose f k[x, y] is not a unit. Then (f) is contained in some maximal ideal (x a, y b) for a, b k. The existence of a chain of prime ideals (f) P 1 P 2 (x a, y b) would contradict that A 2 k is twodimensional. By the lattice isomorphism theorem, k[x, y]/(f) is onedimensional. So V (f) is a onedimensional subvariety (a curve). 2.3 Plane Curves As seen in the previous subsection, any nonconstant polynomial f(x, y) k[x, y] defines a onedimensional subvariety. Definition A plane curve is the vanishing set of a single polynomial in A 2 k. 7
8 The curve V (f) will be irreducible if and only if the polynomial f(x, y) is irreducible in k[x, y]. For example, the polynomial f(x, y) = x 2 + y defines an irreducible curve which we might view as an upsidedown parabola. However, f(x, y) = xy defines the union of the two coordinate axes. However, both curves share a property, called their degree. Definition The degree of a plane curve is the degree of the defining equation of the curve. It can also be defined as the number of intersection points between the curve and a general line, but we would need more machinery to make this alternate definition precise. Notice that the reducible plane curve V (xy) is a union of irreducible degree 1 plane curves, V (x) and V (y). It is always true that degree is additive in this way. The point (0, 0) is special on the curve V (xy). It is where the two irreducible components of the curve meet. We would like a way to detect special points on a curve. First, we look at some examples. Example Let f(x, y) = y 2 x 3 x 2. This is a degree 3 irreducible curve. Despite being irreducible, there is a special point on this curve. (This curve is called a nodal cubic.) Example Let f(x, y) = y 2 x 3. Again, this is an irreducible degree 3 curve. The special point on this curve is different in some sense from the the special point on the curve in the previous example. (This curve is called a cuspidal cubic). How can we measure this. Definition A point P = (a, b) on a curve V (f(x, y)) is a multiple point if f x (P ) = f y (P ) = 0. If we write f(x, y) = f m + + f n where f i is a sum of monomials of degree exactly m, then the multiplicity of f(x, y) at P = (0, 0) is defined to be m. If we write f m = L r 1 1 Lf j j for linear monomials L i, then the lines defined by the L i are called the tangent lines to f(x, y) at P = (0, 0). If f m is the product of m distinct lines, P is called a node of f(x, y). Example When f(x, y) = y 2 x 3 x 2, f x (x, y) = 3x 2 2x and f y (x, y) = 2y. These both vanish only at the point (0, 0). The multiplicity of f at (0, 0) is 2. We can write y 2 x 2 = (y x)(y + x) as the product of two distinct tangent lines, so (0, 0) is a node. Example When f(x, y) = y 2 x 2, again f x and f y both vanish at the point (0, 0). The multiplicity is again 2 at this point. However, f 2 = y 2 is a product of the same line twice, so (0, 0) is not a node. If P = (a, b) is not (0, 0), then we can apply the methods of the previous definition to f(x + a, y + b) to determine is P is a multiple point, and what its tangent lines are. There is another more algebraic way to test for multiple points, which can be applied more generally to points on affine schemes. Definition Let M be the maximal ideal in k[x, y]/i(x) defining the point P in X. Then M/M 2 is the cotangent space to P, and the dimension of M/M 2 as a kvector space is the dimension of the cotangent space at P. Definition A point P on X is a singular point if the dimension of the cotangent space at P is larger than the dimension of X. Example Let f(x, y) = y 2 x 3. The maximal ideal (x, y) + (f) in k[x, y]/(f) defines the point (0, 0). As a k=vector space, (x, y)/(x, y) 2 + (f) is generated by the two elements x and y. On the other hand, if g(x, y) = y x 2 is smooth at (0, 0), the kvector space (x, y)/(x, y) 2 + (g) is already generated by x, since y = x 2 = 0 in this space. 8
ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 2: HILBERT S NULLSTELLENSATZ.
ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 2: HILBERT S NULLSTELLENSATZ. ANDREW SALCH 1. Hilbert s Nullstellensatz. The last lecture left off with the claim that, if J k[x 1,..., x n ] is an ideal, then
More informationSummer Algebraic Geometry Seminar
Summer Algebraic Geometry Seminar Lectures by Bart Snapp About This Document These lectures are based on Chapters 1 and 2 of An Invitation to Algebraic Geometry by Karen Smith et al. 1 Affine Varieties
More information2. Prime and Maximal Ideals
18 Andreas Gathmann 2. Prime and Maximal Ideals There are two special kinds of ideals that are of particular importance, both algebraically and geometrically: the socalled prime and maximal ideals. Let
More informationMath 203A  Solution Set 1
Math 203A  Solution Set 1 Problem 1. Show that the Zariski topology on A 2 is not the product of the Zariski topologies on A 1 A 1. Answer: Clearly, the diagonal Z = {(x, y) : x y = 0} A 2 is closed in
More informationALGEBRAIC GROUPS. Disclaimer: There are millions of errors in these notes!
ALGEBRAIC GROUPS Disclaimer: There are millions of errors in these notes! 1. Some algebraic geometry The subject of algebraic groups depends on the interaction between algebraic geometry and group theory.
More informationExploring the Exotic Setting for Algebraic Geometry
Exploring the Exotic Setting for Algebraic Geometry Victor I. Piercey University of Arizona Integration Workshop Project August 610, 2010 1 Introduction In this project, we will describe the basic topology
More informationMath 203A  Solution Set 1
Math 203A  Solution Set 1 Problem 1. Show that the Zariski topology on A 2 is not the product of the Zariski topologies on A 1 A 1. Answer: Clearly, the diagonal Z = {(x, y) : x y = 0} A 2 is closed in
More information3. The Sheaf of Regular Functions
24 Andreas Gathmann 3. The Sheaf of Regular Functions After having defined affine varieties, our next goal must be to say what kind of maps between them we want to consider as morphisms, i. e. as nice
More informationReid 5.2. Describe the irreducible components of V (J) for J = (y 2 x 4, x 2 2x 3 x 2 y + 2xy + y 2 y) in k[x, y, z]. Here k is algebraically closed.
Reid 5.2. Describe the irreducible components of V (J) for J = (y 2 x 4, x 2 2x 3 x 2 y + 2xy + y 2 y) in k[x, y, z]. Here k is algebraically closed. Answer: Note that the first generator factors as (y
More informationAlgebraic Varieties. Chapter Algebraic Varieties
Chapter 12 Algebraic Varieties 12.1 Algebraic Varieties Let K be a field, n 1 a natural number, and let f 1,..., f m K[X 1,..., X n ] be polynomials with coefficients in K. Then V = {(a 1,..., a n ) :
More informationMATH32062 Notes. 1 Affine algebraic varieties. 1.1 Definition of affine algebraic varieties
MATH32062 Notes 1 Affine algebraic varieties 1.1 Definition of affine algebraic varieties We want to define an algebraic variety as the solution set of a collection of polynomial equations, or equivalently,
More information4.4 Noetherian Rings
4.4 Noetherian Rings Recall that a ring A is Noetherian if it satisfies the following three equivalent conditions: (1) Every nonempty set of ideals of A has a maximal element (the maximal condition); (2)
More informationAlgebraic Varieties. Notes by Mateusz Micha lek for the lecture on April 17, 2018, in the IMPRS Ringvorlesung Introduction to Nonlinear Algebra
Algebraic Varieties Notes by Mateusz Micha lek for the lecture on April 17, 2018, in the IMPRS Ringvorlesung Introduction to Nonlinear Algebra Algebraic varieties represent solutions of a system of polynomial
More informationAlgebraic Geometry. Andreas Gathmann. Class Notes TU Kaiserslautern 2014
Algebraic Geometry Andreas Gathmann Class Notes TU Kaiserslautern 2014 Contents 0. Introduction......................... 3 1. Affine Varieties........................ 9 2. The Zariski Topology......................
