Advanced Studies in Pure Mathematics

Algebraic Shifting

Gil Kalai

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Abstract

Algebraic shifting is a correspondence which associates to a simplicial complex $K$ another simplicial complex $\Delta (K)$ of a special type. In fact, there are two main variants based on symmetric algebra and exterior algebra, respectively. The construction is algebraic and is closely related to "Gröbner bases" and specifically to "generic initial ideals" in commutative algebra.

Algebraic shifting preserves various combinatorial and topological properties of $K$ while others disappear. For example, $\Delta (K)$ has the same Betti numbers as $K$ while the ring structure on cohomology is destroyed as $\Delta (K)$ is always a wedge of spheres. One of the important challenges is to deepen the relation between algebraic shifting and the basic notions and constructions of algebraic topology. Some important progress in this direction was achieved by Duval.

Algebraic shifting also preserves the property that $K$ is Cohen-Macaulay. At the forefront of our knowledge in this direction is a far-reaching extension of this fact achieved by Bayer, Charalambous and Popescu (symmetric shifting) and Aramova and Herzog (exterior shifting). In a different context extensions to Buchsbaum complexes have been made by Schenzel and by Novik (available only for symmetric shifting). These results apply to triangulations of manifolds and have interesting combinatorial consequences. Among the challenges which remain are: To understand algebraic shifting of simplicial spheres and simplicial manifolds, to find relations between shifting and embeddability and to identify intersection homology groups via algebraic shifting.

We will also describe the relation of algebraic shifting to framework rigidity, the connection with the original notion of "combinatorial shifting" which goes back to Erdös, Ko and Rado and some possible applications to extremal combinatorics.

Article information

Source
Computational Commutative Algebra and Combinatorics, T. Hibi, ed. (Tokyo: Mathematical Society of Japan, 2002), 121-163

Dates
Received: 31 January 2001
First available in Project Euclid: 31 December 2018

Permanent link to this document
https://projecteuclid.org/ euclid.aspm/1546230155

Digital Object Identifier
doi:10.2969/aspm/03310121

Mathematical Reviews number (MathSciNet)
MR1890098

Zentralblatt MATH identifier
1034.57021

Subjects
Primary: 05E99: None of the above, but in this section 52B05: Combinatorial properties (number of faces, shortest paths, etc.) [See also 05Cxx] 13Dxx: Homological methods {For noncommutative rings, see 16Exx; for general categories, see 18Gxx} 55Nxx: and 55Uxx for algebraic topology} 05D05: Extremal set theory

Citation

Kalai, Gil. Algebraic Shifting. Computational Commutative Algebra and Combinatorics, 121--163, Mathematical Society of Japan, Tokyo, Japan, 2002. doi:10.2969/aspm/03310121. https://projecteuclid.org/euclid.aspm/1546230155


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