ALGEBRAIC FUNCTIONS ON ALGEBRAIC VARITIES

Size: px
Start display at page:

Download "ALGEBRAIC FUNCTIONS ON ALGEBRAIC VARITIES"

Transcription

1 ALGEBRAIC FUNCTIONS ON ALGEBRAIC VARITIES (1) Let k be an algebraically-closed field throughout this week. (2) The ideals of k [X 1,...,X n ] form a lattice (a) The radical ideals form a sublattice (b) The prime ideals for a sublattice of the radical ideals (c) Every radical ideal is the intersection of a finite number of prime ideals (3) Let E = k n, Euclidean space (a) The polynomials in k [X 1,...,X n ] are functions E k (b) Algebraic subsubsets of E are zeros of one or more functions (i) If S k [X 1,...,X n ],thenv (S) ={x E,s(x) =0all s S} (ii) If S T k [X 1,...,X n ] then V (T ) V (S) (iii) If J =(S) then V (J) =V (S) (iv) For every subset S there exists a finite set T S of polynomials such that V (S) =V (T ). (A) T is a finite set of generators for the ideal generated by S. (v) finite unions and and infinite intersections of algebraic subsets are algebraic subsets (vi) \ V (J k )=V ( P k J k) k (vii) V (J 1 ) V (J 2 )=V (J 1 J 2 ) (viii) V (0) = E, V (k [X 1,...,X n ]) = φ. (c) If W is an algebraic subset of E then I (W )={p k [X 1,...,X n ]:p(x) =0, all x W } (i) If W 1 W 2 are algebraic subsets of E then I (W 2 ) I (W 1 ) (ii) I (W 1 W 2 )=I(W 1 ) I (W 2 ) (iii) p I (W 1 )+I (W 2 )=I(W 1 W 2 ) (iv) I (W ) is a radical ideal (nullstellensatz) (v) V (I (W )) = W (vi) I (V (S)) = p (S), I (V (J)) = J.(nullstellensatz) (vii) W is a single point of E = k n iff I (W ) k [X 1,...,X n ] is a maximal ideal (nullstellensatz) (4) The radical ideals are in 1 1 correspondence with the algebraic subsets of E (5) An algebraic subset is irreducible if it is not the union of two strictly smaller algebraic subsets. Irreducible algebraic subsets are called subvarieties. (a) Otherwise the algebraic subset is reducible (b) Every algebraic subset is the union of a finite number of irreducible algebraic subsets (c) Every radical ideal is the intersection of a finite number of prime ideals (6) A topology on E is a collection of subsets of E, called closed subsets, satisfying certain conditions (a) The collection of algebraic subsets of E are the closed subsets for a topology on E called the Zariski topology. 1

2 2 ALGEBRAIC FUNCTIONS ON ALGEBRAIC VARITIES (b) In the Zariski topology, every infinite intersection of closed sets is a finite intersection (c) In the Zariski topology, an algebraic subset is irreducible in the topological sense (not the union of two proper closed subsets) iff it is an irreducible as an algebraic subset (not the union of two proper algebraic subsets). (i) Proof: algebraic subset = closed set (ii) Corollary: an algebraic subset is irreducible iff all non-empty open subsets are dense iff no two non-empty open subsets have empty intersection (iii) Example: in k 2 the algebraic subset W = V (X 1 X 2 )=V (X 1 ) V (X 2 ) has a non-empty, non-dense open subset U = W V (X 1 ).TheclosureofU is U = V (X 2 ). (d) The Zariski topology on k hasclosedsetsconsistingoffinite sets and all of k (e) Polynomials are continuous functions E k in the Zariski topology (but not the only ones). (7) In everything we have done so far, all algebraic subsets are algebraic subsets of E = k n and all ideals are ideals of k [X 1,...,X n ]. (a) Beware: generalization coming. (8) Remark: since our ground field is infinite, every polynomial represents a different function E k. (9) Useful abstraction: let E be a set and R a ring. The set of R-valued functions on E is the set of functions f : E R. (a) The R-valued functions on E is a ring if we use pointwise addition and multiplication. (10) A polynomial function on an algebraic subset W E is the restriction of a polynomial p : E k to W. (a) If p and q are polynomials, then they represent the same polynomial function on W iff p q I (W ). (b) So different polynomials can represent the same polynomial function on W. (c) Let Γ (W ) be the set of polynomial functions on W. (i) Γ (E) =k [X 1,...,X n ]. (ii) Γ (W ) is a subring of the ring of k-valued functions on W (iii) There is a natural map, restriction, mapping k [X 1,...,X n ] to the k-valued functions on W. The image is Γ (W ) and the kernal is I (W ). π 0 I (W ) k [X 1,...,X n ] Γ (W ) 0 (iv) Thus Γ (W ) = k [X 1,...,X n ] is a ring. I (W ) (v) Γ (W ) is a finitely generated k-algebra. (vi) Γ (W ) has no nilpotent elements (vii) Γ (W ) is an integral domain iff I (W ) is prime iff W is irreducible (viii) Γ (W ) is a field iff I (W ) is maximal iff W is a single point

