Singularities, Monodromy and Zeta Functions Blatt 9
Exercises for discussion in the class on 21.12.2018
Recall that, forAa filteredK-algebra andM a filteredA-module with filtration{Fr(M)}r, we say
M is of type(d, e) :⇐⇒ dimKFr(M) =erd
d! +o(rd).
Aufgabe 1:
Find the type(d, e)of the following filtered modules:
(a) The D-moduleMf with respect to the good filtration discussed in the lecture;
(b) Forxof lengthnandk≤n, theK[x]-moduleK[x]/(x1, ..., xk)(first consider, what is the natural filtration?);
(c) The K[x1, x2]-moduleK[x1, x2]/(x1x2);
(d) TheK[x, y]-moduleK[x, y]/(y2−x3).
Aufgabe 2:
Suppose thatM is a filteredA-algebra, whereA is a filteredK-algebra and consider the associated grading Gr(M)as described in the lecture.
(a) Verify thatGr(M)is indeed aGr(A)-module.
(b) Prove that ifGr(M)is finitely generated as aGr(A)-module, thenM is finitely generated as an A-module.
Aufgabe 3:
Are there alternative filtrations ofK[x]making it of type(2,1) or of type(1,2) as aK[x]-module?
Aufgabe 4:
IfMi are filteredA-modules of type(di, ei)fori= 1,2 respectively, what is the type ofM1⊕M2 with respect to the direct sum of the implicit filtrations?
Aufgabe 5:
(*) For an idealI⊂K[x], give an expression of the type(d, e)ofK[x]/Iin terms of the geometry of the variety defined byI. Conjectural expressions also welcome.
Course website:http://reh.math.uni-duesseldorf.de/~internet/Zeta_WS18/
