# 2. Intersection Multiplicities

Size: px
Start display at page:

Transcription

1 2. Intersection Multiplicities Intersection Multiplicities Let us start our study of curves by introducing the concept of intersection multiplicity, which will be central throughout these notes. It generalizes the well-known notion of multiplicity of a zero of a univariate polynomial: If f K[x] is a polynomial and x 0 K such that f = a(x x 0 ) m for a polynomial a K[x] with a(x 0 ) 0, then f is said to have multiplicity m at x 0. As in the following two pictures on the left, a zero of multiplicity 1 means that the graph of f intersects the x-axis transversely, whereas in the case of multiplicity (at least) 2 it is tangent to it. Roughly speaking, higher multiplicities would correspond to graphs for which the x-axis is an even better approximation around x 0. f x 0 x 0 f G P F G P F multiplicity 1 multiplicity 2 f = a(x x 0 ) f = a(x x 0 ) 2 multiplicity 1 multiplicity 2 In this geometric interpretation, we have already considered how the graph of f intersects the horizontal axis locally at the given point, i. e. how the two curves F = y f and G = y intersect. As in the picture above on the right, this concept should thus also make sense for arbitrary curves F and G at an intersection point P: If they intersect transversely, i. e. with different tangent directions, we want to say that they have an intersection multiplicity of 1 at P, whereas equal tangents correspond to higher multiplicities. But of course, the curves F and G might also have singularities as e. g. the origin in Example 0.1 (b) and (c), in which case it is not clear a priori how their intersection multiplicity can be interpreted or even defined. So our first task must be to actually construct the intersection multiplicity for arbitrary curves. For this we need the following algebraic object that allows us to capture the local geometry of the plane around a point. Definition 2.1 (Local rings of A 2 ). Let P A 2 be a point. (a) The local ring of A 2 at P is defined as O P := O A 2,P := { f g : f,g K[x,y] with g(p) 0 } K(x,y). (b) It admits a well-defined ring homomorphism O P K, f f (P) g g(p) which we will call the evaluation map. Its kernel will be denoted by I P := I A 2,P := { f g : f,g K[x,y] with f (P) = 0 and g(p) 0 } O P. Remark 2.2 (Geometric and algebraic interpretation of local rings). Intuitively, O P describes nice (i. e. rational) functions that have a well-defined value at P (determined by the evaluation map), and thus also in a neighborhood of P. Note however that O P does not admit similar evaluation maps

2 12 Andreas Gathmann at other points Q P since the denominator of the fractions might vanish there. This explains the name local ring from a geometric point of view. The ideal I P in O P describes exactly those local functions that have the value 0 at P. Algebraically, O P is a subring of K(x,y) that contains K[x,y]. As a subring of a field it is an integral domain, and its units are precisely the fractions f g for which both f and g are non-zero at P. Moreover, just like K[x, y] it is a factorial ring, with the irreducible elements being the irreducible polynomials that vanish at P (since the others have become units). For those who know some commutative algebra we should mention that O P is also a local ring in the algebraic sense, i. e. that it contains exactly one maximal ideal, namely I P [G6, Definition 6.9]: If I is any ideal in O P that is not a subset of I P then it must contain an element f g with f (P) 0 and g(p) 0. But this is then a unit since g f O P as well, and hence we have I = O P. In fact, in the algebraic sense O P is just the localization of the polynomial ring K[x,y] at the maximal ideal x x 0,y y 0 associated to the point P = (x 0,y 0 ) which also shows that it is a local ring [G6, Corollary 6.10]. Definition 2.3 (Intersection multiplicities). For a point P A 2 and two curves (or polynomials) F and G we define the intersection multiplicity of F and G at P to be µ P (F,G) := dim O P / F,G N { }, where dim denotes the dimension as a vector space over K. As this definition is rather abstract, we should of course figure out how to compute this number, what its properties are, and why it captures the geometric idea given above. In fact, it is not even obvious whether µ P (F,G) is finite. But let us start with a few simple statements and examples. Remark 2.4. (a) It is clear from the definitions that an invertible affine coordinate transformation from (x, y) to (x,y ) = (ax + by + c,dx + ey + f ) for a,b,c,d,e, f K with ae bd 0 gives us an isomorphism between the local rings O P and O P, where P is the image point of P; and between O P / F,G and O P / F,G, where F and G are F and G expressed in the new coordinates x and y. We will often use this invariance to simplify our calculations by picking suitable coordinates, e. g. such that P = 0 is the origin. (b) The intersection multiplicity is symmetric: We have µ P (F,G) = µ P (G,F) for all F and G. (c) For all F,G,H we have F,G + FH = F,G, and thus µ P (F,G + FH) = µ P (F,G). In Definition 2.3, we have not required a priori that P actually lies on both curves F and G. However, the intersection multiplicity is at least 1 if and only if it does: Lemma 2.5. Let P A 2, and let F and G be two curves (or polynomials). We have: (a) µ P (F,G) 1 if and only if P F G; (b) µ P (F,G) = 1 if and only if I P = F,G in O P. Proof. Assume first that F(P) 0. Then F is a unit in O P, and thus F,G = O P, i. e. µ P (F,G) = 0. Moreover, we then have P / F and F / I P, proving both (a) and (b) in this case. Of course, the case G(P) 0 is analogous. So we may now assume that F(P) = G(P) = 0, i. e. P F G. Then the evaluation map at P induces a well-defined and surjective map O P / F,G K. It follows that µ P (F,G) 1, proving (a) in this case. Moreover, we have µ P (F,G) = 1 if and only if this map is an isomorphism, i. e. if and only if F,G is exactly the kernel I P of the evaluation map.

