The Geometry of Syzygies: A Second Course in Algebraic Geometry and Commutative Algebra

252 Pages·2004·2.35 MB·English

by  David Eisenbud

229
Graduate Texts in Mathematics
Editorial Board
S. Axler F.W. Gehring K.A. Ribet
David Eisenbud
The Geometry of Syzygies
A Second Course in Commutative Algebra
and Algebraic Geometry
With 27 Figures
DavidEisenbud
MathematicalSciencesResearchInstitute
Berkeley,CA94720
USA
[email protected]
EditorialBoard
S.Axler F.W.Gehring K.A.Ribet
MathematicsDepartment MathematicsDepartment MathematicsDepartment
SanFranciscoState EastHall UniversityofCalifornia,
University UniversityofMichigan Berkeley
SanFrancisco,CA94132 AnnArbor,MI48109 Berkeley,CA94720-3840
USA USA USA
[email protected] [email protected] [email protected]
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Contents
Preface: Algebra and Geometry ix
What Are Syzygies? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
The Geometric Content of Syzygies . . . . . . . . . . . . . . . . . . . . xi
What Does Solving Linear Equations Mean? . . . . . . . . . . . . . . . xii
Experiment and Computation . . . . . . . . . . . . . . . . . . . . . . . xiii
What’s In This Book? . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv
Prerequisites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv
How Did This Book Come About? . . . . . . . . . . . . . . . . . . . . . xv
Other Books . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi
Thanks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi
1 Free Resolutions and Hilbert Functions 1
The Generation of Invariants . . . . . . . . . . . . . . . . . . . . . . . . 1
Enter Hilbert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1A The Study of Syzygies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
The Hilbert Function Becomes Polynomial . . . . . . . . . . . . . . . . 4
1B Minimal Free Resolutions . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Describing Resolutions: Betti Diagrams . . . . . . . . . . . . . . . . . . 7
Properties of the Graded Betti Numbers . . . . . . . . . . . . . . . . . 8
The Information in the Hilbert Function . . . . . . . . . . . . . . . . . 9
1C Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 First Examples of Free Resolutions 15
2A Monomial Ideals and Simplicial Complexes . . . . . . . . . . . . . . . . 15
Simplicial Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Labeling by Monomials . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Syzygies of Monomial Ideals . . . . . . . . . . . . . . . . . . . . . . . . 18
vi Contents
2B Bounds on Betti Numbers and Proof of Hilbert’s Syzygy Theorem . . . 20
2C Geometry from Syzygies: Seven Points in P3 . . . . . . . . . . . . . . . 22
The Hilbert Polynomial and Function... . . . . . . . . . . . . . . . . . 23
...and Other Information in the Resolution. . . . . . . . . . . . . . . . 24
2D Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3 Points in P2 31
3A The Ideal of a Finite Set of Points . . . . . . . . . . . . . . . . . . . . . 32
3B Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3C Existence of Sets of Points with Given Invariants . . . . . . . . . . . . . 42
3D Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4 Castelnuovo–Mumford Regularity 55
4A Definition and First Applications. . . . . . . . . . . . . . . . . . . . . . 55
4B Characterizations of Regularity: Cohomology . . . . . . . . . . . . . . . 58
4C The Regularity of a Cohen–Macaulay Module . . . . . . . . . . . . . . 65
4D The Regularity of a Coherent Sheaf . . . . . . . . . . . . . . . . . . . . 67
4E Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5 The Regularity of Projective Curves 73
5A A General Regularity Conjecture . . . . . . . . . . . . . . . . . . . . . . 73
5B Proof of the Gruson–Lazarsfeld–Peskine Theorem . . . . . . . . . . . . 75
5C Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6 Linear Series and 1-Generic Matrices 89
6A Rational Normal Curves. . . . . . . . . . . . . . . . . . . . . . . . . . . 90
6A.1 Where’d That Matrix Come From? . . . . . . . . . . . . . . . . . 91
6B 1-Generic Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
6C Linear Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
6D Elliptic Normal Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6E Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7 Linear Complexes and the Linear Syzygy Theorem 119
7A Linear Syzygies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
7B The Bernstein–Gelfand–Gelfand Correspondence . . . . . . . . . . . . . 124
7C Exterior Minors and Annihilators . . . . . . . . . . . . . . . . . . . . . 130
7D Proof of the Linear Syzygy Theorem . . . . . . . . . . . . . . . . . . . . 135
7E More about the Exterior Algebra and BGG . . . . . . . . . . . . . . . . 136
7F Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
8 Curves of High Degree 145
8A The Cohen–Macaulay Property. . . . . . . . . . . . . . . . . . . . . . . 146
8A.1 The Restricted Tautological Bundle . . . . . . . . . . . . . . . . 148
8B Strands of the Resolution . . . . . . . . . . . . . . . . . . . . . . . . . . 153
8B.1 The Cubic Strand . . . . . . . . . . . . . . . . . . . . . . . . . . 155
8B.2 The Quadratic Strand . . . . . . . . . . . . . . . . . . . . . . . . 159
8C Conjectures and Problems . . . . . . . . . . . . . . . . . . . . . . . . . 169
