Curves of genus zero. In these notes we study noncommutative curves of
genus zero. By a curve we always mean a smooth, projective curve defined over a
field k. A noncommutative curve is given by a small connected k-category H which
shares the properties with the category coh(X) of coherent sheaves over a smooth
projective curve X, listed below:
H is abelian and each object in H is noetherian.
All morphism and extension spaces in H are of finite k-dimension.
There is an autoequivalence τ on H (called Auslander-Reiten translation)
such that Serre duality ExtH(X,
Y ) = D HomH(Y, τX) holds, where D =
Homk(−, k).
H contains an object of infinite length.
It follows from Serre duality that H is a hereditary category, that is,
vanishes for all n 2. Let H0 be the Serre subcategory of H formed by the objects
of finite length. Then H0 =
Ux (for some index set X) where Ux are connected
uniserial categories, called tubes. The objects in Ux are called concentrated in x.
Of course, any curve should also have the following property.
X consists of infinitely many points.
We call X, equipped with H, a noncommutative (smooth, projective) curve.
It follows from the axioms (see [74]) that the quotient category H/H0 is the
category of finite dimensional vector spaces over some skew field k(H), called the
function field. We denote it also by k(X). The dimension over k(H) induces the
rank of objects in H. The full subcategory of H of objects which do not contain
a subobject of finite length is denoted by H+; these objects themselves are called
(vector) bundles. Bundles of rank one are called line bundles. The category H has
the Krull-Remak-Schmidt property, that is, each object is a finite direct sum of
essentially unique indecomposable objects. Moreover, each indecomposable object
lies either in H+ or in H0.
In the classical case where X is a smooth projective curve with structure sheaf
O, the genus of X is zero, that is, dimk ExtX(O,
O) = 0, if and only if the cat-
egory H = coh(X) contains a tilting object [69]. This is an object T H with
T ) = 0 and such that HomH(T, X) = 0 = ExtH(T,
X) only holds for
X = 0.
We therefore say that a noncommutative curve H is of (absolute) genus zero if
H contains a tilting object.
Thus, a noncommutative curve of genus zero is just an exceptional curve as defined
in [68], a term which we will mainly use in these notes. (These curves are called
“exceptional” since the existence of a tilting object is equivalent to the existence
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