Abstract
Let Δ be a triangulated homology ball whose boundary complex is ∂Δ. A result of Hochster asserts that the canonical module of the Stanley–Reisner ring F[Δ] of Δ is isomorphic to the Stanley–Reisner module F[Δ, ∂Δ] of the pair (Δ, ∂Δ). This result implies that an Artinian reduction of F[Δ, ∂Δ] is (up to a shift in grading) isomorphic to the Matlis dual of the corresponding Artinian reduction of F[Δ]. We establish a generalization of this duality to all triangulations of connected orientable homology manifolds with boundary. We also provide an explicit algebraic interpretation of the hʺ-numbers of Buchsbaum complexes and use it to prove the monotonicity of hʺ-numbers for pairs of Buchsbaum complexes as well as the unimodality of hʺ-vectors of barycentric subdivisions of Buchsbaum polyhedral complexes. We close with applications to the algebraic manifold g-conjecture.
| Original language | English |
|---|---|
| Pages (from-to) | 635-656 |
| Number of pages | 22 |
| Journal | Algebra and Number Theory |
| Volume | 11 |
| Issue number | 3 |
| DOIs | |
| Publication status | Published - 2017 Jan 1 |
| Externally published | Yes |
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Keywords
- Buchsbaum rings
- H-vectors
- Stanley–Reisner rings
- Triangulated manifolds
ASJC Scopus subject areas
- Algebra and Number Theory
Cite this
A duality in Buchsbaum rings and triangulated manifolds. / Murai, Satoshi; Novik, Isabella; Yoshida, Ken Ichi.
In: Algebra and Number Theory, Vol. 11, No. 3, 01.01.2017, p. 635-656.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - A duality in Buchsbaum rings and triangulated manifolds
AU - Murai, Satoshi
AU - Novik, Isabella
AU - Yoshida, Ken Ichi
PY - 2017/1/1
Y1 - 2017/1/1
N2 - Let Δ be a triangulated homology ball whose boundary complex is ∂Δ. A result of Hochster asserts that the canonical module of the Stanley–Reisner ring F[Δ] of Δ is isomorphic to the Stanley–Reisner module F[Δ, ∂Δ] of the pair (Δ, ∂Δ). This result implies that an Artinian reduction of F[Δ, ∂Δ] is (up to a shift in grading) isomorphic to the Matlis dual of the corresponding Artinian reduction of F[Δ]. We establish a generalization of this duality to all triangulations of connected orientable homology manifolds with boundary. We also provide an explicit algebraic interpretation of the hʺ-numbers of Buchsbaum complexes and use it to prove the monotonicity of hʺ-numbers for pairs of Buchsbaum complexes as well as the unimodality of hʺ-vectors of barycentric subdivisions of Buchsbaum polyhedral complexes. We close with applications to the algebraic manifold g-conjecture.
AB - Let Δ be a triangulated homology ball whose boundary complex is ∂Δ. A result of Hochster asserts that the canonical module of the Stanley–Reisner ring F[Δ] of Δ is isomorphic to the Stanley–Reisner module F[Δ, ∂Δ] of the pair (Δ, ∂Δ). This result implies that an Artinian reduction of F[Δ, ∂Δ] is (up to a shift in grading) isomorphic to the Matlis dual of the corresponding Artinian reduction of F[Δ]. We establish a generalization of this duality to all triangulations of connected orientable homology manifolds with boundary. We also provide an explicit algebraic interpretation of the hʺ-numbers of Buchsbaum complexes and use it to prove the monotonicity of hʺ-numbers for pairs of Buchsbaum complexes as well as the unimodality of hʺ-vectors of barycentric subdivisions of Buchsbaum polyhedral complexes. We close with applications to the algebraic manifold g-conjecture.
KW - Buchsbaum rings
KW - H-vectors
KW - Stanley–Reisner rings
KW - Triangulated manifolds
UR - http://www.scopus.com/inward/record.url?scp=85019209986&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85019209986&partnerID=8YFLogxK
U2 - 10.2140/ant.2017.11.635
DO - 10.2140/ant.2017.11.635
M3 - Article
AN - SCOPUS:85019209986
VL - 11
SP - 635
EP - 656
JO - Algebra and Number Theory
JF - Algebra and Number Theory
SN - 1937-0652
IS - 3
ER -