8D Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
Contents vii
9 Clifford Index and Canonical Embedding 177
9A The Cohen–Macaulay Property and the Clifford Index . . . . . . . . . . 177
9B Green’s Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
9C Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
Appendix 1 Introduction to Local Cohomology 187
A1A Definitions and Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
A1B Local Cohomology and Sheaf Cohomology . . . . . . . . . . . . . . . . 195
A1C Vanishing and Nonvanishing Theorems . . . . . . . . . . . . . . . . . . 198
A1D Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
Appendix 2 A Jog Through Commutative Algebra 201
A2A Associated Primes and Primary Decomposition . . . . . . . . . . . . . . 202
A2B Dimension and Depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
A2C Projective Dimension and Regular Local Rings . . . . . . . . . . . . . . 208
A2D Normalization: Resolution of Singularities for Curves . . . . . . . . . . 210
A2E The Cohen–Macaulay Property. . . . . . . . . . . . . . . . . . . . . . . 213
A2F The Koszul Complex . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
A2G Fitting Ideals and Other Determinantal Ideals . . . . . . . . . . . . . . 220
A2H The Eagon–Northcott Complex and Scrolls . . . . . . . . . . . . . . . . 222
References 227
Index 237
Preface: Algebra and Geometry
Syzygy [from] Gr. (cid:0) (cid:2) (cid:4) (cid:6) (cid:8)(cid:9)
yoke, pair, copulation, conjunction
— Oxford English Dictionary (etymology)
Implicitin thename “algebraicgeometry” is therelation between geometry and
equations. The qualitative study of systems of polynomial equations is the chief
subject of commutative algebra as well. But when we actually study a ring or
a variety, we often have to know a great deal about it before understanding its
equations. Conversely, given a system of equations, it can be extremely difficult
to analyze its qualitative properties, such as the geometry of the corresponding
variety. The theory of syzygies offers a microscope for looking at systems of
equations, and helps to make their subtle properties visible.
This book is concerned with the qualitative geometric theory of syzygies. It
describes geometric properties of a projective variety that correspond to the
numbers and degrees of its syzygies or to its having some structural property—
such as being determinantal, or having a free resolution with some particularly
simple structure. It is intended as a second course in algebraic geometry and
commutative algebra, such as I have taught at Brandeis University, the Institut
Poincar´e in Paris, and the University of California at Berkeley.
x Preface: Algebra and Geometry
What Are Syzygies?
In algebraic geometry over a field K we study the geometry of varieties through
properties of the polynomial ring
S =K[x ,...,x ]
0 r
anditsideals.Itturnsoutthattostudyidealseffectivelywewealsoneedtostudy
more general graded modules over S. The simplest way to describe a module is
by generators and relations. We may think of a set A⊂M of generators for an
S-module M as a map from a free S-module F =SA onto M, sending the basis
element of F corresponding to a generator m∈A to the element m∈M.
Let M be the kernel of the map F → M; it is called the module of syzygies
1
of M corresponding to the given choice of generators, and a syzygy of M is an
elementofM —alinearrelation,withcoefficientsinS,onthechosengenerators.
1
When we give M by generators and relations, we are choosing generators for M
and generators for the module of syzygies of M.
The use of “syzygy” in this context seems to go back to Sylvester [1853].
The word entered the language of modern science in the seventeenth century,
with the same astronomical meaning it had in ancient Greek: the conjunction
or opposition of heavenly bodies. Its literal derivation is a yoking together, just
like “conjunction”, with which it is cognate.
If r =0, so that we are working over the polynomial ring in one variable, the
module of syzygies is itself a free module, since over a principal ideal domain
every submodule of a free module is free. But when r > 0 it may be the case
thatanysetofgeneratorsofthemoduleofsyzygieshasrelations.Tounderstand
them,weproceedasbefore:wechooseageneratingsetofsyzygiesandusethem
to define a map from a new free module, say F , onto M ; equivalently, we give
1 1
a map φ : F → F whose image is M . Continuing in this way we get a free
1 1 1
resolution of M, that is, a sequence of maps
··· (cid:1) F φ(cid:1)2 F φ(cid:1)1 F (cid:1) M (cid:1) 0,
2 1
where all the modules F are free and each map is a surjection onto the kernel
i
of thefollowingmap. TheimageM of φ is calledthe i-th module of syzygies of
i i
M.
In projective geometry we treat S as a graded ring by giving each variable x
i
degree 1, and we will be interested in the case where M is a finitely generated
graded S-module. In this case we can choose a minimal set of homogeneous
generators for M (that is, one with as few elements as possible), and we choose
the degrees of the generators of F so that the map F → M preserves degrees.
The syzygy module M is then a graded submodule of F, and Hilbert’s Basis
1
Theorem tells us that M is again finitely generated, so we may repeat the
1
procedure. Hilbert’s Syzygy Theorem tells us that the modules M are free as
i
soon as i≥r.
