For a graph G = (V, E) the edge ring k[G] is k[x 1 ,. .. , x n ]/I(G), where n = |V | and I(G) is... more For a graph G = (V, E) the edge ring k[G] is k[x 1 ,. .. , x n ]/I(G), where n = |V | and I(G) is generated by {x i x j ; {i, j} ∈ E}. The conjecture we treat is the following. Conjecture. If k[G] has a 2-linear resolution, then the projective dimension of K[G], pd(k[G]), equals the maximal degree of a vertex in G. As far as we know, this conjecture is first mentioned in a paper by Gitler and Valencia, [7, Conjecture 4.13], and there it is called the Eliahou-Villarreal conjecture. The conjecture is treated in a recent paper by Ahmed, Mafi, and Namiq, [1]. That there are counterexamples was noted already by Moradi and Kiani, [9]. By interpreting k[G] as a Stanley-Reisner ring, we are able to characterize those graphs for which the conjecture holds.
Let k be a field and $$R=k[x_1,\ldots ,x_n]/I=S/I$$ R = k [ x 1 , … , x n ] / I = S / I a graded ... more Let k be a field and $$R=k[x_1,\ldots ,x_n]/I=S/I$$ R = k [ x 1 , … , x n ] / I = S / I a graded ring. Then R has a t-linear resolution if I is generated by homogeneous elements of degree t, and all higher syzygies are linear. Thus, R has a t-linear resolution if $$\mathrm{Tor}^S_{i,j}(S/I,k)=0$$ Tor i , j S ( S / I , k ) = 0 if $$j\ne i+t-1$$ j ≠ i + t - 1 . For a graph G on $$\{1,\ldots ,n\}$$ { 1 , … , n } , the edge algebra is $$k[x_1,\ldots ,x_n]/I$$ k [ x 1 , … , x n ] / I , where I is generated by those $$x_ix_j$$ x i x j for which $$\{ i,j\}$$ { i , j } is an edge in G. We want to determine the Betti numbers of edge rings with 2-linear resolution. But we want to do that by looking at the edge ring as a Stanley–Reisner ring. For a simplicial complex $$\Delta $$ Δ on $$[\mathbf{n}]=\{1,\ldots ,n\}$$ [ n ] = { 1 , … , n } and a field k, the Stanley–Reisner ring $$k[\Delta ]$$ k [ Δ ] is $$k[x_1,\ldots ,x_n]/I$$ k [ x 1 , … , x n ] / I , where I is generated by the squarefree mo...
If S = d1,. .. , dν is a numerical semigroup, we call the ring C[S] = C[t d 1 ,. .. , t dν ] the ... more If S = d1,. .. , dν is a numerical semigroup, we call the ring C[S] = C[t d 1 ,. .. , t dν ] the semigroup ring of S. We study the ring of differential operators on C[S], and its associated graded in the filtration induced by the order of the differential operators. We find that these are easy to describe in case S is a so called Arf semigroup. If I is an ideal in C[S] that is generated by monomials, we also give some results on Der(I, I) (the set of derivations which map I into I).
Two plane analytic branches are topologically equivalent if and only if they have the same multip... more Two plane analytic branches are topologically equivalent if and only if they have the same multiplicity sequence. We show that having same semigroup is equivalent to having same multiplicity sequence, we calculate the semigroup from a parametrization, and we characterize semigroups for plane branches. These results are known, but the proofs are new. Furthermore we characterize multiplicity sequences of plane branches, and we prove that the associated graded ring, with respect to the values, of a plane branch is a complete intersection.
We characterize all Gorenstein rings generated by strongly stable sets of monomials of degree two... more We characterize all Gorenstein rings generated by strongly stable sets of monomials of degree two. We compute their Hilbert series in several cases, which also provides an answer to a question by Migliore and Nagel [10].
Lance Bryant noticed in his thesis [3], that there was a flaw in our paper [2]. It can be fixed b... more Lance Bryant noticed in his thesis [3], that there was a flaw in our paper [2]. It can be fixed by adding a condition, called the BF condition in [3]. We discuss some equivalent conditions, and show that they are fulfilled for some classes of rings, in particular for our motivating example of semigroup rings. Furthermore we discuss the connection to a similar result, stated in more generality, by Cortadella-Zarzuela in [4]. Finally we use our result to conclude when a semigroup ring in embedding dimension at most three has an associated graded which is a complete intersection. 2000 Mathematics Subject Classification: 13A30 If x ∈ R is an element of smallest positive value, i.e. v(x) = e, then xR is a minimal reduction of the maximal ideal, i.e. m n+1 = xm n , for n >> 0. Conversely each minimal reduction of the maximal ideal is a principal ideal generated by an element x of value e. The smallest integer n such that m n+1 = xm n is called the reduction number and we denote it by r. Observe that, if v(x) = e, then Ap e (S) = S \(e+S) = v(R)\v(xR), therefore w j / ∈ v(xR), for j = 0,. .. , e − 1. Consider the m-adic filtration m ⊃ m 2 ⊃ m 3 ⊃. .. . If a ∈ R, we set ord(a) := max{i | a ∈ m i }. If s ∈ S, we consider the semigroup filtration v(m) ⊃ v(m 2) ⊃. .. and set vord(s) := max{i | s ∈ v(m i)}. If a ∈ m i , then v(a) ∈ v(m i) and so ord(a) ≤ vord(v(a)). According to [3], we say that the m-adic filtration is essentially divisible with respect to the minimal reduction xR if, whenever u ∈ v(xR), then there is an a ∈ xR with v(a) = u and ord(a) = vord(u). The m-adic filtration is essentially divisible if there exists a minimal reduction xR such that it is essentially divisible with respect to xR. We fix for all the paper the following notation. Set, for j = 0,. .. , e − 1, b j = max{i|w j ∈ v(m i)}, and let c j = max{i|w j ∈ v(m i + xR)}. Note that the numbers b j 's do not depend on the minimal reduction xR, on the contrary the c j 's depend on xR. Lemma 1.1 If I and J are ideals of R, then v(I +J) = v(I)∪v(J) is equivalent to v(I ∩ J) = v(I) ∩ v(J).
