DOCSLIB.ORG

GEOMETRIC METHODS in COMMUTATIVE ALGEBRA 1. Local

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
GEOMETRIC METHODS in COMMUTATIVE ALGEBRA 1. Local GEOMETRIC METHODS IN COMMUTATIVE ALGEBRA STEVEN DALE CUTKOSKY 1. Local Cohomology Suppose that R is a Noetherian ring, and M is an R-module. Suppose that I is an ideal in R, with generators I = (f1; : : : ; fn). Consider the modified Cech complex C∗ : 0 ! C0 ! C1 !···! Cd ! 0 Where C0 = R and M Ct = R : fi1 fi2 ···fit 1≤i1<i2<···<it≤n The Local Cohomology of M is i i ∗ HI (M) = H (M ⊗R C ): 0 k HI (M) = ff 2 M j I f = 0 for some k ≥ 0g = ΓI (M); the set of elements of M which have support in I. i HI (M) does not depend on the choice of generators of I. Further, Hi(M) = Hpi (M): I I ΓI (∗) is a left exact functor, and the cohomology modules are its right derived functors. If 0 ! A ! B ! C ! 0 is a short exact sequence of R-modules, then there is a long exact sequence 0 0 0 1 0 ! HI (A) ! HI (B) ! HI (C) ! HI (A) ! ::: 1.1. Sheaf Cohomology. Let X = spec(R). Let M = M~ be the quasi coherent sheaf on X associated to M. M~ is determined by the \stalks" (M~ )p = Mp for all p 2 X. We have that for f 2 R, the \sections" over the open set D(f) = X n V (f) are Γ(D(f); M) = Mf : n From the sheaf axioms, and since [i=1D(fi) = U, we have that Γ(U; M), where U = X n V (I), is computed as the kernel of d0: n M d0 M Γ(D(fi); M) ! Γ(D(fj) \ D(fk); M): i=1 j<k This map is just n M d0 M Mfi ! Mfj fk : i=1 j<k Consider the Cech complex F ∗ : F 0 ! F 1 !···! F n−1; 1 where M F t = R : fi1 fi2 ···fit+1 1≤i1<i2<···<it+1≤n The sheaf cohomology of M~ on U = X n V (I) is i i ∗ H (U; M~ ) = H (M ⊗R F ): We have that H0(U; M~ ) = Γ(U; M~ ); the sections of M~ over U = X n V (I). Hi(U; M~ ) depends only on U, so it is independent of the choice of generators of I, even up to the radical of I. Γ(U; M~ ) is a left exact functor on modules, and the cohomology modules are its right derived functors. If 0 ! A ! B ! C ! 0 is a short exact sequence of R-modules, then there is a long exact sequence 0 ! H0(U; A~) ! H0(U; B~) ! H0(U; C~) ! H1(U; A~) ! ::: 1.2. Comparison of Local Cohomology and Sheaf Cohomology. The modified Cech complex C∗ (used to compute local cohomology) is obtained from the Cech complex F ∗ by shifting the Cech complex one to the right, and setting C0 = R. From this we see that there is an exact sequence 0 0 ~ 1 0 ! HI (M) ! M ! H (U; M) ! HI (M) ! 0 and isomorphisms i ~ ∼ i+1 H (U; M) = HI (M) for i ≥ 1; where U = X n V (I). We have the interpretation of 0 ∼ n H (U; M~ ) = lim HomR(I ;M) ! as an \ideal transform". A particularly important case of this is when R = k[x0; : : : ; xn], a polynomial ring over a field with the standard grading, and m = (x0; : : : ; xn). Suppose that M is a graded module over R. Then the local and sheaf cohomology modules Hi(Spec(R) n fmg; M~ ) i and Hm(R) are graded, and the maps of the previous slide are graded. From the natural n surjection of the affine cone Spec(R) n fmg onto the projective space Proj(R) = Pk , we obtain graded isomorphisms i ~ ∼ M i n ~ H (Spec(R) n fmg; M) = H (Pk ; M(j)): j2Z In the first cohomology module, M~ is the sheaf associated to M on Spec(R). In the second ~ n cohomology module, M(j) is the sheaf associated to M(j) on P (M(j)d = Mj+d). We thus have an exact sequence of graded R-modules 0 M 0 n ~ 1 0 ! Hm(M) ! M ! H (P ; M(j)) ! Hm(M) ! 0 j2Z and isomorphisms M i n ~ ∼ i+1 H (P ; M(j)) = Hm (M) for i ≥ 1: j2Z 2 We have the interpretation M 0 n ∼ n H ( ; M~ (j)) = lim HomR(m ;M) P ! j2Z as an ideal transform. 