# π X : X Y X and π Y : X Y Y

Size: px
Start display at page:

Download "π X : X Y X and π Y : X Y Y"

Transcription

1 Math 6130 Notes. Fall Hausdorffness and Compactness. We would like to be able to say that all quasi-projective varieties are Hausdorff and that projective varieties are the only compact varieties. After all, this is the case when the varieties are manifolds and we consider them with their ordinary manifold topologies. But the Zariski topology is a strange topology, and when we take the literal definitions of Hausdorff and compact, we find the exact opposite! All the open subsets U Y of a quasi-projective variety are dense (as in Proposition 2.3(b)) so a variety (other than a point) is never Hausdorff. All open covers Y = U i admit finite subcovers, since the complements Y U 1 Y (U 1 U 2 )... are a descending chain of closed sets, which is eventually stationary (Proposition 2.3(d)). So all varieties are compact. This is a puzzle, until we realize that the ordinary definitions rely on the fact that the product of topological spaces has the product topology. This is not the case with the Zariski topology on a product of varieties, and explains why the correct categorical definitions of Hausdorff and compact differ from the naive ones. With the correct definitions, we will get our desired result. Remark: There are many key moments in the development of the theory of varieties (and later schemes) where category theory saves the day by giving precise meaning to geometric properties of varieties. This is one of the many extraordinary insights due to Grothendieck. Definition: A pair of objects X and Y in a category C has a product, denoted X Y,ifX Y is another object of C, equipped with projections: π X : X Y X and π Y : X Y Y that are universal in the sense that given any object Z and morphisms: p : Z X and q : Z Y there is a unique morphism, which we will denote (p, q) :Z X Y with the property that π X (p, q) =p and π Y (p, q) =q. Observation: Any two product objects are naturally isomorphic. If (X Y ) is another product with projections π X and π Y, then the unique morphisms (π X,π Y ):(X Y ) X Y and (π X,π Y ):X Y (X Y ) are inverses to one another by the uniqueness in the definition. 1

2 Examples: (a) The product of a pair of sets is the Cartesian product. (b) The product of a pair of topological spaces is the Cartesian product with the product topology. (c) The product of a pair of differentiable manifolds is a patched product (i.e. if X = U i and Y = V j then X Y = (U i V j ) with product homeomorphisms to open sets in R n+m ). X Y has the product topology. Proposition 6.1: Every pair of affine varieties has a product affine variety. Proof: Given X C m with C[X] =C[x 1,..., x m ]/P and Y C n with C[Y ]=C[y 1,..., y n ]/Q, consider the Cartesian product: X Y C m+n First of all, X Y is a closed set. If P = {f i } i I for f i C[x 1,..., x m ] and Q = {g j } j J for g j C[y 1,..., y n ], then X Y = V ( {f i } i I {g j } j J ). Secondly, X Y is irreducible. If X Y = Z 1 Z 2 is a union of closed sets, then for each a X, the variety {a} Y = Y is a union of closed sets: {a} Y =(Z 1 ({a} Y )) (Z 2 ({a} Y )) so since Y is irreducible, either {a} Y Z 1 or {a} Y Z 2. But the sets X i := {a X {a} Y Z i } X are also closed since X i = b Y (Z i (X {b})) thought of as an intersection of subsets of X. Since X is irreducible and X 1 X 2 = X, it follows that either X 1 = X (and then Z 1 = X Y ) or else X 2 = X (and then Z 2 = X Y ). The two projections π X (a, b) = a and π Y (a, b) = b are regular maps by Proposition 3.5, and they have the desired universal property, since any regular maps p : Z X and q : Z Y are given by polynomials h 1,..., h m and k 1,..., k n, respectively (Proposition 3.5 again), and then the (unique) map to the Cartesian product (p, q) =(h 1,..., h m,k 1,...k n ) is also regular. Remark: Invoking the Nullstellensatz, we have proved that {f i } {g j } = I(X Y ) C[x 1,.., x m,y 1,..y n ] is prime. The ideal {f i } {g j } itself is prime, but this is left as an exercise. 2

