Published online by Cambridge University Press: 31 January 2011
We have spent many chapters now worrying about ringed spaces. In Chapter 2 we used the example of Ck–manifolds to motivate the definition and convince the reader that ringed spaces, that is spaces with sheaves of rings on them, are natural objects that are worth studying. In Chapter 3 we constructed the ringed spaces that form the object of study of this book, namely the schemes (locally) of finite type over ℂ. Chapters 4, 5 and 6 told us that, if (X, O) is a scheme locally of finite type over ℂ, then there is a natural way to attach to it another ringed space (Xan, Oan). Intuitively we think of 0 as the sheaf of polynomial functions on X, and Oan is the sheaf of holomorphic functions. For every open set U ⊂ X one can speak of the polynomial functions on it, that is the ring Γ(U, O). For any open subset V ⊂ Xan (there are many more such open sets) one can talk of the holomorphic functions on V, that is the elements of Γ(V, Oan).
If the reader glances back to the introduction she will discover that the results we want to prove are not only about spaces and functions on them, but also about vector bundles. It is only natural that we should again carefully define what we mean by an algebraic vector bundle, and what we mean by an analytic vector bundle.
To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.