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The general creation function takes a graded module representing the sheaf and
a scheme X on which it is supported. There are also special constructors for
the structure sheaf of X and the canonical sheaf of X, when X is locally
Cohen-Macaulay and equidimensional. The user may also ask for Serre twists of
a given sheaf. Other constructions giving new sheaves from old are described
in later sections.
Returns the sheaf defined by graded module M on scheme X. X should be
ordinary projective and M a module over the coordinate ring of the ambient
of X which is annihilated by the defining ideal of X.
StructureSheaf(X,n) : Sch, RngIntElt -> ShfCoh
Returns the structure sheaf OX of ordinary projective scheme X,
which is the sheaf defined by the coordinate ring RX of X as a module.
The signature with an additional integer argument n returns the twisted
version of this, Serre's twisting sheaf OX(n), that has RX(n) as its
associated graded module (see Section 5, Chapter II of [Har77]).
These are all invertible sheaves on X and OX(1) is the sheaf OX(H)
corresponding to the class of a hyperplane divisor H on X.
CanonicalSheaf(X,n) : Sch, RngIntElt -> ShfCoh
X should be an ordinary projective scheme which is equidimensional and
locally Cohen-Macaulay. That is, all of the primary components of X should
have the same dimension and its local rings should all be Cohen-Macaulay rings.
These conditions aren't checked as that would involve very computationally heavy
calculations in general. A non-singular variety always satisfies these conditions,
but many singular normal varieties do also. Any curve or normal surface does.
To check the stronger condition of being arithmetically Cohen-Macaulay,
the user can call IsArithmeticallyCohenMacaulay with the structure
sheaf of X as argument.
With these conditions, X has a canonical sheaf KX, defined up to isomorphism,
which acts as a dualising sheaf. See Section 7, Chapter III of [Har77] and
Chapter 21 of [Eis95] for the module-theoretic side. For non-singular
varieties, the canonical sheaf is the usual one: the highest alternating power of the
sheaf of Kahler differentials. The function returns the canonical sheaf of X. It
is computed from the dual complex to the minimal free resolution of the coordinate
ring of X.
The signature with an additional integer argument n returns the nth Serre
twist (see below) of the canonical sheaf KX(n). For a non-singular variety
of dimension d,
the map into projective space corresponding to KX(d - 1) is the
important adjunction map.
The nth Serre twist of Para, Para(n) isomorphic to Para tensor OXOX(n). If
M is a module giving Para, then M(n) gives Para(n).
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