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Commutative Algebra and Algebraic Geometry

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Commutative Algebra and Algebraic Geometry Andrew Hubery Contents I Commutative Rings1 1 Rings and ideals1 1.1 Subrings and factor rings..........................1 1.2 Algebras....................................2 1.3 Operations on subrings and ideals.....................3 2 Maximal, prime and radical ideals5 2.1 Existence of maximal ideals.........................6 2.2 The radical of an ideal............................7 3 Localisation9 3.1 Ideals in a localisation............................ 11 4 Modules 14 4.1 Submodules and factor modules....................... 15 4.2 Operations on modules............................ 16 4.3 The Cayley-Hamilton Theorem....................... 17 5 Exact sequences 19 5.1 Projective and injective modules...................... 21 II Tensor Products 22 6 Adjoint functors 22 7 Tensor products 24 7.1 The construction............................... 24 i 7.2 The hom space isomorphisms........................ 24 7.3 The adjointness property.......................... 25 7.4 Examples................................... 26 7.5 Basic properties................................ 26 7.6 Right exactness and flat modules...................... 28 8 Extension of scalars 29 8.1 Localisation for modules........................... 30 8.2 Local properties................................ 31 9 Tensor products of algebras 33 9.1 Ideals in tensor algebras........................... 34 9.2 Examples................................... 34 III Noetherian Rings 36 10 Noetherian modules 36 11 Noetherian rings 37 IV Integral Extensions 41 12 Integral Extensions 41 12.1 Lying-Over, Going-Up and Going-Down.................. 43 12.2 Noether Normalisation............................ 45 12.3 Krull dimension................................ 47 V Affine Geometry 49 13 Affine varieties 49 13.1 The Nullstellensatz.............................. 51 13.2 Zariski topology............................... 51 13.3 Rational curves................................ 53 14 Regular maps 59 14.1 Regular functions on affine varieties.................... 59 14.2 Regular maps between affine varieties................... 60 ii 14.3 Products of affine varieties.......................... 61 14.4 Noether Normalisation revisited....................... 64 14.5 Automorphisms of affine space....................... 65 15 Local definition of regular maps 66 15.1 Regular maps on quasi-affine varieties................... 66 15.2 Example.................................... 69 VI Projective Geometry 70 16 Projective varieties 70 16.1 Homogeneous ideals and cones....................... 70 16.2 Examples................................... 73 16.3 Open affine covering of projective space.................. 73 16.4 Examples................................... 76 17 Regular maps 78 17.1 Regular maps from projective space.................... 79 17.2 Examples................................... 80 18 Segre varieties and products 81 18.1 Zariski topology on the product....................... 82 18.2 Separatedness................................. 83 19 Completeness of projective varieties 84 VII Appendix 86 A Unique Factorisation Domains 86 B Categories 90 B.1 Initial and terminal objects......................... 91 B.2 Products and coproducts........................... 92 B.3 Kernels and cokernels............................ 93 B.4 Additive and abelian categories....................... 94 C Functors 94 C.1 Natural transformations........................... 95 iii C.2 Equivalence of categories........................... 96 C.3 Representable functors............................ 96 C.4 Contravariant functors............................ 97 C.5 Universal properties............................. 98 D Topological spaces 99 D.1 Continuous maps............................... 100 D.2 Separation axioms.............................. 100 D.3 Irreducibility and locally-closed subsets.................. 100 D.4 Filters and quasi-compact spaces...................... 101 D.5 Separated and complete spaces....................... 