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010 a| 2019033051;

020 a| 9781470436452q| (paperback);

020 z| 9781470453367q| (pdf);

035 a| (WaSeSS)ssj0002227308;

042 a| pcc;

049 a| EREEa| NEHH;

050 00 a| QA614.3b| .J69 2019;

082 00 a| 516.3/62| 23;

084 a| 58A40a| 14A20a| 46E25a| 51K102| msc;

100 1 a| Joyce, Dominic D.=| ^A463160;

245 10 a| Algebraic geometry over C[infinity]-ringsh| [electronic resource] /c| Dominic Joyce.;

260 a| Providence, RI :b| American Mathematical Society,c| [2019];

300 a| v, 139 pages :b| illustrations ;c| 26 cm;

490 0 a| Memoirs of the American Mathematical Society,x| 0065-9266 ;v| number 1256;

500 a| "July 2019, volume 260, number 1256 (fifth of 5 numbers).";

504 a| Includes bibliographical references (pages 131-133) and index.;

505 0 a| 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.;

506 a| Available only to authorized users.;

520 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)"--c| Provided by publisher.;

538 a| Mode of access: World Wide Web;

650 0 a| Differentiable manifolds.=| ^A33725;

650 0 a| Smooth affine curves.=| ^A528417;

650 0 a| Rings (Algebra)=| ^A19672;

650 0 a| Geometry, Algebraic.=| ^A42380;

650 7 a| Global analysis, analysis on manifolds [See also 32Cxx, 32Fxx, 32Wxx, 46-XX, 47Hxx, 53Cxx] {For geometric integration theory, see 49Q15} -- General theory of differentiable manifolds [See also 32Cxx]2| msc;

650 7 a| Algebraic geometry -- Foundations -- Generalizations (algebraic spaces, stacks).2| msc;

650 7 a| Functional analysis {For manifolds modeled on topological linear spaces, see 57Nxx, 58Bxx} -- Linear function spaces and their duals [See also 30H05, 32A38, 46F05] {For function algebras, see 46J10} -2| msc;

650 7 a| Geometry {For algebraic geometry, see 14-XX} -- Distance geometry -- Synthetic differential geometry.2| msc;

655 0 a| Electronic books.=| ^A491897;

856 40 z| Full text available from Ebook Central - Academic Completeu| https://ebookcentral.proquest.com/lib/eastcarolina/detail.action?docID=5904555;

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999 a| CLICK ON WEB ADDRESSw| ASISc| 1i| 5640840-3001l| MNETm| JMUSICr| Ys| Yt| MNESSBKu| 7/19/2021x| EBOOKz| JERESOURCE;

Algebraic geometry over C[infinity]-rings / Dominic Joyce.

Author/creator	Joyce, Dominic D.
Format	Electronic
Publication Info	Providence, RI : American Mathematical Society, [2019]
Description	v, 139 pages : illustrations ; 26 cm
Supplemental Content	Full text available from Ebook Central - Academic Complete
Subjects	Differentiable manifolds. Smooth affine curves. Rings (Algebra) Geometry, Algebraic.

Series	Memoirs of the American Mathematical Society, 0065-9266 ; number 1256
Contents	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.
Abstract	"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.
General note	"July 2019, volume 260, number 1256 (fifth of 5 numbers)."
Bibliography note	Includes bibliographical references (pages 131-133) and index.
Access restriction	Available only to authorized users.
Technical details	Mode of access: World Wide Web
Genre/form	Electronic books.
LCCN	2019033051
ISBN	9781470436452 (paperback)
ISBN	(pdf)

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