Table Of ContentUlrich Görtz | Torsten Wedhorn
Algebraic Geometry I
Ulrich Görtz | Torsten Wedhorn
Algebraic
Geometry I
Schemes
With Examples and Exercises
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The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie;
detailed bibliographic data are available in the Internet at http://dnb.d-nb.de.
Prof. Dr. Ulrich Görtz
Institute of Experimental Mathematics
University Duisburg-Essen
Ellernstraße 29
45326 Essen
[email protected]
Prof. Dr. Torsten Wedhorn
Department of Mathematics
University of Paderborn
Warburger Straße 100
33098 Paderborn
[email protected]
1st Edition 2010
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V
Contents
Introduction 1
1 Prevarieties 7
Affine algebraic sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Affine algebraic sets as spaces with functions . . . . . . . . . . . . . . . . . . 17
Prevarieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Projective varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2 Spectrum of a Ring 40
Spectrum of a ring as a topological space . . . . . . . . . . . . . . . . . . . . 41
Excursion: Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Spectrum of a ring as a locally ringed space . . . . . . . . . . . . . . . . . . . 57
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3 Schemes 66
Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
Examples of schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Basic properties of schemes and morphisms of schemes . . . . . . . . . . . . . 74
Prevarieties as Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Subschemes and Immersions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4 Fiber products 93
Schemes as functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Fiber products of schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Base change, Fibers of a morphism . . . . . . . . . . . . . . . . . . . . . . . . 105
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5 Schemes over fields 118
Schemes over a field which is not algebraically closed . . . . . . . . . . . . . . 118
Dimension of schemes over a field . . . . . . . . . . . . . . . . . . . . . . . . . 120
Schemes over fields and extensions of the base field . . . . . . . . . . . . . . . 133
Intersections of plane curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6 Local Properties of Schemes 145
The tangent space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
Smooth morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
Regular schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
Normal schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
VI Contents
7 Quasi-coherent modules 169
Excursion: O -modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
X
Quasi-coherent modules on a scheme . . . . . . . . . . . . . . . . . . . . . . . 181
Properties of quasi-coherent modules . . . . . . . . . . . . . . . . . . . . . . . 189
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
8 Representable Functors 205
Representable Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
The example of the Grassmannian . . . . . . . . . . . . . . . . . . . . . . . . 209
Brauer-Severi schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
9 Separated morphisms 226
Diagonal of scheme morphisms and separated morphisms . . . . . . . . . . . 226
Rational maps and function fields. . . . . . . . . . . . . . . . . . . . . . . . . 232
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
10 Finiteness Conditions 241
Finiteness conditions (noetherian case) . . . . . . . . . . . . . . . . . . . . . . 242
Finiteness conditions in the non-noetherian case . . . . . . . . . . . . . . . . 249
Schemes over inductive limits of rings . . . . . . . . . . . . . . . . . . . . . . 258
Constructible properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278
11 Vector bundles 286
Vector bundles and locally free modules . . . . . . . . . . . . . . . . . . . . . 286
Flattening stratification for modules . . . . . . . . . . . . . . . . . . . . . . . 296
Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298
Vector bundles on P1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
12 Affine and proper morphisms 320
Affine morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320
Finite and quasi-finite morphisms . . . . . . . . . . . . . . . . . . . . . . . . . 324
Serre’s and Chevalley’s criteria to be affine . . . . . . . . . . . . . . . . . . . 334
Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
Proper morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
Zariski’s main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361
13 Projective morphisms 366
Projective spectrum of a graded algebra . . . . . . . . . . . . . . . . . . . . . 367
Embeddings into projective space . . . . . . . . . . . . . . . . . . . . . . . . . 384
Blowing-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418
VII
14 Flat morphisms and dimension 423
Flat morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
Properties of flat morphisms. . . . . . . . . . . . . . . . . . . . . . . . . . . . 429
Faithfully flat descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
Dimension and fibers of morphisms . . . . . . . . . . . . . . . . . . . . . . . . 463
Dimension and regularity conditions . . . . . . . . . . . . . . . . . . . . . . . 473
Hilbert schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 478
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 480
15 One-dimensional schemes 485
Morphisms into and from one-dimensional schemes . . . . . . . . . . . . . . . 485
Valuative criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 487
Curves over fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491
Divisors on curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 500
16 Examples 503
Determinantal varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503
Cubic surfaces and a Hilbert modular surface . . . . . . . . . . . . . . . . . . 520
Cyclic quotient singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 528
Abelian varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539
A The language of categories 541
B Commutative Algebra 547
C Permanence for properties of morphisms of schemes 573
D Relations between properties of morphisms of schemes 576
E Constructible and open properties 578
Bibliography 583
Detailed List of Contents 588
Index of Symbols 598
Index 602
1
Introduction
Algebraic geometry has its origin in the study of systems of polynomial equations
f (x ,...,x )=0,
1 1 n
.
.
