Veronese Rings Research Articles

Equivariant resolutions over Veronese rings

AbstractWorking in a polynomial ring , where is an arbitrary commutative ring with 1, we consider the th Veronese subalgebras , as well as natural ‐submodules inside . We develop and use characteristic‐free theory of Schur functors associated to ribbon skew diagrams as a tool to construct simple ‐equivariant minimal free ‐resolutions for the quotient ring and for these modules . These also lead to elegant descriptions of for all and for any pair of these modules .

Homological properties of pinched Veronese rings

Pinched Veronese rings are formed by removing an algebra generator from a Veronese subring of a polynomial ring. We study the homological properties of such rings, including the Cohen-Macaulay, Gorenstein, and complete intersection properties. Greco and Martino classified Cohen-Macaulayness of pinched Veronese rings by the maximum entry of the exponent vector of the pinched monomial; we re-prove their results with semigroup methods and correct an omission of a small class of examples of Cohen-Macaulay pinched Veronese rings. When the underlying field is of prime characteristic, we show that pinched Veronese rings exhibit a variety of F-singularities, including F-regular, F-injective, and F-nilpotent. We also compute upper bounds on the Frobenius test exponents of pinched Veronese rings, a computational invariant which controls the Frobenius closure of all parameter ideals simultaneously.

Cohen–Macaulay property and linearity of pinched Veronese rings

In this work, we study the Betti numbers of pinched Veronese rings, by means of the reduced homology of squarefree divisor complexes. We characterize when these rings are Cohen–Macaulay and we study the shape of the Betti tables for the pinched Veronese in the two variables. As a byproduct we obtain information on the linearity of such rings. Moreover, in the last section we compute the canonical modules of the Veronese modules.

Homogeneous algebras via homogeneous triples

To study s-homogeneous algebras, we introduce the category of quivers with s-homogeneous corelations and the category of s-homogeneous triples. We show that both of these categories are equivalent to the category of s-homogeneous algebras. We prove some properties of the elements of s-homogeneous triples and give some consequences for s-Koszul algebras. Then we discuss the relations between the s-Koszulity and the Hilbert series of s-homogeneous triples. We give some application of the obtained results to s-homogeneous algebras with simple zero component. We describe all s-Koszul algebras with one relation recovering the result of Berger and all s-Koszul algebras with one dimensional s-th component. We show that if the s-th Veronese ring of an s-homogeneous algebra has two generators, then it has at least two relations. Finally, we classify all s-homogeneous algebras with s-th Veronese rings k〈x,y〉/(xy,yx) and k〈x,y〉/(x2,y2). In particular, we show that all of these algebras are not s-Koszul while their s-homogeneous duals are s-Koszul.

Some noncommutative minimal surfaces

In the ongoing programme to classify noncommutative projective surfaces (connected graded noetherian domains of Gelfand-Kirillov dimension three) a natural question is to determine the minimal models within any birational class.In this paper we show that the generic noncommutative projective plane (corresponding to the three dimensional Sklyanin algebra) as well as noncommutative analogues of P1×P1 and the more general Van den Bergh quadrics satisfy very strong minimality conditions. Translated into an algebraic question, where one is interested in a maximality condition, we prove the following result.Theorem A:Let R be a Sklyanin algebra or a Van den Bergh quadric that is infinite dimensional over its centre and letA⊇Rbe any connected graded noetherian maximal order, with the same graded quotient ring as R. Then, up to taking Veronese rings, A is isomorphic to R.Let T be an elliptic algebra (that is, the coordinate ring of a noncommutative surface containing an elliptic curve). Then, under an appropriate homological condition, we prove that every connected graded noetherian overring of T is obtained by blowing down finitely many lines (line modules) of self-intersection(−1).

Syzygies of the Veronese Modules

We study the minimal free resolution of the Veronese modules, Sn, d, k = ⊕i≥0Sk+id, where S = 𝕂[x1,…, xn], by giving a formula for the Betti numbers in terms of the reduced homology of some skeleton of a simplicial complex. We prove that Sn, d, k is Cohen–Macaulay if and only if k < d, and that its minimal resolution is pure and has some linearity features when k > d(n − 1) − n. We prove combinatorially that the resolution of S2, d, k is pure. We show that . As an application, we calculate the complete Betti diagrams of the Veronese rings 𝕂[x, y, z](d), for d = 4, 5, and 𝕂[x, y, z, u](3).