1.3. Regularity. We continue to study the graded polynomial ring R = k[x0; : : : ; xn], and assume that M is a finitely generated graded R-module. supfj j Hi (M) 6= 0g if Hi (M) 6= 0 ai(M) = m j m −∞ otherwise The regularity of M is defined to be reg(M) = maxfai(M) + ig: i n Interpreting R as the coordinate ring of P = Proj(R), and considering the sheaf M~ on n P associate to M, we can define the regularity of M~ to be i n reg(M~ ) = maxfm j H (P ; M~ (m − i − 1)) 6= 0 for some i ≥ 1g i = maxi≥2fa (M) + ig: Thus reg(M~ ) ≤ reg(M) 1.4. Geometric Consequences of Regularity of Sheaves of Modules. The classical interpretations are for sections of line bundles on a projective variety X. Theorem 1.1. (Geometric Regularity Theorem) (Mumford [29]) Suppose that F is a n i n i coherent sheaf on P = Proj(R), and m 2 Z satisfies H (P ; F(m−i)) = H (F(m−i)) = 0 for all i ≥ 1. Then a) H0(F(k)) is spanned by H0(F(k − 1)) ⊗ H0(O(1)) if k > m. b) Hi(F(k)) = 0 whenever i > 0, k + i ≥ m. c) M H0(F(d)) d2Z is minimally generated as an R-module in degrees ≤ m. 1.5. Interpretations of Regularity of Modules. Let F∗ : 0 !···! Fj !···! F1 ! F0 ! M ! 0 be a minimal free resolution of M as a graded R-module. Let bj be the maximum degree of the generators of Fj. Then reg(M) = maxfbj − j j j ≥ 0g: In fact, we have (Eisenbud and Goto [14]) that reg(M) = maxfbj − j j j ≥ 0g R = maxfn j 9j such that Torj (k; M)n+j 6= 0g j = maxfn j 9j such that Hm(M)n−j 6= 0g: R The equality of the first and second of these conditions follows since Torj (k; M) = Hj(F∗ ⊗ R=m), and as F∗ is minimal, we have that the maps of the complex F∗ ⊗ R=m are 3 all zero. To obtain the equality of the first and third conditions, we take the cohomology j of the dual of F∗, to compute ExtR(M; R), and then apply graded local duality. We will now give another proof, which brings out the role of sheaf cohomology, of the most interesting part of these equivalent conditions, namely, that for a finitely generated graded R-module M, reg(M) ≤ m implies bj ≤ m + j for all j. Here bj is the maximal degree of a generator of Fj in the minimal free resolution of M (our proof is from Bayer and Mumford [1]). maxfai(M) + ig = reg(M) ≤ m implies ai(M) ≤ m − i for i ≥ 2 which implies that c) of the geometric regularity theorem holds, so L H0(M~ (d)) is minimally generated in d2Z degree ≤ m. 0 1 0 a (M) ≤ m, a (M) ≤ m − 1 implies Md ! H (M~ (d)) is an isomorphism if d > m and a surjection if d = m. Thus M is minimally generated in degree ≤ m. We thus have a surjection with kernel K1, M 0 ! K1 ! F0 = R(−dα) ! M ! 0 α with dα ≤ m for all α. Sheafify to get M ~ n ~ 0 ! K1 ! OP (−dα) ! M ! 0: α Consider the long exact cohomology sequence: 0 0 1 L n ~ ~ α H (OP (m − dα)) ! H (M(m)) ! H (K1(m)) ! 1 1 2 L n ~ ~ α H (OP (m − dα)) ! H (M(m)) ! H (K1(m)) !··· i n H (OP (j)) = 0 for i ≥ 1 and j ≥ 1 − i. By b) of the the geometric regularity theorem, Hi(M~ (j)) = 0 for i ≥ 1 and j ≥ m − i. Thus i H (K~1(m + 1 − i)) = 0 for i ≥ 1: Induction Assumption: Assume we have constructed Fk !···! F0 ! M ! 0 such that the minimal generators of Fi have degree ≤ m + i for 0 ≤ i ≤ k and i H (K~k(m + k − i)) = 0 for i ≥ 1; where Kk = Kernel(Fk ! Fk−1): Then we will construct such a sequence of length k + 1. By c) of the geometric regularity theorem, we have a surjection with kernel Kk+1, M 0 ! Kk+1 ! Fk+1 = R(−dα) ! Kk ! 0 α where dα ≤ m + k for all α. 4 Sheafify, and compute the long exact cohomology sequence, to get i H (K~k+1(m + k + 1 − i)) = 0 for i ≥ 1: We define R regi(M) = maxfn j Tori (k; M)n 6= 0g − i: Then reg(M) = maxfregi(M) j i ≥ 0g: We have that reg0(M) is the maximum degree of a homogeneous generator of M. 2. Regularity of Powers of Ideals We outline the proof of {, Herzog and Trung [12] showing that reg(In) is a linear polynomial for large n. Let F1;:::;Fs be homogeneous generators of I ⊂ R = k[x0; : : : ; xn], with deg(Fi) = di. The map yi 7! Fi induces a surjection of bigraded R-algebras M m S = R[y1; : : : ; ys] ! R(I) = I m≥0 where we have bideg(xi) = (1; 0) for 0 ≤ i ≤ n, and bideg(yj) = (dj; 1) for 1 ≤ j ≤ s.