More informationAlgebraic varieties and schemes over any scheme. Non singular varieties
Algebraic varieties and schemes over any scheme. Non singular varieties Trang June 16, 2010 1 Lecture 1 Let k be a field and k[x 1,..., x n ] the polynomial ring with coefficients in k. Then we have two
More informationMath 203A  Solution Set 1
Math 203A  Solution Set 1 Problem 1. Show that the Zariski topology on A 2 is not the product of the Zariski topologies on A 1 A 1. Answer: Clearly, the diagonal Z = {(x, y) : x y = 0} A 2 is closed in
More informationABSTRACT NONSINGULAR CURVES
ABSTRACT NONSINGULAR CURVES Affine Varieties Notation. Let k be a field, such as the rational numbers Q or the complex numbers C. We call affine nspace the collection A n k of points P = a 1, a,..., a
More informationINTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 14
INTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 14 RAVI VAKIL Contents 1. Dimension 1 1.1. Last time 1 1.2. An algebraic definition of dimension. 3 1.3. Other facts that are not hard to prove 4 2. Nonsingularity:
More informationTHE ALGEBRAIC GEOMETRY DICTIONARY FOR BEGINNERS. Contents
THE ALGEBRAIC GEOMETRY DICTIONARY FOR BEGINNERS ALICE MARK Abstract. This paper is a simple summary of the first most basic definitions in Algebraic Geometry as they are presented in Dummit and Foote ([1]),
More information10. Noether Normalization and Hilbert s Nullstellensatz
10. Noether Normalization and Hilbert s Nullstellensatz 91 10. Noether Normalization and Hilbert s Nullstellensatz In the last chapter we have gained much understanding for integral and finite ring extensions.
More informationCommutative Algebra. Andreas Gathmann. Class Notes TU Kaiserslautern 2013/14
Commutative Algebra Andreas Gathmann Class Notes TU Kaiserslautern 2013/14 Contents 0. Introduction......................... 3 1. Ideals........................... 9 2. Prime and Maximal Ideals.....................
More information8. Prime Factorization and Primary Decompositions
70 Andreas Gathmann 8. Prime Factorization and Primary Decompositions 13 When it comes to actual computations, Euclidean domains (or more generally principal ideal domains) are probably the nicest rings
More informationMATH 8253 ALGEBRAIC GEOMETRY WEEK 12
MATH 8253 ALGEBRAIC GEOMETRY WEEK 2 CİHAN BAHRAN 3.2.. Let Y be a Noetherian scheme. Show that any Y scheme X of finite type is Noetherian. Moreover, if Y is of finite dimension, then so is X. Write f
More informationPROBLEMS, MATH 214A. Affine and quasiaffine varieties
PROBLEMS, MATH 214A k is an algebraically closed field Basic notions Affine and quasiaffine varieties 1. Let X A 2 be defined by x 2 + y 2 = 1 and x = 1. Find the ideal I(X). 2. Prove that the subset
More informationRings and groups. Ya. Sysak
Rings and groups. Ya. Sysak 1 Noetherian rings Let R be a ring. A (right) R module M is called noetherian if it satisfies the maximum condition for its submodules. In other words, if M 1... M i M i+1...
More information12. Hilbert Polynomials and Bézout s Theorem
12. Hilbert Polynomials and Bézout s Theorem 95 12. Hilbert Polynomials and Bézout s Theorem After our study of smooth cubic surfaces in the last chapter, let us now come back to the general theory of
More informationYuriy Drozd. Intriduction to Algebraic Geometry. Kaiserslautern 1998/99
Yuriy Drozd Intriduction to Algebraic Geometry Kaiserslautern 1998/99 CHAPTER 1 Affine Varieties 1.1. Ideals and varieties. Hilbert s Basis Theorem Let K be an algebraically closed field. We denote by
More information10. Smooth Varieties. 82 Andreas Gathmann
82 Andreas Gathmann 10. Smooth Varieties Let a be a point on a variety X. In the last chapter we have introduced the tangent cone C a X as a way to study X locally around a (see Construction 9.20). It
More informationCourse 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra
Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra D. R. Wilkins Contents 3 Topics in Commutative Algebra 2 3.1 Rings and Fields......................... 2 3.2 Ideals...............................