Veronese transform and Castelnuovo--Mumford regularity of modules

Veronese rings, Segre embeddings, or more generally Segre{Veronese embeddings are very important rings in algebraic geometry. In this paper we present an original, elementary way to compute the Hilbert{Poincar e series of these rings; as a consequence we compute their Castelnuovo{Mumford regularity and also the highest graded Betti number. Moreover, using the Castelnuovo{Mumford regularity of a Cohen{Macaulay nitely generated graded module,

Classifying orders in the Sklyanin algebra

One of the major open problems in noncommutative algebraic geometry is the classification of noncommutative surfaces, and this paper resolves a significant case of this problem. Specifically, let S denote the 3-dimensional Sklyanin algebra over an algebraically closed field k and assume that S is not a finite module over its centre. (This algebra corresponds to a generic noncommutative P^2.) Let A be any connected graded k-algebra that is contained in and has the same quotient ring as a Veronese ring S^(3n). Then we give a reasonably complete description of the structure of A. This is most satisfactory when A is a maximal order, in which case we prove, subject to a minor technical condition, that A is a noncommutative blowup of S^(3n) at a (possibly non-effective) divisor on the associated elliptic curve E. It follows that A has surprisingly pleasant properties; for example it is automatically noetherian, indeed strongly noetherian, and has a dualizing complex.

Toric fiber products versus Segre products

The toric fiber product is an operation that combines two ideals that are homogeneous with respect to a grading by an affine monoid. The Segre product is a related construction that combines two multigraded rings. The quotient ring by a toric fiber product of two ideals is a subring of the Segre product, but in general this inclusion is strict. We contrast the two constructions and show that any Segre product can be presented as a toric fiber product without changing the involved quotient rings. This allows to apply previous results about toric fiber products to the study of Segre products. We give criteria for the Segre product of two affine toric varieties to be dense in their toric fiber product, and for the map from the Segre product to the toric fiber product to be finite. We give an example that shows that the quotient ring of a toric fiber product of normal ideals need not be normal. In rings with Veronese type gradings, we find examples of toric fiber products that are always Segre products, and we show that iterated toric fiber products of Veronese ideals over Veronese rings are normal.

Shellability of the higher pinched Veronese posets

The pinched Veronese poset $${\mathcal {V}}^{\bullet }_n$$ V n ? is the poset with ground set consisting of all nonnegative integer vectors of length $$n$$ n such that the sum of their coordinates is divisible by $$n$$ n with exception of the vector $$(1,\ldots ,1)$$ ( 1 , ? , 1 ) . For two vectors $$\mathbf {a}$$ a and $$\mathbf {b}$$ b in $${\mathcal {V}}^{\bullet }_n$$ V n ? , we have $$\mathbf {a}\preceq \mathbf {b}$$ a ? b if and only if $$\mathbf {b}- \mathbf {a}$$ b - a belongs to the ground set of $${\mathcal {V}}^{\bullet }_n$$ V n ? . We show that every interval in $${\mathcal {V}}^{\bullet }_n$$ V n ? is shellable for $$n \ge 4$$ n ? 4 . In order to obtain the result, we develop a new method for showing that a poset is shellable. This method differs from classical lexicographic shellability. Shellability of intervals in $${\mathcal {V}}^{\bullet }_n$$ V n ? has consequences in commutative algebra. As a corollary, we obtain a combinatorial proof of the fact that the pinched Veronese ring is Koszul for $$n \ge 4$$ n ? 4 . (This also follows from a result by Conca, Herzog, Trung, and Valla.)

Castelnuovo–Mumford regularity of seminormal simplicial affine semigroup rings

We show that the Eisenbud–Goto conjecture holds for (homogeneous) seminormal simplicial affine semigroup rings. Moreover, we prove an upper bound for the Castelnuovo–Mumford regularity in terms of the dimension, which is similar as in the normal case. Finally, we compute explicitly the regularity of full Veronese rings.

