Computational Commutative Algebra and Combinatorics

Abstract We equate various Euler classes of algebraic vector bundles, including those of [12] and one suggested by M. J. Hopkins, A. Raksit, and J.-P. Serre. We establish integrality results for this Euler class and give formulas for local indices at isolated zeros, both in terms of the six-functors formalism of coherent sheaves and as an explicit recipe in the commutative algebra of Scheja and Storch. As an application, we compute the Euler classes enriched in bilinear forms associated to arithmetic counts of d-planes on complete intersections in $\mathbb P^n$ in terms of topological Euler numbers over $\mathbb {R}$ and $\mathbb {C}$ .

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Jordan–Hölder, modularity and distributivity in non-commutative algebra

Journal of Pure and Applied Algebra ◽

10.1016/j.jpaa.2006.03.004 ◽

2007 ◽

Vol 208 (2) ◽

pp. 665-689 ◽

Cited By ~ 1

Author(s):

Francis Borceux ◽

Marco Grandis

Keyword(s):

Commutative Algebra

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ON FINITE TYPE 3-MANIFOLD INVARIANTS I

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216596000278 ◽

1996 ◽

Vol 05 (04) ◽

pp. 441-461 ◽

Cited By ~ 11

Author(s):

STAVROS GAROUFALIDIS

Keyword(s):

Vector Space ◽

Finite Type ◽

Commutative Algebra ◽

Integral Homology ◽

Knot Invariants ◽

Finite Type Invariants ◽

Definition Of ◽

Type 3

Recently Ohtsuki [Oh2], motivated by the notion of finite type knot invariants, introduced the notion of finite type invariants for oriented, integral homology 3-spheres. In the present paper we propose another definition of finite type invariants of integral homology 3-spheres and give equivalent reformulations of our notion. We show that our invariants form a filtered commutative algebra. We compare the two induced filtrations on the vector space on the set of integral homology 3-spheres. As an observation, we discover a new set of restrictions that finite type invariants in the sense of Ohtsuki satisfy and give a set of axioms that characterize the Casson invariant. Finally, we pose a set of questions relating the finite type 3-manifold invariants with the (Vassiliev) knot invariants.

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