3 Proposition 6.2: Every pair of varieties has a product variety. Proof: We nearly proved this already for a pair of projective spaces CP m, CP n in Proposition 5.4. The Segre variety S m,n has the desired properties, but we haven t actually proved that S m,n is irreducible. To see this, as in Proposition 6.1, we only need to know that the sets: {a} CP n S m,n and CP m {b} S m,n are closed and isomorphic to CP n and CP m respectively. But not only are they closed, they are intersections of S n,m with the linear subspaces of CP (m+1)(n+1) 1 with respective ideals: {a i x kl a k x il } and {b j x kl b l x kj } and it follows that the bijections CP n {a} CP n and CP m CP m {b} are isomorphisms. For a general pair of projective varieties X CP m and Z CP n, let I(X) = F 1,..., F r C[x 0,..., x m ] and I(Z) = G 1,..., G s C[y 0,..., y n ]. Then the Cartesian product X Z S m,n is the closed set: V ( {F α (x 0l,..., x ml ),G β (x i0,..., x in )} ) S m,n And if W X and Y Z are open subsets (i.e. quasi-projective varieties) whose complements have ideals H γ and K δ, respectively, then: W Y = X Z (X V ( K δ ) (V ( H γ ) Z) = X Z V ( {H γ (x 0l,..., x nl )K δ (x i0,..., x im )} ) and irreduciblity is proved as for projective space, and the universal property with respect to regular maps follows immediately from Proposition 4.4. Before we get to some peculiarities and examples of this product, here is a consistency result which isn t immediately obvious. Corollary 6.3: The products of affine varieties in Propositions 6.1 and 6.2 yield isomorphic quasi-projective varieties. Proof: By the universal property, all we need to do is to prove that the quasi-projective product from Proposition 6.2 is an affine variety! But if W = W V (x 0 ) and Y = Y V (y 0 ) as in Proposition 4.7, then: W Y = W Y V ( {x 0l x i0 } ) by Proposition 6.2. But the equations x 00 x il = x 0l x i0 (for the Segre variety) tell us V ( {x 0l x i0 } ) =V (x 00 )sow Y = W Y V (x 00 ), is isomorphic to an affine variety by Proposition

4 Examples: (a) In the quadric CP 1 CP 1 = V (x 00 x 11 x 01 x 10 ) CP 3, each further V (x ij ) is a pair of intersecting projective lines on the quadric: V (x 00 )={(0 : a 01 :0:a 11 )} {(0:0:a 01 : a 11 )} V (x 01 )={(a 00 :0:a 01 :0)} {(0 : 0 : a 01 : a 11 )} V (x 10 )={(a 00 : a 01 :0:0)} {(0 : a 01 :0:a 11 )} V (x 11 )={(a 00 : a 01 :0:0)} {(a 00 :0:a 01 :0)} and CP 1 CP 1 V (x 00 )={(1 : a 01 : a 10 : a 01 a 10 )} (setting a 00 = 1 and using the equation) is isomorphic to C 2 = C 1 C 1. (b) On the other hand, the intersection of the quadric with a plane V ( c ij x ij )isnot, in general, a pair of intersecting lines. For instance: 1 C 1 := V (x 00 x 11 ) V (x 00 )={(1 : a 01 : :1)} a 01 is a hyperbola,and V (x 00 x 11 )=C 1 {(0 :1 :0 :0)} {(0: 0 :1 :0)} and C 2 := V (x 10 x 01 ) V (x 00 )={(1 : a 01 : a 01 : a 2 01 )} is a parabola (in C 2 = {(1 :a :a :b)}) with V (x 10 x 01 )=C 2 {(0 :0 :0 :1)} (c) According to the prescription of Proposition 6.2, the equations for the point (1 : 0) (1 : 0) = V (x 1 ) V (y 1 ) CP 1 CP 1 CP 3 are: V (x 00 x 11 x 10 x 01,x 10,x 11,x 01,x 11 )=(1:0:0:0) which are more equations than we need, but that doesn t matter. Crucial Fact: Products of varieties do not have the product topology. Recall that a set Z X Y is closed for the product topology when Z is an intersection of sets of the form Z X Z Y. In C 1, for example, the closed sets are all finite (or C 1 ), so the only closed sets in C 2 for the product topology would be of the form (S X Y ) (X S Y ) S where S X,S Y and S are all finite. But there are plenty of other closed sets in the Zariski topology, such as the diagonal: =V (x y) C 2 4

