Derived Algebraic Geometry

This text is a survey of derived algebraic geometry. It covers a variety of general notions and results from the subject with a view on the recent developments at the interface with deformation quantization.

[1]  D. Mumford,et al.  The irreducibility of the space of curves of given genus , 1969 .

[2]  M. Schlessinger Functors of Artin rings , 1968 .

[3]  R. M. Vogt,et al.  Topological Hochschild Homology , 2000 .

[4]  G. Vezzosi,et al.  Brave new algebraic geometry and global derived moduli spaces of ring spectra , 2003, math/0309145.

[5]  J. Rognes Galois Extensions of Structured Ring Spectra/Stably Dualizable Groups , 2008 .

[6]  A. Grothendieck,et al.  Théorie des Topos et Cohomologie Etale des Schémas , 1972 .

[7]  M. Spivakovsky A new proof of D. Popescu’s theorem on smoothing of ring homomorphisms , 1999 .

[8]  Locally complete intersection homomorphisms and a conjecture of Quillen on the vanishing of cotangent homology , 1999, math/9909192.

[9]  D. Arinkin,et al.  When is the self-intersection of a subvariety a fibration? , 2010, 1007.1671.

[10]  Enumeration of Rational Curves Via Torus Actions , 1994, hep-th/9405035.

[11]  On the variety of complexes , 1981 .

[12]  K. Costello Notes on supersymmetric and holomorphic field theories in dimensions 2 and 4 , 2011, 1111.4234.

[13]  Ib Madsen,et al.  The cyclotomic trace and algebraic K-theory of spaces , 1993 .

[14]  Douglas C. Ravenel,et al.  Nilpotence and Periodicity in Stable Homotopy Theory. , 1992 .

[15]  W. Dwyer,et al.  Simplicial localizations of categories , 1980 .

[16]  Monique Hakim,et al.  Topos annelés et schémas relatifs , 1972 .

[17]  Topological Automorphic Forms , 2007, math/0702719.

[18]  J. Lurie A Survey of Elliptic Cohomology , 2009 .

[19]  Michael Cole,et al.  Rings, Modules, and Algebras in Stable Homotopy Theory , 2007 .

[20]  L. Illusie Complexe cotangent et déformations II , 1971 .

[21]  B. Shipley,et al.  Equivalences of monoidal model categories , 2002, math/0209342.

[22]  D. Harrison Commutative algebras and cohomology , 1962 .

[23]  D. Quillen On the (co)homology of commutative rings , 1970 .

[24]  Peter B. Shalen,et al.  Varieties of group representations and splittings of 3-manifolds , 1983 .

[25]  J. P. May,et al.  The geometry of iterated loop spaces , 1972 .

[26]  J.P.Pridham Representability of derived stacks , 2010, 1011.2189.

[27]  B. Fantechi,et al.  The intrinsic normal cone , 1996, alg-geom/9601010.

[28]  M. Hopkins Algebraic topology and modular forms , 2002, math/0212397.

[29]  C. Simpson GEOMETRICITY OF THE HODGE FILTRATION ON THE ∞-STACK OF PERFECT COMPLEXES OVER X , 2005 .

[30]  Yan Soibelman,et al.  Homological mirror symmetry and torus fibrations , 2000 .

[31]  B. Toën Higher and derived stacks: a global overview , 2009 .

[32]  Edward C. Hook Equivariant cobordism and duality , 1973 .

[33]  M. Kapranov Rozansky–Witten invariants via Atiyah classes , 1997, Compositio Mathematica.

[34]  S. Lichtenbaum,et al.  THE COTANGENT COMPLEX OF A MORPHISM , 1967 .

[35]  M. Artin,et al.  Versal deformations and algebraic stacks , 1974 .

[36]  Carlos T. Simpson,et al.  Homotopy Theory of Higher Categories: THE MODEL STRUCTURE , 2010, 1001.4071.

[37]  T. Pirashvili Hodge decomposition for higher order Hochschild homology , 2000 .

[38]  G. Vezzosi,et al.  Homotopical Algebraic Geometry II: Geometric Stacks and Applications , 2004, math/0404373.

[39]  D. Nadler,et al.  Loop spaces and connections , 2010, 1002.3636.

[40]  Craig Huneke,et al.  Commutative Algebra I , 2012 .

[41]  J. Millson,et al.  The deformation theory of representations of fundamental groups of compact Kähler manifolds , 1988 .

[42]  DG coalgebras as formal stacks , 1998, math/9812034.

[43]  M. Kapranov,et al.  Derived Hilbert schemes , 2000, math/0005155.

[44]  D. M. Kan,et al.  Homotopy Limits, Completions and Localizations , 1987 .

[45]  Bertrand Toën,et al.  Algèbres simpliciales S1-équivariantes, théorie de de Rham et théorèmes HKR multiplicatifs , 2011, Compositio Mathematica.

[46]  Jim Stasheff,et al.  Homotopy associativity of $H$-spaces. II , 1963 .

[47]  Segal topoi and stacks over Segal categories , 2002, math/0212330.

[48]  A. Haefliger Structures feuilletées et cohomologie à valeur dans un faisceau de groupoïdes , 1958 .