3 ALGEBRAIC FUNCTIONS ON ALGEBRAIC VARITIES 3 (11) Every finitely-generated k-algebra without nilpotents is the ring of polynomial functions for some algebraic subset (a) Given k [x 1,...,x n ],define π : k [X 1,...,X n ] k [x 1,...,x n ].Then ker (π) is a radical ideal, and if W is its algebraic subset then Γ (W ) = k [x 1,...,x n ]. (12) Fix an algebraic subset W,of E = k n. (a) There are 1 1 correspondences between the following sets: (i) radical ideals of Γ (W ) (ii) radical ideals of k [X 1,...,X n ] containing I (V ) (iii) algebraic subsets of E contained in W (b) The restriction of the Zariski topology to W is called the Zariski topology on W. The closed sets are the subvarieties of W. (c) Polynomial functions in Γ (W ) are continuous functions W k in the Zariski topology (d) If X W then I W (X) ={p Γ (W ):p(x) =0, all x X} (i) I W (X) =π (I (X)). (ii) If X Y W then I W (Y ) I W (X). (iii) I W (X) is a radical ideal (e) If S Γ (W ) is an ideal, define V W (S) ={x W : p (x) =0all p S} (i) V W (S) =V π 1 (S) (ii) if S T Γ (W ) then V W (T ) V W (S) (iii) If X is an algebraic subset then V W (I W (X)) = X (iv) If S is a radical ideal I W (V W (S)) = S (f) If X W are algebraic subsets of E, then there is a natural map Γ (W ) Γ (X). Infact: 0 I W (X) Γ (W ) Γ (X) 0 so Γ (X) = Γ (W ) I W (X). (g) X is a single point of W iff I W (X) is a maximal ideal of Γ (W ) (i) Serre s bright idea: even if k is not algebraically closed, you can use the set of maximal ideals of a, finitely generated k-algebra without nilpotents A as the ring of polynomial functions on a topological space whose point set is the collection of maximal ideals of A. (ii) Grothendieck extended this idea to include all of the prime ideals of A in the point set. Only the maximal ideals are closed points; the other prime ideals are non-closed points in the set. (h) X is irreducible in W (not the union of two proper closed subsets = not the union of two proper algebraic subsets) iff I W (X) is a prime ideal of Γ (W ). (i) Example: in k [x, y] = k [X, Y ],considertheideal(x). Theringisthe (XY ) ring of polynomial functions on the two axes in k 2,and(x) is the ideal of one of the axes. (13) Let s do something very general, the apply the idea to algebraic subsets. (a) If X and Y are sets, define Γ (X, Y ) as the set of all functions from X to Y. (i) If Y = R is a ring then Γ (X i,r) is a ring and an R-algebra.

