We say that M is a (tame) bimodule of (dimension) type (2, 2), (1, 4) or (4, 1) if
this pair is ([M : F ], [M : G]). We call the number ε {1, 2} the numerical type of
M (or of X), which is defined by
ε =
1 if M is of type (2, 2).
2 if M is of type (1, 4) or (4, 1).
The numerical type is an invariant of the curve X.
With κ := [L], [L] , for the normalization factor c = [Z : K0(X), w ] as above
we have c = κε.
0.3.17 (Automorphism groups). Let X be an exceptional curve with associ-
ated abelian hereditary category H and structure sheaf L. Denote by Aut(H) the
automorphism class group of H, that is, the group of isomorphism classes of au-
toequivalences of H (in the literature sometimes also called the Picard group [8],
which has a different meaning in our presentation). We call this group the auto-
morphism group of H and call the elements automorphisms. (If there is need to
emphasize the base field k, we also write Autk(X) and use a similar notation in
analogue situations.)
By a slight abuse of terminology, we will also call the autoequivalences them-
selves automorphisms, that is, the representatives of such classes; if F is an autoe-
quivalence, then its class in the automorphism group is also denoted by F .
The subgroup of elements of Aut(H) fixing L (up to isomorphism) is denoted
by Aut(X), the automorphism group of X. (We will later see that this group does
not dependent on L.)
Each element φ Aut(H) induces a bijective map φ on the points of X by
φ(Ux) = Uφ(x) for all x X. We call φ the shadow of φ. If φ lies in the kernel of the
homomorphism Aut(H) −→ Bij(X), φ φ, then we call φ point fixing (or invisible
on X). If φ(x) = x we also say (by a slight abuse of terminology) that the point x
is fixed by φ. Similarly, if φ(x) = y we also write φ(x) = y.
Denote by Aut0(H) the (normal) subgroup of Aut(H) given by the point fixing
Non-trivial elements of Aut(X) which are point fixing are called ghost automor-
phisms, or just ghosts. The subgroup G of Aut(X) formed by the ghosts is called the
ghost group. It is a normal subgroup of Aut(H). We have G = Aut(X) Aut0(H).
We call the factor group Aut(X)/G the geometric automorphism group of X, its
elements geometric automorphisms. By a slight abuse of terminology, we also call
the elements in Aut(X) which are not ghosts geometric.
Denote by
the group of isomorphism classes of exact autoequiva-
lences of the triangulated category
called the automorphism group of
(Compare also [9]. There is also the related notion of the derived Picard group [82].)
0.3.18 (Projective coordinate algebras). Let H be a finitely generated abelian
group of rank one, which is equipped with a partial order ≤, compatible with the
group structure. Let R =
Rh be an H-graded k-algebra, such that each
homogeneous component Rh is finite dimensional over k and such that Rh = 0 for
0 h. Assume moreover that R is a finitely generated k-algebra and noetherian.
Note that we do not require that R is commutative.
Denote by
(R) the category of finitely generated right H-graded R-
modules, and by mod0
(R) the full subcategory of graded modules of finite length
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