MATH 631: ALGEBRAIC GEOMETRY I
                                   INTRODUCTION TO ALGEBRAIC VARIETIES
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                          LECTURESBYPROF.MIRCEAMUSTAT¸A;NOTESBYALEKSANDERHORAWA
                These are notes from Math 631: Algebraic geometry I taught by Professor Mircea Musta¸t˘a
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                in Fall 2017, LT X’ed by Aleksander Horawa (who is the only person responsible for any
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                mistakes that may be found in them).
                This version is from January 5, 2018. Check for the latest version of these notes at
                                   http://www-personal.umich.edu/~ahorawa/index.html
                If you find any typos or mistakes, please let me know at ahorawa@umich.edu.
                References for the first part of the course:
                    (1) Mumford, The Red Book of varieties and schemes [Mum99], Chapter I
                    (2) Shafarevich, Basic Algebraic Geometry [Sha13], Chapter I,
                    (3) Hartshorne, Algebraic Geometry [Har77], Chapter I.
                The problem sets, homeworks, and official notes can be found on the course website:
                                http://www-personal.umich.edu/~mmustata/631_2017.html
                                                              Contents
                   1.   Affine algebraic varieties                                                                      3
                   1.1.   Algebraic sets and ideals                                                                   3
                                               n
                   1.2.   The topology on A                                                                           9
                   1.3.   Regular functions and morphisms                                                            12
                   1.4.   Local rings                                                                                19
                   1.5.   Rational functions and maps                                                                20
                   1.6.   Products of affine and quasi-affine varieties                                                  24
                   1.7.   Affine toric varieties                                                                       30
                   1.8.   Normal varieties                                                                           37
                   2.   Dimension of algebraic varieties                                                             38
                   2.1.   Krull dimension                                                                            38
                   Date: January 5, 2018.
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                   2.2.   Finite morphisms between affine varieties                                                    39
                   2.3.   The Principal Ideal Theorem (Krull)                                                        43
                   2.4.   Dimension of fibers of morphisms                                                            48
                   3.   General algebraic varieties                                                                  50
                   3.1.   Presheaves and sheaves                                                                     50
                   3.2.   Prevarieties                                                                               53
                   3.3.   Subvarieties                                                                               55
                   3.4.   Fibered products of prevarieties                                                           58
                   3.5.   Separated prevarieties                                                                     60
                   4.   Projective varieties                                                                         63
                   4.1.   The projective space                                                                       63
                   4.2.   Projective varieties                                                                       68
                   5.   Classes of morphisms                                                                         76
                   5.1.   Proper morphisms and complete varieties                                                    76
                   5.2.   Finite morphisms                                                                           81
                   5.3.   Flat morphisms                                                                             84
                   6.   Smooth varieties                                                                             88
                   6.1.   The tangent space                                                                          89
                   6.2.   Smooth points and varieties                                                                92
                   6.3.   Blow ups (of affine varieties)                                                               93
                   6.4.   Back to smooth points                                                                      99
                   6.5.   Bertini’s Theorem                                                                         103
                   6.6.   Smooth morphisms between smooth varieties                                                 105
                   6.7.   Resolution of singularities                                                               107
                   7.   Quasicoherent and coherent sheaves                                                          107
                   7.1.   Operations with sheaves                                                                   107
                   7.2.   O -modules                                                                                114
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                   7.3.   Quasicoherent sheaves on affine varieties                                                   117
                   References                                                                                       120
                                               MATH 631: ALGEBRAIC GEOMETRY I                                     3
                                               1. Affine algebraic varieties
                1.1. Algebraic sets and ideals. The goal is to establish a correspondence
                                   geometric objects defined  ←→             ideals in a    .
                                     by polynomial equations                polynomial ring
                Let k be a fixed algebraically closed field of arbitrary characteristic. For example, k could
                be C, Q, or F for p prime.
                               p
                                                                     n      n               n
                Say n is fixed. The n-dimensional affine space A or Ak is (as a set) k and the polynomial
                ring in n variables over k is R = k[x ,...,x ].
                                                        1       n
                How do these two objects correspond to each other? First, note that if f ∈ R and u =
                (u ,...,u ) ∈ kn, we can evaluate f at u to get f(u) ∈ k. More specifically, for u =
                  1       n
                (u ,...,u ) ∈ kn, we get a homomorphism
                  1       n
                                                R=k[x ,...,x ] → k, x 7→ u
                                                         1       n          i     i
                which is surjective with kernel (x − u ,...,x −u ).
                                                    1    1       n     n
                Definition 1.1.1. Given a subset S ⊆ R, define
                                   V(S) = {u = (u ,...,u ) ∈ An | f(u) = 0 for all f ∈ S}
                                                    1       n
                                                               n
                called the zero locus of S or the subset of A defined by S.
                                          n
                An algebraic subset of A is a subset of the form V (S) for some S ⊆ R.
                Example 1.1.2. A linear subspace of kn is an algebraic subset. We can take S to a finite
                collection of linear polynomials:
                                                 ( n                          )
                                                   Xax :a ∈k,notall 0 .
                                                         i i   i
                                                    i=1
                More generally, any translation of a linear subspace is an algebraic subset (called an affine
                subspace).
                Example 1.1.3. A union of 2 lines in A2
                                                                                 V(x1x2)
                                                      2
                Example 1.1.4. A hyperbola in A .
                                                                         ˘
                4                                       MIRCEAMUSTAT¸A
                                                     x2
                                                                      x1     V(x1x2 −1)
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                Remark 1.1.5. The pictures are drawn in R . For obvious reasons, it is impossible to draw
                              2                  2      2                               2
                pictures in C (or especially Q or Fp ), but even the pictures in R can be used to develop
                a geometric intuition.
                Proposition 1.1.6.
                   (1) ∅ = V(1) = V(R) (hence ∅ is an algebraic subset).
                         n                    n
                   (2) A =V(0) (hence A is an algebraic subset).
                   (3) If I is the ideal generated by S, then V (S) = V (I).
                   (4) If I ⊆ J are ideals, then V (J) ⊆ V (I).
                   (5) If (I )   is a family of ideals, then
                            α α                                    !               !
                                            \V(I )=V [I              =V XI .
                                                   α              α               α
                                             α                α               α
                   (6) If I,J are ideals, then
                                             V(I)∪V(J)=V(I·J)=V(I∩J).
                Proof. Properties (1) and (2) are trivial. For (3), recall that
                                       I = {g f +···+g f | m≥0,f ∈S,g ∈R}
                                               1 1          m m             i       i
                and
                                   (g f +···+g f )(u) = g (u)f (u)+···+g (u)f (u).
                                     1 1          m m          1     1              m     m
                Properties (4) and (5) follow easily from definitions. For (6), note first that
                                              V(I)∪V(J)⊆V(I∩J)⊆V(I·J)
                by property (4). To show V(I · J) ⊆ V(I) ∪ V(J), suppose u ∈ V(I · J) \ (V(I) ∪ V(J)).
                Then there exist f ∈ I and g ∈ J such that f(u) 6= 0 and g(u) 6= 0. But then f · g ∈ I · J
                and (f · g)(u) = f(u)g(u) 6= 0, which is a contradiction.                                       
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                Remark 1.1.7. By Proposition 1.1.6, the algebraic subsets of A form the closed sets of a
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                topology on A , the Zariski topology.
                Suppose now that W ⊆ An is any subset. Then
                                        I(W)={f ∈R|f(u)=0for all u∈W}⊆R
                is an ideal in R.