3 2. Intersection Multiplicities 13 Example 2.6 (Intersection multiplicity of coordinate axes). The kernel I 0 of the evaluation map at 0 consists exactly of the fractions f g such that f does not have a constant term, which is just the ideal x,y in O 0. By Lemma 2.5 (b) this means that µ 0 (x,y) = 1, i. e. (as expected) that the two coordinate lines have intersection multiplicity 1 at the origin. Another basic case (which we did not exclude in the definition) is when the two curves actually agree, or more generally if they have a common irreducible component through P. Although this is clearly not the main case we are interested in, it is reassuring to know that in this case the intersection multiplicity is infinite since the curves touch at P to infinite order : Exercise 2.7. Let F and G be two curves through a point P A 2. Show: (a) If F and G have no common component then the family (F n ) n N is linearly independent in O P / G. (b) If F and G have a common component through P then µ P (F,G) =. For the last important basic property of intersection multiplicities we first need another easy algebraic tool. Construction 2.8 ((Short) exact sequences). We say that a sequence 0 U ϕ V ψ W 0 of linear maps between vector spaces (where 0 denotes the zero vector space) is exact if the image of each map equals the kernel of the next, i. e. if (a) kerϕ = 0 (i. e. ϕ is injective); (b) imϕ = kerψ; and (c) imψ = W (i. e. ψ is surjective). In this case, we get a dimension formula dimu + dimw (a),(c) = dimimϕ + dimimψ = dimimϕ + dimv /kerψ (b) = dimimϕ + dimv /imϕ = dimv. Proposition 2.9 (Additivity of intersection numbers). Let P A 2, and let F,G,H be any three curves (or polynomials). Proof. (a) If F and G have no common component through P there is an exact sequence 0 O P / F,H G π O P / F,GH O P / F,G 0, where π is the natural quotient map. (b) We have µ P (F,GH) = µ P (F,G) + µ P (F,H). (a) We may assume that F and G have no common component at all, since components that do not pass through P are units in O P and can therefore be dropped in the ideals. It is checked immediately that both non-trivial maps in this sequence are well-defined, and that conditions (b) and (c) of Construction 2.8 hold. Hence we just have to show that the first multiplication map is injective: Assume that f g is in the kernel of this map, i. e. that f g G = f g F + f GH g for certain f, f,g,g K[x,y] with g (P) and g (P) non-zero. We may assume without loss of generality that all three fractions have the same denominator, and multiply by it to obtain the equation f G = f F + f GH in K[x,y]. Now G clearly divides f G and f GH, hence also f F, and consequently f as F and G have no common component. So we have

4 14 Andreas Gathmann f = ag for some a K[x,y], and we see that f G = afg + f GH. Dividing by G, it follows that f = af + f H, so that f and hence also f g are zero in O P/ F,H. This shows the injectivity of the first map. (b) If F and G have no common component through P the statement follows immediately from (a) by taking dimensions as in Construction 2.8. Otherwise the equation is true as = by Exercise 2.7 (b). Example 2.10 (Intersection multiplicity with a coordinate axis). Let F be a curve that does not have the x-axis y as a component. We want to compute its intersection multiplicity µ 0 (y,f) with this axis at the origin. First of all, by Remark 2.4 (c) we may remove all multiples of y from F, i. e. replace F by the polynomial F(x,0) K[x], which is not the zero polynomial since y is not a component of F. We can write F(x,0) = x m g where g K[x] is non-zero at the origin, so that m is the multiplicity of 0 in F(x,0). Hence we obtain µ 0 (y,f) = µ 0 (y,f(x,0)) (Remark 2.4 (c)) = µ 0 (y,x m g) = m µ 0 (y,x) + µ 0 (y,g) (Proposition 2.9 (b)) = m (Example 2.6 and Lemma 2.5 (a)). Note that this coincides with the expectation from the beginning of this chapter: If f K[x] is a univariate polynomial with a zero x 0 of multiplicity m (which is just x 0 = 0 in our current case) then the intersection multiplicity of its graph y f with the x-axis at the point (x 0,0) is m. We are now ready to compute the intersection multiplicity of two arbitrary curves F and G. By Remark 2.4 (a) it suffices to do this at the origin. For future reference, we formulate the following recursive algorithm for any two curves F and G through 0. Afterwards, we will prove that the algorithm actually terminates and gives a finite answer for µ 0 (F,G) if F and G have no common component through 0. Algorithm 2.11 (Computation of the intersection multiplicity µ 0 (F,G)). Let F and G be two curves (or polynomials) with F(0) = G(0) = 0. We then repeat the following procedure recursively: (a) If F and G both contain a monomial independent of y, we write F = ax m + (terms involving y or with a lower power of x), G = bx n + (terms involving y or with a lower power of x) for some a,b K and m,n N, where we may assume (by possibly swapping F and G) that m n. We then set F := F a b xm n G, hence canceling the x m -term in F. By Remark 2.4 (c) we then have µ 0 (F,G) = µ 0 (F,G). As F (0) = G(0) = 0, we can repeat the algorithm recursively with F and G to compute µ 0 (F,G). (b) If one of the polynomials F and G, say F, does not contain a monomial independent of y, we can factor F = yf and obtain by Proposition 2.9 (b) µ 0 (F,G) = µ 0 (y,g) + µ 0 (F,G). The multiplicity µ 0 (y,g) can be computed directly: By Example 2.10, it is the lowest power of x in a term of G independent of y (note that G contains such a term as otherwise y would be a common component of F and G through 0). As for µ 0 (F,G), if F does not vanish at 0 then µ 0 (F,G) = 0 by Lemma 2.5 (a). So we have then computed µ 0 (F,G) and stop the algorithm. Otherwise, we have F (0) = G(0) = 0, and we can repeat the algorithm recursively with F and G to compute µ 0 (F,G).