Load more

Recommended publications
  • ALGEBRAIC GEOMETRY (PART III) EXAMPLE SHEET 3 (1) Let X Be a Scheme and Z a Closed Subset of X. Show That There Is a Closed Imme
    ALGEBRAIC GEOMETRY (PART III) EXAMPLE SHEET 3 CAUCHER BIRKAR (1) Let X be a scheme and Z a closed subset of X. Show that there is a closed immersion f : Y ! X which is a homeomorphism of Y onto Z. (2) Show that a scheme X is affine iff there are b1; : : : ; bn 2 OX (X) such that each D(bi) is affine and the ideal generated by all the bi is OX (X) (see [Hartshorne, II, exercise 2.17]). (3) Let X be a topological space and let OX to be the constant sheaf associated to Z. Show that (X; OX ) is a ringed space and that every sheaf on X is in a natural way an OX -module. (4) Let (X; OX ) be a ringed space. Show that the kernel and image of a morphism of OX -modules are also OX -modules. Show that the direct sum, direct product, direct limit and inverse limit of OX -modules are OX -modules. If F and G are O H F G O L X -modules, show that omOX ( ; ) is also an X -module. Finally, if is a O F F O subsheaf of an X -module , show that the quotient sheaf L is also an X - module. O F O O F ' (5) Let (X; X ) be a ringed space and an X -module. Prove that HomOX ( X ; ) F(X). (6) Let f :(X; OX ) ! (Y; OY ) be a morphism of ringed spaces. Show that for any O F O G ∗G F ' G F X -module and Y -module we have HomOX (f ; ) HomOY ( ; f∗ ).
    [Show full text]
  • SHEAVES of MODULES 01AC Contents 1. Introduction 1 2
    SHEAVES OF MODULES 01AC Contents 1. Introduction 1 2. Pathology 2 3. The abelian category of sheaves of modules 2 4. Sections of sheaves of modules 4 5. Supports of modules and sections 6 6. Closed immersions and abelian sheaves 6 7. A canonical exact sequence 7 8. Modules locally generated by sections 8 9. Modules of finite type 9 10. Quasi-coherent modules 10 11. Modules of finite presentation 13 12. Coherent modules 15 13. Closed immersions of ringed spaces 18 14. Locally free sheaves 20 15. Bilinear maps 21 16. Tensor product 22 17. Flat modules 24 18. Duals 26 19. Constructible sheaves of sets 27 20. Flat morphisms of ringed spaces 29 21. Symmetric and exterior powers 29 22. Internal Hom 31 23. Koszul complexes 33 24. Invertible modules 33 25. Rank and determinant 36 26. Localizing sheaves of rings 38 27. Modules of differentials 39 28. Finite order differential operators 43 29. The de Rham complex 46 30. The naive cotangent complex 47 31. Other chapters 50 References 52 1. Introduction 01AD This is a chapter of the Stacks Project, version 77243390, compiled on Sep 28, 2021. 1 SHEAVES OF MODULES 2 In this chapter we work out basic notions of sheaves of modules. This in particular includes the case of abelian sheaves, since these may be viewed as sheaves of Z- modules. Basic references are [Ser55], [DG67] and [AGV71]. We work out what happens for sheaves of modules on ringed topoi in another chap- ter (see Modules on Sites, Section 1), although there we will mostly just duplicate the discussion from this chapter.
