O.

In the next example, we introduce the Euler characteristic a useful arithmetic invariant of a graded vector space. 9. Let A* be a locally finite graded vector space over k, that is, climk An is finite for all n. The Poincaré series for A* is the formal power series given by P(A*,t) = (dimk An)tn. n=0 On the face of it, the Poincaré series seems to carry very little information, especially if we treat A* simply as a graded vector space. 14* , t) is a polynomial. 1-b3 •—2-• We can define an Euler characteristic for A* by x (A*) = P(A* , —1), when this expression makes sense.

Notice that a2 y could hit w by d4. However, d4 commutes with r* and so d4 (a2 y) = a2 d4 (y) = 0. Thus w survives since it cannot be hit by any F*-multiple of x or y. The element x, however, could be mapped to aw by d9. Since bx = 0, we can compute 0 = d2(bx) = bd2(x) = baw 0. 4. Working backwards. 19 bz • • • • • ba2y i ak •• •• •• •• a4y • • • • • bay az •• • • •• a3y f f • • •• • z • • • • • • • • • • • • • • • • • * • • • • a4u, • • a2x • baw • • • • a 3w • • ax • bit'. • by • • • a2w a • ay • x • • • • • • • • aw • * ay • • • • • • • • • • w a y —,•—e—sa—a—e • • Thus (12(x) = con leads to a contradiction and so (12(x) = 0.