More informationA course in. Algebraic Geometry. Taught by Prof. Xinwen Zhu. Fall 2011
A course in Algebraic Geometry Taught by Prof. Xinwen Zhu Fall 2011 1 Contents 1. September 1 3 2. September 6 6 3. September 8 11 4. September 20 16 5. September 22 21 6. September 27 25 7. September
More informationwhere m is the maximal ideal of O X,p. Note that m/m 2 is a vector space. Suppose that we are given a morphism
8. Smoothness and the Zariski tangent space We want to give an algebraic notion of the tangent space. In differential geometry, tangent vectors are equivalence classes of maps of intervals in R into the
More informationAN INTRODUCTION TO AFFINE SCHEMES
AN INTRODUCTION TO AFFINE SCHEMES BROOKE ULLERY Abstract. This paper gives a basic introduction to modern algebraic geometry. The goal of this paper is to present the basic concepts of algebraic geometry,
More informationALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 4: MORE ABOUT VARIETIES AND REGULAR FUNCTIONS.
ALGERAIC GEOMETRY COURSE NOTES, LECTURE 4: MORE AOUT VARIETIES AND REGULAR FUNCTIONS. ANDREW SALCH. More about some claims from the last lecture. Perhaps you have noticed by now that the Zariski topology
More informationSpring 2016, lecture notes by Maksym Fedorchuk 51
Spring 2016, lecture notes by Maksym Fedorchuk 51 10.2. Problem Set 2 Solution Problem. Prove the following statements. (1) The nilradical of a ring R is the intersection of all prime ideals of R. (2)
More informationMATH 326: RINGS AND MODULES STEFAN GILLE
MATH 326: RINGS AND MODULES STEFAN GILLE 1 2 STEFAN GILLE 1. Rings We recall first the definition of a group. 1.1. Definition. Let G be a non empty set. The set G is called a group if there is a map called
More informationCHAPTER 1. AFFINE ALGEBRAIC VARIETIES
CHAPTER 1. AFFINE ALGEBRAIC VARIETIES During this first part of the course, we will establish a correspondence between various geometric notions and algebraic ones. Some references for this part of the
More informationALGEBRAIC GEOMETRY (NMAG401) Contents. 2. Polynomial and rational maps 9 3. Hilbert s Nullstellensatz and consequences 23 References 30
ALGEBRAIC GEOMETRY (NMAG401) JAN ŠŤOVÍČEK Contents 1. Affine varieties 1 2. Polynomial and rational maps 9 3. Hilbert s Nullstellensatz and consequences 23 References 30 1. Affine varieties The basic objects
More informationHomogeneous Coordinate Ring
Students: Kaiserslautern University Algebraic Group June 14, 2013 Outline Quotients in Algebraic Geometry 1 Quotients in Algebraic Geometry 2 3 4 Outline Quotients in Algebraic Geometry 1 Quotients in
More informationCOMPLEX VARIETIES AND THE ANALYTIC TOPOLOGY
COMPLEX VARIETIES AND THE ANALYTIC TOPOLOGY BRIAN OSSERMAN Classical algebraic geometers studied algebraic varieties over the complex numbers. In this setting, they didn t have to worry about the Zariski
More information2. Intersection Multiplicities
2. Intersection Multiplicities 11 2. Intersection Multiplicities Let us start our study of curves by introducing the concept of intersection multiplicity, which will be central throughout these notes.
More informationHARTSHORNE EXERCISES
HARTSHORNE EXERCISES J. WARNER Hartshorne, Exercise I.5.6. Blowing Up Curve Singularities (a) Let Y be the cusp x 3 = y 2 + x 4 + y 4 or the node xy = x 6 + y 6. Show that the curve Ỹ obtained by blowing
More informationBEZOUT S THEOREM CHRISTIAN KLEVDAL
BEZOUT S THEOREM CHRISTIAN KLEVDAL A weaker version of Bézout s theorem states that if C, D are projective plane curves of degrees c and d that intersect transversally, then C D = cd. The goal of this
More informationHere is another way to understand what a scheme is 1.GivenaschemeX, and a commutative ring R, the set of Rvalued points
Chapter 7 Schemes III 7.1 Functor of points Here is another way to understand what a scheme is 1.GivenaschemeX, and a commutative ring R, the set of Rvalued points X(R) =Hom Schemes (Spec R, X) This is
More informationdiv(f ) = D and deg(d) = deg(f ) = d i deg(f i ) (compare this with the definitions for smooth curves). Let:
Algebraic Curves/Fall 015 Aaron Bertram 4. Projective Plane Curves are hypersurfaces in the plane CP. When nonsingular, they are Riemann surfaces, but we will also consider plane curves with singularities.