    [Show full text]
  • 4. Coherent Sheaves Definition 4.1. If (X,O X) Is a Locally Ringed Space
    4. Coherent Sheaves Definition 4.1. If (X; OX ) is a locally ringed space, then we say that an OX -module F is locally free if there is an open affine cover fUig of X such that FjUi is isomorphic to a direct sum of copies of OUi . If the number of copies r is finite and constant, then F is called locally free of rank r (aka a vector bundle). If F is locally free of rank one then we way say that F is invertible (aka a line bundle). The group of all invertible sheaves under tensor product, denoted Pic(X), is called the Picard group of X. A sheaf of ideals I is any OX -submodule of OX . Definition 4.2. Let X = Spec A be an affine scheme and let M be an A-module. M~ is the sheaf which assigns to every open subset U ⊂ X, the set of functions a s: U −! Mp; p2U which can be locally represented at p as a=g, a 2 M, g 2 R, p 2= Ug ⊂ U. Lemma 4.3. Let A be a ring and let M be an A-module. Let X = Spec A. ~ (1) M is a OX -module. ~ (2) If p 2 X then Mp is isomorphic to Mp. ~ (3) If f 2 A then M(Uf ) is isomorphic to Mf . Proof. (1) is clear and the rest is proved mutatis mutandis as for the structure sheaf. Definition 4.4. An OX -module F on a scheme X is called quasi- coherent if there is an open cover fUi = Spec Aig by affines and ~ isomorphisms FjUi ' Mi, where Mi is an Ai-module.
    [Show full text]
  • Generalized Divisors and Reflexive Sheaves
    GENERALIZED DIVISORS AND REFLEXIVE SHEAVES KARL SCHWEDE 1. Preliminary Commutative Algebra We introduce the notions of depth and of when a local ring (or module over a local ring) is \S2". These notions are found in most books on commutative algebra, see for example [Mat89, Section 16] or [Eis95, Section 18]. Another excellent book that is focussed on these ideas is [BH93]. We won't be focussing on the commutative algebra, but one should be aware, at the very least, that this background exists. In particular, we won't really use any of the theory on Cohen-Macaulay rings. However, since one ought to build up the same machinery in order to define S2, we include these definitions as well. All rings will be assumed to be noetherian. Definition 1.1. Let (R; m) be a local ring and let M be a finite R-module (which means finitely generated). An element r 2 R is said to be M-regular if rx 6= 0 for all x 2 M, x 6= 0 (in other words, r is not a zero divisor on M). A sequence of elements r1; : : : ; rn 2 R is said to be M-regular if (i) ri is a regular M=(r1; : : : ; ri−1) element for all i ≥ 1 (in particular r1 is M-regular) (ii) (r1; : : : ; rn)M ( M (that is, the containment is proper). Remark 1.2. If condition (i) is satisfied, but condition (ii) is not necessarily satisfied then such a sequence is called weakly M-regular. Remark 1.3. Notice that condition (ii) implies in a regular sequence, all ri 2 m 2 R.
    [Show full text]
  • 4 Sheaves of Modules, Vector Bundles, and (Quasi-)Coherent Sheaves
    4 Sheaves of modules, vector bundles, and (quasi-)coherent sheaves “If you believe a ring can be understood geometrically as functions its spec- trum, then modules help you by providing more functions with which to measure and characterize its spectrum.” – Andrew Critch, from MathOver- flow.net So far we discussed general properties of sheaves, in particular, of rings. Similar as in the module theory in abstract algebra, the notion of sheaves of modules allows us to increase our understanding of a given ringed space (or a scheme), and to provide further techniques to play with functions, or function-like objects. There are particularly important notions, namely, quasi-coherent and coherent sheaves. They are analogous notions of the usual modules (respectively, finitely generated modules) over a given ring. They also generalize the notion of vector bundles. Definition 38. Let (X, ) be a ringed space. A sheaf of -modules, or simply an OX OX -module, is a sheaf on X such that OX F (i) the group (U) is an (U)-module for each open set U X; F OX ✓ (ii) the restriction map (U) (V ) is compatible with the module structure via the F !F ring homomorphism (U) (V ). OX !OX A morphism of -modules is a morphism of sheaves such that the map (U) F!G OX F ! (U) is an (U)-module homomorphism for every open U X. G OX ✓ Example 39. Let (X, ) be a ringed space, , be -modules, and let ' : OX F G OX F!G be a morphism. Then ker ', im ', coker ' are again -modules. If is an - OX F 0 ✓F OX submodule, then the quotient sheaf / is an -module.