5 2. Intersection Multiplicities 15 Example Let us compute the intersection multiplicity µ 0 (F,G) at the origin of the two curves F = y 2 x 3 and G = x 2 y 3 as in the picture below on the right. We follow Algorithm 2.11 and indicate by (a) and (b) which step we performed each time: µ 0 (y 2 x 3,x 2 y 3 ) (a) = µ 0 (y 2 x 3 + x(x 2 y 3 ),x 2 y 3 ) = µ 0 (y 2 xy 3,x 2 y 3 ) (b) = µ 0 (y,x 2 y 3 ) +µ 0 (y xy 2,x 2 y 3 ) }{{} =2 by 2.10 (b) = 2 + µ 0 (y,x 2 y 3 ) + µ 0 (1 xy,x 2 y 3 ) }{{}}{{} =2 by 2.10 =0 by 2.5 (a) = 4. Fact 2.13 (Noetherian rings). A ring R is called Noetherian if there is no infinite strictly ascending chain of ideals I 0 I 1 I 2 in R. It can be shown that the polynomial ring K[x,y] is Noetherian [G6, Proposition 7.13], and that this property is inherited by the local rings O P [G6, Exercise 7.23]. Proposition 2.14 (Finiteness of the intersection multiplicity). Let F and G be two curves (or polynomials) that have no common component through the origin. Then Algorithm 2.11 terminates, and thus computes the intersection multiplicity µ 0 (F,G). In particular, intersection multiplicities µ P (F,G) are always finite as long as F and G have no common component through P. G F 02 Proof. First note that the property of not having a common component through 0 is preserved in the algorithm: In (a) the common components of F and G are the same as those of F and G, and in (b) F and G clearly cannot have a common component through 0 if F = yf and G do not. To prove termination, we now consider the ideal F,G in O 0 during the algorithm. In case (a) of the algorithm we have F,G = F,G, and the degree of the highest monomial of F independent of y strictly decreases. Hence, just as in the usual Euclidean algorithm [G1, Proposition 10.25], this case (a) can only happen finitely many times in a row before we must be in case (b). In case (b) we then have F = yf, and hence F,G F,G. In fact, this inclusion is strict: Otherwise, in the exact sequence 0 O P / y,g F O P / F,G O P / F,G 0 of Proposition 2.9 (a) the map π would be an isomorphism, and hence the first term O P / y,g would have to be zero, i. e. µ P (y,g) = 0 in contradiction to Lemma 2.5 (a). So if the algorithm did not stop, we would get an infinite strictly ascending chain of ideals in O P, which does not exist by Fact Remark If F and G have a common component through 0, Algorithm 2.11 might not terminate, or we might get the polynomial 0 during the algorithm: For example, for the curves F = x 2 and G = xy x it yields µ 0 (x 2,xy x) (a) = µ 0 (x 2 + x(xy x),xy x) π = µ 0 (x 2 y,xy x) (b) = µ 0 (y,xy x) +µ 0 (x 2,xy x), }{{} =1 by 2.10 leading to an infinite loop. However, Algorithm 2.11 is correct in any case, so if it does terminate (with a finite answer), then by Exercise 2.7 (b) we have proven simultaneously with this computation that F and G have no common component through P.