More informationCommutative Algebra. Contents. B Totaro. Michaelmas Basics Rings & homomorphisms Modules Prime & maximal ideals...
Commutative Algebra B Totaro Michaelmas 2011 Contents 1 Basics 4 1.1 Rings & homomorphisms.............................. 4 1.2 Modules........................................ 6 1.3 Prime & maximal ideals...............................
More informationAlgebraic geometry of the ring of continuous functions
Algebraic geometry of the ring of continuous functions Nicolas Addington October 27 Abstract Maximal ideals of the ring of continuous functions on a compact space correspond to points of the space. For
More informationAlgebraic Geometry. Instructor: Stephen Diaz & Typist: Caleb McWhorter. Spring 2015
Algebraic Geometry Instructor: Stephen Diaz & Typist: Caleb McWhorter Spring 2015 Contents 1 Varieties 2 1.1 Affine Varieties....................................... 2 1.50 Projective Varieties.....................................
More informationLECTURE Affine Space & the Zariski Topology. It is easy to check that Z(S)=Z((S)) with (S) denoting the ideal generated by elements of S.
LECTURE 10 1. Affine Space & the Zariski Topology Definition 1.1. Let k a field. Take S a set of polynomials in k[t 1,..., T n ]. Then Z(S) ={x k n f(x) =0, f S}. It is easy to check that Z(S)=Z((S)) with
More informationTROPICAL SCHEME THEORY
TROPICAL SCHEME THEORY 5. Commutative algebra over idempotent semirings II Quotients of semirings When we work with rings, a quotient object is specified by an ideal. When dealing with semirings (and lattices),
More informationMath 203A  Solution Set 3
Math 03A  Solution Set 3 Problem 1 Which of the following algebraic sets are isomorphic: (i) A 1 (ii) Z(xy) A (iii) Z(x + y ) A (iv) Z(x y 5 ) A (v) Z(y x, z x 3 ) A Answer: We claim that (i) and (v)
More informationProjective Varieties. Chapter Projective Space and Algebraic Sets
Chapter 1 Projective Varieties 1.1 Projective Space and Algebraic Sets 1.1.1 Definition. Consider A n+1 = A n+1 (k). The set of all lines in A n+1 passing through the origin 0 = (0,..., 0) is called the
More informationMATH 631: ALGEBRAIC GEOMETRY: HOMEWORK 1 SOLUTIONS
MATH 63: ALGEBRAIC GEOMETRY: HOMEWORK SOLUTIONS Problem. (a.) The (t + ) (t + ) minors m (A),..., m k (A) of an n m matrix A are polynomials in the entries of A, and m i (A) = 0 for all i =,..., k if and
More informationAlgebraic Geometry. Andreas Gathmann. Notes for a class. taught at the University of Kaiserslautern 2002/2003
Algebraic Geometry Andreas Gathmann Notes for a class taught at the University of Kaiserslautern 2002/2003 CONTENTS 0. Introduction 1 0.1. What is algebraic geometry? 1 0.2. Exercises 6 1. Affine varieties
More informationThis is a closed subset of X Y, by Proposition 6.5(b), since it is equal to the inverse image of the diagonal under the regular map:
Math 6130 Notes. Fall 2002. 7. Basic Maps. Recall from 3 that a regular map of affine varieties is the same as a homomorphism of coordinate rings (going the other way). Here, we look at how algebraic properties
More informationAlgebraic Geometry. Question: What regular polygons can be inscribed in an ellipse?
Algebraic Geometry Question: What regular polygons can be inscribed in an ellipse? 1. Varieties, Ideals, Nullstellensatz Let K be a field. We shall work over K, meaning, our coefficients of polynomials
More informationMath 145. Codimension
Math 145. Codimension 1. Main result and some interesting examples In class we have seen that the dimension theory of an affine variety (irreducible!) is linked to the structure of the function field in
More information12. Linear systems Theorem Let X be a scheme over a ring A. (1) If φ: X P n A is an Amorphism then L = φ O P n
12. Linear systems Theorem 12.1. Let X be a scheme over a ring A. (1) If φ: X P n A is an Amorphism then L = φ O P n A (1) is an invertible sheaf on X, which is generated by the global sections s 0, s
More informationExtended Index. 89f depth (of a prime ideal) 121f ArtinRees Lemma. 107f descending chain condition 74f Artinian module
Extended Index cokernel 19f for Atiyah and MacDonald's Introduction to Commutative Algebra colon operator 8f Key: comaximal ideals 7f  listings ending in f give the page where the term is defined commutative
More informationInstitutionen för matematik, KTH.
