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On Algebraic Geometry.] Mumford, D MATHEMATICS: CONCEPTS, AND FOUNDATIONS – Vol. I - Algebraic Geometry and Applications - Tadao ODA ALGEBRAIC GEOMETRY AND APPLICATIONS Tadao ODA Tohoku University, Japan Keywords: algebraic set, maximal ideal space, Hilbert’s Nullstellensatz, Noether’s normalization theorem, affine, projective, separated, algebraic variety, sheaf, scheme, product, local ring, stalk, local homomorphism, morphism, Zariski tangent space, singular, nonsingular, Jacobian matrix, complex analytic space, resolution, alteration, valuation ring, discrete valuation ring, Mori’s Minimal Model Program, function field, complete, proper, normal, birational, divisor, complete linear system, Riemann-Roch theorem, algebraic curve, algebraic surface, error-correcting code Contents 1. Affine Algebraic Varieties 2. Projective Algebraic Varieties 3. Sheaves and General Algebraic Varieties 4. Properties of Algebraic Varieties 5. Divisors 6. Algebraic Geometry over Algebraically Closed Fields 7. Schemes 8. Applications Glossary Bibliography Biographical Sketch Summary Algebraic geometry deals with geometric objects defined algebraically. The set of solutions (in complex numbers) of a system of algebraic equations, called an affine algebraic set, is first given an intrinsic formulation as the maximal ideal space of a finitely generated algebra over complex numbers. This formulation and the notion of sheaves enable one to glue affine algebraic sets together and obtain algebraic sets. Interesting among them are algebraic varieties whose basic geometric properties are explained. More general notions of schemes and algebraic varieties over arbitrary fields are introduced. As one of the applications, algebro-geometric coding system is sketched. Due to space limitation, practically no examples are given to illustrate the rich geometry behind the abstract formulation. The readers are referred to the references listed at the end for illuminating self-contained description as well as for good references at more advanced level. Results on matrices and linear algebra in Matrices, Vectors, Determinants and Linear Algebra, on groups in Groups and Applications, on rings and modules in Rings and Modules and on fields and algebraic equations in Fields and Algebraic Equations will be freely used. 248 ©Encyclopedia of Life Support Systems (EOLSS) MATHEMATICS: CONCEPTS, AND FOUNDATIONS – Vol. I - Algebraic Geometry and Applications - Tadao ODA 1. Affine Algebraic Varieties A very basic geometric object defined algebraically is the set of solutions of a system of algebraic equations. To fix the ideas, let us consider the n -dimensional complex affine space n :={(aa…aa12 , , ,nj ) ∈ , j = 1 , 2 , …n , } . Given polynomials f112()()()tt…t,,,nnln , ftt…t…ftt…t 212 ,,, ,, 12 ,,, in n variables tt…t12,,,n with coefficients in the field of complex numbers, the system of algebraic equations ⎧ ftt…t112()0,,,n = ⎪ ⎪ ftt…t()0,,,n = ⎨ 212 ⎪ ⎪ ⎩ ftt…tln()012,,, = gives rise to the set X ⊂ n of its solutions n Xaaa…a:={ = (12 , , ,nl ) ∈ fafa…fa 1 ()0()0 = , 2 = , , ()0} = , where we use the simplified notation f j ()afaa…a:=jn (12 , , , ) for j =12,,…l ,. Subsets of n of this form are called affine algebraic subsets. Consider the polynomial ring []tt…t12, ,,n in n variables tt…t12, ,,n with complex coefficients. The notation []ttt…t:= [12 , , , n ] will be used for simplicity when there is no fear of confusion. In the description of an algebraic set X above, consider the ideal I :={ϕϕ11f + 2ff…t 2 + + ϕϕϕϕll 1 , 2 , , l ∈[]} . of []t generated by f12,,,f…fl . Obviously, one has Xa=∈{()0}n fa =,∀∈. fI 249 ©Encyclopedia of Life Support Systems (EOLSS) MATHEMATICS: CONCEPTS, AND FOUNDATIONS – Vol. I - Algebraic Geometry and Applications - Tadao ODA By Hilbert’s basis theorem, any ideal J of []tt…t= [1,,n ] is necessarily finitely generated, that is, there exist gg…g12,,,∈s [] t such that J =+++,,∈.{ψψ11gg 2 2 ψψψss g… 1 s [] t} Thus one could have defined algebraic subsets of n to be those of the form n Xaa…afafI :={( =1 , ,n ) ∈ ()0 = , ∀ ∈ I } for an ideal I ⊂ []t . For two ideals I and J of [t ] with IJ⊂ , one obviously has X I ⊃ XJ . The radical I of an ideal I ⊂ [t ] is defined to be II:={gtg ∈[ ]r ∈ for a positive integer r} . An important theorem known as Hilbert’s Nullstellensatz says that for ideals I and J of [t ] one has XXI =⇐⇒=.J IJ To motivate further considerations of algebraic subsets, consider, for instance, 22 2 Xaaaa:={(12 , ) 1 + 2 = 1} ⊂ , which is an important geometric object with its “real locus” X ∩ 2 equal to the circle of radius one centered at the origin. The change of coordinates ⎧ ⎪utt11=+−1 2 ⎨ ⎪utt ⎩ 2 =−−121 in 2 turns X into an algebraic subset of different shape 2 Ybbbb:={(12 , ) ∈ 12 = 1} , 22 2 2 since ttuu12+= 12. Although the real loci X ∩ and Y ∩ have completely different geometric properties, one would like to regard X and Y to be intrinsically the same at least as geometric objects in n . 