6 16 Andreas Gathmann Exercise Draw the real curves F = x 2 + y 2 + 2y and G = y 3 x 6 y 6 x 2, determine their irreducible decompositions, their intersection points, and their intersection multiplicities at these points. Exercise (a) For the curves F = y x 3 and G = y 3 x 4 1, find a polynomial representative of x+1 1 in O 0 / F,G, i. e. compute a polynomial f K[x,y] whose class equals that of x+1 in O 0 / F,G. (b) Prove for arbitrary coprime F and G and any P A 2 that every element of O P / F,G has a polynomial representative. Is this statement still true if F and G are not coprime? Following our algorithm, we can also give an easy and important criterion for when the intersection multiplicity is 1. Notation 2.18 (Homogeneous parts of polynomials). For a polynomial F K[x, y] of degree d and i = 0,...,d, we define the degree-i part of F to be the sum of all terms of F of degree i. Hence all F i are homogeneous, and we have F = F F d. We call F 0 the constant part, F 1 the linear part, and F d the leading part of F. Proposition 2.19 (Intersection multiplicity 1). Let F and G be two curves (or polynomials) through the origin. Then µ 0 (F,G) = 1 if and only if the linear parts F 1 and G 1 are linearly independent. Proof. We prove the statement following Algorithm 2.11, using the notation from there. In case (a), note that F 1 and G 1 are linearly independent if and only if F 1 and G 1 are, as either F 1 = F 1 (if m > n) or F 1 = F 1 a b G 1 (if m = n). Hence we can consider the first time we reach case (b). As µ 0 (y,g) > 0 by Lemma 2.5 (a), we have µ 0 (F,G) = 1 µ 0 (y,g) = 1 and µ(f,g) = 0 G contains a monomial x 1 y 0 and F contains a constant term (by Example 2.10 and Lemma 2.5 (a)) G 1 = ax + by for some a K,b K, and F 1 = cy for some c K F 1 and G 1 are linearly independent, where the last implication follows since F = yf does not contain a monomial x 1 y 0. In fact, Proposition 2.19 has an easy geometric interpretation in the spirit of the beginning of this chapter: F 1 and G 1 can be thought of as the linear approximations of F and G around the origin. If these approximations are non-zero, hence lines, they can be thought of as the tangents to the curves as in the picture on the right, and the proposition states that the intersection multiplicity is 1 if and only if these tangent directions are not the same. In general, it is the lowest non-zero terms of a curve F that can be considered as the best local approximation of F around 0. We can use this idea to define tangents to arbitrary curves (i. e. even if F 1 vanishes) as follows. Definition 2.20 (Tangents and multiplicities of points). Let F be a curve. (a) The smallest m N for which the homogeneous part F m is non-zero is called the multiplicity m 0 (F) of F at the origin. Any linear factor of F m is called a tangent to F at the origin; the multiplicity of this linear factor is then also called the multiplicity of the tangent. (b) For a general point P = (x 0,y 0 ) A 2, tangents at P and the multiplicity m P (F) are defined by first shifting coordinates to x = x + x 0 and y = y + y 0, and then applying (a) to the origin (x,y ) = (0,0). G G 1 0 F F 1

7 2. Intersection Multiplicities 17 Exercise Given a linear coordinate transformation that maps the origin to itself and a curve F to F, show that m 0 (F) = m 0 (F ), and that the transformation maps any tangent of F to a tangent of F. In particular, despite its appearance, Definition 2.20 is independent of the choice of coordinates on A 2. Lemma 2.22 (Alternative characterization of tangents). Let P be a point of multiplicity m = m P (F) on a curve F, and let L be a line through P. (a) µ P (F,L) m. (b) We have µ P (F,L) > m if and only if L is a tangent to F at P. Proof. By Exercise 2.21 we can choose coordinates such that P = 0 and L = y. Moreover, by assumption we have F = F m + F m F d with d = degf. Hence the lowest monomial in x occurring in F can be x m, which means µ P (F,L) m by Example For the same reason, we have µ P (F,L) > m if and only if F m contains no monomial x m, which is the same as saying that y F m, i. e. that y is a tangent to F. By definition, we clearly have m P (F) > 0 if and only if P F. The most important case of Definition 2.20 is when m P (F) = 1, i. e. if there is a non-zero local linear approximation for F around P. There is a special terminology for this case. Definition 2.23 (Smooth and singular points). Let F be a curve. (a) A point P on F is called smooth or regular if m P (F) = 1. Note that F has then a unique tangent at P, which we will denote by T P F. For P = 0, it is simply given by the linear part F 1 of F. If P is not a smooth point, i. e. if m P (F) > 1, we say that P is a singular point or a singularity of F. As a special case, a singularity with m P (F) = 2 such that F has two different tangents there is called a node. (b) The curve F is said to be smooth or regular if all its points are smooth. Otherwise, F is called singular. Example Let us consider the origin in the real curves in the following picture. y x 2 y 2 x 2 x 3 y 2 x 3 x 2 + y 2 (a) (b) (c) (d) For the case (a), the curve F = y x 2 in (a) has (no constant but) a linear term y. Hence, we have m 0 (F) = 1, the origin is a smooth point of the curve, and its tangent there is T 0 F = y. For the other three curves, the origin is a singular point of multiplicity 2. In (b), this singularity is a node, since the quadratic term is y 2 x 2 = (y x)(y + x), and thus we have the two tangents y x and y + x, shown as dashed lines in the picture. The curve in (c) has only one tangent y which is of multiplicity 2. Finally, in (d) there is no tangent at all since x 2 + y 2 does not contain a linear factor over R. Note that, in any case, knowing the tangents of F at the origin (which are easy to compute) tells us to some extent what the curve looks like locally around 0. With these notations we can now reformulate Proposition 2.19.