5 Definitions: Let Y be an object in a category of topological spaces where products always exist and are always Cartesian products (as sets). (a) Consider the diagonal map δ : Y Y Y applying the universal property to the pair of identity morphisms p, q : Y Y, and let := δ(y ). Then Y Y is the diagonal, and Y is separated if Y Y is closed. (b) Y is proper if Y is separated and if for every object W, the projection morphism π W : W Y W is closed (i.e. maps closed sets to closed sets). Proposition 6.4: Y as above is separated if and only if for every X and pair of morphisms p, q : X Y, the following subset is closed: {p = q} := {x X p(x) =q(x)} X Proof: Notice first that = {π 1 = π 2 } Y Y, for the two projections π i : Y Y Y. Next, notice that the unique morphism (p, q) :X Y Y has the property that {p = q} =(p, q) 1 ( ). If is closed, then this is always closed since the morphisms are all continuous. Proposition 6.5: (a) If products have the product topology in the category, then Y is Hausdorff if and only if Y is separated. (b) Quasi-projective varieties are always separated. Proof: (a) Hausdorff means that if x y Y then there are open sets x U x,y U y Y with empty intersection. If Y is Hausdorff and Y Y has the product topology, then U x U y is open, so Y Y is covered by open sets U x U y and is closed. Conversely, if Y Y has the product topology and x y Y, then if is closed, it follows that (x, y) U Y Y and U is a union of open sets of the form U 1 U 2. One (or more) of these contains (x, y), which then can be declared to be U x U y. (b) It suffices to prove CP n is separated, since once CP n CP n is closed, then the diagonal in Y Y is closed for any Y CP n since Y Y has the induced topology. But the diagonal is given by explicit equations: =V ( {x ij x ji } ) S n,n = V ({x ij x kl x il x kj }) CP 2n+1 so of course it is closed. 5

6 Example: The closure of the parabola in the earlier example is the diagonal: =V (x 01 x 10 ) V (x 00 x 11 x 01 x 10 )=CP 1 CP 1 CP 3 and V (x 01 x 10 )=V(x 00 x 11 z 2 ) is the smooth projective conic from 4 (in the plane CP 2 = V (x01 x 10 ) with coordinates x 00,x 11,z = x 01 = x 10 ). Non-example: If X is any topological space together with a sheaf O X of regular functions that admits an open cover by finitely many affine varieties, then X is often called a prevariety. A quasi-projective variety is one example, but there are other prevarieties. For example, the affine line with 2 origins: X := (C 1 {0}) {0 } {0 } can be given a topology and sheaf O X so that the subsets U = X {0 } and U = X {0 } are open and isomorphic to C 1 with 0 (respectively 0 ) replacing 0 as the origin. X is not separated because (Proposition 6.4) the maps p : C 1 U X and q : C 1 U X give {p = q} = C 1 {0}, which isn t closed in C 1.SoXis not isomorphic to a quasi-projective variety. Let s begin our discussion of properness with a simple observation: Proposition 6.6: If Y CP n is a quasi-projective variety and Y Y, then Y isn t proper. Proof: Recall that all quasi-projective varieties are separated. If Y Y, consider: Z := (Y Y ) Y Y Then Z is closed in Y Y but π Y (Z) =Y Y isn t closed in Y because it is an open subset and Y is irreducible. So Y isn t proper. Remark: Recall that a basic open subset U Y of an affine variety is affine (Proposition 3.6). When we prove below that projective varieties are proper, it follows from Proposition 6.6 that no open subset U X of a projective variety (except X itself) is isomorphic to a projective variety. For this reason, proper varieties are often called complete. The two notions of compactness and properness agree when products have the product topology, once we dispose of some pathological cases. For example, in the category of sets with the discrete topology, all objects are proper because all subsets are closed, but no such infinite set is compact! 6

