Derived Functors and Hilbert Polynomials

Topological Noetherianity of polynomial functors

We prove that any finite-degree polynomial functor is topologically Noetherian. This theorem is motivated by the recent resolution of Stillman's conjecture and a recent Noetherianity proof for the space of cubics. Via work by Erman-Sam-Snowden, our theorem implies Stillman's conjecture and indeed boundedness of a wider class of invariants of ideals in polynomial rings with a fixed number of generators of prescribed degrees.

Highest weight categories and strict polynomial functors

Highest weight categories are described in terms of standard objects and recollements of abelian categories, working over an arbitrary commutative base ring. Then the highest weight structure for categories of strict polynomial functors is explained, using the theory of Schur and Weyl functors. A consequence is the well-known fact that Schur algebras are quasi-hereditary.

Topological Noetherianity of polynomial functors II: base rings with Noetherian spectrum

In a previous paper, the third author proved that finite-degree polynomial functors over infinite fields are topologically Noetherian. In this paper, we prove that the same holds for polynomial functors from free $R$-modules to finitely generated $R$-modules, for any commutative ring $R$ whose spectrum is Noetherian. As Erman-Sam-Snowden pointed out, when applying this with $R = \mathbb{Z}$ to direct sums of symmetric powers, one of their proofs of a conjecture by Stillman becomes characteristic-independent. Our paper advertises and further develops the beautiful but not so well-known machiner…

Complete intersections with given Hilbert polynomials

The Hilbert polynomial of a homogeneous complete intersection is determined by the degrees of the generators of the defining ideal. The degrees of the generators are not, in general, determined by the Hilbert polynomial -- but sometimes they are. When? We give some general criteria and completely answer the question up to codimension 6.

Sharp upper bounds of the Betti numbers for a given Hilbert polynomial

We show that there exists a saturated graded ideal in a standard graded polynomial ring which has the largest total Betti numbers among all saturated graded ideals for a fixed Hilbert polynomial.

On the functoriality of marked families

The application of methods of computational algebra has recently introduced new tools for the study of Hilbert schemes. The key idea is to define flat families of ideals endowed with a scheme structure whose defining equations can be determined by algorithmic procedures. For this reason, several authors developed new methods, based on the combinatorial properties of Borel-fixed ideals, that allow to associate to each ideal $J$ of this type a scheme $\mathbf{Mf}_{J}$, called $J$-marked scheme. In this paper we provide a solid functorial foundation to marked schemes and show that the algorithmic…

Normal Hilbert Polynomials : A survey

We survey some of the major results about normal Hilbert polynomials of ideals. We discuss a formula due to Lipman for complete ideals in regular local rings of dimension two, theorems of Huneke, Itoh, Huckaba, Marley and Rees in Cohen-Macaulay analytically unramified local rings. We also discuss recent works of Goto-Hong-Mandal and Mandal-Singh-Verma concerning the positivity of the first coefficient of the normal Hilbert polynomial in unmixed analytically unramified local rings. Results of Moral{é}s and Villarreal linking normal Hilbert polynomial of monomial ideals with Ehrhart polynomials …

Witt Vectors, Polynomial Maps, and Real Topological Hochschild Homology

We show that various flavors of Witt vectors are functorial with respect to multiplicative polynomial laws of finite degree. We then deduce that the $p$-typical Witt vectors are functorial in multiplicative polynomial maps of degree at most $p-1$. This extra functoriality allows us to extend the $p$-typical Witt vectors functor from commutative rings to $\mathbb{Z}/2$-Tambara functors, for odd primes $p$. We use these Witt vectors for Tambara functors to describe the components of the dihedral fixed-points of the real topological Hochschild homology spectrum at odd primes.

正特征下多项式函子的理想论非诺特性

Ideal-theoretic non-noetherianity of polynomial functors in positive characteristic

本文利用不变量理论，在正特征域上构造了严格多项式函子范畴中一个有限生成但非诺特的交换代数，从而否定了该范畴中诺特性质的长期猜想。

Hilbert polynomials of the algebras of $SL_2$-invariants

We offer a Maple-procedure for computing of the Hilbert polynomials of the algebras of $SL_2$-invariants

On the sign patterns of the coefficients of Hilbert polynomials

We show that all sign patterns of the coefficients Hilbert polynomials of standard graded $k$-algebras are possible.

自由Tambara函子的分层有限生成

The Levelwise Finite Generation of Free Tambara Functors

研究有限戴德金群\(G\)时自由多项式\(G\)-Tambara函子的分层有限生成，通过确立Tambara函子相关有限性条件的持久性，包括弱希尔伯特基定理来实现。

On generalized Hilbert-Kunz function and multiplicity

Let $(R,\mathfrak m)$ be a local ring of characteristic $p>0$ and $M$ a finitely generated $R$-module. In this note we consider the limit: $\lim_{n\to \infty} \frac{\ell(H^0_{\mathfrak m}(F^n(M)))}{p^{n\dim R}} $ where $F(-)$ is the Peskine-Szpiro functor. A consequence of our main results shows that the limit always exists when $R$ is excellent and has isolated singularity. Furthermore, if $R$ is a complete intersection, then the limit is 0 if and only if the projective dimension of $M$ is less than the Krull dimension of $R$. We exploit this fact to give a quick proof that if $R$ is a comple…

Torsion functors with monomial support

The dependence of torsion functors on their supporting ideals is investigated, especially in the case of monomial ideals of certain subrings of polynomial algebras over not necessarily Noetherian rings. As an application it is shown how flatness of quasicoherent sheaves on toric schemes is related to graded local cohomology.

Hilbert polynomial of the Kimura 3-parameter model

Buczyńska and Wiśniewski showed that for the Jukes Cantor binary model of a 3-valent tree the Hilbert polynomial depends only on the number of leaves of the tree and not on its shape. We ask if this can be generalized to other group-based models. In this paper we consider the Kimura 3-parameter model and show that the generalization of the statement about the Hilbert polynomials to the Kimura 3-parameter model is not possible as the Hilbert polynomial depends on the shape of a 3-valent tree.

Hilbert quasi-polynomial for order domains and application to coding theory

We present an application of Hilbert quasi-polynomials to order domains, allowing the effective check of the second order-domain condition in a direct way. We also provide an improved algorithm for the computation of the related Hilbert quasi-polynomials. This allows to identify order domain codes more easily.