4 4 ALGEBRAIC FUNCTIONS ON ALGEBRAIC VARITIES (ii) In many contexts X and Y will have some special properties or structure, and we will only consider maps respecting that structure. (iii) For example, if X and Y are topological spaces, we may restrict Γ (X, Y ) to continuous maps (iv) If X is a manifold and Y = R we may restrict Γ (X, Y ) to differentiable functions. (v) In the following, X will be an algebraic set and Y = k and Γ (X, k) =Γ (X) will be the set of polynomial functions on X. (b) Suppose X 1,X 2 and Y are sets and f : X 1 X 2 is a function. Then f induces a function f : Γ (X 2,Y) Γ (X 1,Y) as follows: f (α) =α f f g (i) If X 1 X 2 X 3 then (g f) = g f (draw diagram) (ii) If Y = R isaringthenγ (X i,r) are R-algebras and f is an R-algebra homomorphism. (iii) If Γ (X i,y) is a set of special functions, we have to be sure that f maps special functions to special functions. (A) If X i,y are topological spaces and f is continuous, then f maps continuous functions to continuous functions because the composition of continuous functions is continuous. (iv) If Γ (X, k) means the polynomial functions on X only, then what kind of function f : X 1 X 2 has the property that f takes polynomial functions on X 2 into polynomial functions on X 1? (14) A polynomial map P : k n k m is an orders set P =(p 1,...,p m ) of polynomials p i k [X 1,...,X n ].Thatis,ifx k n then P (x) =y =(y 1,...,y m ) where y i = p i (x) =p i (x 1,...,x n ).Inotherwords,P k [X 1,...,X n ] m. (a) Fix a point y k m. Then P 1 (y) ={x k n : P (x) =y} is an algebraic subset of k n,possiblyempty. (b) If W is an algebraic subset of k m then P 1 (W )={x k n : P (x) W }.is an algebraic subset of k n,possiblyempty. (c) Example: P : k k 3, k (x) = x, x 2,x 3, the image is a twisted cubic. (d) A polynomial map P induces a k-algebra homomorphism P : k [Y 1,...,Y m ] k [X 1,...,X n ].Infact,P (Y i )=p i, and that s all you have to know. (i) Proof: let F (k n ) be the k-valued functions on k n.thenk[y 1,...,Y m ] is a subring of F (k m ) and k [X 1,...,X n ] is a subring of F (k n ). We know that P : F (k m ) F (k n ) is a k-algebra homomorphism (13.b.i), so it suffices to show that P maps polynomials topolynomials. Forthatitsuffices to show that P (Y i ) is a polynomial. But P (Y i )=Y i P = p i. (e) Corollary: if k n P k m Q k t then (Q P ) = Q P (f) Every homomorphism ϕ : k [Y 1,...,Y m ] k [X 1,...,X n ] comes from a polynomial map P : k n k m.

5 ALGEBRAIC FUNCTIONS ON ALGEBRAIC VARITIES 5 (i) Given ϕ, considerp =(ϕ (Y 1 ),...,ϕ(y m )). This is a polynomial map, and we will show that P = ϕ. Letq k [Y 1,...,Y m ]: P (q(y i ) = q (P (Y i )) = q (Y i P ) = q (ϕ (Y i )) = ϕ (q (Y i )) (ii) Corollary: the set of homomorphisms k [Y 1,...,Y m ] k [X 1,...,X n ] is equivalent to the set of polynomial maps k n k m. (15) Let W 1 be an algebraic subset of k n and W 2 an algebraic subset of k m and P : k n k m a polynomial map. If P (W 1 ) W 2, then we say (what else) that P : W 1 W 2 is an algebraic map from W 1 to W 2. (a) Note that an algebraic map must be defined on the Euclidean spaces containing W 1 and W 2. (b) If we think of the polynomial functions on W i as Γ (W i ) where Γ (W 1 )= k [x 1,...,x n ] = k [X 1,...,X n ] and Γ (W 2 )=k[y 1,...,y m ] I (W 1 ) = k [Y 1,...,Y m ], I (W 2 ) then P (y i )=p i (x 1,...,x n ). (c) The map P induces a k-algebra homomorphism: P : Γ (W 2 ) Γ (W 1 ). That is, the map P defined above from all k-valued functions on W 2 to all k-valued functions on W 1 restricts to a k-algebra homomorphism Γ (W 2 ) Γ (W 1 ). (i) The proof must be obvious (ii) Proof: let F (W i ) be the k-valued functions on W i.thenk[y 1,...,y m ] is a subring of F (W 2 ) and k [x 1,...,x n ] is a subring of F (W 1 ). We know that P : F (W 2 ) F (W 1 ) is a k-algebra homomorphism (13.b.i), so it suffices to show that P sends polynomial maps to polynomial maps. For that it suffices to show that P (y i ) is a polynomial map. But P (y i )=p i (x 1,...,x n ),a polynomial map. (iii) You have to read all this very carefully and remember that all the symbols X i,x i,y i,y i are functions or maps from their respective domains to k for it to make sense. Otherwise it is just abstract nonsense or (in the words of the textbook author) rubbish. But it s not rubbish these ideas are simple, deep and important. (16) In 15 we proved that a polynomial map P : W 1 W 2 induces a homomorphism P : Γ (W 2 ) Γ (W 1 ). Nowwewanttousemorepowerful algebraic tools to prove that the set of polynomial maps is equivalent to the set of homomorphisms. That is, the assigment P Ã P is 1 1 and onto from the set of polynomial maps W 1 W 2 to the set of homomorphisms Γ (W 2 ) Γ (W 1 ). In the process of demonstrating this result we will construct the inverse equivalence. (a) Theorem: Let P : k n k m be a polynomial map, and let W 1 k n, W 2 k m be algebraic subsets. Then P (W 1 ) W 2 iff P (I (W 2 )) I (W 1 ).