7 Proposition 6.7: (a) If products in a given category have the product topology, then compact implies proper. If there some object W with a chain W W 1 of open sets so that W i W isn t open, and if every open cover of Y has a countable subcover, then proper implies compact for Y. (b) Projective varieties are always proper. Proof of (a) Y is compact if it is Hausdorff and if every open cover Y = λ Λ U λ has a finite subcover. Hausdorff and separated agree by Proposition 6.5. Given a compact Y and a closed subset Z Y W, pick w W π W (Z). Then for each point y Y, we can find a product neighborhood U y W y of (y,w) which is disjoint from the closed set Z. Thus Y = y Y U y, so since Y is compact, a finite number U y1,..., U yr will also cover, and it follows that w r i=1 W y i, and moreover that this open subset of W is in the complement of π W (Z). Thus π W (Z) is closed and Y is proper. Conversely, suppose Y is proper and that the extra conditions hold. Given an open cover Y = λ Λ U λ, take a countable subcover Y = n=1 U λ n and consider the ascending chain of open sets V n = U λ1... U λn. Wewantto prove that this chain is eventually stationary, filling Y. In the product Y W (for the W in the Proposition) consider the set Z := Y W n=1 V n W n. I claim that if V n is not eventually Y, then π W (Z) = W W i, which isn t closed, contradicting the properness of Y. To see this, suppose that w W n. For each y Y, we can find a V n from ascending chain so that y V n, and then (w, y) W n V n so (w, y) Z. Since we can do this for every y, it follows that w π W (Z) and π W (Z) W W n. On the other hand, if w W n for some (minimal) n and if the increasing chain of V s isn t eventually stationary, it follows in particular that that there is a y Y V n for this n, and that y V m for all m n, so for this choice of y, (w, y) W n V n for all n. Thus(w, y) Z and w π W (Z). Remark: The conditions in (a) are very mild. In most reasonable settings, there are chains of open sets W W 1... with W n = {w} for any w W, so unless points are open (the discrete topology case!) almost any W will do. And every open cover has a countable subcover in most reasonable settings, such as open subsets of R n or manifolds patched from finitely many of them. For the proof of (b), we will need a new result from commutative algebra which will have many geometric consequences when applied to varieties. There will be several versions of this result. Here is the first one: 7

8 Nakayama s Lemma I: Suppose: A is a commutative ring with 1, M is a finitely generated A-module, and I A is an ideal that stabilizes M, in the sense that M = IM. Then there is an a =1+b A with b I such that am = 0 for all m M. (i.e. a annihilates the module M). Proof: Let m 1,..., m n generate M. By assumption, we can solve: n m i = b ij m j j=1 with b ij B. This tells us that the matrix I n B annilates each m j, hence (I n B)m = 0 for each m M (writing m = a j m j ). By Cramer s rule, a = det(i n B) satisfies am = 0 for all m M, and evidently a =1+b for some b I. Proof of (b) (Due to Grothendieck) If X CP n is a projective variety, then closed subsets of W X are closed in W CP n, so it suffices to prove that CP n is proper. We may also assume W is affine. If π W (Z) weren t closed for some Z W CP n, then π U (Z (U CP n )) = π W (Z) U, wouldn t be closed in U for some U in an affine open cover of W. Finally, if W C m is closed, we may replace W by C m, since subsets of W CP n and W are closed if and only if they are closed in C m CP n and C m respectively. We will label points of C m CP n by: (a, b) :=((a 1,..., a m ), (b 0 :... : b n )) C m CP n and the coordinate rings by C[x] =C[x 1,..., x m ] and C[y] =C[y 0,..., y n ]. We are trying to show that if Z C m CP n is closed, then: π C m(z) ={a C m (a, b) Z for some b CP n } C m is also closed. We will do this in two steps. 8

9 Step 1: Define a partial grading on C[x, y] by degree in the y-variables, so: C[x, y] = C[x, y] d := C[x] C C[y] d d=0 d=0 and homogeneous polynomials and ideals in the usual way for this grading. For each F C[x, y] d, it makes sense to ask whether F (a, b) = 0 or not, so we may define V (I) C m CP n for homogeneous ideals I. Our first step is to show that every closed subset Z C m CP n is a V (I) for some such I. Proof: Cover CP n by open sets U i = C m C n, with coordinate rings: C[U i ] = C[x 1,..., x m, y 0 y i,..., y n y i ], Define the homogeneous ideal I associated to Z by: I d := {F C[x, y] d F y d i I(Z U i ) C[U i ] for all i} and define I := d=0 I d. This is the homogeneous ideal we will use. It is immediate from the definition that Z V (I), since F I(Z U yi d i ) if and only if F vanishes at every point of Z U i. For the other inclusion, take f I(Z U i ). Pick d large enough so that G := yi df C[x, y] d. We do not know that G I d, but each of the polynomials f j := G vanishes on the set Z U yj d i U j,so y i y j f j I(Z U j ) because this function vanishes on Z U i U j and on Z (U j U i ). So for any e>d,wedohavef = y e i f I e satisfying F (x 1,..., x m, y 0 y i,..., yn y i )=f. If (a, b) Z then (a, b) U i for some i, and so there is an f I(Z U i ) such that f(a, b 0 bi,..., b n bi ) 0. AnyF I e constructed above will also satisfy F (a, b) 0, so we have proved that Z = V (I). Step 2. For the homogeneous ideal I from Step 1, π C m(z) =V (I 0 ) so in particular the projection is closed, proving (b). 9

