Course title: Sheaves and Cohomology in Algebraic Geometry (2 credits) Course goals: Cover the basics of sheaf cohomology, including a statement of Serre Duality for projective schemes Participants in the course will be graded on two or more 90 minute presentations (70% of grade), participation in discussion (10% of grade), and a small selection of weekly homework problems (20% of grade). The course will meet once a week for a two-hour time block, Wednesdays from 11:30 to 1:30 in RCB 8104. Course texts are: [EH] Eisenbud and Harris, The Geometry of Schemes [H] Hartshorne, Algebraic Geometry Prerequisites are: ** Math 818 or equivalent ** Review of I.1-I.3 of [EH] prior to begin of the semester. ** Instructor approval Outline: Jan 9 (Ahmad): Sheaves [EH I.1.3] and [H II.1]. Basic definitions and examples. Sheafification. Stalks. Kernels, images, quotients, and exactness. Left exactness of global section functor. Exactness via exactness on stalks. Homework: II.1.8, II.1.16 Optional: II.1.14, II.1.18 Jan 16 (Sharon): Schemes [EH I.1-I.2] and [H II.2]. Spec A as a topological space. Structure sheaf for an affine scheme. Examples. Schemes in general. Morphisms. Proj construction. Integral and noetherian schemes [H II.3]. Homework: II.2.3, II.2.13 Optional: II.2.11, II.2.19 Jan 23 (Eugene): Sheaves of Modules [H II.5]. Sheaves of O_X modules. Locally free sheaves, ideal sheaves. f_* and f^*. M^~ and its basic properties. (Quasi)coherent sheaves. Exactness of global section functor for affine schemes. Quasicoherence preserved for kernels, pushforwards, etc. Homework: II.5.1, II.5.2 Optional: II.5.7, II.5.18 Jan 30 (Reza): Sheaves of Modules, cont'd [H II.5]. Ideal scheaf of closed subschemes. M^~ for graded modules. Twisting sheaves. \Gamma_* construction. Very ample line bundles. Serre's theorem on global generation and consequences. Homework: II.5.9, II.5.10 Optional: II.5.11, II.5.12 Feb 6 (Aven): Derived Functors [H III.1]. Complexes. Cohomology of complexes. Homotopies. Additive and (right/left) exact functors. Injective and acyclic resolutions. (Universal) delta-functors and effeaceable functors. Homework: Problem 1. Show that R-mod has enough injectives as follows: (i)show that divisable abelian groups are injectives in Z-mod (ii)show that every abelian group has an injection to a divisable group (iii)show that for a ring homomorphism R->S and injective R-module Q, Hom_R(S,Q) is an injective S-module, and (iv)R-mod has enough injectives. For (iv), first find an injection to an injective Z-mod, then use (iii). Problem 2. Show that any morphism f:M->N (e.g. in R-mod) lifts to a morphism of co-chain complexes between injective resolutions of M,N. Feb 13 (Dan): Sheaf Cohomology [H III.2]. Injective resolutions for sheaves. Flasque sheaves and cohomology vanishing. Indiference of working in Ab(X) or Mod(X). Grothendieck vanishing. Cohomology on closed subsets. Homework: III.2.1(a), III.2.2, III.2.7 Optional III.2.3, III.2.4 Feb 27 (Peter): Cohomology on affine schemes [H III.3] Serre's cohomological characterization of noetherian affine schemes (and partial proof). Homework: III.3.1, III.3.2, Optional III.3.3 Mar 6 (Sasha): Cech Cohomology [H III.4] Cech complex. Examples. Sheafified Cech complex. Vanishing for flasque sheaves. Functorial map to sheaf cohomology. Isomorphisms in good situations (Theorem 4.5 and exercise 4.11) Homework: see http://www.sfu.ca/~nilten/teaching/simplicial.pdf Mar 13 (Ahmad and Sharon): Cohomology of Projective Space [H III.5] Cohomology for line bundles via Cech cohomology (and surjectivity of section multiplication). Cohomology for arbitrary coherent sheaves. Cohomological characterization of ampleness. Homework: III.5.1, III.5.2, III.5.3(a,b) Optional III.5.5 III.5.7 Mar 20 (Eugene and Aven): Sheaf Ext [H III.6] Definition of Ext and sheaf Ext. Basic properties. Long exact sequence for Ext(-,G). Calculating Ext via locally free resolutions. Ext and the tensor product. Stalks of sheaf Ext. Relationship between Ext and sheaf Ext on projective schemes. Homework: III.6.1 and III.6.7 Mar 27 (Reza and Dan): Sheaves of Differentials [H II.8] Derivations and Kaehler differentials. Sheaves of differentials. Euler sequence (8.13). Criteria for non-singularity. Tangent sheaf and canonical sheaf. Normal sheaf. Adjunction formula (8.20). Homework: II.8.3 Apr 3) (Peter and Sasha): Serre Duality [H III.7] Duality of projective space. Dualizing sheaves. Cohen-Macaulayness. Statement of Serre duality for projective schemes. Simplification for vector bundles. Dualizing sheaf for smooth projective varieties. Poincare symmetry of Hodge numbers (7.13). No Homework! Hooray! Presentation topics will be assigned based on discussion with course participants. One week prior to presenting, each student is required to submit a detailed outline of the presentation to me. Piazza forum for course: piazza.com/sfu.ca/spring2019/math894g100/home