6 6 ALGEBRAIC FUNCTIONS ON ALGEBRAIC VARITIES (i) Proof: P (W 1 ) W 2 P (x) W 2,allx W 1 q(p (x)) = 0, allq I (W 2 ),allx W 1 (P (q)) (x) =0,allq I (W 2 ),allx W 1 P (q) I (W 1 ),allq I(W 2 ) P (I (W 2 )) I (W 1 ) (b) Corollary: P (W 1 ) W 2 iff P lifts a homomorphism Γ (W 2 ) Γ (W 1 ). (i) Proof: 0 I (W 2 ) k [Y 1,...,Y m ] Γ (W 2 ) 0 P P ϕ 0 I (W 1 ) k [X 1,...,X n ] Γ (W 1 ) 0 By standard theorems of algebra, P induces a homomorphism ϕ iff P (I (W 2 )) I (W 1 ). (c) So every polynomial map P : W 1 W 2 induces a k-algebra homomorphism P : Γ (W 1 ) Γ (W 1 ), and every such homomorphism comes from a polynomial map. It remains to show that the assignment (polynomial maps) to (k-algebra homomorphisms) is 1 1. The choice of P in the argument above is not unique, and it is not obvious that P is unique. That is what we have to show. To do this we construct an inverse map from (k-algebra homomorphisms) to (polynomial maps) viaathirdobject. (d) If A is a finitely generated k-algebra, let MaxSpec(A) be the set of maximal ideals of A. Sincek is algebraically closed, by the Nullstellensatz there is a 1 1 correspondence between MaxSpec(Γ (W )) and W given by (a 1,...,a n ) (x 1 a 1,...,x n a n ). (i) Let W 1 k n, W 2 k m be algebraic subsets, and let ϕ : Γ (W 2 ) Γ (W 1 ) be a k-algebra homomorphism. Then ϕ induces a map MaxSpec(Γ (W 1 )) MaxSpec(Γ (W 2 )) as follows: (A) Let m MaxSpec(Γ (W 1 )). Iclaimϕ 1 (m) MaxSpec(W 2 ). because we have the induced injection: k Γ (W 2) ϕ 1 (m) Γ (W 1) m = k Therefore Γ (W 2) ϕ 1 (m) = k is a field so ϕ 1 (m) is a maximal ideal of Γ (W 2 ). Therefore the homomorphism ϕ induces amapϕ 1 : MaxSpec(Γ (W 1 )) MaxSpec(W 2 ).Next we show that from ϕ 1 we can construct an algebraic map Φ : W 1 W 2 (B) Suppose (a 1,...,a n ) W 1.Ifϕ 1 ((x 1 a 1,...,x n a n )) = (y 1 b 1,...,y m b m ) for some point (b 1,...,b m ) W 2 then ϕ (y i b i ) (x 1 a 1,...,x n a n ) or ϕ (y i )(a 1,...,a n )= b i. Thus Φ =(ϕ (y 1 ),...,ϕ(y m )) : W 1 W 2. Moreover Φ is an algebraic map. For each ϕ (y i ) k [x 1,...,x n ] lifts to a polynomial p i k [X 1,...,X n ],andp =(p 1,...,p n ):

7 ALGEBRAIC FUNCTIONS ON ALGEBRAIC VARITIES 7 k n k m is an algebraic map that restricts to Φ : W 1 W 2.ThemapP is not unique, but Φ is the unique algebraic map W 1 W 2 induced by ϕ. (e) We leave it to the diligent reader to verify that the correspondences between algebraic maps and homomorphisms: P Ã P ϕ Ã Φ are inverse to each other. That is, if you perform one transformation followed by the other, you get back where you started from.

Summer Algebraic Geometry Seminar

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

More information

ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 2: HILBERT S NULLSTELLENSATZ.

