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-pure threshold, and the diagonal
-threshold are three important invariants of a graded
-algebra. Hirose, Watanabe, and Yoshida have conjectured relations among these invariants for strongly
-regular rings. In this article, we prove that these relations hold only assuming that the algebra is
-pure. In addition, we present an interpretation of the
-pure Gorenstein graded
-algebras in terms of regular sequences that preserve
-purity. This result is in the spirit of Bertini theorems for projective varieties. Moreover, we show connections with projective dimension, Castelnuovo–Mumford regularity, and Serre’s condition
. We also present analogous results and questions in characteristic zero.
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