03195nam 2200469 450 991079389020332120191015173613.01-4704-5336-3(CKB)4100000009374627(MiAaPQ)EBC5904555(RPAM)21255944(PPN)240205979(EXLCZ)99410000000937462720191015h20192019 uy| 0engurcnu||||||||rdacontentrdamediardacarrierAlgebraic geometry over C[infinity]-rings /Dominic JoyceProvidence, RI :American Mathematical Society,[2019]©20191 online resource (152 pages) illustrationsMemoirs of the American Mathematical Society,0065-9266 ;number 1256"July 2019, volume 260, number 1256 (third of 5 numbers)."1-4704-3645-0 Includes bibliographical references and index.C[infinity]-rings -- The C[infinity]-ring C[infinity](X) of a manifold X -- C[infinity]-ringed spaces and C[infinity]-schemes -- Modules over C[infinity]-rings and C[infinity]-schemes -- C[infinity]-stacks -- Deligne-Mumford C[infinity]-stacks -- Sheaves on Deligne-Mumford C[infinity]-stacks -- Orbifold strata of C[infinity]-stacks."If X is a manifold then the R-algebra C[infinity](X) of smooth functions C : X [right arrow] R is a C[infinity]-ring. That is, for each smooth function f : Rn [right arrow] R there is an n-fold operation ]Phi]f : C[infinity](X)n [right arrow] C[infinity](X) acting by [Phi]f : (c1, . . . , cn) [right arrow] f(c1, . . . , cn), and these operations [Phi]f satisfy many natural identities. Thus, C[infinity](X) actually has a far richer structure than the obvious R-algebra structure. We explain the foundations of a version of algebraic geometry in which rings or algebras are replaced by C[infinity]-rings. As schemes are the basic objects in algebraic geometry, the new basic objects are C[infinity]-schemes, a category of geometric objects which generalize manifolds, and whose morphisms generalize smooth maps. We also study quasicoherent sheaves on C[infinity]-schemes, and C[infinity]-stacks, in particular Deligne- Mumford C[infinity]-stacks, a 2-category of geometric objects generalizing orbifolds. Many of these ideas are not new: C[infinity]-rings and C[infinity]-schemes have long been part of synthetic differential geometry. But we develop them in new directions. In Joyce (2014, 2012, 2012 preprint), the author uses these tools to define d-manifolds and d-orbifolds, 'derived' versions of manifolds and orbifolds related to Spivak's 'derived manifolds' (2010)"--Provided by publisher.Memoirs of the American Mathematical Society ;no. 1256.Geometry, AlgebraicGeometry, Algebraic.516.3/658A4014A2046E2551K10mscJoyce Dominic D.66989MiAaPQMiAaPQMiAaPQBOOK9910793890203321Algebraic geometry over C-rings3753849UNINA