Local Cohomology: An Algebraic Introduction with Geometric Applications

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10 Οκτ 2013 (πριν από 3 χρόνια και 8 μήνες)

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Markus P. Brodmann & Rodney Y. Sharp


Local Cohomology:

An Algebraic Introduction with Geometric Applications


Cambridge Studies in Advanced Mathematics 136, Cambridge University Press (2012)

ISBN: 9780521513630




This Second Edition of a successful gr
aduate text provides a careful and detailed algebraic
introduction to Grothendieck's local cohomology theory, including in multi
-
graded situations, and
provides many illustrations of applications of the theory in commutative algebra and in the
geometry of
quasi
-
affine and quasi
-
projective varieties. Topics covered include Serre's Affineness
Criterion, the Lichtenbaum
-
Hartshorne Vanishing Theorem, Grothendieck's Finiteness Theorem and
Faltings' Annihilator Theorem, local duality and canonical modules, Castel
nuovo
-
Mumford
regularity, the Fulton
-
Hansen Connectedness theorem for projective varieties, and connections
between local cohomology and both reductions of ideals and sheaf cohomology.


The book is designed for graduate students who have some experience of

basic commutative
algebra and homological algebra, and also for experts in commutative algebra and algebraic
geometry. Over 300 exercises are interspersed among the text; these range in difficulty from routine
to challenging, and hints are provided for so
me of the more difficult ones.


505 pages, 330 exercices.


About the Book:



... Brodmann and Sharp have produced an excellent book; it is clearly, carefully and
enthusiastically written: it covers all important aspects and main uses of the subject; and it

gives a
thorough and well
-
rounded appreciation of the topic's geometric and algebraic interrelationships...

I am sure, that this will be a standard text and reference book for years to come.”

Liam O'Caroll,

Bull. London Mathematical Society



The book is
well organized, very nicely written, and reades very well... a very good overview of
local cohomology theory.”

European Mathematical Society




Contents:


1. The local cohomology functors

1.1 Torsion functors

1.2 Local cohomology modules

1.3 Connected sequences of functors



2. Torsion modules and ideal transforms

2.1 Torsion modules

2.1 Ideal transforms and generalized ideal transforms

2.
3 Geometrical significance


3. The Mayer
-
Vietoris Sequence

3.1 Comparison of systems of ideals

3.2 Construction of the sequence

3.3 Arithmetic rank

3.4 Direct limits


4. Change of rings

4.1 Some acyclic modules

4
.2 The Independence Theorem

4.3 The Flat Base Change Theorem


5. Other approaches

5.1 Use of Ĉech complexes

5.2 Use of Koszul complexes

5.3 Local cohomology in prime characteristic


6. Fundamental vanishing theorems

6
.1 Grothendieck's Vanishing Theorem

6.2 Connections with grade

6.3 Exactness of ideal transforms

6.4 An Affineness Criterion due to Serre

6.5 Applications to local algebra in prime characteristic



7. Artinian local cohomol
ogy modules

7.1 Artinian modules

7.2 Secondary presentations

7.3 The Non
-
Vanishing Theorem again


8. The Lichtenbaum
-
Hartshorne Theorem

8.1 Preparatory lemmas

8.2 The main theorem


9. The Annihilator and Finiteness

Theorems

9.1 Finiteness dimensions

9.2 Adjusted depths

9.3 The first inequality

9.4 The second inequality

9.5 The main theorems

9.6 Extensions


10. Matlis duality

10.1 Indecomposable injective modules

10.2 Matlis
duality


11. Local duality

11.1 Minimal injective resolutions

11.2 Local Duality Theorems


12. Canonical modules


12.1 Definition and basic properties

12.2 The endomorphism ring

12.3
S_2
-
ifications



13. Foundations in
the graded case

13.1 Basic multi
-
graded commutative algebra

13.2 *Injective modules

13.3 The *restriction property

13.4 The reconciliation

13.5 Some examples and applications


14. Graded versions of basic theorems

14.1 Fundamen
tal theorems

14.2 *Indecomposable *injective modules

14.3 A graded version of the Annihilator Theorem

14.4 Graded local duality

14.5 *Canonical modules


15.
Links with projective varieties

15.1 Affine algebraic cones

15.2 Projective

varieties


16. Castelnuovo regularity

16.1 Finitely generated components

16.2 The basics of Castelnuovo regularity

16.3 Degrees of generators


17. Hilbert polynomials

17.1 The characteristic function

17.2 The significance of reg^2


17.3 Bounds on reg^2 in terms of Hilbert coefficients

17.4 Bounds on reg^1 and reg^0


18. Applications to reductions of ideals

18.1 Reductions and integral closures

18.2 The analytic spread

18.3 Links with Castelnuovo regularity


19.

Connectivity in algebraic varieties

19.1 The connectedness dimension

19.2 Complete local rings and connectivity

19.3 Some local dimensions

19.4 Connectivity of affine algebraic cones

19.5 Connectivity of projective varieties

19.6 Con
nectivity of intersections

19.7 The projective spectrum and connectedness


20. Links with sheaf cohomology

20.1 The Deligne Isomorphism

20.2 The Graded Deligne Isomorphism

20.3 Links with sheaf theory

20.4 Applications to projective sch
emes

20.5 Locally free sheaves