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 information

Math 418 Algebraic Geometry Notes

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

More information

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 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 information

ALGEBRAIC GROUPS. Disclaimer: There are millions of errors in these notes!

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.

More information

Algebraic varieties. Chapter A ne varieties

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

More information

2. Prime and Maximal Ideals

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

More information

10. Noether Normalization and Hilbert s Nullstellensatz

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.

More information

Yuriy Drozd. Intriduction to Algebraic Geometry. Kaiserslautern 1998/99

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

More information

MATH 631: ALGEBRAIC GEOMETRY: HOMEWORK 1 SOLUTIONS

MATH 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 information

Math 203A - Solution Set 1

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

More information

Math 203A - Solution Set 1

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

More information

4.4 Noetherian Rings

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)

More information

Math 120 HW 9 Solutions

Math 120 HW 9 Solutions Math 120 HW 9 Solutions June 8, 2018 Question 1 Write down a ring homomorphism (no proof required) f from R = Z[ 11] = {a + b 11 a, b Z} to S = Z/35Z. The main difficulty is to find an element x Z/35Z

More information

Algebraic geometry of the ring of continuous functions

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

More information

ne varieties (continued)

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

More information

The most important result in this section is undoubtedly the following theorem.

The 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 information

ALGEBRAIC GEOMETRY (NMAG401) Contents. 2. Polynomial and rational maps 9 3. Hilbert s Nullstellensatz and consequences 23 References 30

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

More information

Extended Index. 89f depth (of a prime ideal) 121f Artin-Rees Lemma. 107f descending chain condition 74f Artinian module

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

More information

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. 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 information

MATH 8253 ALGEBRAIC GEOMETRY WEEK 12

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

More information

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:

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

More information

Math 203A - Solution Set 3

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)

More information

4.5 Hilbert s Nullstellensatz (Zeros Theorem)

4.5 Hilbert s Nullstellensatz (Zeros Theorem) 4.5 Hilbert s Nullstellensatz (Zeros Theorem) We develop a deep result of Hilbert s, relating solutions of polynomial equations to ideals of polynomial rings in many variables. Notation: Put A = F[x 1,...,x

More information

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 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 information

Honors 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 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 information

Course 311: Michaelmas Term 2005 Part III: Topics in Commutative Algebra

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...............................

More information

1. Group Theory Permutations.

1. Group Theory Permutations. 1.1. Permutations. 1. Group Theory Problem 1.1. Let G be a subgroup of S n of index 2. Show that G = A n. Problem 1.2. Find two elements of S 7 that have the same order but are not conjugate. Let π S 7

More information

Rings and Fields Theorems

Rings and Fields Theorems Rings and Fields Theorems Rajesh Kumar PMATH 334 Intro to Rings and Fields Fall 2009 October 25, 2009 12 Rings and Fields 12.1 Definition Groups and Abelian Groups Let R be a non-empty set. Let + and (multiplication)

More information

π X : X Y X and π Y : X Y Y

π 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 quasi-projective varieties are Hausdorff and that projective varieties are the only compact varieties.

More information

Commutative Algebra. Andreas Gathmann. Class Notes TU Kaiserslautern 2013/14

Commutative 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 information

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 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

More information

LECTURE 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 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 information

Math 203A - Solution Set 1

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

More information

ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 4: MORE ABOUT VARIETIES AND REGULAR FUNCTIONS.

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

More information

Constrained Zeros and the Ritt Nullstellensatz

Constrained Zeros and the Ritt Nullstellensatz Constrained Zeros and the Ritt Nullstellensatz Phyllis Joan Cassidy City College of CUNY The Kolchin Seminar in Di erential Algebra The Graduate Center of CUNY November 9, 2007 G := - eld, H := extension

More information

= Spec(Rf ) R p. 2 R m f gives a section a of the stalk bundle over X f as follows. For any [p] 2 X f (f /2 p), let a([p]) = ([p], a) wherea =

= Spec(Rf ) R p. 2 R m f gives a section a of the stalk bundle over X f as follows. For any [p] 2 X f (f /2 p), let a([p]) = ([p], a) wherea = LECTURES ON ALGEBRAIC GEOMETRY MATH 202A 41 5. Affine schemes The definition of an a ne scheme is very abstract. We will bring it down to Earth. However, we will concentrate on the definitions. Properties

More information

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 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.