Rings. Fields, and Ideals. Monomial Ideals. Grobner Bases. Algebraic Sets. Primary Decomposition ... more Rings. Fields, and Ideals. Monomial Ideals. Grobner Bases. Algebraic Sets. Primary Decomposition of Ideals. Solving Systems of Polynomial Equations. Applications of Grobner Bases. Homogeneous Algebras. Projective Varieties. The Associated Graded Ring. Hilbert Series. Variations of Grobner Bases. Improvements to Buchberger's Algorithm. Software. Hints to Some Exercises. Answers to Exercises. Bibliography. Index.
If k[x 1 ,. .. , x n ]/I = R = i≥0 R i , k a field, is a standard graded algebra, the Hilbert ser... more If k[x 1 ,. .. , x n ]/I = R = i≥0 R i , k a field, is a standard graded algebra, the Hilbert series of R is the formal power series i≥0 dim k R i t i. It is known already since Macaulay which power series are Hilbert series of graded algebras [12]. A much harder question is which series are Hilbert series if we fix the number of generators of I and their degrees, say for ideals I = (
Moreno studies the following question. Let I be an ideal in k½x 1 , :::, x n generated minimally ... more Moreno studies the following question. Let I be an ideal in k½x 1 , :::, x n generated minimally by elements of degree d, d þ 1, d þ 2, …. How long can such a sequence of generators be? Later he also studies the opposite question.
Given an ideal I = (f1,. .. , fr) in C[x1,. .. , xn] generated by forms of degree d, and an integ... more Given an ideal I = (f1,. .. , fr) in C[x1,. .. , xn] generated by forms of degree d, and an integer k > 1, how large can the ideal I k be, i.e., how small can the Hilbert function of C[x1,. .. , xn]/I k be? If r ≤ n the smallest Hilbert function is achieved by any complete intersection, but for r > n, the question is in general very hard to answer. We study the problem for r = n + 1, where the result is known for k = 1. We also study a closely related problem, the Weak Lefschetz property, for S/I k , where I is the ideal generated by the d'th powers of the variables.
In what follows, we present a large number of questions which were posed on the problem solving s... more In what follows, we present a large number of questions which were posed on the problem solving seminar in algebra at Stockholm University during the period Fall 2014-Spring 2017 along with a number of results related to these problems. Many of the results were obtained by participants of the latter seminar. Keywords Waring problem for forms • Generic and maximal ranks • Ideals of generic forms • Power ideals • Lefschetz properties • Symbolic powers 1 The Waring Problem for Complex-Valued Forms The following famous result on binary forms was proven by Sylvester in 1851. Below we use the terms "forms" and "homogeneous polynomials" as synonyms.
Let R = k[t n 1 ,. .. , t n s ] = k[x 1 ,. .. , x s ]/P be a numerical semigroup ring and let P (... more Let R = k[t n 1 ,. .. , t n s ] = k[x 1 ,. .. , x s ]/P be a numerical semigroup ring and let P (n) = P n R P ∩ R be the symbolic power of P and R s (P) = ⊕ i≥0 P (n) t n the symbolic Rees ring of P. It is hard to determine symbolic powers of P ; there are even non-Noetherian symbolic Rees rings for 3-generated semigroups. We determine the primary decomposition of powers of P for some classes of 3-generated numerical semigroups.
Given an ideal of forms in an algebra (polynomial ring, tensor algebra, exterior algebra, Lie alg... more Given an ideal of forms in an algebra (polynomial ring, tensor algebra, exterior algebra, Lie algebra, bigraded polynomial ring), we consider the Hilbert series of the factor ring. We concentrate on the minimal Hilbert series, which is achieved when the forms are generic. In the polynomial ring we also consider the opposite case of maximal series. This is mainly a survey article, but we give a lot of problems and conjectures. The only novel results concern the maximal series in the polynomial ring.
A hypergraph $H=(V,E)$, where $V=\{x_1,\ldots,x_n\}$ and $E\subseteq 2^V$ defines a hypergraph al... more A hypergraph $H=(V,E)$, where $V=\{x_1,\ldots,x_n\}$ and $E\subseteq 2^V$ defines a hypergraph algebra $R_H=k[x_1,\ldots, x_n]/(x_{i_1}\cdots x_{i_k}; \{i_1,\ldots,i_k\}\in E)$. All our hypergraphs are $d$-uniform, i.e., $|e_i|=d$ for all $e_i\in E$. We determine the Poincaré series $P_{R_H}(t)=\sum_{i=1}^\infty\dim_k\mathrm{Tor}_i^{R_H}(k,k)t^i$ for some hypergraphs generalizing lines, cycles, and stars. We finish by calculating the graded Betti numbers and the Poincaré series of the graph algebra of the wheel graph.
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Papers by Ralf Fröberg