4
ERNST DIETERICH
sets of points of A{K) and A{K') . Moreover, it induces an isomor-
2
phism of quivers A(K) = A{K' ) if and only if ker#v c rad (X,Y) ,
A, Y A.
for all X,Y indK . )
For any commutative ring S and any pair (m,n) !Nn x I N we de-
note by S the set of all mxn - matrices over S . (In the text S
will be one of the rings R , for some n I N . ) Some special types of
matrices will be denoted by special symbols:
E = identity matrix in S (Usually, we simply write E = E .)
T
. . . .
cS
lxO
J = unique matrix i n
. c.Oxl
i—I = unique
matri•
x i n S
i n = Frobenius matrix of A in k , m = n»deg A , for all
A
A J\{oo} , n I N .
In matrices with block structure, blocks carrying no symbol are under-
stood to be zero blocks. For any matrix M , M denotes its transposed
matrix. If p : X —» Y is a morphism in modR then we denote by (p)
the matrix corresponding to p with respect to chosen bases in X and
Y . Throughout, matrices will be viewed within the column calculus.
That is to say, vectors correspond to columns and linear maps corres-
pond to sets of columns. (Note that this convention fits to the above
agreement on the composition of morphisms: if p : X Y and
ijj : Y Z are morphisms in modR then {\fjp) = (0)(p) , for any fixed
choice of bases of X,Y,Z .)
0.2. GENERALIZED FACTORSPACE CATEGORIES.
Let (K,| | ) be a Krull-Schmidt category K together with an
R-additive functor | | : K modR , for some n D M .With the
n co
pair (K,| | ) we associate the generalized factorspace category
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