Abstract
In this chapter we shall describe two constructions—the associated graded ring and the blowup algebra—that are made from a descending multiplicative filtration of a ring R; that is, from a sequence of ideals $R = I_0\supset I_1\supset I_2\supset ...\ {\rm satisfying}\ I_iJ_j\supset I_{i+j}\quad {\rm for\ all}\ i,j.$ A third such construction, the Rees algebra, is treated at the end of the next chapter, and sheds some light on the results we shall prove about the associated graded ring. Chapter 7 will be devoted to a fourth example, the completion. Each is used to get information about R by comparing it with a closely related ring that is simpler in some way.
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