More information

CHEVALLEY S THEOREM AND COMPLETE VARIETIES

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

More information

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 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

More information

Extension theorems for homomorphisms

Extension theorems for homomorphisms Algebraic Geometry Fall 2009 Extension theorems for homomorphisms In this note, we prove some extension theorems for homomorphisms from rings to algebraically closed fields. The prototype is the following

More information

Introduction to Algebraic Geometry. Jilong Tong

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................................

More information

Institutionen för matematik, KTH.

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...............................

More information

Pure Math 764, Winter 2014

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

More information

INTRODUCTION TO ALGEBRAIC GEOMETRY. Throughout these notes all rings will be commutative with identity. k will be an algebraically

INTRODUCTION TO ALGEBRAIC GEOMETRY. Throughout these notes all rings will be commutative with identity. k will be an algebraically INTRODUCTION TO ALGEBRAIC GEOMETRY STEVEN DALE CUTKOSKY Throughout these notes all rings will be commutative with identity. k will be an algebraically closed field. 1. Preliminaries on Ring Homomorphisms

More information

AN INTRODUCTION TO AFFINE SCHEMES

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,

More information

10. Smooth Varieties. 82 Andreas Gathmann

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

More information

Math 145. Codimension

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

More information

Dimension Theory. Mathematics 683, Fall 2013

Dimension 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 information

Rings and groups. Ya. Sysak

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...

More information

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

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

More information

Lecture 4. Corollary 1.2. If the set of all nonunits is an ideal in A, then A is local and this ideal is the maximal one.

Lecture 4. Corollary 1.2. If the set of all nonunits is an ideal in A, then A is local and this ideal is the maximal one. Lecture 4 1. General facts Proposition 1.1. Let A be a commutative ring, and m a maximal ideal. Then TFAE: (1) A has only one maximal ideal (i.e., A is local); (2) A \ m consists of units in A; (3) For

More information

Math 210B. Artin Rees and completions

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

More information

CHAPTER 1. AFFINE ALGEBRAIC VARIETIES

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

More information

TROPICAL SCHEME THEORY

TROPICAL 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 information

Lecture 7: Etale Fundamental Group - Examples

Lecture 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 information

Commutative Algebra. Contents. B Totaro. Michaelmas Basics Rings & homomorphisms Modules Prime & maximal ideals...

Commutative 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 information

Algebraic Geometry. Andreas Gathmann. Class Notes TU Kaiserslautern 2014

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......................

More information

Algebra Homework, Edition 2 9 September 2010

Algebra Homework, Edition 2 9 September 2010 Algebra Homework, Edition 2 9 September 2010 Problem 6. (1) Let I and J be ideals of a commutative ring R with I + J = R. Prove that IJ = I J. (2) Let I, J, and K be ideals of a principal ideal domain.

More information

Chapter 3. Rings. The basic commutative rings in mathematics are the integers Z, the. Examples

Chapter 3. Rings. The basic commutative rings in mathematics are the integers Z, the. Examples Chapter 3 Rings Rings are additive abelian groups with a second operation called multiplication. The connection between the two operations is provided by the distributive law. Assuming the results of Chapter

More information

Chapter 1. Affine algebraic geometry. 1.1 The Zariski topology on A n

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

More information

A MODEL-THEORETIC PROOF OF HILBERT S NULLSTELLENSATZ

A MODEL-THEORETIC PROOF OF HILBERT S NULLSTELLENSATZ A MODEL-THEORETIC PROOF OF HILBERT S NULLSTELLENSATZ NICOLAS FORD Abstract. The goal of this paper is to present a proof of the Nullstellensatz using tools from a branch of logic called model theory. In

More information

Commutative Algebra. B Totaro. Michaelmas Basics Rings & homomorphisms Modules Prime & maximal ideals...

Commutative 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 information

Spring 2016, lecture notes by Maksym Fedorchuk 51

Spring 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 information

Exploring the Exotic Setting for Algebraic Geometry

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

More information

Algebraic Geometry: MIDTERM SOLUTIONS

Algebraic Geometry: MIDTERM SOLUTIONS Algebraic Geometry: MIDTERM SOLUTIONS C.P. Anil Kumar Abstract. Algebraic Geometry: MIDTERM 6 th March 2013. We give terse solutions to this Midterm Exam. 1. Problem 1: Problem 1 (Geometry 1). When is