.
f (x ,...,x )=0.
r 1 n
Here the f ∈k[X ,...,X ] are polynomials in n variables with coefficients in a field k.
i 1 n
ThesetofsolutionsisasubsetV(f ,...,f )ofkn.Polynomialequationsareomnipresent
1 r
inandoutsidemathematics,andhavebeenstudiedsinceantiquity.Thefocusofalgebraic
geometry is studying the geometric structure of their solution sets.
If the polynomials f are linear, then V(f ,...,f ) is a subvector space of kn. Its
i 1 r
“size” is measured by its dimension and it can be described as the kernel of the linear
map kn →kr, x=(x ,...,x )(cid:4)→(f (x),...,f (x)).
1 n 1 r
For arbitrary polynomials, V(f ,...,f ) is in general not a subvector space. To study
1 r
it, one uses the close connection of geometry and algebra which is a key property of
algebraic geometry, and whose first manifestation is the following: If g =g f +...g f
1 1 r r
is a linear combination of the f (with coefficients g ∈ k[T ,...,T ]), then we have
i i 1 n
V(f ,...,f ) =V (g,f ,...,f ). Thus the set of solutions depends only on the ideal
1 r 1 r
a ⊆ k[T ,...,T ] generated by the f . On the other hand we may consider polynomials
1 n i
f ∈k[T ,...,T ], view them as polynomial functions kn →k, and define the ideal a(cid:2) of
1 n
thosef suchthatf vanishesonV(f ,...,f ).Itisoneofthefirstmainresults(Hilbert’s
1 r
Nullstellensatz) that if k is algebraically closed, then a(cid:2) is closely related to a, more
precisely
a(cid:2) =rad(a):={f ∈k[T ,...,T ] ; ∃m>0:fm ∈a}.
1 n
The quotient k[T ,...,T ]/a(cid:2) may be considered as thek -algebra of polynomial func-
1 n
tions on the affine variety V(f ,...,f ). We obtain a close relation between ideals a of
1 r
k[T ,...,T ] (or, equivalently, quotient algebras of k[T ,...,T ]) and affine varieties in
1 n 1 n
kn –atleastifk isalgebraicallyclosed.Fornotalgebraicallyclosedfieldsk thisapproach
is too naive.
Besides this algebraic description, one can endow the sets V(f ,...,f ) with a “geo-
1 r
metricstructure”.Theonlyreasonabletopologywhichcanbedefinedpurelyinalgebraic
terms, i.e., without appealing to analytic notions as convergence, is the Zariski topology
which is the coarsest topology (on kn, say) such that all zero sets of polynomials are
closed. Not surprisingly, it is very coarse and therefore is not sufficient to determine
the “geometric structure” of the spaces in question. The right way to remedy this, is to
considereachspacetogetherwiththeentiretyoffunctionsonthespace.Similarlyasadif-
ferentiable(orholomorphic)manifoldisdeterminedbyitstopologicalstructuretogether
with the entirety of differentiable (or holomorphic) functions on all its open subsets, we
obtain a satisfactory notion of algebraic geometric objects, “affine varieties over k”, by
considering closed subsets of kn together with the entirety of functions on them, which
in this case means all functions defined by rational polynomials.
2 Introduction
Polynomial equations also arise in number theory, and especially in the last decades
algebraic-geometricmethodshavebecomeextremelyfruitfulforsolvingnumber-theoretic
problems. In this case the polynomials have coefficients in Q or Z (or more generally in
numberfields,finitefields,orp-adicrings).OneofthemostfamousexamplesisFermat’s
equation xm +ym = zm with x,y,z ∈ Z. The proof of Fermat’s Last Theorem which
asserts that this equations has no solutions for m≥3, xyz (cid:8)=0, by Wiles and Taylor in
1995 relies heavily on modern algebraic geometry.
The unifying approach to study polynomial equations f ,...,f over arbitrary (com-
1 r
mutative) rings R is the theory of schemes developed by Grothendieck and his school. It
allowstoattachtoanarbitrarycommutativeringA(e.g.,A=R[T ,...,T ]/(f ,...,f )
1 n 1 r
or A = Z[X,Y,Z]/(Xm +Ym −Zm)) a geometric object SpecA consisting of a topo-
logical space X and a datum O of “systems of functions” on this space such that the
X
ring of “globally defined functions” on SpecA is the ring A itself. Such a pair (X,O ) is
X
a so-called locally ringed space. This allows us to view commutative rings as geometric
objectscalledaffineschemes.TheaffinevarietyV(f ,...,f )⊆kn canberecoveredfrom
1 r
the affine scheme Speck[T ,...,T ]/(f ,...,f ).
1 n 1 r
Asinelementarygeometrysomeproblemsonlyhaveasatisfyingsolutionifweconsider
them not in affine space but in projective space. For instance two different lines in the
affine plane intersect always in one point except if they are parallel. By adding points
at infinity (the “horizon”) we obtain the projective plane, where any two different lines
intersect in precisely one point. The projective space can be obtained by gluing affine
spaces. Vastly generalizing this process we arrive at the central notion of this book: a
scheme. It is defined as a locally ringed space that is locally isomorphic to an affine
scheme. Note the similarity to the definition of a smooth n-dimensional manifold which
is a geometric object that is locally isomorphic – within the right category – to an open
subset of Rn.