102 iv Part I Commutative Rings 1 Rings and ideals We recall that an (associative, unital) ring R is given by an abelian group under addition, with identity element 0 (called the zero element), together with an associative multiplication, with identity element 1 (called the identity or unit), satisfying the distributivity rule x(y + z) = xy + xz and (x + y)z = xz + yz for all x; y; z 2 R: A commutative ring is a ring R for which the multiplication is commutative xy = yx for all x; y 2 R: In this course, nearly all rings will be commutative, so we will just use the term ring. Examples. The integers Z and all fields K provide examples of rings. If R is a ring, then we can form the ring of polynomials over R, denoted R[X] and formal power series, denoted R X . Also, given a family of rings Ri, we can form their Q J K direct product i Ri, which is a ring via pointwise addition and multiplication, so (ri) + (si) = (ri + si) and (ri)(si) := (risi). If 1 = 0 in a ring R, then R = f0g and we call R the trivial ring. A ring homomorphism f : R ! S is an additive group homomorphism which pre- serves the identity f(1R) = 1S and respects multiplication f(xy) = f(x)f(y) for all x; y 2 R: In particular, the identity map idR : R ! R is a ring homomorphism, and if f : R ! S and g : S ! T are ring homomorphisms, then so is the composition gf : R ! T . Moreover, composition is associative, so h(gf) = (hg)f whenever this makes sense. Remark. It follows that there is a category having objects the commutative rings and morphisms the ring homomorphisms. 1.1 Subrings and factor rings A subring S ≤ R is an additive subgroup containing the identity element and closed under multiplication 1 2 S; and xy 2 S for all x; y 2 S; in which case the inclusion map S ! R is a ring homomorphism. 1 An ideal I C R is an additive subgroup closed under multiplication by elements of R rx 2 I for all r 2 R and x 2 I: In this case there is a unique multiplication on the factor group R=I such that the canonical map R R=I is a ring homomorphism. Lemma 1.1. Let f : R ! S be a ring homomorphism. Then the image of f Im(f) := ff(x) 2 S : x 2 Rg is a subring of S, and the kernel of f is an ideal of R, Ker(f) := fx 2 R : f(x) = 0g: Lemma 1.2 (Factor Lemma). Let f : R ! S be a ring homomorphism, and I C R an ideal. If I ≤ Ker(f), then there is a unique ring homomorphism f¯: R=I ! S; r + I 7! f(r); satisfying f = fπ¯ . Thus f factors through the canonical map π. We call f¯ the homomorphism induced by f. Corollary 1.3 (First Isomorphism Theorem). A ring homomorphism f : R ! S yields an isomorphism R= Ker(f) −!∼ Im(f); r + Ker(f) 7! f(r): 1.2 Algebras Let k be a ring. A k-algebra is a ring R together with a ring homomorphism i: k ! R, called the structure map. In the special case when i is injective, we often identify k with its image, and so view k as a subring of R. Let R and S be two k-algebras, with structure maps iR : k ! R and iS : k ! S.A k-algebra homomorphism is a ring homomorphism f : R ! S which commutes with the structure maps, so fiR = iS. In fact, there is a category having objects the k-algebras and morphisms the k-algebra homomorphisms. Let R be a k-algebra. If I C R is an ideal, then R=I becomes a k-algebra via the composition k ! R ! R=I, in which case R ! R=I is a k-algebra homomorphism. Similarly, a k-subalgebra of R is a subring S containing the image of k, in which case the inclusion map S ! R is a k-algebra homomorphism. For example, let k be a field. The only ideals of k are f0g and k, so if R is a non-trivial ring, then every ring homomorphism k ! R is injective. It follows that every k-algebra is a k-vector space and every k-algebra homomorphism is k-linear. On the other hand, there is a unique ring homomorphism Z ! R for each ring R. Thus every ring R is a Z-algebra, and every ring homomorphism is necessarily a Z-algebra homomorphism. For this (and other) reasons, it is common always to work in the `relative' setting by fixing a base ring k and considering only k-algebras. 2 1.3 Operations on subrings and ideals Given a family of subalgebras Si of an algebra R, their intersection is again a subalgebra \ Si := fx 2 R : x 2 Si for all ig: i We therefore define for each subset X ⊂ R the subalgebra generated by X, denoted k[X], which is the smallest subalgebra containing X \ k[X] := S: X⊂S The elements of k[X] are k-linear sums of products of elements of X. Warning: this is not to be confused with the polynomial ring. A subalgebra S of R is called finitely generated if S = k[X] for some finite set X.1 Similarly, given a family of ideals Ii in a ring R, their intersection is again an ideal \ Ii := fx 2 R : x 2 Ii for all ig: i We therefore define for each subset X ⊂ R the ideal generated by X, denoted (X), which is the smallest ideal containing X \ (X) := I: X⊂I The elements of (X) are those which can be expressed in the form r1x1 +···+rnxn for some ri 2 R and xi 2 X. An ideal I is called finitely generated if I = (x1; : : : ; xn), and principal if I = (x). For example, the unit ideal R = (1) and the trivial ideal, or zero ideal, 0 = (0) are both principal. An ideal I is called proper provided I 6= R, or equivalently R=I 6= 0. S We define the sum of ideals Ii to be the ideal generated by the union i Ii, so X n X 2o Ii = xi : xi 2 Ii almost all zero : i i Similarly we define the product of two ideals to be the ideal generated by all products xy with x 2 I and y 2 J, so IJ = fx1y1 + ··· + xnyn : n ≥ 1; xi 2 I; yi 2 Jg: The product of ideals is associative, so I(JK) = (IJ)K.
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