For a commutative ring $R$, the $F$-signature was defined by Huneke and Leuschke \cite{H-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 $F^e$.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 $P(R,t)$. \[ s(R)\le\frac{(-a(R))^d}{2^{d-1}d!}\lim_{t\rightarrow 1}(1-t)^dP(R,t) \]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.
We introduce the cohomological blow up of a graded Artinian Gorenstein (AG) algebra along a surjective map, which we term BUG (Blow Up Gorenstein) for short. This is intended to translate to an algebraic context the cohomology ring of a blow up of a projective manifold along a projective submanifold. We show, among other things, that a BUG is a connected sum, that it is the general fiber in a flat family of algebras, and that it preserves the strong Lefschetz property. We also show that standard graded compressed algebras are rarely BUGs, and we classify those BUGs that are complete intersecti…
Let $R$ be a polynomial ring over a field and $M= \bigoplus_n M_n$ a finitely generated graded $R$-module, minimally generated by homogeneous elements of degree zero with a graded $R$-minimal free resolution $\mathbf{F}$. A Cohen-Macaulay module $M$ is Gorenstein when the graded resolution is symmetric. We give an upper bound for the first Hilbert coefficient, $e_1$ in terms of the shifts in the graded resolution of $M$. When $M = R/I$, a Gorenstein algebra, this bound agrees with the bound obtained in \cite{ES} in Gorenstein algebras with quasi-pure resolution. We conjecture a similar bound f…
Goto Numbers of a Numerical Semigroup Ring and the Gorensteiness of Associated Graded Rings
Noether局部环（R，m）中参数理想Q的Goto数是使得Q：m^q是Q上整数的最大整数q. R = k[[x^{a_1}，x^{a_2}，.，x_{a_ν} ]]用R的半群刻画.这有助于计算它们的类的数值半群环，以及在个案的基础上。探讨了R的最小Goto数及其与其他不变量的关系。再次利用R的半群给出了R和R/x^{a_1}R的相伴分次环是Gorenstein的充要条件.
Let R be a graded ring and n > 1 an integer. In this paper, We introduce the notions of Ding n-gr-injective and Ding n-gr-flat modules by using of special finitely presented graded modules . Then, some properties of Ding n-gr-injective and Ding n-gr-flat modules are obtained. On n-gr-coherent rings, we investigate the relationships among Ding n-gr-injective and Ding n-gr-flat modules and also, we prove that any graded module in R-gr (resp. gr-R) admits an Ding n-gr-injective (resp. Ding n-gr-flat) cover and preenvelope.
A connected sum construction for local rings was introduced in a paper by H. Ananthnarayan, L. Avramov, and W.F. Moore. In the graded Artinian Gorenstein case, this can be viewed as an algebraic analogue of the topological construction of the same name. We give two alternative description of this algebraic connected sum: the first uses algebraic analogues of Thom classes of vector bundles and Gysin homomorphisms, the second is in terms of Macaulay dual generators. We also investigate the extent to which the connected sum construction preserves the weak or strong Lefschetz property, thus provid…
Levelness and nearly Gorensteinness are well-studied properties of graded rings as a generalized notion of Gorensteinness. In this paper, we compare the strength of these properties. For any Cohen-Macaulay homogeneous affine semigroup ring R, we give a necessary condition for R to be non-Gorenstein and nearly Gorenstein in terms of the h-vector of R and we show that if R is nearly Gorenstein with Cohen-Macaulay type 2, then it is level. We also show that if Cohen-Macaulay type is more than 2, there are 2-dimensional counterexamples. Moreover, we characterize nearly Gorensteinness of Stanley-Re…
In this paper, we give a criterion of the nearly Gorenstein property of the Ehrhart ring of the stable set polytope of an h-perfect graph: the Ehrhart ring of the stable set polytope of an h-perfect graph $G$ with connected components $G^{(1)}, \ldots, G^{(\ell)}$ is nearly Gorenstein if and only if (1) for each $i$, the Ehrhart ring of the stable set polytope of $G^{(i)}$ is Gorenstein and (2) $|ω(G^{(i)})-ω(G^{(j)})|\leq 1$ for any $i$ and $j$, where $ω(G^{(i)})$ is the clique number of $G^{(i)}$. We also show that the Segre product of Cohen-Macaulay graded rings with linear non-zerodivisor …
Let $R$ be a standard graded Gorenstein algebra over a field presented by quadrics. Conca, Rossi, and Valla have shown that such a ring is Koszul if $\mathrm{reg}\, R \leq 2$ or if $\mathrm{reg}\, = 3$ and $c= \mathrm{codim}\, R \leq 4$, and they ask whether this is true for $\mathrm{reg}\, R = 3$ in general. We determine sufficient conditions on a non-Koszul quadratic Cohen-Macaulay ring $R$ that guarantee the Nagata idealization $\tilde{R} = R \ltimes ω_R(-a-1)$ is a non-Koszul quadratic Gorenstein ring. We use this to negatively answer the question of Conca-Rossi-Valla, constructing non-Kos…
