LEADER 03195nam 2200469 450 001 9910793890203321 005 20191015173613.0 010 $a1-4704-5336-3 035 $a(CKB)4100000009374627 035 $a(MiAaPQ)EBC5904555 035 $a(RPAM)21255944 035 $a(PPN)240205979 035 $a(EXLCZ)994100000009374627 100 $a20191015h20192019 uy| 0 101 0 $aeng 135 $aurcnu|||||||| 181 $2rdacontent 182 $2rdamedia 183 $2rdacarrier 200 10$aAlgebraic geometry over C[infinity]-rings /$fDominic Joyce 210 1$aProvidence, RI :$cAmerican Mathematical Society,$d[2019] 210 4$dİ2019 215 $a1 online resource (152 pages) $cillustrations 225 1 $aMemoirs of the American Mathematical Society,$x0065-9266 ;$vnumber 1256 300 $a"July 2019, volume 260, number 1256 (third of 5 numbers)." 311 $a1-4704-3645-0 320 $aIncludes bibliographical references and index. 327 $aC[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. 330 $a"If X is a manifold then the R-algebra C[infinity](X) of smooth functions C : X [right arrow] R is a C[infinity]-ring. That is, for each smooth function f : Rn [right arrow] R there is an n-fold operation ]Phi]f : C[infinity](X)n [right arrow] C[infinity](X) acting by [Phi]f : (c1, . . . , cn) [right arrow] f(c1, . . . , cn), and these operations [Phi]f satisfy many natural identities. Thus, C[infinity](X) actually has a far richer structure than the obvious R-algebra structure. We explain the foundations of a version of algebraic geometry in which rings or algebras are replaced by C[infinity]-rings. As schemes are the basic objects in algebraic geometry, the new basic objects are C[infinity]-schemes, a category of geometric objects which generalize manifolds, and whose morphisms generalize smooth maps. We also study quasicoherent sheaves on C[infinity]-schemes, and C[infinity]-stacks, in particular Deligne- Mumford C[infinity]-stacks, a 2-category of geometric objects generalizing orbifolds. Many of these ideas are not new: C[infinity]-rings and C[infinity]-schemes have long been part of synthetic differential geometry. But we develop them in new directions. In Joyce (2014, 2012, 2012 preprint), the author uses these tools to define d-manifolds and d-orbifolds, 'derived' versions of manifolds and orbifolds related to Spivak's 'derived manifolds' (2010)"--$cProvided by publisher. 410 0$aMemoirs of the American Mathematical Society ;$vno. 1256. 606 $aGeometry, Algebraic 615 0$aGeometry, Algebraic. 676 $a516.3/6 686 $a58A40$a14A20$a46E25$a51K10$2msc 700 $aJoyce$b Dominic D.$066989 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910793890203321 996 $aAlgebraic geometry over C-rings$93753849 997 $aUNINA