Given a monomial $m$ in a polynomial ring and a subset $L$ of the variables of the polynomial ring, the principal $L$-Borel ideal generated by $m$ is the ideal generated by all monomials which can be obtained from $m$ by successively replacing variables of $m$ by those which are in $L$ and have smaller index. Given a collection $\mathcal{I}=\{I_1,\ldots,I_r\}$ where $I_i$ is $L_i$-Borel for $i=1,\ldots,r$ (where the subsets $L_1,\ldots,L_r$ may be different for each ideal), we prove in essence that if the bipartite incidence graph among the subsets $L_1,\ldots,L_r$ is chordal bipartite, then the defining equations of the multi-Rees algebra of $\mathcal{I}$ has a Gröbner basis of quadrics with squarefree lead terms under lexicographic order. Thus the multi-Rees algebra of such a collection of ideals is Koszul, Cohen-Macaulay, and normal. This significantly generalizes a theorem of Ohsugi and Hibi on Koszul bipartite graphs. As a corollary we obtain that the multi-Rees algebra of a collection of principal Borel ideals is Koszul. To prove our main result we use a fiber-wise Gröbner basis criterion for the kernel of a toric map and we introduce a modification of Sturmfels' sorting algorithm.
Let $M$ and $N$ be two monomials of the same degree, and let $I$ be the smallest Borel ideal containing $M$ and $N$. We show that the toric ring of $I$ is Koszul by constructing a quadratic Gröbner basis for the associated toric ideal. Our proofs use the construction of graphs corresponding to fibers of the toric map. As a consequence, we conclude that the Rees algebra is also Koszul.
We introduce (weighted) Segre and Rees products for posets and show that these constructions preserve the Cohen-Macaulay property over a field $k$ and homotopically. As an application we show that the weighted Segre product of two affine semigroup rings that are Koszul is again Koszul. This result generalizes previous results by Crona on weighted Segre products of polynomial rings. We also give a new proof of the fact that the Rees ring of a Koszul affine semigroup ring is again Koszul. The paper ends with a list of some open problems in the area.
Generalizing techniques that prove that Veronese subrings are Koszul, we show that Rees and multi-Rees algebras of certain types of principal strongly stable ideals are Koszul. We provide explicit Gröbner basis for the defining ideals of these multi-Rees algebras with squarefree initial monomials, to show that they are also normal Cohen-Macaulay domains.
We investigate the Rees algebra and the toric ring of the squarefree monomial ideal associated to the three-dimensional Ferrers diagram. Under the projection property condition, we describe explicitly the presentation ideals of the Rees algebra and the toric ring. We show that the toric ring is a Koszul Cohen--Macaulay normal domain, while the Rees algebra is Koszul and the defining ideal is of fiber type.
We prove that the multi-Rees algebra $\mathcal{R}(I_1 \oplus \cdots \oplus I_r)$ of a collection of strongly stable ideals $I_1, \ldots, I_r$ is of fiber type. In particular, we provide a Gröbner basis for its defining ideal as a union of a Gröbner basis for its special fiber and binomial syzygies. We also study the Koszulness of $\mathcal{R}(I_1 \oplus \cdots \oplus I_r)$ based on parameters associated to the collection. Furthermore, we establish a quadratic Gröbner basis of the defining ideal of $\mathcal{R}(I_1 \oplus I_2)$ where each of the strongly stable ideals has two quadric Borel gene…
In this paper we describe the defining equations of the Rees algebra and the special fiber ring of a truncation I of a complete intersection ideal in a polynomial ring over a field with homogeneous maximal ideal m. To describe explicitly the Rees algebra R(I) in terms of generators and relations we map another Rees ring R(M) onto it, where M is the direct sum of powers of m. We compute a Groebner basis of the ideal defining R(M). It turns out that the normal domain R(M) is a Koszul algebra and from this we deduce that in many instances R(I) is a Koszul algebra as well.
We show that diagonal subalgebras and generalized Veronese subrings of a bigraded Koszul algebra are Koszul. We give upper bounds for the regularity of sidediagonal and relative Veronese modules and apply the results to symmetric algebras and Rees rings.
It gives a class of $p$-Borel principal ideals of a polynomial algebra over a field $K$ for which the graded Betti numbers do not depend on the characteristic of $K$ and the Koszul homology modules have monomial cyclic basis. Also it shows that all principal $p$-Borel ideals have binomial cycle basis on Koszul homology modules.
We determine the defining equations of the Rees algebra and of the special fiber ring of the ideal of maximal minors of a $2\times n$ sparse matrix. We prove that their initial algebras are ladder determinantal rings. This allows us to show that the Rees algebra and the special fiber ring are Cohen-Macaulay domains, they are Koszul, they have rational singularities in characteristic zero and are F-rational in positive characteristic.