Institutionen för matematik, KTH. Contents 7 Affine Varieties 1 7.1 The polynomial ring....................... 1 7.2 Hypersurfaces........................... 1 7.3 Ideals...............................
More information1 Flat, Smooth, Unramified, and Étale Morphisms
1 Flat, Smooth, Unramified, and Étale Morphisms 1.1 Flat morphisms Definition 1.1. An Amodule M is flat if the (rightexact) functor A M is exact. It is faithfully flat if a complex of Amodules P N Q
More informationMath 40510, Algebraic Geometry
Math 40510, Algebraic Geometry Problem Set 1, due February 10, 2016 1. Let k = Z p, the field with p elements, where p is a prime. Find a polynomial f k[x, y] that vanishes at every point of k 2. [Hint:
More informationCommutative Algebra. B Totaro. Michaelmas Basics Rings & homomorphisms Modules Prime & maximal ideals...
Commutative Algebra B Totaro Michaelmas 2011 Contents 1 Basics 2 1.1 Rings & homomorphisms................... 2 1.2 Modules............................. 4 1.3 Prime & maximal ideals....................
More informationLecture 2. (1) Every P L A (M) has a maximal element, (2) Every ascending chain of submodules stabilizes (ACC).
Lecture 2 1. Noetherian and Artinian rings and modules Let A be a commutative ring with identity, A M a module, and φ : M N an Alinear map. Then ker φ = {m M : φ(m) = 0} is a submodule of M and im φ is
More informationMA 206 notes: introduction to resolution of singularities
MA 206 notes: introduction to resolution of singularities Dan Abramovich Brown University March 4, 2018 Abramovich Introduction to resolution of singularities 1 / 31 Resolution of singularities Let k be
More informationThe most important result in this section is undoubtedly the following theorem.
28 COMMUTATIVE ALGEBRA 6.4. Examples of Noetherian rings. So far the only rings we can easily prove are Noetherian are principal ideal domains, like Z and k[x], or finite. Our goal now is to develop theorems
More informationChapter 1. Affine algebraic geometry. 1.1 The Zariski topology on A n
Chapter 1 Affine algebraic geometry We shall restrict our attention to affine algebraic geometry, meaning that the algebraic varieties we consider are precisely the closed subvarieties of affine n space
More information11. Dimension. 96 Andreas Gathmann
96 Andreas Gathmann 11. Dimension We have already met several situations in this course in which it seemed to be desirable to have a notion of dimension (of a variety, or more generally of a ring): for
More informationWEAK NULLSTELLENSATZ
WEAK NULLSTELLENSATZ YIFAN WU, wuyifan@umich.edu Abstract. We prove weak Nullstellensatz which states if a finitely generated k algebra is a field, then it is a finite algebraic field extension of k. We
More informationCHEVALLEY S THEOREM AND COMPLETE VARIETIES
CHEVALLEY S THEOREM AND COMPLETE VARIETIES BRIAN OSSERMAN In this note, we introduce the concept which plays the role of compactness for varieties completeness. We prove that completeness can be characterized
More informationDimension Theory. Mathematics 683, Fall 2013
Dimension Theory Mathematics 683, Fall 2013 In this note we prove some of the standard results of commutative ring theory that lead up to proofs of the main theorem of dimension theory and of the Nullstellensatz.
More informationπ X : X Y X and π Y : X Y Y
Math 6130 Notes. Fall 2002. 6. Hausdorffness and Compactness. We would like to be able to say that all quasiprojective varieties are Hausdorff and that projective varieties are the only compact varieties.
More information0.1 Spec of a monoid
These notes were prepared to accompany the first lecture in a seminar on logarithmic geometry. As we shall see in later lectures, logarithmic geometry offers a natural approach to study semistable schemes.
