A Beilinson-type theorem for coherent sheaves on weighted projective spaces
✍ Scribed by Alberto Canonaco
- Publisher
- Elsevier Science
- Year
- 2000
- Tongue
- English
- Weight
- 162 KB
- Volume
- 225
- Category
- Article
- ISSN
- 0021-8693
No coin nor oath required. For personal study only.
✦ Synopsis
a weighted beilinson theorem 29 theorem to the coherent sheaf on n π * (π: n → Q is the projection), thus getting two resolutions E • S and E • of π * , and then applying the invariant direct image functor π µ Q * yields the desired resolutions of . The main part of the present paper is dedicated to the proof that this procedure actually produces a similar result: in Section 1 we establish some properties of the functors π * and π µ Q * (in particular, we prove that π µ Q * is exact and that ∼ = π µ Q * π * ), and in Section 3 we verify that the resolutions E •
S and E • are compatible with the action of µ Q on π * (to this purpose, we have to examine some parts of the proof of Beilinson's theorem). Moreover, since each term in the resolutions of is a finite direct sum of sheaves of the form
in Sections 1 and 2 we obtain a description of these two types of sheaves, which allows us to formulate our main result (Theorem 4.1).
We were not able to find (as in the case of n ) categories equivalent to D b Coh Q (see Remark 4.3). In a forthcoming paper we will give applications of our main result to the canonical rings of algebraic surfaces (cf. ).