10 Proof: Again, it is clear that π C m(z) V (I 0 ). Suppose a π C m(z) and let m a = x 1 a 1,..., x m a m. The closed set πc 1 m(a) ={a} CPn is disjoint from Z, so intersecting with U i gives disjoint closed sets Z U i and {a} U i in U i with ideals I(Z U i ) and m a C[U i ] respectively. Thus I(Z U i )+ m a = C[U i ] by the Hilbert Nullstellensatz. In other words, for each i =0,..., n there exist f i I(Z U i ),m i,j m a and g i,j C[U i ] such that f i + j g i,j m i,j =1. Multiplying through by sufficiently large powers e i of the y i, we get: F i + j G i,j m i,j = y e i i for F i I ei,g i,j C[x, y] ei (and F i I ei as in Step 1). If we further take d e i, then this gives us I d + m a C[x, y] d = C[x, y] d since every monomial of degree d contains at least one y i to the power e i. Now consider the C[x]-module M := C[x, y] d /I d. The equality above tells us m a M = M, hence by Nakayama s Lemma there is an f C[x] m a that annihilates M, i.e. so that fc[x, y] d I d. Thus fyi d I d for all i and it follows that f I 0. Thus we have found an f I 0 with f(a) 0, as desired. Example: In C 2 CP 1, consider: Z = V ( x 1 y 1 y 0 x 2, y 0 y 1 ) When we write this as a union of the two open sets U 0,U 1, we get: Z = {(0,a 2, 1, 0)} {(a 1, 0, 0, 1)} so π C 2(Z) =V (x 1 x 2 ) which isn t readily apparent from the equations for Z. But the point is that: x 2 1 (y 0y 1 ) y 0 x 1 (x 1 y 1 y 0 x 2 )=y 2 0 x 1x 2, and so x 1 x 2 I 0. x 2 2 (y 0y 1 )+y 1 x 2 (x 1 y 1 y 0 x 2 )=y 2 1 x 1x 2 10

11 Exercises If C[X] = C[x 1,..., x m ]/ {f i } and C[Y ] = C[y 1,..., y n ]/ {g j } as in Proposition 6.1, finish the proof that I(X Y )= {f i } {g j } and conclude that: C[X Y ] = C[X] C C[Y ] for any pair X, Y of affine varieties. 2. If X CP n is a projective variety, then the affine cone C(X) C n+1 is the union of the lines parametrized by the points of X. In other words: (b 0,..., b n ) C(X) (b 0 :... : b n ) X or (b 0,..., b n )=(0,..., 0) Prove that if x i is chosen so that X V (x i ), then: C(X) V (x i ) = (X V (x i )) (C 1 0) hence that C(X) 0 is covered by such open affine sets. 3. (a) Prove that CP m CP n CP m+n when m, n > 0. (b) Given homogeneous coordinate rings C[x 0,..., x n ] and C[y 0,..., y n ]on CP m and CP n, prove that the closed subsets of CP m CP n are all of the form V (I), where I C[x 0,..., x m,y 0,..., y m ]isabihomogeneous ideal, i.e. I is generated by bihomogeneous polynomials F (x, y) (of bidegrees (d, e)) for the double grading: C[x 0,..., x m,y 0,..., y m ]= C[x 0,..., x m ] d C C[y 0,..., y n ] e d=0 e=0 (c) Express the twisted cubic curve as a bihomogeneous hypersurface in CP 1 CP 1. (d) Which of the irreducible bihomogeneous hypersurfaces V (F ) CP 1 CP 1 are intersections of CP 1 CP 1 = S 1,1 CP 3 with hypersurfaces in CP 3? 11