More informationCHAPTER 0 PRELIMINARY MATERIAL. Paul Vojta. University of California, Berkeley. 18 February 1998
CHAPTER 0 PRELIMINARY MATERIAL Paul Vojta University of California, Berkeley 18 February 1998 This chapter gives some preliminary material on number theory and algebraic geometry. Section 1 gives basic
More information(dim Z j dim Z j 1 ) 1 j i
Math 210B. Codimension 1. Main result and some interesting examples Let k be a field, and A a domain finitely generated kalgebra. In class we have seen that the dimension theory of A is linked to the
More information214A HOMEWORK KIM, SUNGJIN
214A HOMEWORK KIM, SUNGJIN 1.1 Let A = k[[t ]] be the ring of formal power series with coefficients in a field k. Determine SpecA. Proof. We begin with a claim that A = { a i T i A : a i k, and a 0 k }.
More informationNOTES FOR DRAGOS: MATH 210 CLASS 12, THURS. FEB. 22
NOTES FOR DRAGOS: MATH 210 CLASS 12, THURS. FEB. 22 RAVI VAKIL Hi Dragos The class is in 381T, 1:15 2:30. This is the very end of Galois theory; you ll also start commutative ring theory. Tell them: midterm
More informationAlgebraic varieties. Chapter A ne varieties
Chapter 4 Algebraic varieties 4.1 A ne varieties Let k be a field. A ne nspace A n = A n k = kn. It s coordinate ring is simply the ring R = k[x 1,...,x n ]. Any polynomial can be evaluated at a point
More informationLecture 7: Etale Fundamental Group  Examples
Lecture 7: Etale Fundamental Group  Examples October 15, 2014 In this lecture our only goal is to give lots of examples of etale fundamental groups so that the reader gets some feel for them. Some of
More informationIntroduction to Algebraic Geometry. Jilong Tong
Introduction to Algebraic Geometry Jilong Tong December 6, 2012 2 Contents 1 Algebraic sets and morphisms 11 1.1 Affine algebraic sets.................................. 11 1.1.1 Some definitions................................
More informationExercises of the Algebraic Geometry course held by Prof. Ugo Bruzzo. Alex Massarenti
Exercises of the Algebraic Geometry course held by Prof. Ugo Bruzzo Alex Massarenti SISSA, VIA BONOMEA 265, 34136 TRIESTE, ITALY Email address: alex.massarenti@sissa.it These notes collect a series of
More informationAlgebraic Geometry I Lectures 14 and 15
Algebraic Geometry I Lectures 14 and 15 October 22, 2008 Recall from the last lecture the following correspondences {points on an affine variety Y } {maximal ideals of A(Y )} SpecA A P Z(a) maximal ideal
More informationIntroduction to Arithmetic Geometry Fall 2013 Lecture #18 11/07/2013
18.782 Introduction to Arithmetic Geometry Fall 2013 Lecture #18 11/07/2013 As usual, all the rings we consider are commutative rings with an identity element. 18.1 Regular local rings Consider a local
More information4. Images of Varieties Given a morphism f : X Y of quasiprojective varieties, a basic question might be to ask what is the image of a closed subset
4. Images of Varieties Given a morphism f : X Y of quasiprojective varieties, a basic question might be to ask what is the image of a closed subset Z X. Replacing X by Z we might as well assume that Z
More informationThe GeometryAlgebra Dictionary
Chapter 1 The GeometryAlgebra Dictionary This chapter is an introduction to affine algebraic geometry. Working over a field k, we will write A n (k) for the affine nspace over k and k[x 1,..., x n ]
More informationCOMMUNICATIONS IN ALGEBRA, 15(3), (1987) A NOTE ON PRIME IDEALS WHICH TEST INJECTIVITY. John A. Beachy and William D.
COMMUNICATIONS IN ALGEBRA, 15(3), 471 478 (1987) A NOTE ON PRIME IDEALS WHICH TEST INJECTIVITY John A. Beachy and William D. Weakley Department of Mathematical Sciences Northern Illinois University DeKalb,
More informationFILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS.
