2011/04/21 by Akiyoshi Sannai, Sannai, Akiyoshi, Kei-ichi Watanabe +1
Mathematics · #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1104.4236
openalex publication_date 2011/04/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a commutative ring R, the F-signature was defined by Huneke and Leuschke \citeH-L. It is an invariant that measures the order of the rank of the free direct summand of R(e). Here, R(e) is R itself, regarded as an R-module through e-times Frobenius action Fe.In this paper, we show a connection of the F-signature of a graded ring with other invariants. More precisely, for a graded F-finite Gorenstein ring R of dimension d, we give an inequality among the F-signature s(R), a-invariant a(R) and Poincaré polynomial P(R,t). s(R)≤\frac(-a(R))d2d-1d!limt→ 1(1-t)dP(R,t)Moreover, we show that R(e) has only one free direct summand for any e, if and only if R is F-pure and a(R)=0. This gives a characterization of such rings.