Nonunimodality of Graded Gorenstein Artin Algebras

Abstract

Abstract.We give an explicit expression for the Hilbert function of a large class of graded Gorenstein Artin algebras and give a criterion for this function to be unimodal. As a result we obtain an abundance of graded Gorenstein Artin algebras with nonunimodal Hilbert function.

Artinian Gorenstein algebras of embedding dimension four: components of PGor(H) for H=(1Iarrobino1, Srinivasan2 2005Journal of Pure and Applied Algebra23000View full textAdd to dashboardCite

“…This example shows that it is not the inclusion of a Z [6] term in G that keys the simpler case 3.23. When j = 8, G = X [3] Y [5] + X [2] Y [4] Z [2] + Y [5] Z [3] , then…”

“…The subsequent Lemma 3.24 explains some of the observations. Changing G by adding a Z [6] term, we have G 1 = X [4] Z [2] − X [4] Y Z + Z [6] , F 1 = G 1 + W Z [5] , J (1) = Ann (G 1 ) = (w, y 2 , yz 2 , xyz + xz 2 , x 4 y + z 5 , x 5 ), so (J (1)) = 5, and I (1) = Ann (F 1 ) = (w 2 , wx, wy, y 2 , yz 2 , xyz + xz 2 , x 4 y + wz 4 , x 5 , wz 5 − z 6 ). Also H (R/J (1)) = (1, 3, 5, 5, 5, 3, 1), and H (R/I (1)) = (1, 4, 6, 6, 6, 4, 1) = H (R/J (1)) + H 0 .…”

“…In this example, we chose G = (Z + X) [6] + (Z + 2X) [6] + (Z + Y ) [6] + (Z + 2Y ) [6] + (Z + X + Y ) [6] + (Z + 2X + 2Y ) [6] , the sum of 6 divided powers, and let J = Ann (G). Then H (R/J ) has the expected value H (R/J ) = (1, 3, 6, 6, 6, 3, 1) (see [20]), and (J ) = 3.…”

“…Omitting the pure Z [6] term from G and F, to obtain G 1 , F 1 we have H (R/Ann (G 1 )) = (1, 3, 6, 7, 6, 3, 1), (Ann (G 1 )) = 4 and…”

“…For embedding dimensions five or greater, it is known that a Gorenstein sequence may be nonunimodal: that is, it may have several maxima separated by a smaller local minimum [2,6].…”