[49]  route de Chartres,et al.  Deformation quantization of algebraic varieties , 2008 .

[50]  W. Fulton,et al.  Intersection Theory: Ergebnisse Der Mathematik Und Ihrer Grenzgebiete; 3 Folge; Band 2 , 1984 .

[51]  M. André Homologie des algèbres commutatives , 1974 .

[52]  M. Kontsevich,et al.  The Geometry of the Master Equation and Topological Quantum Field Theory , 1995, hep-th/9502010.

[53]  A. Alekseev,et al.  Lie group valued moment maps , 1997, dg-ga/9707021.

[54]  G. G. Stokes "J." , 1890, The New Yale Book of Quotations.

[55]  L.Katzarkov,et al.  Hodge theoretic aspects of mirror symmetry , 2008, 0806.0107.

[56]  A. Grothendieck Techniques de construction en géométrie analytique. VI. Étude locale des morphismes : germes d'espaces analytiques, platitude, morphismes simples , 1961 .

[57]  Miss A.O. Penney (b) , 1974, The New Yale Book of Quotations.

[58]  D. Spencer,et al.  On Deformations of Complex Analytic Structures, II , 1958 .

[59]  Peter Gabriel,et al.  Calculus of Fractions and Homotopy Theory , 1967 .

[60]  B. Toën Derived Azumaya algebras and generators for twisted derived categories , 2012, Inventiones mathematicae.

[61]  Timo Schurg Deriving Deligne-Mumford Stacks with Perfect Obstruction Theories , 2010, 1005.3945.

[62]  J. Lurie Higher Topos Theory , 2006, math/0608040.

[63]  Bernhard Keller,et al.  On differential graded categories , 2006, math/0601185.

[64]  Julia E. Bergner,et al.  A Survey of (∞, 1)-Categories , 2010 .

[65]  B. Kostant,et al.  Graded manifolds, graded Lie theory, and prequantization , 1977 .

[66]  Jean-Pierre Serre,et al.  Algèbre locale : multiplicités , 1957 .

[67]  From Hag To Dag: Derived Moduli Stacks , 2002, math/0210407.

[68]  J. Stasheff,et al.  The Lie algebra structure of tangent cohomology and deformation theory , 1985 .

[69]  Pu Zhang,et al.  Triangulated Categories , 2021, Homological Theory of Representations.

[70]  B. Toen Derived Algebraic Geometry and Deformation Quantization , 2014, 1403.6995.

[71]  A. Schwarz Geometry of Batalin-Vilkovisky quantization , 1992, hep-th/9205088.

[72]  Z. Fiedorowicz,et al.  MacLane homology and topological Hochschild homology , 1992 .

[73]  M. Kontsevich Operads and Motives in Deformation Quantization , 1999, math/9904055.

[74]  B. Conrad,et al.  Approximation of versal deformations , 2002 .

[75]  A. Grothendieck Catégories cofibrées additives et complexe cotangent relatif , 1968 .

[76]  Tom Leinster A Survey of Definitions of n-Category , 2001 .

[77]  G. Carlsson Derived Completions in Stable Homotopy theory , 2007, 0707.2585.

[78]  G. Vezzosi,et al.  Derived algebraic geometry, determinants of perfect complexes, and applications to obstruction theories for maps and complexes , 2011, 1102.1150.

[79]  I. Batalin,et al.  Gauge Algebra and Quantization , 1981 .

[80]  Virtual fundamental classes via dg-manifolds , 2007, math/0703214.

[81]  문정진 § 19 , 2000 .

[82]  T. Goodwillie On the general linear group and Hochschild homology , 1985 .

[83]  Bertrand Toën,et al.  Au-dessous de SpecZ . , 2009 .

[84]  G. Vezzosi,et al.  A remark on K-theory and S-categories , 2002, math/0210125.

[85]  A. Beilinson p-adic periods and derived de Rham cohomology , 2011, 1102.1294.

[86]  HKR Theorem for Smooth S-algebras , 2003, math/0306243.

[87]  B. Mitchell A quick proof of the Gabriel-Popesco theorem , 1981 .

[88]  MATT STEVENSON,et al.  ON THE DE RHAM COHOMOLOGY OF ALGEBRAIC VARIETIES , 2015 .

[89]  Moduli of objects in dg-categories , 2005, math/0503269.

[90]  R. Hartshorne Residues And Duality , 1966 .

[91]  B. Feigin,et al.  Additive K-theory and crystalline cohomology , 1985 .

[92]  C. Rezk,et al.  A cartesian presentation of weak n–categories , 2009, 0901.3602.

[93]  W. Dwyer,et al.  Function complexes in homotopical algebra , 1980 .

[94]  A. Kapustin Topological Field Theory, Higher Categories, and Their Applications , 2010, 1004.2307.

[95]  J. Lurie Moduli Problems for Ring Spectra , 2011 .

[96]  J. Francis The tangent complex and Hochschild cohomology of $\mathcal {E}_n$-rings , 2011, Compositio Mathematica.

[97]  A. Grothendieck Technique de descente et théorèmes d'existence en géométrie algébrique. I. Généralités. Descente par morphismes fidèlement plats , 1960 .

[98]  B. Fresse On the homotopy of simplicial algebras over an operad , 1999 .