Computing Global Extension Modules for Coherent Sheaves on a Projective Scheme
نویسنده
چکیده
Let X be a projective scheme; let M and N be two coherent OX modules. Given an integer m, we present an algorithm for computing the global extension module Ext(X;M,N ). In particular, this allows one to calculate the sheaf cohomology H(X,N ) and to construct the sheaf corresponding to an element of the module Ext(X;M,N ). This algorithm can be implemented using only the computation of Gröbner bases and syzygies, and it has been implemented in the computer algebra system Macaulay2.
منابع مشابه
Coherent Algebras and Noncommutative Projective Lines
A well-known conjecture says that every one-relator group is coherent. We state and partly prove a similar statement for graded associative algebras. In particular, we show that every Gorenstein algebra A of global dimension 2 is graded coherent. This allows us to define a noncommutative analogue of the projective line P as a noncommutative scheme based on the coherent noncommutative spectrum q...
متن کاملComputing Global Extension Modules
Let X be a projective scheme and let M and N be two coherent OX -modules. Given an integer m, we present an algorithm for computing the global extension module ExtX (M,N ). As a consequence, one may calculate the sheaf cohomology Hm(X,N ) and construct the sheaf corresponding to an element of the module ExtX(M,N ). This algorithm depends only on the computation of Gröbner bases and syzygies and...
متن کاملComputing Noncommutative Deformations of Presheaves and Sheaves of Modules
We describe a noncommutative deformation theory for presheaves and sheaves of modules that generalizes the commutative deformation theory of these global algebraic structures, and the noncommutative deformation theory of modules over algebras due to Laudal. In the first part of the paper, we describe a noncommutative deformation functor for presheaves of modules on a small category, and an obst...
متن کاملAuslander-reiten Sequences on Schemes
Let X be a smooth projective scheme of dimension d ≥ 1 over the field k, and let C be an indecomposable coherent sheaf on X . Then there is an Auslander-Reiten sequence in the category of quasi-coherent sheaves on X , 0 → (ΣC)⊗ ω −→ B −→ C → 0. Here ΣC is the (d−1)’st syzygy in a minimal injective resolution of C, and ω is the dualizing sheaf of X . 0. Introduction This note shows that Auslande...
متن کاملDescribing Coherent Sheaves on Projective Spaces via Koszul Duality
It is well known that there is a close connection between coherent sheaves on a projective space P(V ) where V is a a vector space over a field k, and finitely generated graded modules over the symmetric algebra S(V ). The Bernstein-Gelfand-Gelfand (BGG) correspondence [5] from 1978 relates coherent sheaves on P(V ) with graded modules over the exterior algebra Λ(V ∗) = ⊕Λi(V ∗) where V ∗ is th...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 1999