For schemes geometric notions as dimension or smoothness are defined. As schemes
are locally given by commutative rings, many of these notions are defined in terms of
rings and ideals. Conversely, every definition or result in commutative algebra has its
geometric counterpart in the theory of schemes. Thus algebra and geometry become two
aspects of the same theory.
Anotherreasonfortheimportanceofschemesisthatitisoftenpossibletoparameterize
interesting objects by schemes. An example is the Grassmannian which is a scheme that
parameterizes subvector spaces of a fixed dimension in a given finite-dimensional vector
space. The general concept behind schemes as parameter spaces is the point of view of
schemes representing certain functors. This plays an important role in modern algebraic
geometry and beyond. It will be one of the main focuses in this book.
Grothendieck’s theory of schemes is technically demanding but essential in modern
algebraic geometry even for applications in classical complex algebraic geometry. Even
more so it is indispensable in arithmetic geometry. For instance, the Fermat problem
was solved in 1994 by Wiles and Taylor using techniques from algebraic geometry in its
mostmodernform.Moreoveralgebraicgeometryhasalsobecomeanimportanttoolwith
manyapplicationsinotherfieldsofmathematicssuchastopology,representationtheory,
Lie theory, group theory, string theory, or cryptography.
The goal of this book is to provide its reader with the background in algebraic geometry
to go on to current research in algebraic geometry itself, in number theory, or in other
fieldsofmathematics.Itstrivesforthenecessarygeneralitytobeastablesteppingstone
for most of these fields.
3
There is a wealth of literature on algebraic geometry from which we learned a lot.
It is a pleasure to acknowledge the overwhelming influence of the pioneering work of
Grothendieck and Dieudonn´e ([EGAI],[EGAInew], [EGAII],[EGAIII], [EGAIV]). Other
sources are Mumford’s red book [Mu1], and the books by Shafarevich [Sh], Hartshorne
[Ha3]andPerrin[Per].Furthermorewelistthemorespecializedbooks,eachwithitsown
focus, by Mumford [Mu2], Griffiths and Harris [GH], Liu [Liu], Harris [Har], Eisenbud
and Harris [EH], and Harder [Ha].
Further sources had a more local impact. We followed Kurke [Ku1], [Ku2] and Pe-
skine [Pes] quite closely in our proof of Zariski’s main theorem. In our treatment of
geometric properties of schemes over a field one of our main references was Jouanolou’s
book [Jo]. Our main source for determinantal varieties was the book [BV] by Bruns and
Vetter.FortheexampleofcubicsurfacesweprofitedmuchfromBeauville[Bea]and[Ge]
and for the example of Brauer-Severi varieties from Gille and Szamuely [GS].
Leitfaden
The notion of scheme which is the main object of investigation of the whole book is
introduced in Chapter 3, using the affine schemes defined in Chapter 2 as local building
blocks. These two chapters are therefore indispensable for all of the book. In Chapter
1 we discuss a precursor of schemes, namely prevarieties (over an algebraically closed
field). These prevarieties are much closer to geometric intuition, and on the other hand
comprise a large number of interesting schemes. However, besides other defects of this
notion, prevarieties are not suitable for discussing arithmetic questions because it is not
easilypossibletolinkobjectslivingoverbasefieldsofdifferentcharacteristics.InChapter
4weintroducefiberproductsofschemeswhichareubiquitousinalloftheremainder.In
particularfiberproductsallowustoviewthefibersofmorphismsofschemesasschemes,
sothatwecanmakeprecisethephilosophythatamorphismf: X →S ofschemesshould
beseenasafamily(f−1(s))s∈S ofschemes.Forbeginnersinalgebraicgeometry,working
through all of Chapters 1 to 4 is therefore recommended. For those with a background
in classical algebraic geometry, Chapter 1 can probably be skipped, and all readers with
someknowledgeaboutschemesshouldbeabletostartwithChapter5withouttoomany
problems.
After this first part of the book, some choices can be made. In Chapter 5, the part
on dimension of schemes over a field should be read in any case—not only since it is
used at many places, but because the dimension of a scheme is a fundamental notion in
algebraicgeometryasawhole.Thepartsonschemesovernon-algebraicallyclosedfields,
andonbasechangeofthegroundfield,aremorespecializedandcanbeskippedatafirst
reading. References to the latter can usually be avoided by assuming that the base field
in question is perfect or algebraically closed. The part on intersections of plane curves
with a proof of B´ezout’s theorem is one of the first applications of the theory developed
sofar,butisnotstrictlynecessaryfortherestofthetext;theonlyplacewhereitisused
again is the discussion of elliptic curves in Chapter 16.
The topic of Chapter 6 are local properties of schemes, in particular the notions of
tangent space, smooth, regular and singular points and of normality. We make essential
use of the notion of normal scheme in Chapter 12 when we discuss normalizations and
Zariski’s main theorem.
Chapter 7 provides definitions and results on (quasi-coherent) O -modules. Its first
X