### This 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

### Summer 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

### Math 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

### 9. Birational Maps and Blowing Up

72 Andreas Gathmann 9. Birational Maps and Blowing Up In the course of this class we have already seen many examples of varieties that are almost the same in the sense that they contain isomorphic dense

### Institutionen 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...............................

### 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

### Math 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

### 10. 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

### Exploring the Exotic Setting for Algebraic Geometry

Exploring the Exotic Setting for Algebraic Geometry Victor I. Piercey University of Arizona Integration Workshop Project August 6-10, 2010 1 Introduction In this project, we will describe the basic topology

### Course 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...............................

### Yuriy 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

### Math 418 Algebraic Geometry Notes

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

### ALGEBRAIC 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.

### 3. 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

### div(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.

### Algebraic 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 ) :

### Algebraic Geometry (Math 6130)

Algebraic Geometry (Math 6130) Utah/Fall 2016. 2. Projective Varieties. Classically, projective space was obtained by adding points at infinity to n. Here we start with projective space and remove a hyperplane,

### Resolution of Singularities in Algebraic Varieties

Resolution of Singularities in Algebraic Varieties Emma Whitten Summer 28 Introduction Recall that algebraic geometry is the study of objects which are or locally resemble solution sets of polynomial equations.

### D-MATH Algebraic Geometry FS 2018 Prof. Emmanuel Kowalski. Solutions Sheet 1. Classical Varieties

D-MATH Algebraic Geometry FS 2018 Prof. Emmanuel Kowalski Solutions Sheet 1 Classical Varieties Let K be an algebraically closed field. All algebraic sets below are defined over K, unless specified otherwise.

### 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.

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

### Algebraic 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

### ALGEBRAIC 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

### Algebraic 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

### COMPLEX 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

### CHEVALLEY 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

### MATH 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

### 12. 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

### Basic facts and definitions

Synopsis Thursday, September 27 Basic facts and definitions We have one one hand ideals I in the polynomial ring k[x 1,... x n ] and subsets V of k n. There is a natural correspondence. I V (I) = {(k 1,

### Algebraic 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......................

### Algebraic 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

### FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 27

FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 27 RAVI VAKIL CONTENTS 1. Proper morphisms 1 2. Scheme-theoretic closure, and scheme-theoretic image 2 3. Rational maps 3 4. Examples of rational maps 5 Last day:

### Math 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

### CHAPTER 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

### Boolean Algebras, Boolean Rings and Stone s Representation Theorem

Boolean Algebras, Boolean Rings and Stone s Representation Theorem Hongtaek Jung December 27, 2017 Abstract This is a part of a supplementary note for a Logic and Set Theory course. The main goal is to

### 1. Algebraic vector bundles. Affine Varieties

0. Brief overview Cycles and bundles are intrinsic invariants of algebraic varieties Close connections going back to Grothendieck Work with quasi-projective varieties over a field k Affine Varieties 1.

### Algebraic varieties. Chapter A ne varieties

Chapter 4 Algebraic varieties 4.1 A ne varieties Let k be a field. A ne n-space 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

### 3 Lecture 3: Spectral spaces and constructible sets

3 Lecture 3: Spectral spaces and constructible sets 3.1 Introduction We want to analyze quasi-compactness properties of the valuation spectrum of a commutative ring, and to do so a digression on constructible

### 10. 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.

### MAT4210 Algebraic geometry I: Notes 2

MAT4210 Algebraic geometry I: Notes 2 The Zariski topology and irreducible sets 26th January 2018 Hot themes in notes 2: The Zariski topology on closed algebraic subsets irreducible topological spaces

### 4. Images of Varieties Given a morphism f : X Y of quasi-projective 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 quasi-projective 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

### Algebraic 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

### Lecture 3: Flat Morphisms

Lecture 3: Flat Morphisms September 29, 2014 1 A crash course on Properties of Schemes For more details on these properties, see [Hartshorne, II, 1-5]. 1.1 Open and Closed Subschemes If (X, O X ) is a