    [Show full text]
  • AG for NT Week 6
    AG for NT Week 6 Recap 1. A ring, X = Spec A, M an A-module which we associated a sheaf Mf of OX -modules. Have shown: Any quasi-coherent sheafs on Spec A has this form. 2. S graded ring, X = PrjS, M graded S-module, associated to it Mf a quasi-coherent sheaf. Mf(D+(f)) = fhomogeneous elements of degree 0 in Mf g. Want to show: Given any quasi-coherent sheaf on X = ProjS, we have f =∼ Mf for some graded S-module M. We won't (can't?) say this in general, but Proposition 0.1. S graded ring, nitely generated by S1 as an S0-algebra (S = ⊕a≥0Sa). Let X = ProjS and f a quasi coherent sheaf on X. Then there is a graded S-module M such that f =∼ Mf. Remark. 1. Applies to for some ring , i.e., to n S = A[x0; : : : ; xn] A PA 2. Don't have an equivalence of categories between graded S-modules and quasi-coherent sheafs on ProjS. Indeed, 2 dierent graded modules can give the same sheaf. Example: and for some ⊕n≥0Mn ⊕n≥n0 Mn n0 ≥ 0 3. How to dene M given f? Cannot just set M = Γ(X; f) as in the ane case. Solution is to use twists. Example. , Proj r . S = A[x0; : : : ; xr] X = S = PA Claim. Γ(X; OX (n)) = Sn = fhomogenous polynomials of degree n}. In ∼ particular, Γ(X; OX ) = A. Proof. The sets D+(xi) cover X. A section t 2 Γ(X; OX (n)) is the same as an (r + 1)-tuple of sections (t0; : : : ; tr) with ti 2 OX (n)(D+(xi)) com- patible on D+(xi) \ D+(xj) = D+(xixj) Now, homogeneous elements in and OX (n)(D+(xi)) = fdeg n Sxi g OX (n)(D+(xixj)) = homogeneous elements in .
    [Show full text]
  • K-Stability of Relative Flag Varieties
    K-stability of relative flag varieties Anton Isopoussu University of Cambridge Department of Pure Mathematics and Mathematical Statistics Churchill College arXiv:1307.7638v2 [math.AG] 10 Nov 2015 April 2015 This dissertation is submitted for the degree of Doctor of Philosophy Abstract We generalise partial results about the Yau-Tian-Donaldson correspondence on ruled manifolds to bundles whose fibre is a classical flag variety. This is done using Chern class computations involving the combinatorics of Schur functors. The strongest results are obtained when working over a Riemann surface. Weaker partial results are obtained for adiabatic polarisations in the general case. We develop the notion of relative K-stability which embeds the idea of working over a base variety into the theory of K-stability. We equip the set of equivalence classes of test configuration with the structure of a convex space fibred over the cone of rational polarisations. From this, we deduce the open- ness of the K-unstable locus. We illustrate our new algebraic constructions with several examples. Declaration This dissertation is the result of my own work and includes nothing which is the outcome of work done in collaboration except as declared in the Preface and specified in the text. It is not substantially the same as any that I have submitted, or, is being concurrently submitted for a degree or diploma or other qualification at the University of Cambridge or any other University or similar institution except as declared in the Preface and specified in the text. I further state that no substantial part of my dissertation has already been submitted, or, is being concurrently submitted for any such degree, diploma or other qualification at the University of Cambridge or any other University of similar institution except as specified in the text Anton Isopoussu November 11, 2015 Acknowledgements I gratefully acknowledge the patient guidance of my supervisor Dr.