More information

Algebraic varieties and schemes over any scheme. Non singular varieties

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

More information

Structure of rings. Chapter Algebras

Structure of rings. Chapter Algebras Chapter 5 Structure of rings 5.1 Algebras It is time to introduce the notion of an algebra over a commutative ring. So let R be a commutative ring. An R-algebra is a ring A (unital as always) together

More information

R S. with the property that for every s S, φ(s) is a unit in R S, which is universal amongst all such rings. That is given any morphism

R S. with the property that for every s S, φ(s) is a unit in R S, which is universal amongst all such rings. That is given any morphism 8. Nullstellensatz We will need the notion of localisation, which is a straightforward generalisation of the notion of the field of fractions. Definition 8.1. Let R be a ring. We say that a subset S of

More information

The Geometry-Algebra Dictionary

The Geometry-Algebra Dictionary Chapter 1 The Geometry-Algebra Dictionary This chapter is an introduction to affine algebraic geometry. Working over a field k, we will write A n (k) for the affine n-space over k and k[x 1,..., x n ]

More information

3. The Sheaf of Regular Functions

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

More information

Advanced Algebra II. Mar. 2, 2007 (Fri.) 1. commutative ring theory In this chapter, rings are assume to be commutative with identity.

Advanced Algebra II. Mar. 2, 2007 (Fri.) 1. commutative ring theory In this chapter, rings are assume to be commutative with identity. Advanced Algebra II Mar. 2, 2007 (Fri.) 1. commutative ring theory In this chapter, rings are assume to be commutative with identity. 1.1. basic definitions. We recall some basic definitions in the section.

More information

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 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

More information

Lecture 2. (1) Every P L A (M) has a maximal element, (2) Every ascending chain of submodules stabilizes (ACC).

Lecture 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 A-linear map. Then ker φ = {m M : φ(m) = 0} is a submodule of M and im φ is

More information

Commutative Banach algebras 79

Commutative Banach algebras 79 8. Commutative Banach algebras In this chapter, we analyze commutative Banach algebras in greater detail. So we always assume that xy = yx for all x, y A here. Definition 8.1. Let A be a (commutative)

More information

3.1. Derivations. Let A be a commutative k-algebra. Let M be a left A-module. A derivation of A in M is a linear map D : A M such that

3.1. Derivations. Let A be a commutative k-algebra. Let M be a left A-module. A derivation of A in M is a linear map D : A M such that ALGEBRAIC GROUPS 33 3. Lie algebras Now we introduce the Lie algebra of an algebraic group. First, we need to do some more algebraic geometry to understand the tangent space to an algebraic variety at

More information

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

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......................................................................

More information

BEZOUT S THEOREM CHRISTIAN KLEVDAL

BEZOUT 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 information

Definitions. Notations. Injective, Surjective and Bijective. Divides. Cartesian Product. Relations. Equivalence Relations

Definitions. Notations. Injective, Surjective and Bijective. Divides. Cartesian Product. Relations. Equivalence Relations Page 1 Definitions Tuesday, May 8, 2018 12:23 AM Notations " " means "equals, by definition" the set of all real numbers the set of integers Denote a function from a set to a set by Denote the image of

More information

2a 2 4ac), provided there is an element r in our

2a 2 4ac), provided there is an element r in our MTH 310002 Test II Review Spring 2012 Absractions versus examples The purpose of abstraction is to reduce ideas to their essentials, uncluttered by the details of a specific situation Our lectures built

More information

MATH 326: RINGS AND MODULES STEFAN GILLE

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

More information

NOTES FOR DRAGOS: MATH 210 CLASS 12, THURS. FEB. 22

NOTES 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 381-T, 1:15 2:30. This is the very end of Galois theory; you ll also start commutative ring theory. Tell them: midterm

More information

WEAK NULLSTELLENSATZ

WEAK 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 information

Algebraic Varieties. Chapter Algebraic Varieties

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 ) :

More information

ADVANCED COMMUTATIVE ALGEBRA: PROBLEM SETS

ADVANCED COMMUTATIVE ALGEBRA: PROBLEM SETS ADVANCED COMMUTATIVE ALGEBRA: PROBLEM SETS UZI VISHNE The 11 problem sets below were composed by Michael Schein, according to his course. Take into account that we are covering slightly different material.