Veronese algebras and modules of rings with straightening laws

Do the Veronese rings of an algebra with straightening laws (ASL) still have an ASL structure? We give positive answers to this question in some particular cases, namely for the second Veronese algebra of Hibi rings and of discrete ASLs. We also prove that the Veronese modules of the polynomial ring have a structure of module with straightening laws. In dimension at most three we present a poset construction that has the required combinatorial properties to support such a structure.

Hilbert schemes and maximal Betti numbers over veronese rings

Macaulay’s Theorem (Macaulay in Proc. Lond Math Soc 26:531–555, 1927) characterizes the Hilbert functions of graded ideals in a polynomial ring over a field. We characterize the Hilbert functions of graded ideals in a Veronese ring R (the coordinate ring of a Veronese embedding of Pr-1). We also prove that the Hilbert scheme, which parametrizes all graded ideals in R with a fixed Hilbert function, is connected; this is an analogue of Hartshorne’s Theorem (Hartshorne in Math. IHES 29:5–48, 1966) that Hilbert schemes over a polynomial ring are connected. Furthermore, we prove that each lex ideal in R has the greatest Betti numbers among all graded ideals with the same Hilbert function.

Vertex Algebroids over Veronese Rings

We find a canonical quantization of Courant algebroids over Veronese rings. Part of our approach allows a semi-infinite cohomology interpretation, and the latter can be used to define sheaves of chiral differential operators on some homogeneous spaces including the space of pure spinors punctured at a point.

On equations defining Veronese rings

For a standard graded noetherian algebra S that is of weakly linear type, the defining equations of the Veronesian subrings S (d) are described explicitly, for d sufficiently large.

Topology of bounded-degree graph complexes

The bounded-degree graph complexes were first introduced by Reiner and Roberts [J. Algebraic Combin. 11 (2000) 135–154]. They arise from the finite free resolution of quadratic Veronese rings and modules. We prove various results about the homotopy types of these complexes, and deduce corresponding characteristic-free results about the quadratic Veronese resolutions. In particular, we characterize the set of multidegrees which support at least one higher syzygy in this resolution. The answer turns out to be independent of the field characteristic.

Semiprime Graded Algebras of Dimension Two

Semiprime, noetherian, connected graded k-algebras R of quadratic growth are described in terms of geometric data. A typical example of such a ring is obtained as follows: Let Y be a projective variety of dimension at most one over the base field k and let E be an OY-order in a finite dimensional semisimple algebra A over K=k(Y). Then, for any automorphism τ of A that restricts to an automorphism σ of Y and any ample, invertible E-bimodule B, Van den Bergh constructs a noetherian, “twisted homogeneous coordinate ring” B=⊕H0(Y,B⊗···⊗Bτn−1). We show that R is noetherian if and only if some Veronese ring R(m) of R has the form k+I, where I is a left ideal of such a ring B and where I=B at each point p∈Y at which σ has finite order. This allows one to give detailed information about the structure of R and its modules.

Minimal Resolutions and the Homology of Matching and Chessboard Complexes

We generalize work of Lascoux and Jozefiak-Pragacz-Weyman on Betti numbers for minimal free resolutions of ideals generated by 2 × 2 minors of generic matrices and generic symmetric matrices, respectively. Quotients of polynomial rings by these ideals are the classical Segre and quadratic Veronese subalgebras, and we compute the analogous Betti numbers for some natural modules over these Segre and quadratic Veronese subalgebras. Our motivation is two-fold: • We immediately deduce from these results the irreducible decomposition for the symmetric group action on the rational homology of all chessboard complexes and complete graph matching complexes as studied by Bjorner, Lovasz, Vrecica and Zivaljevic. This follows from an old observation on Betti numbers of semigroup modules over semigroup rings described in terms of simplicial complexes. • The class of modules over the Segre rings and quadratic Veronese rings which we consider is closed under the operation of taking canonical modules, and hence exposes a pleasant symmetry inherent in these Betti numbers.