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Register a new userA note on the subadditivity problem for maximal shifts in free resolutions
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We present some partial results regarding subadditivity of maximal shifts in finite graded free resolutions.
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Let $R=S/I$ be a graded algebra with $t_i$ and $T_i$ being the minimal and maximal shifts in the minimal $S$ resolution of $R$ at degree $i$. In this paper we prove that $t_nleq t_1+T_{n-1}$, for all $n$ and as a consequence, we show that for Gorenstein algebras of codimension $h$, the subadditivity of maximal shifts $T_i$ in the minimal resolution holds for $i geq h-1$, i.e, we show that $T_i leq T_a+T_{i-a}$ for $igeq h-1$.
Under reasonable assumptions, a group action on a module extends to the minimal free resolutions of the module. Explicit descriptions of these actions can lead to a better understanding of free resolutions by providing, for example, convenient expressions for their differentials or alternative characterizations of their Betti numbers. This article introduces an algorithm for computing characters of finite groups acting on minimal free resolutions of finitely generated graded modules over polynomial rings.
Let $A$ be a semigroup whose only invertible element is 0. For an $A$-homogeneous ideal we discuss the notions of simple $i$-syzygies and simple minimal free resolutions of $R/I$. When $I$ is a lattice ideal, the simple 0-syzygies of $R/I$ are the binomials in $I$. We show that for an appropriate choice of bases every $A$-homogeneous minimal free resolution of $R/I$ is simple. We introduce the gcd-complex $D_{gcd}(bf b)$ for a degree $mathbf{b}in A$. We show that the homology of $D_{gcd}(bf b)$ determines the $i$-Betti numbers of degree $bf b$. We discuss the notion of an indispensable complex of $R/I$. We show that the Koszul complex of a complete intersection lattice ideal $I$ is the indispensable resolution of $R/I$ when the $A$-degrees of the elements of the generating $R$-sequence are incomparable.
We have proved the following Problem:{it Let $R$ be a $mathbb{C}$-affine domain, let $T$ be an element in $R setminus mathbb{C}$ and let $i : mathbb{C}[T] hookrightarrow R$ be the inclusion. Assume that $R/TR cong_{mathbb{C}} mathbb{C}^{[n-1]}$ and that $R_T cong_{mathbb{C}[T]} mathbb{C}[T]_T^{[n-1]}$. Then $R cong_{mathbb{C}} mathbb{C}^{[n]}$.} This result leads to the negative solution of the candidate counter-example of V.Arno den Lessen : Conjecture E : {it Let $A:=mathbb{C}[t,u,x,y,z]$ denote a polynomial ring, and let $f(u):=u^3-3u, g(u):=u^4-4u^2$ and $h(u):=u^5-10u$ be the polynomials in $mathbb{C}[u]$. Let $D:= f(u)partial_x + g(u)partial_y + h(u)partial_z + tpartial_u$ (which is easily seen to be a locally nilpotent derivation on $A$). Then $A^D otcong_{mathbb{C}} mathbb{C}^{[4]}$.} Consequently our result in this short paper guarantees that the conjectures : the Cancellation Problem for affine spaces, the Linearization Problem, the Embedding Problem and the affine $mathbb{A}^n$-Fibration Problem are still open.
Let $k$ be an arbitrary field. In this note, we show that if a sequence of relatively prime positive integers ${bf a}=(a_1,a_2,a_3,a_4)$ defines a Gorenstein non complete intersection monomial curve ${mathcal C}({bf a})$ in ${mathbb A}_k^4$, then there exist two vectors ${bf u}$ and ${bf v}$ such that ${mathcal C}({bf a}+t{bf u})$ and ${mathcal C}({bf a}+t{bf v})$ are also Gorenstein non complete intersection affine monomial curves for almost all $tgeq 0$.
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