FILTERED RINGS AND MODULES. GRADINGS AND COMPLETIONS. Let A be a ring, for simplicity assumed commutative. A filtering, or filtration, of an A module M means a descending sequence of submodules M = M 0
More informationMATH 233B, FLATNESS AND SMOOTHNESS.
MATH 233B, FLATNESS AND SMOOTHNESS. The discussion of smooth morphisms is one place were Hartshorne doesn t do a very good job. Here s a summary of this week s material. I ll also insert some (optional)
More informationV (f) :={[x] 2 P n ( ) f(x) =0}. If (x) ( x) thenf( x) =
20 KIYOSHI IGUSA BRANDEIS UNIVERSITY 2. Projective varieties For any field F, the standard definition of projective space P n (F ) is that it is the set of one dimensional F vector subspaces of F n+.
More informationPure Math 764, Winter 2014
Compact course notes Pure Math 764, Winter 2014 Introduction to Algebraic Geometry Lecturer: R. Moraru transcribed by: J. Lazovskis University of Waterloo April 20, 2014 Contents 1 Basic geometric objects
More informationk k would be reducible. But the zero locus of f in A n+1
Math 145. Bezout s Theorem Let be an algebraically closed field. The purpose of this handout is to prove Bezout s Theorem and some related facts of general interest in projective geometry that arise along
More informationne varieties (continued)
Chapter 2 A ne varieties (continued) 2.1 Products For some problems its not very natural to restrict to irreducible varieties. So we broaden the previous story. Given an a ne algebraic set X A n k, we
More informationMath 6140 Notes. Spring Codimension One Phenomena. Definition: Examples: Properties:
Math 6140 Notes. Spring 2003. 11. Codimension One Phenomena. A property of the points of a variety X holds in codimension one if the locus of points for which the property fails to hold is contained in
More informationAtiyah/Macdonald Commutative Algebra
Atiyah/Macdonald Commutative Algebra Linus Setiabrata ls823@cornell.edu http://pi.math.cornell.edu/ ls823 I m not sure if these arguments are correct! Most things in [square brackets] are corrections to
More informationCRing Project, Chapter 7
Contents 7 Integrality and valuation rings 3 1 Integrality......................................... 3 1.1 Fundamentals................................... 3 1.2 Le sorite for integral extensions.........................
More informationArithmetic Algebraic Geometry
Arithmetic Algebraic Geometry 2 Arithmetic Algebraic Geometry Travis Dirle December 4, 2016 2 Contents 1 Preliminaries 1 1.1 Affine Varieties.......................... 1 1.2 Projective Varieties........................
More informationLECTURES ON ALGEBRAIC GEOMETRY MATH 202A
LECTURES ON ALGEBRAIC GEOMETRY MATH 202A KIYOSHI IGUSA BRANDEIS UNIVERSITY Contents Introduction 1 Overfield 1 1. A ne varieties 2 1.1. Lecture 1: Weak Nullstellensatz 2 1.2. Lecture 2: Noether s normalization
More informationAN EXPOSITION OF THE RIEMANN ROCH THEOREM FOR CURVES
AN EXPOSITION OF THE RIEMANN ROCH THEOREM FOR CURVES DOMINIC L. WYNTER Abstract. We introduce the concepts of divisors on nonsingular irreducible projective algebraic curves, the genus of such a curve,
More informationGLUING SCHEMES AND A SCHEME WITHOUT CLOSED POINTS
GLUING SCHEMES AND A SCHEME WITHOUT CLOSED POINTS KARL SCHWEDE Abstract. We first construct and give basic properties of the fibered coproduct in the category of ringed spaces. We then look at some special
More informationHonors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35
Honors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35 1. Let R be a commutative ring with 1 0. (a) Prove that the nilradical of R is equal to the intersection of the prime
More informationNOTES ON ALGEBRAIC GEOMETRY MATH 202A. Contents Introduction Affine varieties 22
NOTES ON ALGEBRAIC GEOMETRY MATH 202A KIYOSHI IGUSA BRANDEIS UNIVERSITY Contents Introduction 1 1. Affine varieties 2 1.1. Weak Nullstellensatz 2 1.2. Noether s normalization theorem 2 1.3. Nullstellensatz
More information