    [Show full text]
  • Geometry of the Projectivization of Ideals and Applications to Problems of Birationality Remi Bignalet-Cazalet
    Geometry of the projectivization of ideals and applications to problems of birationality Remi Bignalet-Cazalet To cite this version: Remi Bignalet-Cazalet. Geometry of the projectivization of ideals and applications to problems of birationality. Algebraic Geometry [math.AG]. Université Bourgogne Franche-Comté, 2018. English. NNT : 2018UBFCK038. tel-02004276 HAL Id: tel-02004276 https://tel.archives-ouvertes.fr/tel-02004276 Submitted on 1 Feb 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. These` de doctorat de l'Universite´ de Bourgogne Franche-Comte´ pr´epar´ee`al'Institut de Math´ematiquesde Bourgogne Ecole´ doctorale Carnot Pasteur (ED553) Doctorat de Math´ematiques par R´emiBignalet-Cazalet Geom´ etrie´ de la projectivisation des ideaux´ et applications aux problemes` de birationalite´ Directeurs de th`ese: Adrien Dubouloz et Daniele Faenzi Th`esesoutenue le 24 octobre 2018 `aDijon Composition du jury Laurent Buse´ INRIA - Sophia Antipolis Examinateur Julie Deserti´ IMJ-PRG - Universit´eParis Diderot Examinatrice Adrien Dubouloz IMB - Universit´ede
    [Show full text]
  • The Cohomology of Coherent Sheaves
    CHAPTER VII The cohomology of coherent sheaves 1. Basic Cechˇ cohomology We begin with the general set-up. (i) X any topological space = U an open covering of X U { α}α∈S a presheaf of abelian groups on X. F Define: (ii) Ci( , ) = group of i-cochains with values in U F F = (U U ). F α0 ∩···∩ αi α0,...,αYi∈S We will write an i-cochain s = s(α0,...,αi), i.e., s(α ,...,α ) = the component of s in (U U ). 0 i F α0 ∩··· αi (iii) δ : Ci( , ) Ci+1( , ) by U F → U F i+1 δs(α ,...,α )= ( 1)j res s(α ,..., α ,...,α ), 0 i+1 − 0 j i+1 Xj=0 b where res is the restriction map (U U ) (U U ) F α ∩···∩ Uαj ∩···∩ αi+1 −→ F α0 ∩··· αi+1 and means “omit”. Forb i = 0, 1, 2, this comes out as δs(cα , α )= s(α ) s(α ) if s C0 0 1 1 − 0 ∈ δs(α , α , α )= s(α , α ) s(α , α )+ s(α , α ) if s C1 0 1 2 1 2 − 0 2 0 1 ∈ δs(α , α , α , α )= s(α , α , α ) s(α , α , α )+ s(α , α , α ) s(α , α , α ) if s C2. 0 1 2 3 1 2 3 − 0 2 3 0 1 3 − 0 1 2 ∈ One checks very easily that the composition δ2: Ci( , ) δ Ci+1( , ) δ Ci+2( , ) U F −→ U F −→ U F is 0. Hence we define: 211 212 VII.THECOHOMOLOGYOFCOHERENTSHEAVES s(σβ0, σβ1) defined here U σβ0 Uσβ1 Vβ1 Vβ0 ref s(β0, β1) defined here Figure VII.1 (iv) Zi( , ) = Ker δ : Ci( , ) Ci+1( , ) U F U F −→ U F = group of i-cocycles, Bi( , ) = Image δ : Ci−1( , ) Ci( , ) U F U F −→ U F = group of i-coboundaries Hi( , )= Zi( , )/Bi( , ) U F U F U F = i-th Cech-cohomologyˇ group with respect to .