8 18 Andreas Gathmann Corollary 2.25 (Transverse intersections). Let P be a point in the intersection of two curves F and G. Then µ P (F,G) = 1 if and only if P is a smooth point of both F and G, and T P F T P G. We say in this case that F and G intersect transversely at P. Remark 2.26 (Additivity of point multiplicities). Note that m P (FG) = m P (F) + m P (G). Hence, any point that lies on at least two (not necessarily distinct) irreducible components has multiplicity at least 2, and is thus a singular point. In particular, all points on a component of multiplicity at least 2 (in the sense of Definition 1.5 (c)) are always singular. To check if a given curve F is smooth, i. e. whether every point P F is a smooth point of F, there is a simple criterion that does not require to shift P to the origin first. It uses the (partial) derivatives F F x and y of F, which can be defined purely formally over an arbitrary ground field and then satisfy the usual rules of differentiation [G1, Exercise 9.11]. Proposition 2.27 (Affine Jacobi Criterion). Let P = (x 0,y 0 ) be a point on an affine curve F. (a) P is a singular point of F if and only if F x F (P) = y (P) = 0. (b) If P is a smooth point of F the tangent to F at P is given by T P F = F x (P) (x x 0) + F y (P) (y y 0). Proof. Substituting x = x + x 0 and y = y + y 0, we can consider F as a polynomial in x and y. If we expand F = ax + by + (higher order terms in x and y ), then by definition F is singular at (x,y ) = (0,0), i. e. at P, if and only if a = b = 0. But by the chain rule of differentiation we have F F (P) = x x (0) = a and F y F (P) = (0) = b, y so that (a) follows. Moreover, if F is smooth at P then its tangent is just the term of F linear in x and y, i. e. ax + by = F x (P) (x x 0) + F y (P) (y y 0), as claimed in (b). Example Consider again the real curve F = y 2 x 2 x 3 from Example 2.24 (b). To determine its singular points, we compute the partial derivatives F x = 2x 3x2 and F y = 2y. Its common zeros are (0,0) and ( 2 3,0). But the latter does not lie on the curve, and so we conclude that the origin is the only singular point of F. Smoothness of a curve F at a point P has another important algebraic consequence: It means that the containment of ideals containing F in O P can be checked using intersection multiplicities. Proposition 2.29 (Comparing ideals using intersection multiplicities). Let P be a smooth point on a curve F. Then for any two curves G and H that do not have a common component with F through P we have F,G F,H in O P µ P (F,G) µ P (F,H). Proof. : Clearly, if F,G F,H then µ P (F,G) = dimo P / F,G dimo P / F,H = µ P (F,H).