In this paper we study graded Bourbaki ideals. It is a well-known fact that for torsionfree modules over Noetherian normal domains, Bourbaki sequences exist. We give criteria in terms of certain attached matrices for a homomorphism of modules to induce a Bourbaki sequence. Special attention is given to graded Bourbaki sequences. In the second part of the paper, we apply these results to the Koszul cycles of the residue class field and determine particular Bourbaki ideals explicitly. We also obtain in a special case the relationship between the structure of the Rees algebra of a Koszul cycle an…
We study a large family of products of Borel fixed ideals of maximal minors. We compute their initial ideals and primary decompositions, and show that they have linear free resolutions. The main tools are an extension of straightening law and a very surprising primary decomposition formula. We study also the homological properties of associated multi-Rees algebra which are shown to be Cohen-Macaulay, Koszul and defined by a Gröbner basis of quadrics.
We study the module of Koszul cycles $Z_t(I,M)$ of a homogeneous ideal $I$ in a polynomial ring $S$ with respect to a graded module $M$. Under mild assumptions on the base field we prove that the regularity of $Z_t(I,S)$ is a subadditive function of the homological position t when I is 0-dimensional. For Borel-fixed ideals $I$ and $J$ we prove that the regularity of $Z_t(I,S/J)$ is bounded above by $t(1+\reg I)+\reg S/J$.
In this paper, we study the family of determinantal ideals of "close" cuts of Hankel matrices, say $ \mathcal{f} $. We show that the multi-Rees algebra of ideals in $ \mathcal{f} $ is defined by a quadratic Gröbner basis, it is Koszul, normal Cohen-Macaulay domain and it has a nice Sagbi basis. As a consequence of Koszulness, we prove that every product $ I_1\ldots I_l $ of ideals of $ \mathcal{f} $ has linear resolution. Moreover, we show that natural generators of every product $ I_1\ldots I_l $ form a Gröbner basis.
A new approach is established to computing the image of a rational map, whereby the use of approximation complexes is complemented with a detailed analysis of the torsion of the symmetric algebra in certain degrees. In the case the map is everywhere defined this analysis provides free resolutions of graded parts of the Rees algebra of the base ideal in degrees where it does not coincide with the corresponding symmetric algebra. A surprising fact is that the torsion in those degrees only contributes to the first free module in the resolution of the symmetric algebra modulo torsion. An additiona…
In \cite{rees} Rees gave a characterization for the normal joint reduction number zero of two $\m$-primary ideals in an analytically unramified Cohen-Macaulay local ring of dimension two. Rees' result is a generalization of Zariski's product theorem for complete ideals in a regular local ring of dimension two. The aim of this paper is to extend Rees' theorem for the ordinary powers of $\m$-primary ideals $I$ and $J$ in a Cohen-Macaulay local ring of dimension two. Following Rees' approach, we define the modified Koszul homology modules $M^1_{r,s}(a^k,b^k)$ for a joint reduction $(a,b)$ of $I$ …
Let ${\sf k}$ be a field, $S$ be a bigraded ${\sf k}$-algebra, and $S_Δ$ denote the diagonal subalgebra of $S$ corresponding to $Δ= \{ (cs,es) \; | \; s \in \mathbb{Z} \}$. It is know that the $S_Δ$ is Koszul for $c,e \gg 0$. In this article, we find bounds for $c,e$ for $S_Δ$ to be Koszul, when $S$ is a geometric residual intersection. Furthermore, we also study the Cohen-Macaulay property of these algebras. Finally, as an application, we look at classes of linearly presented perfect ideals of height two in a polynomial ring, show that all their powers have a linear resolution, and study the …
We develop the theory of integrable representations for an arbitrary maximal parabolic subalgebra of an affine Lie algebra. We see that such subalgebras can be thought of as arising in a natural way from a Borel--de Siebenthal pair of semisimple Lie algebras. We see that although there are similarities with the represenation thery of the standard maximal parabolic subalgebra there are also very interesting and non--trivial differences; including the fact that there are examples of non--trivial global Weyl modules which are irreducible and finite--dimensional. We also give a presentation of the…
We study the ideal of maximal minors in Littlewood varieties, a class of quadratic complete intersections in spaces of matrices. We give a geometric construction for a large class of modules, including all powers of this ideal, and show that they have a linear free resolution over the complete intersection and that their Koszul dual is an infinite-dimensional irreducible representation of the orthosymplectic Lie superalgebra. We calculate the algebra of cohomology operators acting on this free resolution. We prove analogous results for powers of the ideals of maximal minors in the variety of l…
We investigate Rees algebras and special fiber rings obtained by blowing up specialized Ferrers ideals. This class of monomial ideals includes strongly stable monomial ideals generated in degree two and edge ideals of prominent classes of graphs. We identify the equations of these blow-up algebras. They generate determinantal ideals associated to subregions of a generic symmetric matrix, which may have holes. Exhibiting Gröbner bases for these ideals and using methods from Gorenstein liaison theory, we show that these determinantal rings are normal Cohen-Macaulay domains that are Koszul, that …