The free resolution of M appears to depend strongly on our initial choice of
generators for M, as well as the subsequent choices of generators of M , and so
1
Preface: Algebra and Geometry xi
on. But if M is a finitely generated graded module and we choose a minimal
set of generators for M, then M is, up to isomorphism, independent of the
1
minimal set of generators chosen. It follows that if we choose minimal sets of
generatorsateachstageintheconstructionofafreeresolutionwegetaminimal
free resolution of M that is, up to isomorphism, independent of all the choices
made. Since,by theHilbertSyzygyTheorem,M is freefor i>r, we see thatin
i
the minimal free resolution F = 0 for i > r+1. In this sense the minimal free
i
resolution is finite: it has length at most r+1. Moreover, any free resolution of
M can be derived from the minimal one in a simple way (see Section 1B).
The Geometric Content of Syzygies
The minimal free resolution of a module M is a good tool for extracting infor-
mation about M. For example, Hilbert’s motivation for his results just quoted
was to devise a simple formula for the dimension of the d-th graded component
of M as a function of d. He showed that the function d(cid:5)→dimKMd, now called
the Hilbert function of M, agrees for large d with a polynomial function of d.
The coefficients of this polynomial are among the most important invariants of
the module. If X ⊂ Pr is a curve, the Hilbert polynomial of the homogeneous
coordinate ring S of X is
X
(degX)d+(1−genusX),
whose coefficients degX and 1−genusX give a topological classification of the
embeddedcurve.Hilbertoriginallystudiedfreeresolutionsbecausetheirdiscrete
invariants,thegradedBettinumbers,determinetheHilbertfunction(seeChapter
1).
ButthegradedBettinumberscontainmoreinformationthantheHilbertfunc-
tion.AtypicalexampleisthecaseofsevenpointsinP3,describedinSection2C:
everysetof7pointsinP3 inlinearlygeneralpositionhasthesameHilbertfunc-
tion, but the graded Betti numbers of the ideal of the points tell us whether the
points lie on a rational normal curve.
Most of this bookis concernedwith examples onedimension higher:westudy
the graded Betti numbers of the ideals of a projective curve, and relate them to
the geometric properties of the curve. To take just one example from those we
will explore, Green’s Conjecture (still open) says that the graded Betti numbers
of the ideal of a canonically embedded curve tell us the curve’s Clifford index
(most of thetime this indexis2 lessthan theminimal degreeof a map fromthe
curve to P1). This circle of ideas is described in Chapter 9.
Some work has been done on syzygies of higher-dimensional varieties too,
though this subject is less well-developed. Syzygies are important in the study
of embeddings of abelian varieties, and thus in the study of moduli of abelian
varieties (for example [Gross and Popescu 2001]). They currently play a part
in the study of surfaces of low codimension (for example [Decker and Schreyer
2000]), and other questions about surfaces (for example [Gallego and Purnapra-
xii Preface: Algebra and Geometry
jna 1999]). They have also been used in the study of Calabi–Yau varieties (for
example [Gallego and Purnaprajna 1998]).
What Does Solving Linear Equations Mean?
A free resolution may be thought of as the result of fully solving a system of
linearequationswithpolynomialcoefficients.Tosetthestage,considerasystem
of linear equations AX = 0, where A is a p×q matrix of elements of K, which
we may think of as a linear transformation
F =Kq A(cid:1) Kp =F .
1 0
Suppose we find some solution vectors X ,...,X . These vectors constitute a
1 n
complete solution to the equations if every solution vector can be expressed as
a linear combination of them. Elementary linear algebra shows that there are
completesolutionsconsisting ofq−rankA independentvectors. Moreover, there
is a powerful test for completeness: A given set of solutions {X } is complete if
i
and only if it contains q−rankA independent vectors.
A set of solutions can be interpreted as the columns of a matrix X defining a
map X :F →F such that
2 1
X(cid:1) A(cid:1)
F F F
2 1 0
is a complex. The test for completeness says that this complex is exact if and
only if rankA+rankX = rankF . If the solutions are linearly independent as
1
well as forming a complete system, we get an exact sequence
0→F X(cid:1) F A(cid:1) F .
2 1 0
Suppose now that the elements of A vary as polynomial functions of some
parameters x ,...,x , and we need to find solution vectors whose entries also
0 r
vary as polynomial functions. Given a set X ,...,X of vectors of polynomials
1 n
that are solutions to the equations AX = 0, we ask whether every solution can
bewrittenasalinearcombinationoftheX withpolynomialcoefficients.Ifsowe
i
say that the set of solutions is complete. The solutions are once again elements
ofthekernelofthemapA:F =Sq →F =Sp,andacompletesetofsolutions
1 0
is a set of generators of the kernel. Thus Hilbert’s Basis Theorem implies that
there do exist finite complete sets of solutions. However, it might be that every
complete set of solutions is linearly dependent: the syzygy module M = kerA
1
is not free. Thus to understand the solutions we must compute the dependency
relations on them, and then the dependency relations on these. This is precisely
afreeresolutionofthecokernelofA.Whenwethinkofsolvingasystemoflinear
equations, we should think of the whole free resolution.
Onerewardforthispointofviewisacriterionanalogoustotherankcriterion
givenaboveforthecompletenessofasetofsolutions.Weknownosimplecriterion