More information

9. Integral Ring Extensions

9. Integral Ring Extensions 80 Andreas Gathmann 9. Integral ing Extensions In this chapter we want to discuss a concept in commutative algebra that has its original motivation in algebra, but turns out to have surprisingly many applications

More information

Algebraic Geometry. Instructor: Stephen Diaz & Typist: Caleb McWhorter. Spring 2015

Algebraic 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 information

12. Hilbert Polynomials and Bézout s Theorem

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

More information

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

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

More information

ADELIC VERSION OF MARGULIS ARITHMETICITY THEOREM. Hee Oh

ADELIC VERSION OF MARGULIS ARITHMETICITY THEOREM. Hee Oh ADELIC VERSION OF MARGULIS ARITHMETICITY THEOREM Hee Oh Abstract. In this paper, we generalize Margulis s S-arithmeticity theorem to the case when S can be taken as an infinite set of primes. Let R be

More information

TWO IDEAS FROM INTERSECTION THEORY

TWO IDEAS FROM INTERSECTION THEORY TWO IDEAS FROM INTERSECTION THEORY ALEX PETROV This is an expository paper based on Serre s Local Algebra (denoted throughout by [Ser]). The goal is to describe simple cases of two powerful ideas in intersection

More information

Honors Algebra 4, MATH 371 Winter 2010 Assignment 3 Due Friday, February 5 at 08:35

Honors Algebra 4, MATH 371 Winter 2010 Assignment 3 Due Friday, February 5 at 08:35 Honors Algebra 4, MATH 371 Winter 2010 Assignment 3 Due Friday, February 5 at 08:35 1. Let R 0 be a commutative ring with 1 and let S R be the subset of nonzero elements which are not zero divisors. (a)

More information

MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA

MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA These are notes for our first unit on the algebraic side of homological algebra. While this is the last topic (Chap XX) in the book, it makes sense to

More information

INTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 1. Contents 1. Commutative algebra 2 2. Algebraic sets 2 3. Nullstellensatz (theorem of zeroes) 4

INTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 1. Contents 1. Commutative algebra 2 2. Algebraic sets 2 3. Nullstellensatz (theorem of zeroes) 4 INTRODUCTION TO ALGEBRAIC GEOMETRY, CLASS 1 RAVI VAKIL Contents 1. Commutative algebra 2 2. Algebraic sets 2 3. Nullstellensatz (theorem of zeroes) 4 I m going to start by telling you about this course,

More information

Integral Extensions. Chapter Integral Elements Definitions and Comments Lemma

Integral Extensions. Chapter Integral Elements Definitions and Comments Lemma Chapter 2 Integral Extensions 2.1 Integral Elements 2.1.1 Definitions and Comments Let R be a subring of the ring S, and let α S. We say that α is integral over R if α isarootofamonic polynomial with coefficients

More information

ALGEBRAIC GROUPS J. WARNER

ALGEBRAIC GROUPS J. WARNER ALGEBRAIC GROUPS J. WARNER Let k be an algebraically closed field. varieties unless otherwise stated. 1. Definitions and Examples For simplicity we will work strictly with affine Definition 1.1. An algebraic

More information

4.2 Chain Conditions

4.2 Chain Conditions 4.2 Chain Conditions Imposing chain conditions on the or on the poset of submodules of a module, poset of ideals of a ring, makes a module or ring more tractable and facilitates the proofs of deep theorems.

More information

RINGS: SUMMARY OF MATERIAL

RINGS: SUMMARY OF MATERIAL RINGS: SUMMARY OF MATERIAL BRIAN OSSERMAN This is a summary of terms used and main results proved in the subject of rings, from Chapters 11-13 of Artin. Definitions not included here may be considered

More information

Introduction to Arithmetic Geometry Fall 2013 Lecture #15 10/29/2013

Introduction to Arithmetic Geometry Fall 2013 Lecture #15 10/29/2013 18.782 Introduction to Arithmetic Geometry Fall 2013 Lecture #15 10/29/2013 As usual, k is a perfect field and k is a fixed algebraic closure of k. Recall that an affine (resp. projective) variety is an

More information

EXERCISES. = {1, 4}, and. The zero coset is J. Thus, by (***), to say that J 4- a iu not zero, is to

EXERCISES. = {1, 4}, and. The zero coset is J. Thus, by (***), to say that J 4- a iu not zero, is to 19 CHAPTER NINETEEN Whenever J is a prime ideal of a commutative ring with unity A, the quotient ring A/J is an integral domain. (The details are left as an exercise.) An ideal of a ring is called proper

More information