### SECTION 2: THE COMPACT-OPEN TOPOLOGY AND LOOP SPACES

SECTION 2: THE COMPACT-OPEN TOPOLOGY AND LOOP SPACES In this section we will give the important constructions of loop spaces and reduced suspensions associated to pointed spaces. For this purpose there

### MA 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

### Introduction 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................................

### Algebraic 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

### Math 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

### NONSINGULAR CURVES BRIAN OSSERMAN

NONSINGULAR CURVES BRIAN OSSERMAN The primary goal of this note is to prove that every abstract nonsingular curve can be realized as an open subset of a (unique) nonsingular projective curve. Note that

### Introduction 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

### Math 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

### 14. Rational maps It is often the case that we are given a variety X and a morphism defined on an open subset U of X. As open sets in the Zariski

14. Rational maps It is often the case that we are given a variety X and a morphism defined on an open subset U of X. As open sets in the Zariski topology are very large, it is natural to view this as

### ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 9: SCHEMES AND THEIR MODULES.

ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 9: SCHEMES AND THEIR MODULES. ANDREW SALCH 1. Affine schemes. About notation: I am in the habit of writing f (U) instead of f 1 (U) for the preimage of a subset

### MATH 221 NOTES BRENT HO. Date: January 3, 2009.

MATH 22 NOTES BRENT HO Date: January 3, 2009. 0 Table of Contents. Localizations......................................................................... 2 2. Zariski Topology......................................................................

### PROBLEMS, MATH 214A. Affine and quasi-affine varieties

PROBLEMS, MATH 214A k is an algebraically closed field Basic notions Affine and quasi-affine 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

### A ne Algebraic Varieties Undergraduate Seminars: Toric Varieties

A ne Algebraic Varieties Undergraduate Seminars: Toric Varieties Lena Ji February 3, 2016 Contents 1. Algebraic Sets 1 2. The Zariski Topology 3 3. Morphisms of A ne Algebraic Sets 5 4. Dimension 6 References

### A 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

### Topological properties

CHAPTER 4 Topological properties 1. Connectedness Definitions and examples Basic properties Connected components Connected versus path connected, again 2. Compactness Definition and first examples Topological

### where 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

### Rings 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...

### k 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

### 1 Topology Definition of a topology Basis (Base) of a topology The subspace topology & the product topology on X Y 3

Index Page 1 Topology 2 1.1 Definition of a topology 2 1.2 Basis (Base) of a topology 2 1.3 The subspace topology & the product topology on X Y 3 1.4 Basic topology concepts: limit points, closed sets,

### ALGEBRAIC 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

### Formal power series rings, inverse limits, and I-adic completions of rings

Formal power series rings, inverse limits, and I-adic completions of rings Formal semigroup rings and formal power series rings We next want to explore the notion of a (formal) power series ring in finitely

### 4.4 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)

### Algebraic Geometry Spring 2009

MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry

### MATH32062 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,

### My way to Algebraic Geometry. Varieties and Schemes from the point of view of a PhD student. Marco Lo Giudice

My way to Algebraic Geometry Varieties and Schemes from the point of view of a PhD student Marco Lo Giudice July 29, 2005 Introduction I began writing these notes during the last weeks of year 2002, collecting

### 3 Hausdorff and Connected Spaces

3 Hausdorff and Connected Spaces In this chapter we address the question of when two spaces are homeomorphic. This is done by examining two properties that are shared by any pair of homeomorphic spaces.

### Algebraic Geometry Spring 2009

MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry

### Projective 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

### Pure 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

### GLUING 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

### QUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday 10 February 2004 (Day 1)

Tuesday 10 February 2004 (Day 1) 1a. Prove the following theorem of Banach and Saks: Theorem. Given in L 2 a sequence {f n } which weakly converges to 0, we can select a subsequence {f nk } such that the

### 12. Linear systems Theorem Let X be a scheme over a ring A. (1) If φ: X P n A is an A-morphism 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 A-morphism then L = φ O P n A (1) is an invertible sheaf on X, which is generated by the global sections s 0, s

### F 1 =. Setting F 1 = F i0 we have that. j=1 F i j

Topology Exercise Sheet 5 Prof. Dr. Alessandro Sisto Due to 28 March Question 1: Let T be the following topology on the real line R: T ; for each finite set F R, we declare R F T. (a) Check that T is a