9 2. Intersection Multiplicities 19 : Let L be a line through P which is not the tangent T P F. Then µ P (F,L) = 1 by Corollary 2.25, and hence µ P (F,L n ) = n for all n N by Proposition 2.9. Note that F,G O P = F,L 0 and F,G F,L n for n > µ P (F,G) by the direction that we have already shown. There must therefore by a maximal n with F,G F,L n in O P. We claim that then F,G = F,L n in O P, i. e. that L n F,G. To see this, note that F,G F,L n implies G = af + bl n for some a,b O P. If we had b(p) = 0 it would follow that b I P = F,L by Lemma 2.5 (b), i. e. b = cf + dl for some c,d O P, which means that G = af + (cf + dl)l n F,L n+1 and thus contradicts the maximality of n. Hence b(p) 0, i. e. b is a unit in O P, and we obtain L n = 1 b (G af) F,G as desired. Of course, now F,G = F,L n implies that µ P (F,G) = µ P (F,L n ) = n, so that we obtain F,G = F,L µ P(F,G). But the same holds for H instead of G, and so the inequality µ P (F,G) µ P (F,H) yields F,G = F,L µ P(F,G) F,L µ P(F,H) = F,H. Example Proposition 2.29 is false without the smoothness assumption on F: For the real curve F = x 2 y 2 = (x y)(x+y) (i. e. the union of the two diagonals in A 2, with singular point 0), G = x, and H = y, we have F,G = x,y 2 and F,H = y,x 2. Hence µ 0 (F,G) = µ 0 (F,H) = 2, but F,G = F,H (since y / x,y 2, as otherwise we would have x,y 2 = x,y, in contradiction to µ 0 (x,y 2 ) = 2 1 = µ 0 (x,y)). Remark 2.31 (Smooth curves over R). For the ground field K = R, our results on smooth curves have an intuitive interpretation: (a) The Jacobi Criterion of Proposition 2.27 (a) states that P is a smooth point of the curve F if and only if the Implicit Function Theorem [G2, Proposition 27.9] can be applied to the equation F = 0 around P, so that V (F) is a 1-dimensional submanifold of R 2 [G2, Definition 27.17]. Hence, in this case V (F) is locally the graph of a differentiable function (expressing y as a function of x or vice versa), and thus we arrive at the intuitive interpretation of smoothness as having no sharp corners. (b) To interpret Proposition 2.29, let us continue the picture of (a) and consider a local (analytic) coordinate z around P on the 1-dimensional manifold V (F). In accordance with the idea of intersection multiplicity at the beginning of this chapter, a curve G should have intersection multiplicity n with F at P if on F it is locally a function of the form az n in this coordinate, with a non-zero at P (corresponding to a unit in O P ). Now if n = µ P (F,G) µ P (F,H) = m then in the same way H is of the form bz m, so that bz m = H divides az n = G. This means that G H in O P / F (i. e. as functions on F, a point of view that we will discuss in detail starting in Chapter 6) and thus that F,G F,H in O P. Exercise (a) Find all singular points of the curve F = (x 2 +y 2 1) 3 +10x 2 y 2 R[x,y], and determine the multiplicities and tangents to F at these points. (b) Show that an irreducible curve F over a field of characteristic 0 has only finitely many singular points. Can you find weaker assumptions on F that also imply that F has only finitely many singular points? Exercise 2.33 (Cusps). Let P be a point on an affine curve F. We say that P is a cusp if m P (F) = 2, there is exactly one tangent L to F at P, and µ P (F,L) = 3. (a) Give an example of a real curve with a cusp, and draw a picture of it. (b) If F has a cusp at P, prove that F has only one irreducible component passing through P. (c) If F and G have a cusp at P, what is the minimum possible value for the intersection multiplicity µ P (F,G)? 03

### Plane Algebraic Curves

Plane Algebraic Curves Andreas Gathmann Class Notes TU Kaiserslautern 2018 Contents 0. Introduction......................... 3 1. Affine Curves......................... 6 2. Intersection Multiplicities.....................

### 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

### 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

### 8. Prime Factorization and Primary Decompositions

70 Andreas Gathmann 8. Prime Factorization and Primary Decompositions 13 When it comes to actual computations, Euclidean domains (or more generally principal ideal domains) are probably the nicest rings

### 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

### 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

### 11. Dimension. 96 Andreas Gathmann

96 Andreas Gathmann 11. Dimension We have already met several situations in this course in which it seemed to be desirable to have a notion of dimension (of a variety, or more generally of a ring): for

### 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

### 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

### 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

### 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.

### 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

### 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......................

### 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

### 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 ) :

### 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...............................

### 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

### be any ring homomorphism and let s S be any element of S. Then there is a unique ring homomorphism

21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UFD. Therefore

### 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.....................

### 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

### (dim Z j dim Z j 1 ) 1 j i

Math 210B. Codimension 1. Main result and some interesting examples Let k be a field, and A a domain finitely generated k-algebra. In class we have seen that the dimension theory of A is linked to the

### 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 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

### MATH 233B, FLATNESS AND SMOOTHNESS.

MATH 233B, FLATNESS AND SMOOTHNESS. The discussion of smooth morphisms is one place were Hartshorne doesn t do a very good job. Here s a summary of this week s material. I ll also insert some (optional)

### Theorem 5.3. Let E/F, E = F (u), be a simple field extension. Then u is algebraic if and only if E/F is finite. In this case, [E : F ] = deg f u.

5. Fields 5.1. Field extensions. Let F E be a subfield of the field E. We also describe this situation by saying that E is an extension field of F, and we write E/F to express this fact. If E/F is a field

### 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

### n P say, then (X A Y ) P

COMMUTATIVE ALGEBRA 35 7.2. The Picard group of a ring. Definition. A line bundle over a ring A is a finitely generated projective A-module such that the rank function Spec A N is constant with value 1.

### INTERSECTION NUMBER OF PLANE CURVES

INTERSECTION NUMBER OF PLANE CURVES MARGARET E. NICHOLS 1. Introduction One of the most familiar objects in algebraic geometry is the plane curve. A plane curve is the vanishing set of a polynomial in

### 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

### 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

### 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

### 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

### 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),

### 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

### NOTES ON FINITE FIELDS

NOTES ON FINITE FIELDS AARON LANDESMAN CONTENTS 1. Introduction to finite fields 2 2. Definition and constructions of fields 3 2.1. The definition of a field 3 2.2. Constructing field extensions by adjoining

### 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

### 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,

### 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 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

### 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:

### 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

### (1) is an invertible sheaf on X, which is generated by the global sections

7. Linear systems First a word about the base scheme. We would lie to wor in enough generality to cover the general case. On the other hand, it taes some wor to state properly the general results if one

### Polynomials, Ideals, and Gröbner Bases

Polynomials, Ideals, and Gröbner Bases Notes by Bernd Sturmfels for the lecture on April 10, 2018, in the IMPRS Ringvorlesung Introduction to Nonlinear Algebra We fix a field K. Some examples of fields

### 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

### 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

### 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

### 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

### 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,

### 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

### 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

### where c R and the content of f is one. 1

9. Gauss Lemma Obviously it would be nice to have some more general methods of proving that a given polynomial is irreducible. The first is rather beautiful and due to Gauss. The basic idea is as follows.

