I'm looking for an introductory (or rather, nonadvanced) book on manifolds as locally ringed spaces, i.e., from the algebraic geometric viewpoint. Most introductory texts only introduce manifolds from the differentiable viewpoint; I wonder if a text introducing differential manifolds from an algebraic viewpoint exists.

$\begingroup$ Locally ringed spaces are not algebraic objects, they are algebraicgeometric objects. $\endgroup$– Martin BrandenburgFeb 10, 2020 at 20:19

1$\begingroup$ @MartinBrandenburg That's what I meant, perhaps I was imprecise. I fixed my wording $\endgroup$– xuq01Feb 10, 2020 at 20:45

1$\begingroup$ Locally ring spaces are neither algebraic nor algebraicgeometric objects (they have nothing to do specifically with algebraic geometry); they are sheaftheoretic objects. $\endgroup$– Dmitri PavlovFeb 11, 2020 at 4:31

$\begingroup$ There is a serious mismatch between the title and the question asked in the body of the post. The author should clarify whether he is looking specifically for a treatment using locally ringed spaces, as opposed to treatments that largely parallel those found in algebraic geometry books. Nestruev's book is a prime example of the latter, but not the former. $\endgroup$– Dmitri PavlovFeb 11, 2020 at 4:34

$\begingroup$ @DmitriPavlov Thanks for pointing this out. Since I'm looking for an introductory text, apparently this is a field which I know little about. Yes, I mean the former, not the algebraic geometry of complex manifolds covered in, e.g., Griffiths and Harris. $\endgroup$– xuq01Feb 11, 2020 at 16:28
4 Answers
I'm not very familiar with this book (in particular, I don't know how introductory or not it is), but I think
Torsten Wedhorn, Manifolds, Sheaves, and Cohomology. Springer Studium Mathematik—Master. Springer Spektrum, Wiesbaden, 2016. xvi+354 pp. ISBN: 9783658106324; 9783658106331
would fit your description.
Ramanan's "Global Calculus" is a very nice introductory text which defines manifolds by their sheaves of differentiable functions. I don't know if this is what you're looking for: I don't see it as very algebraic, and tools from analysis have an important role.
Jet Nestruev (a collective author, I think) does this in "Smooth Manifolds and Observables".

$\begingroup$ In that book manifolds are defined by charts in the usual way and regarded as formal dual of certain commutative $\mathbb{R}$algebras. But is a ringed space POV also present in that book? $\endgroup$– QfwfqFeb 10, 2020 at 22:57

$\begingroup$ This is the ringed space POV, no? (Although now that you mention it I don't think they prove that the $\mathbb R$algebras they consider are obtained from sections of the structure sheaf of suchandsuch ringed space.) $\endgroup$ Feb 11, 2020 at 0:32

1$\begingroup$ @Qfwfq For real smooth manifolds the ringed space POV is redundant, in that it suffices to deal with global sections. Bump functions and other special properties make everything affine, so to speak. $\endgroup$ Feb 11, 2020 at 8:13

$\begingroup$ @Michael Bächtold: yes, I'm aware of that, I was just considering the OP $\endgroup$– QfwfqFeb 11, 2020 at 20:31
In Introduction to differential geometry (see the review) by R.Sikorski the author introduces the concept of (what is now called) Sikorski space. Sikorski spaces are "affine, reduced differential spaces" and hence they can be approached algebraically by looking at their coordinate rings. Differentiable manifolds are important examples Sikorski spaces. Unfortunately the book was not translated to english. Luckily there are some publications (in english and perhaps in french) by Sikorski in which he explains this very natural concept. One of them is Differential modules.
A book $C^{\infty}$ Differentiable Spaces by Navarro González and Sancho de Salas develop theory of differentiable spaces by first constructing real spectra for smooth algebras and then glue them in order to obtain general spaces. This is analogical to the development of algebraic geometry (scheme theory) by Grothendieck and his school. The book might be a bit advanced.