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Medientyp:
E-Artikel
Titel:
Differential operators on Stanley-Reisner rings
Beteiligte:
Tripp, J.
Erschienen:
American Mathematical Society (AMS), 1997
Erschienen in:
Transactions of the American Mathematical Society, 349 (1997) 6, Seite 2507-2523
Sprache:
Englisch
DOI:
10.1090/s0002-9947-97-01749-2
ISSN:
0002-9947;
1088-6850
Entstehung:
Anmerkungen:
Beschreibung:
Let k k be an algebraically closed field of characteristic zero, and let R = k [ x 1 , … , x n ] R=k[x_{1},\dots ,x_{n}] be a polynomial ring. Suppose that I I is an ideal in R R that may be generated by monomials. We investigate the ring of differential operators D ( R / I ) \mathcal {D}(R/I) on the ring R / I R/I , and I R ( I ) \mathcal {I}_{R}(I) , the idealiser of I I in R R . We show that D ( R / I ) \mathcal {D}(R/I) and I R ( I ) \mathcal {I}_{R}(I) are always right Noetherian rings. If I I is a square-free monomial ideal then we also identify all the two-sided ideals of I R ( I ) \mathcal {I}_{R}(I) . To each simplicial complex Δ \Delta on V = { v 1 , … , v n } V=\{v_{1},\dots ,v_{n}\} there is a corresponding square-free monomial ideal I Δ I_{\Delta } , and the Stanley-Reisner ring associated to Δ \Delta is defined to be k [ Δ ] = R / I Δ k[\Delta ]=R/I_{\Delta } . We find necessary and sufficient conditions on Δ \Delta for D ( k [ Δ ] ) \mathcal {D}(k[\Delta ]) to be left Noetherian.