    [Show full text]
  • Multiplier Ideals.Pdf
    Multiplier Ideals Matt Stevenson∗ May 16{20, 2016 Abstract Multiplier ideals are an important tool in complex algebraic/analytic geometry. They are often used to study e.g. the singularities of divisors, or metrics on holomorphic line bundles. In this mini course, I will give an overview of the analytic and algebraic theories, along with plenty of examples. Topics will include: 1. Analytic multiplier ideals; 2. Review of Q-divisors and log resolutions; 3. Algebraic multiplier ideals: basic properties, and examples; 4. Positivity and vanishing theorems; 5. Metrics on line bundles and their multiplier ideals; 6. Zariski decompositions and the Fujita Approximation theorem. Lecture 1 References. • [Algebraic] xx9{10 in: Robert Lazarsfeld. Positivity in algebraic geometry. II. Positivity for vector bundles, and multiplier ideals. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] 49. Berlin: Springer-Verlag, 2004. isbn: 3-540-22534-X. doi: 10.1007/978-3-642-18808-4. mr: 2095472. • [Analytic] Jean-Pierre Demailly. \Multiplier ideal sheaves and analytic methods in algebraic geometry." School on Vanishing Theorems and Effective Results in Algebraic Geometry (Trieste, 2000). ICTP Lect. Notes 6. Trieste: Abdus Salam Int. Cent. Theoret. Phys., 2001, pp. 1{148. mr: 1919457. Plan. • [Monday{Wednesday] General theory; • [Thursday{Friday] \Fancy" analytic theory and applications, in particular { Koll´ar'sresults on singularities of θ-divisors on principally polarized abelian varieties and { the Fujita approximation theorem. Note that there is a notion of an asymptotic multiplier ideal that can be used to derive these results purely algebraically, but we will instead be using the analytic theory for multiplier ideals.
    [Show full text]
  • Sheaf Cohomology and Its Applications in Analytic Geometry
    Sheaf Cohomology and its Applications in Analytic Geometry by Jean Michel Menjanahary ([email protected]) 17th May 2020 A THESIS SUBMITTED TO CENTRAL EUROPEAN UNIVERSITY IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE AWARD OF A MASTER OF SCIENCE IN MATHEMATICS AND ITS APPLICATIONS CEU eTD Collection Declaration As a work carried out at Central European University in partial fulfilment of the requirements for a Master of Science in Mathematics, I hereby declare that the work contained in this thesis is my original work. The work done by others has been acknowledged and referenced accordingly. Student: Jean Michel Menjanahary Supervisor: Prof. András Némethi CEU eTD Collection i Acknowledgments I would like to appreciate the CEU founder and philanthropist, George Soros and the CEU community, who trusted and gave me the opportunity to continue my studies at CEU. My appreciation goes particularly to our head of department, Prof Károly Böröczky and our MSc program director, Prof Pál Hegedüs. They helped me a lot to join this program. I am so grateful for all that they have done for me. Many thanks also to our department coordinators Elvira Kadvány and Melinda Balázs for their great helps during my stay in Budapest. I would also like to express my thanks to my supervisor Prof András Némethi. He agreed to supervise me on this wonderful topic. He worked a lot in guiding and teaching me throughout the academic year. He also helped me a lot to write this work in a better way. I am really thankful to him. Finally, it has been a wonderful and fun time with my Malagasy friends Manana, Mahenina, Mahefa, Ny Aina and Masera Vicky in Budapest for two years.
    [Show full text]
  • GAGA TALK NOTES This Note Gives a Mostly Complete Proof of The
    GAGA TALK NOTES BY CONNOR HALLECK-DUBE´ This note gives a mostly complete proof of the standard GAGA theorems for projective schemes over C. I closely Serre's original paper [Ser56] and some OCW lecture notes by Kedlaya [Ked], the latter of which discuss the proof in a more modern language, but only in the case of projective space. Hartshorne also has a GAGA appendix, but I haven't looked at it. I tried to balance the classical and explicit perspective of the original paper with a more modern perspective. Disclaimer: I cite without proof standard (but not necessarily easy) facts from both algebraic and analytic complex geometry. I. Analytification of spaces The analytification functor is intuitively an extremely straightforward idea. The Zariski topology is n extremely weak, but we could also give the C-valued locus of some polynomial equations V (f1; : : : ; fn) ⊆ C the induced topology as a subspace of Cn. Patching these together, we have a natural notion of the analytic topology on a scheme locally of finite type over C. Since polynomial functions are continuous in the classical topology, this should be a finer topology than the Zariski topology. If the scheme is not a smooth variety, then the resulting space will not be a complex manifold, so we need a notion of complex analytic space that can allow for such degeneracies. Definition 1. An affine complex analytic space is the locally ringed space (Z; OZ ) consisting of a closed n subspace Z of an open U ⊆ C such that Z is cut out by holomorphic functions f1; : : : ; fk on U, with structure sheaf OU =I(Z) (the holomorphic functions modulo the analytic ideal sheaf).
    [Show full text]