### 2. 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 so-called prime and maximal ideals. Let

### MATH 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

### Math 210B. Artin Rees and completions

Math 210B. Artin Rees and completions 1. Definitions and an example Let A be a ring, I an ideal, and M an A-module. In class we defined the I-adic completion of M to be M = lim M/I n M. We will soon show

### Notas de Aula Grupos Profinitos. Martino Garonzi. Universidade de Brasília. Primeiro semestre 2018

Notas de Aula Grupos Profinitos Martino Garonzi Universidade de Brasília Primeiro semestre 2018 1 Le risposte uccidono le domande. 2 Contents 1 Topology 4 2 Profinite spaces 6 3 Topological groups 10 4

### Synopsis of material from EGA Chapter II, 5

Synopsis of material from EGA Chapter II, 5 5. Quasi-affine, quasi-projective, proper and projective morphisms 5.1. Quasi-affine morphisms. Definition (5.1.1). A scheme is quasi-affine if it is isomorphic

### Extended Index. 89f depth (of a prime ideal) 121f Artin-Rees 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

### FUNCTIONAL ANALYSIS LECTURE NOTES: COMPACT SETS AND FINITE-DIMENSIONAL SPACES. 1. Compact Sets

FUNCTIONAL ANALYSIS LECTURE NOTES: COMPACT SETS AND FINITE-DIMENSIONAL SPACES CHRISTOPHER HEIL 1. Compact Sets Definition 1.1 (Compact and Totally Bounded Sets). Let X be a metric space, and let E X be

### FILTERED 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

### Topological vectorspaces

(July 25, 2011) Topological vectorspaces Paul Garrett garrett@math.umn.edu http://www.math.umn.edu/ garrett/ Natural non-fréchet spaces Topological vector spaces Quotients and linear maps More topological

### Math General Topology Fall 2012 Homework 6 Solutions

Math 535 - General Topology Fall 202 Homework 6 Solutions Problem. Let F be the field R or C of real or complex numbers. Let n and denote by F[x, x 2,..., x n ] the set of all polynomials in n variables

### AN 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,

### 2. 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.

### 1 Introduction. 2 Categories. Mitchell Faulk June 22, 2014 Equivalence of Categories for Affine Varieties

Mitchell Faulk June 22, 2014 Equivalence of Categories for Affine Varieties 1 Introduction Recall from last time that every affine algebraic variety V A n determines a unique finitely generated, reduced

### 1 Categorical Background

1 Categorical Background 1.1 Categories and Functors Definition 1.1.1 A category C is given by a class of objects, often denoted by ob C, and for any two objects A, B of C a proper set of morphisms C(A,

### Math 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)

### Chapter 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

### THE CLOSED-POINT ZARISKI TOPOLOGY FOR IRREDUCIBLE REPRESENTATIONS. K. R. Goodearl and E. S. Letzter

THE CLOSED-POINT ZARISKI TOPOLOGY FOR IRREDUCIBLE REPRESENTATIONS K. R. Goodearl and E. S. Letzter Abstract. In previous work, the second author introduced a topology, for spaces of irreducible representations,

### SCHEMES. David Harari. Tsinghua, February-March 2005

SCHEMES David Harari Tsinghua, February-March 2005 Contents 1. Basic notions on schemes 2 1.1. First definitions and examples.................. 2 1.2. Morphisms of schemes : first properties.............

### MAS 6396 Algebraic Curves Spring Semester 2016 Notes based on Algebraic Curves by Fulton. Timothy J. Ford April 4, 2016

MAS 6396 Algebraic Curves Spring Semester 2016 Notes based on Algebraic Curves by Fulton Timothy J. Ford April 4, 2016 FLORIDA ATLANTIC UNIVERSITY, BOCA RATON, FLORIDA 33431 E-mail address: ford@fau.edu

### NOTES 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

### INVERSE LIMITS AND PROFINITE GROUPS

INVERSE LIMITS AND PROFINITE GROUPS BRIAN OSSERMAN We discuss the inverse limit construction, and consider the special case of inverse limits of finite groups, which should best be considered as topological

### ne 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

### CALCULUS ON MANIFOLDS

CALCULUS ON MANIFOLDS 1. Manifolds Morally, manifolds are topological spaces which locally look like open balls of the Euclidean space R n. One can construct them by piecing together such balls ( cells