### HARTSHORNE EXERCISES

HARTSHORNE EXERCISES J. WARNER Hartshorne, Exercise I.5.6. Blowing Up Curve Singularities (a) Let Y be the cusp x 3 = y 2 + x 4 + y 4 or the node xy = x 6 + y 6. Show that the curve Ỹ obtained by blowing

### 15. Polynomial rings Definition-Lemma Let R be a ring and let x be an indeterminate.

15. Polynomial rings Definition-Lemma 15.1. Let R be a ring and let x be an indeterminate. The polynomial ring R[x] is defined to be the set of all formal sums a n x n + a n 1 x n +... a 1 x + a 0 = a

### 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

### 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

### (1) A frac = b : a, b A, b 0. We can define addition and multiplication of fractions as we normally would. a b + c d

The Algebraic Method 0.1. Integral Domains. Emmy Noether and others quickly realized that the classical algebraic number theory of Dedekind could be abstracted completely. In particular, rings of integers

### 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

### Dedekind Domains. Mathematics 601

Dedekind Domains Mathematics 601 In this note we prove several facts about Dedekind domains that we will use in the course of proving the Riemann-Roch theorem. The main theorem shows that if K/F is a finite

### AN EXPOSITION OF THE RIEMANN ROCH THEOREM FOR CURVES

AN EXPOSITION OF THE RIEMANN ROCH THEOREM FOR CURVES DOMINIC L. WYNTER Abstract. We introduce the concepts of divisors on nonsingular irreducible projective algebraic curves, the genus of such a curve,

### 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

### 0.1 Spec of a monoid

These notes were prepared to accompany the first lecture in a seminar on logarithmic geometry. As we shall see in later lectures, logarithmic geometry offers a natural approach to study semistable schemes.

### 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)

### Exercises for algebraic curves

Exercises for algebraic curves Christophe Ritzenthaler February 18, 2019 1 Exercise Lecture 1 1.1 Exercise Show that V = {(x, y) C 2 s.t. y = sin x} is not an algebraic set. Solutions. Let us assume that

### 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

### 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

### 5 Quiver Representations

5 Quiver Representations 5. Problems Problem 5.. Field embeddings. Recall that k(y,..., y m ) denotes the field of rational functions of y,..., y m over a field k. Let f : k[x,..., x n ] k(y,..., y m )

### 214A HOMEWORK KIM, SUNGJIN

214A HOMEWORK KIM, SUNGJIN 1.1 Let A = k[[t ]] be the ring of formal power series with coefficients in a field k. Determine SpecA. Proof. We begin with a claim that A = { a i T i A : a i k, and a 0 k }.

### Homework 2 - Math 603 Fall 05 Solutions

Homework 2 - Math 603 Fall 05 Solutions 1. (a): In the notation of Atiyah-Macdonald, Prop. 5.17, we have B n j=1 Av j. Since A is Noetherian, this implies that B is f.g. as an A-module. (b): By Noether

### 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

### Thus, the integral closure A i of A in F i is a finitely generated (and torsion-free) A-module. It is not a priori clear if the A i s are locally

Math 248A. Discriminants and étale algebras Let A be a noetherian domain with fraction field F. Let B be an A-algebra that is finitely generated and torsion-free as an A-module with B also locally free

### Local Parametrization and Puiseux Expansion

Chapter 9 Local Parametrization and Puiseux Expansion Let us first give an example of what we want to do in this section. Example 9.0.1. Consider the plane algebraic curve C A (C) defined by the equation

### Math 110, Spring 2015: Midterm Solutions

Math 11, Spring 215: Midterm Solutions These are not intended as model answers ; in many cases far more explanation is provided than would be necessary to receive full credit. The goal here is to make

### Introduction Non-uniqueness of factorization in A[x]... 66

Abstract In this work, we study the factorization in A[x], where A is an Artinian local principal ideal ring (briefly SPIR), whose maximal ideal, (t), has nilpotency h: this is not a Unique Factorization

### Math 249B. Nilpotence of connected solvable groups

Math 249B. Nilpotence of connected solvable groups 1. Motivation and examples In abstract group theory, the descending central series {C i (G)} of a group G is defined recursively by C 0 (G) = G and C

### 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.

### ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA

ALGEBRA QUALIFYING EXAM PROBLEMS LINEAR ALGEBRA Kent State University Department of Mathematical Sciences Compiled and Maintained by Donald L. White Version: August 29, 2017 CONTENTS LINEAR ALGEBRA AND

### A Harvard Sampler. Evan Chen. February 23, I crashed a few math classes at Harvard on February 21, Here are notes from the classes.

