# A remark on generators of $$\mathtt D(\hbox {X})$$ D ( X ) and flags

A remark on generators of $$\mathtt D(\hbox {X})$$ D ( X ) and flags We give a simple proof of the following fact. Let $$\hbox {X}$$ X be an n-dimensional, smooth, projective variety with ample or anti-ample canonical bundle, over an algebraically closed base field. Let $$\hbox {Y}_0 \subset \hbox {Y}_{1} \subset \cdots \subset \hbox {Y}_n = \hbox {X}$$ Y 0 ⊂ Y 1 ⊂ ⋯ ⊂ Y n = X be a complete flag of closed smooth subvarieties, where $$\hbox {Y}_{j+1} {\setminus } \hbox {Y}_{j}$$ Y j + 1 \ Y j is affine. Then $$\hbox {G} = \bigoplus _{j=0}^n \mathcal O_{\mathrm{Y}_{j}}$$ G = ⨁ j = 0 n O Y j is a generator of the (bounded coherent) derived category $$\mathtt D(\hbox {X})$$ D ( X ) . Moreover, from the endomorphism dg-algebra $${{\mathrm{REnd}}}_{\mathrm{X}}(\hbox {G})$$ REnd X ( G ) one can recover not only $$\hbox {X}$$ X but also the flag $$\hbox {Y}_0 \subset \hbox {Y}_{1} \subset \cdots \subset \hbox {Y}_n$$ Y 0 ⊂ Y 1 ⊂ ⋯ ⊂ Y n . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Manuscripta Mathematica Springer Journals

# A remark on generators of $$\mathtt D(\hbox {X})$$ D ( X ) and flags

, Volume 154 (2) – Nov 25, 2016
4 pages

/lp/springer_journal/a-remark-on-generators-of-mathtt-d-hbox-x-d-x-and-flags-v1lkQ9eDrJ
Publisher
Springer Berlin Heidelberg
Subject
Mathematics; Mathematics, general; Algebraic Geometry; Topological Groups, Lie Groups; Geometry; Number Theory; Calculus of Variations and Optimal Control; Optimization
ISSN
0025-2611
eISSN
1432-1785
D.O.I.
10.1007/s00229-016-0902-7
Publisher site
See Article on Publisher Site

### Abstract

We give a simple proof of the following fact. Let $$\hbox {X}$$ X be an n-dimensional, smooth, projective variety with ample or anti-ample canonical bundle, over an algebraically closed base field. Let $$\hbox {Y}_0 \subset \hbox {Y}_{1} \subset \cdots \subset \hbox {Y}_n = \hbox {X}$$ Y 0 ⊂ Y 1 ⊂ ⋯ ⊂ Y n = X be a complete flag of closed smooth subvarieties, where $$\hbox {Y}_{j+1} {\setminus } \hbox {Y}_{j}$$ Y j + 1 \ Y j is affine. Then $$\hbox {G} = \bigoplus _{j=0}^n \mathcal O_{\mathrm{Y}_{j}}$$ G = ⨁ j = 0 n O Y j is a generator of the (bounded coherent) derived category $$\mathtt D(\hbox {X})$$ D ( X ) . Moreover, from the endomorphism dg-algebra $${{\mathrm{REnd}}}_{\mathrm{X}}(\hbox {G})$$ REnd X ( G ) one can recover not only $$\hbox {X}$$ X but also the flag $$\hbox {Y}_0 \subset \hbox {Y}_{1} \subset \cdots \subset \hbox {Y}_n$$ Y 0 ⊂ Y 1 ⊂ ⋯ ⊂ Y n .

### Journal

Manuscripta MathematicaSpringer Journals

Published: Nov 25, 2016

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 12 million articles from more than
10,000 peer-reviewed journals.

All for just $49/month ### Explore the DeepDyve Library ### Unlimited reading Read as many articles as you need. Full articles with original layout, charts and figures. Read online, from anywhere. ### Stay up to date Keep up with your field with Personalized Recommendations and Follow Journals to get automatic updates. ### Organize your research It’s easy to organize your research with our built-in tools. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. ### DeepDyve Freelancer ### DeepDyve Pro Price FREE$49/month

\$360/year
Save searches from