MA 310: Algebraic Geometry I
Credits: 3:0
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Refresher on Commutative Algebra: localization, local rings, integral closure, Krull dimension.
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Zariski topology, Hochster’s characterization of Zariski topology, spectral spaces.
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Zariski spectrum as a frame
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Refresher on categories : Categories, functors, Yoneda Lemma, equivalence of categories, adjoints.
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Grothendieck sites : Zariski, '{e}tale and Nisnevich sites.
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Presheaves and Sheaves
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Locally ringed spaces and schemes
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Separated schemes, proper schemes, irreducible schemes, reduced schemes, integral schemes, noetherian schemes.
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Morphisms : separated, proper, finite morphisms, finite type morphisms, affine morphisms
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Sheaves of algebras : affine morphisms as sheaves of algebras
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Sheaves of modules over a scheme, Quasi-coherent and coherent sheaves
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Divisors and Line Bundles, Weil divisors, Cartier divisors, Line bundles on Projective spaces, Serre sheaves.
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Projective morphisms, ample and very ample line bundles
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Formal schemes
Suggested books and references:
- Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977.
- Robin Hartshorne, Residues and duality, Lecture notes of a seminar on the work of A.~Grothendieck, given at Harvard 1963/64. With an appendix by P.~Deligne. Lecture Notes in Mathematics, No. 20 Springer-Verlag, Berlin-New York 1966.