By Dominique Arlettaz

ISBN-10: 082183696X

ISBN-13: 9780821836965

ISBN-10: 3019815835

ISBN-13: 9783019815834

ISBN-10: 7119964534

ISBN-13: 9787119964539

ISBN-10: 8619866036

ISBN-13: 9788619866033

The second one Arolla convention on algebraic topology introduced jointly experts protecting a variety of homotopy idea and $K$-theory. those court cases mirror either the diversity of talks given on the convention and the variety of promising examine instructions in homotopy idea. The articles contained during this quantity contain major contributions to classical risky homotopy conception, version classification conception, equivariant homotopy concept, and the homotopy conception of fusion structures, in addition to to $K$-theory of either neighborhood fields and $C^*$-algebras

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**Additional resources for An Alpine Anthology of Homotopy Theory**

**Example text**

The reduction of ideals was ﬁrst introduced by Northcott and Rees [NR] and Rees [R] extended the notion to modules. Since then the reduction of ideals and modules have been discussed extensively. It is known that the ideals with reduction relation have the same integral closure and that they, if m-primary, have the same Hilbert-Samuel multiplicity. The main aim of this paper is to discuss the reductions of monomial ideals in a twodimensional localized polynomial ring R = k[x, y](x,y) over an inﬁnite ﬁeld.

Fm ∈ k[x, y]. If I is a monomial ideal containing I, then it is clear that I contains Γ(f1 ) ∪ · · · ∪ Γ(fm ). Hence the smallest monomial ideal containing I is generated by Γ(f1 ) ∪ · · · ∪ Γ(fm ). We denote this monomial ideal by I ∗ . We are interested in conditions under which I ∗ becomes integral over I. 2. Under what conditions is I a reduction of I ∗ ? 18 6 C-Y. 3 states a suﬃcient condition, in terms of monomials in Γ(f1 ) ∪ · · · ∪ Γ(fm ), for I to be a reduction of I ∗ . 7 provides a minimal reduction of a given monomial ideal.

From this we observe that not only does the Buchsbaum-Rim multiplicity generalize the Hilbert-Samuel multiplicity by deﬁnition and share parallel properties in the reduction theory as described earlier but also the two multiplicities are connected in such a special case. Such a relation was generalized in [J] when I and J are monomial ideals with small number of generators. In Section 5, we take the results in [J] one step further by formulating its outcome. 3 also motivates the work in [CLU] for arbitrary modules over a two dimensional Gorenstein local ring.

### An Alpine Anthology of Homotopy Theory by Dominique Arlettaz

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