A Harvard Sampler Evan Chen February 23, 2014 I crashed a few math classes at Harvard on February 21, 2014. Here are notes from the classes. 1 MATH 123: Algebra II In this lecture we will make two assumptions.

### Introduction to Arithmetic Geometry Fall 2013 Lecture #23 11/26/2013

18.782 Introduction to Arithmetic Geometry Fall 2013 Lecture #23 11/26/2013 As usual, a curve is a smooth projective (geometrically irreducible) variety of dimension one and k is a perfect field. 23.1

### MATH 205B NOTES 2010 COMMUTATIVE ALGEBRA 53

MATH 205B NOTES 2010 COMMUTATIVE ALGEBRA 53 10. Completion The real numbers are the completion of the rational numbers with respect to the usual absolute value norm. This means that any Cauchy sequence

### 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.

### 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...

### 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

### 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

### Exercise Sheet 3 - Solutions

Algebraic Geometry D-MATH, FS 2016 Prof. Pandharipande Exercise Sheet 3 - Solutions 1. Prove the following basic facts about algebraic maps. a) For f : X Y and g : Y Z algebraic morphisms of quasi-projective

### 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)

### 4 Hilbert s Basis Theorem and Gröbner basis

4 Hilbert s Basis Theorem and Gröbner basis We define Gröbner bases of ideals in multivariate polynomial rings and see how they work in tandem with the division algorithm. We look again at the standard

### 1 Flat, Smooth, Unramified, and Étale Morphisms

1 Flat, Smooth, Unramified, and Étale Morphisms 1.1 Flat morphisms Definition 1.1. An A-module M is flat if the (right-exact) functor A M is exact. It is faithfully flat if a complex of A-modules P N Q

### are Banach algebras. f(x)g(x) max Example 7.4. Similarly, A = L and A = l with the pointwise multiplication

7. Banach algebras Definition 7.1. A is called a Banach algebra (with unit) if: (1) A is a Banach space; (2) There is a multiplication A A A that has the following properties: (xy)z = x(yz), (x + y)z =

### LINEAR ALGEBRA BOOT CAMP WEEK 1: THE BASICS

LINEAR ALGEBRA BOOT CAMP WEEK 1: THE BASICS Unless otherwise stated, all vector spaces in this worksheet are finite dimensional and the scalar field F has characteristic zero. The following are facts (in

### Chapter 8. P-adic numbers. 8.1 Absolute values

Chapter 8 P-adic numbers Literature: N. Koblitz, p-adic Numbers, p-adic Analysis, and Zeta-Functions, 2nd edition, Graduate Texts in Mathematics 58, Springer Verlag 1984, corrected 2nd printing 1996, Chap.

### LECTURE 16: LIE GROUPS AND THEIR LIE ALGEBRAS. 1. Lie groups

LECTURE 16: LIE GROUPS AND THEIR LIE ALGEBRAS 1. Lie groups A Lie group is a special smooth manifold on which there is a group structure, and moreover, the two structures are compatible. Lie groups are

### 7 Orders in Dedekind domains, primes in Galois extensions

18.785 Number theory I Lecture #7 Fall 2015 10/01/2015 7 Orders in Dedekind domains, primes in Galois extensions 7.1 Orders in Dedekind domains Let S/R be an extension of rings. The conductor c of R (in

### 2 Lecture 2: Logical statements and proof by contradiction Lecture 10: More on Permutations, Group Homomorphisms 31

Contents 1 Lecture 1: Introduction 2 2 Lecture 2: Logical statements and proof by contradiction 7 3 Lecture 3: Induction and Well-Ordering Principle 11 4 Lecture 4: Definition of a Group and examples 15

### ABSOLUTE VALUES AND VALUATIONS

ABSOLUTE VALUES AND VALUATIONS YIFAN WU, wuyifan@umich.edu Abstract. We introduce the basis notions, properties and results of absolute values, valuations, discrete valuation rings and higher unit groups.

### Tangent spaces, normals and extrema

Chapter 3 Tangent spaces, normals and extrema If S is a surface in 3-space, with a point a S where S looks smooth, i.e., without any fold or cusp or self-crossing, we can intuitively define the tangent

### 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

### Holomorphic line bundles

Chapter 2 Holomorphic line bundles In the absence of non-constant holomorphic functions X! C on a compact complex manifold, we turn to the next best thing, holomorphic sections of line bundles (i.e., rank

### MAKSYM FEDORCHUK. n ) = z1 d 1 zn d 1.

DIRECT SUM DECOMPOSABILITY OF SMOOTH POLYNOMIALS AND FACTORIZATION OF ASSOCIATED FORMS MAKSYM FEDORCHUK Abstract. We prove an if-and-only-if criterion for direct sum decomposability of a smooth homogeneous