250 ©Encyclopedia of Life Support Systems (EOLSS) MATHEMATICS: CONCEPTS, AND FOUNDATIONS – Vol. I - Algebraic Geometry and Applications - Tadao ODA For that purpose, regard an element f ()tftt…tt= (12,,,n ) ∈ [] as a “polynomial function” on n defined by n ∋=aaa…afa()()12 ,,,n ∈. Given an algebraic subset X I for an ideal I ⊂ []t , the restrictions to X I of polynomial functions gt( ),∈ ht ( ) [ t ] give rise to the same function if the difference gt()− ht () belongs to the ideal I . Thus the residue class ring AX()I := [] t /I can be regarded as the set of “polynomial functions” on the algebraic subset X I . Here are two important observations which enable one to recover X I from the ring AX()I of its polynomial functions: • Through the inclusion map → [t ] sending c ∈ to the constant polynomial c , one regards [t ] as a -algebra. A -algebra homomorphism α :→[t ] (i.e., a ring homomorphism whose restriction to ⊂ [t ] is the identity map to ) is uniquely determined by α()tt…t12,αα ( ),, (n ) ∈ , and one has a bijective correspondence ~ n Hom-alg ( [tt…t 1,,, 2 n ] , ) ⎯⎯→ n sending a -algebra homomorphism α to (()ααtt12,,,∈ () α ()) tn . Here and ′ elsewhere, Hom-alg (B, B ) denotes the set of -algebra homomorphisms from a - algebra B to another -algebra B′. For the residue class ring AAX:=(I ) = [ t ] /I , the set Hom-alg (A, ) of -algebra homomorphisms can be identified with the subset of Hom-alg ( [t ], ) consisting of those α ’s such that α(I )= 0 . Thus one has a bijective correspondence ~ n Hom-alg ( [tAXX ], )⊃,⎯⎯ Hom -alg ( (I ) ) →⊂.I • The kernel of a -algebra homomorphism α :→[]t is obviously a maximal ideal of [t ]. As a consequence of an important theorem known as Noether’s normalization theorem together with the fact that is algebraically closed (the fundamental theorem of algebra), one sees that all maximal ideals of [t ] arise in this way, and one has a bijective correspondence ~ Hom-alg ( [tt ], )⎯⎯→, Max( [ ])α ker(α ) , 251 ©Encyclopedia of Life Support Systems (EOLSS) MATHEMATICS: CONCEPTS, AND FOUNDATIONS – Vol. I - Algebraic Geometry and Applications - Tadao ODA where Max( [t ]) denotes the set of maximal ideals of []t . A maximal ideal of AAX==/()I [] tI is of the form MI/ for a maximal ideal M of [t ] satisfying MI⊃ . Hence the set Max(A ) of maximal ideals of A may be regarded as a subset of Max( [t ]) consisting of those maximal ideals M satisfying MI⊃ . Thus one has a bijective correspondence ~ Hom-alg ( [tAXAXt ],⊃ ) Hom -alg ( (I ) ,⎯⎯ )→⊂. Max( (I )) Max( [ ]) In terms of the -algebra AX(I ):= [ t ] /I , one thus has bijections ~~ XAXAXI ←⎯⎯,⎯⎯Hom-alg ( (II ) )→. Max( ( )) n X I is given originally as a subset of . This information can be recovered from AX()I as the choice of a system of -algebra generators t…t1,,n ∈ AX()I , which are the images of t…t1,,n under the canonical surjection [tAX ]→ (I ) . The nilradical {0} of the ring At= [ ]/I , which is the ideal consisting of nilpotent elements of A , is nothing but II/ . One denotes AAred := /{0} = [ t ] /I . As a consequence of Hilbert’s Nullstellensatz, one thus has Max(AA )= Max ()red . In conclusion, the very basic among algebraically defined geometric objects of interest can now be intrinsically defined as the sets Max(A )=, Max()AAred for finitely generated -algebras which are called affine algebraic sets. (Finitely generated -algebras are also called -algebras of finite type.) An affine algebraic set X A := Max(A ) is not just a point set but has rich geometry hidden inside the -algebra A . Let us denote by m(x )∈ Max(A ) the maximal ideal corresponding to each x∈ X A . First of all, it is a topological space with the Zariski topology in which the closed subsets of X are defined as those subsets of the form X AA()IxXxI:= { ∈m () ⊃ } forideals IA ⊂ . 252 ©Encyclopedia of Life Support Systems (EOLSS) MATHEMATICS: CONCEPTS, AND FOUNDATIONS – Vol. I - Algebraic Geometry and Applications - Tadao ODA One sees that X A(IAI )=/ Max( ) , hence closed subsets of X A are affine algebraic sets as well. By definition, the open subsets of X A are those subsets of the form UIXXIxXxIU():=AA () = { ∈ Amm () ⊃// } =∪ f , with UxXxf f := { ∈ A () ∋ } . fI∈ Open sets of the form U f for f ∈ A are called distinguished open sets, and they thus form an open basis for the Zariski topology of X A . U f =∅ if f is a nilpotent element. Otherwise, consider the ring of quotients Af[1/ ] of A with respect to the multiplicatively closed set Sfff…:={1 , ,23 , , } . Since maximal ideals m()x with f ∈/ m()x are in one-to-one correspondence with the maximal ideals of A[1/f ] , one has a natural identification Max(A )⊃=UAff Max( [1 /. ]) n If X A is defined as before as an algebraic subset of by At…t= C[]1,,n /I , then X A n is the closed subset of =,,,Max( [tt…t12 n ]) defined by the ideal I with respect to the Zariski topology of the latter.
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