The affine class group of a normal scheme

Generic Bernstein-Sato polynomial on an irreducible affine scheme

Given $p$ polynomials with coefficients in a commutative unitary integral ring $\mathcal{C}$ containing $\mathbb{Q}$, we define the notion of a generic Bernstein-Sato polynomial on an irreducible affine scheme $V \subset \text{Spec}(\mathcal{C})$. We prove the existence of such a non zero rational polynomial which covers and generalizes previous existing results by H. Biosca. When $\mathcal{C}$ is the ring of an algebraic or analytic space, we deduce a stratification of the space of the parameters such that on each stratum, there is a non zero rational polynomial which is a Bernstein-Sato poly…

Groebner bases for families of affine or projective schemes

Let $I$ be an ideal of the polynomial ring $A[x]=A[x_1,...,x_n]$ over the commutative, noetherian ring $A$. Geometrically $I$ defines a family of affine schemes over $\Spec(A)$: For $\p\in\Spec(A)$, the fibre over $\p$ is the closed subscheme of affine space over the residue field $k(\p)$, which is determined by the extension of $I$ under the canonical map $σ_\p:A[x]\to k(\p)[x]$. If $I$ is homogeneous there is an analogous projective setting, but again the ideal defining the fibre is $\sigI$. For a chosen term order this ideal has a unique reduced Gröbner basis which is known to contain consi…

Zur Affinitaet von Hyperflaechenkomplementen in affinen und projektiven Schemata

We study complements of hypersurfaces in schemes with respect to the property being affine.

On cancellation of variables of the form $bT^n-a$ over affine normal domains

In this article we extend a cancellation theorem of D. Wright to the case of affine normal domains. We shall show that if $A$ is an algebra over a Noetherian normal domain $R$ containing a field $k$ and if $A[T ] = R^{[3]}$, then $A = R^{[2]}$ if and only if $A[T]$ has a variable of the form $bT^n - a$ for some $a, b \in A$ with $n \ge 2$ and $ch(k) \nmid n$.

Differential Graded Schemes I: Perfect Resolving Algebras

We introduce perfect resolving algebras and study their fundamental properties. These algebras are basic for our theory of differential graded schemes, as they give rise to affine differential graded schemes. We also introduce etale morphisms. The purpose for studying these, is that they will be used to glue differential graded schemes from affine ones with respect to an etale topology.

特殊边界基概形的重嵌入

Re-Embeddings of Special Border Basis Schemes

本文将近年来开发的仿射概形重嵌入到低维空间的技术结合起来，研究了它们对 MaxDeg、L-shape、单纯及平面等特殊边界基概形的有效性，并致力于使重嵌入高效可计算且验证其与仿射空间同构的条件。

Standard pairs of monomial ideals over non-normal affine semigroups in SageMath

We present stdPairs.spyx, a SageMath library to compute standard pairs of a monomial ideal over a pointed (non-normal) affine semigroup ring. Moreover, stdPairs.spyx provides the associated prime ideals, the corresponding multiplicities, and an irredundant irreducible primary decomposition of a monomial ideal. The library expands on the standardPairs function on Macaulay2 over polynomial rings, and is based on algorithms from arXiv:2005.10968v2 [math.AC]. We also provide methods that allow the outputs from this library to be compatible with the Normaliz package of Macaulay2 and SageMath.

Cohen-Macaulayness of non-affine normal semigroups

In this paper, we study the Cohen-Macaulayness of non-affine normal semigroups in $\mathbb{Z}^n$. We do this by establishing the following four statements each of independent interest: 1) a Lazard type result on $I$-supported elements of $\prod_{\mathbb{N}}\mathbb{Q}_{\geq0}$ for an index set $I\subset\mathbb{N}$; 2) a criterion of regularity of sequences of elements of the ring via projective dimension; 3) a direct limit of polynomial rings with toric maps; 4) any direct summand of rings of the third item is Cohen-Macaulay. To illustrate the idea, we give many examples.

仿射 Hensel 概形与 Hensel 仿射体的结构层上同调非消失性

Non-vanishing of the structure-sheaf cohomology of affine henselian schemes and henselian affinoids

本文证明特征零 Hensel 对上的平坦仿射 Hensel 概形在纤维正维时第一结构层上同调非零，否定 Devadas 混合特征问题，并推广至 Hensel 仿射体。

Affine schemes and topological closures in the Zariski-Riemann space of valuation rings

Let $F$ be a field, let $D$ be a subring of $F$, and let ${\mathfrak{X}}$ be the Zariski-Riemann space of valuation rings containing $D$ and having quotient field $F$. We consider the Zariski, inverse and patch topologies on ${\mathfrak{X}}$ when viewed as a projective limit of projective integral schemes having function field contained in $F$, and we characterize the locally ringed subspaces of ${\mathfrak{X}}$ that are affine schemes.

Non-normal affine monoids

We give a geometric description of the set of holes in a non-normal affine monoid $Q$. The set of holes turns out to be related to the non-trivial graded components of the local cohomology of $k[Q]$. From this, we see how various properties of $k[Q]$ like local normality and Serre's conditions $(R_1)$ and $(S_2)$ are encoded in the geometry of the holes. A combinatorial upper bound for the depth the monoid algebra $k[Q]$ is obtained and some cases where equality holds are identified. We apply this results to seminormal affine monoids.

Measuring the non-Gorenstein locus of Hibi rings and normal affine semigroup rings

The trace of the canonical module of a Cohen-Macaulay ring describes its non-Gorenstein locus. We study the trace of the canonical module of a Segre product of algebras, and we apply our results to compute the non-Gorenstein locus of toric rings. We provide several sufficient and necessary conditions for Hibi rings and normal semigroup rings to be Gorenstein on the punctured spectrum.

On the canonical, fpqc, and finite topologies on affine schemes. The state of the art

This is a systematic study of the behaviour of finite coverings of (affine) schemes with regard to two Grothendieck topologies: the canonical topology and the fpqc topology. The history of the problem takes roots in the foundations of Grothendieck topologies, passes through main strides in Commutative Algebra and leads to new Mathematics up to perfectoids and prisms. We first review the canonical topology of affine schemes and show, keeping with Olivier's lost work, that it coincides with the effective descent topology; covering maps are given by universally injective ring maps, which we discu…

关于三维仿射概形中光滑曲线的集合论完全交

On set-theoretic complete intersections for smooth curves in three-dimensional affine schemes

本文证明了三维诺特环仿射概形中的局部完全交曲线是集合论完全交，并将类似结果推广到四维仿射代数中的曲面及平凡余法丛情形。

Ulrich elements in normal simplicial affine semigroups

Let $H\subseteq \mathbb{N}^d$ be a normal affine semigroup, $R=K[H]$ its semigroup ring over the field $K$ and $ω_R$ its canonical module. The Ulrich elements for $H$ are those $h$ in $H$ such that for the multiplication map by $\mathbf{x}^h$ from $R$ into $ω_R$, the cokernel is an Ulrich module. We say that the ring $R$ is almost Gorenstein if Ulrich elements exist in $H$. For the class of slim semigroups that we introduce, we provide an algebraic criterion for testing the Ulrich propery. When $d=2$, all normal affine semigroups are slim. Here we have a simpler combinatorial description of th…