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In this paper it is shown that a polyomino is balanced if and only if it is simple. As a consequence one obtains that the coordinate ring of a simple polyomino is a normal Cohen-Macaulay domain.
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Motivated by questions in algebra and combinatorics we study two ideals associated to a simple graph G: --> the Lovasz-Saks-Schrijver ideal defining the d-dimensional orthogonal representations of the graph complementary to G and --> the determinantal ideal of the (d+1)-minors of a generic symmetric with 0s in positions prescribed by the graph G. In characteristic 0 these two ideals turns out to be closely related and algebraic properties such as being radical, prime or a complete intersection transfer from the Lovasz-Saks-Schrijver ideal to the determinantal ideal. For Lovasz-Saks-Schrijver ideals we link these properties to combinatorial properties of G and show that they always hold for d large enough. For specific classes of graph, such a forests, we can give a complete picture and classify the radical, prime and complete intersection Lovasz-Saks-Schrijver ideals.
This work concerns the Koszul complex $K$ of a commutative noetherian local ring $R$, with its natural structure as differential graded $R$-algebra. It is proved that under diverse conditions, involving the multiplicative structure of $H(K)$, any dg $R$-algebra automorphism of $K$ induces the identity map on $H(K)$. In such cases, it is possible to define an action of the automorphism group of $R$ on $H(K)$. On the other hand, numerous rings are described for which $K$ has automorphisms that do not induce the identity on $H(K)$. For any $R$, it is shown that the group of automorphisms of $H(K)$ induced by automorphisms of $K$ is abelian.
Let $R=K[X_1,dots, X_n]$ be a polynomial ring in $n$ variables over a field $K$ of charactersitic zero and $d$ a $K$-derivation of $R$. Consider the isotropy group if $d$: $ text{Aut}(R)_d :={rho in text{Aut}_K(R)|; rho d rho^{-1}=d}$. In his doctoral thesis, Baltazar proved that if $d$ is a simple Shamsuddin derivation of $K[X_1,X_2]$, then its isotropy group is trivial. He also gave an example of a non-simple derivation whose isotropy group is infinite. Recently, Mendes and Pan generalized this result to an arbitrary derivation of $K[X_1,X_2]$ proving that a derivation of $K[X_1,X_2]$ is simple if, and only if, its isotropy group is trivial. In this paper, we prove that the isotropy group of a simple Shamsuddin derivation of the polynomial ring $R=K[X_1,dots, X_n]$ is trivial. We also calculate other isotropy groups of (not necessarily simple) derivations of $K[X_1,X_2]$ and prove that they are finite cyclic groups.
In this paper, we answer a question of Dwyer, Greenlees, and Iyengar by proving a local ring $R$ is a complete intersection if and only if every complex of $R$-modules with finitely generated homology is proxy small. Moreover, we establish that a commutative noetherian ring $R$ is locally a complete intersection if and only if every complex of $R$-modules with finitely generated homology is virtually small.
In this paper we introduce the concept of clique disjoint edge sets in graphs. Then, for a graph $G$, we define the invariant $eta(G)$ as the maximum size of a clique disjoint edge set in $G$. We show that the regularity of the binomial edge ideal of $G$ is bounded above by $eta(G)$. This, in particular, settles a conjecture on the regularity of binomial edge ideals in full generality.
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