We introduce the notion of functionally compact sets into the theory of nonlinear generalized functions in the sense of Colombeau. The motivation behind our construction is to transfer, as far as possible, properties enjoyed by standard smooth functions on compact sets into the framework of generalized functions. Based on this concept, we introduce spaces of compactly supported generalized smooth functions that are close analogues to the test function spaces of distribution theory. We then develop the topological and functional–analytic foundations of these spaces.

Let $\mathbb{K}$ be a field and S = ${\mathbb{K}}$ [x 1, . . ., xn ] be the polynomial ring in n variables over the field $\mathbb{K}$ . For every monomial ideal I ⊂ S, we provide a recursive formula to determine a lower bound for the Stanley depth of S/I. We use this formula to prove the inequality sdepth(S/I) ≥ size(I) for a particular class of monomial ideals.

We show that any Kawamata log terminal del Pezzo surface over an algebraically closed field of large characteristic is globally $F$ -regular or it admits a log resolution which lifts to characteristic zero. As a consequence, we prove the Kawamata–Viehweg vanishing theorem for klt del Pezzo surfaces of large characteristic.

To a pair $P$ and $Q$ of finite posets we attach the toric ring $K[P,Q]$ whose generators are in bijection to the isotone maps from $P$ to $Q$ . This class of algebras, called isotonian, are natural generalizations of the so-called Hibi rings. We determine the Krull dimension of these algebras and for particular classes of posets $P$ and $Q$ we show that $K[P,Q]$ is normal and that their defining ideal admits a quadratic Gröbner basis.

Let $S|_{R}$ be a groupoid Galois extension with Galois groupoid $G$ such that $E_{g}^{G_{r(g)}}\subseteq C1_{g}$ , for all $g\in G$ , where $C$ is the centre of $S$ , $G_{r(g)}$ is the principal group associated to $r(g)$ and $\{E_{g}\}_{g\in G}$ are the ideals of $S$ . We give a complete characterisation in terms of a partial isomorphism groupoid for such extensions, showing that $G=\dot{\bigcup }_{g\in G}\text{Isom}_{R}(E_{g^{-1}},E_{g})$ if and only if $E_{g}$ is a connected commutative algebra or $E_{g}=E_{g}^{G_{r(g)}}\oplus E_{g}^{G_{r(g)}}$ , where $E_{g}^{G_{r(g)}}$ is connected, for all $g\in G$ .

If $(G,V)$ is a polar representation with Cartan subspace $\mathfrak{c}$ and Weyl group $W$ , it is shown that there is a natural morphism of Poisson schemes $\mathfrak{c}\oplus \mathfrak{c}^{\ast }/W\rightarrow V\oplus V^{\ast }/\!\!/\!\!/G$ . This morphism is conjectured to be an isomorphism of the underlying reduced varieties if $(G,V)$ is visible. The conjecture is proved for visible stable locally free polar representations and some other examples.

Let (A, ${\mathfrak{m}$ ) be a Cohen–Macaulay local ring of dimension d and let I ⊆ J be two ${\mathfrak{m}$ -primary ideals with I a reduction of J. For i = 0,. . .,d, let e i J (A) (e i I (A)) be the ith Hilbert coefficient of J (I), respectively. We call the number c i (I, J) = e i J (A) − e i I (A) the ith relative Hilbert coefficient of J with respect to I. If G I (A) is Cohen–Macaulay, then c i (I, J) satisfy various constraints. We also show that vanishing of some c i (I, J) has strong implications on depth G J n (A) for n ≫ 0.

Asymptotic triangulations can be viewed as limits of triangulations under the action of the mapping class group. In the case of the annulus, such triangulations have been introduced in K. Baur and G. Dupont (Compactifying exchange graphs: Annuli and tubes, Ann. Comb.3(18) (2014), 797–839). We construct an alternative method of obtaining these asymptotic triangulations using Coxeter transformations. This provides us with an algebraic and combinatorial framework for studying these limits via the associated quivers.

This paper concerns a generalization of the Rees algebra of ideals due to Eisenbud, Huneke and Ulrich that works for any finitely generated module over a noetherian ring. Their definition is in terms of maps to free modules. We give an intrinsic definition using divided powers.

This work concerns the linearity defect of a module $M$ over a Noetherian local ring $R$ , introduced by Herzog and Iyengar in 2005, and denoted $\text{ld}_{R}M$ . Roughly speaking, $\text{ld}_{R}M$ is the homological degree beyond which the minimal free resolution of $M$ is linear. It is proved that for any ideal $I$ in a regular local ring $R$ and for any finitely generated $R$ -module $M$ , each of the sequences $(\text{ld}_{R}(I^{n}M))_{n}$ and $(\text{ld}_{R}(M/I^{n}M))_{n}$ is eventually constant. The first statement follows from a more general result about the eventual constancy of the sequence $(\text{ld}_{R}C_{n})_{n}$ where $C$ is a finitely generated graded module over a standard graded algebra over $R$ .

We prove the Green–Lazarsfeld secant conjecture [Green and Lazarsfeld, On the projective normality of complete linear series on an algebraic curve, Invent. Math. 83 (1986), 73–90; Conjecture (3.4)] for extremal line bundles on curves of arbitrary gonality, subject to explicit genericity assumptions.

In this paper, we give an explicit realization of the universal SL2-representation rings of free groups by using ‘the ring of component functions’ of SL(2, ℂ)-representations of free groups. We introduce a descending filtration of the ring, and determine the structure of its graded quotients. Then we study the natural action of the automorphism group of a free group on the graded quotients, and introduce a generalized Johnson homomorphism. In the latter part of this paper, we investigate some properties of these homomorphisms from a viewpoint of twisted cohomologies of the automorphism group of a free group.

In this paper we study certain sheaves of $p$-adically complete rings of differential operators on semistable models of the projective line over the ring of integers in a finite extension $L$ of $\mathbb{Q}_{p}$. The global sections of these sheaves can be identified with (central reductions of) analytic distribution algebras of wide open congruence subgroups. It is shown that the global sections functor furnishes an equivalence between the categories of coherent module sheaves and finitely presented modules over the distribution algebras. Using the work of M. Emerton, we then describe admissible representations of $\text{GL}_{2}(L)$ in terms of sheaves on the projective limit of these formal schemes. As an application, we show that representations coming from certain equivariant line bundles on Drinfeld’s first étale covering of the $p$-adic upper half plane are admissible.

We study the question of which Henselian fields admit definable Henselian valuations (with or without parameters). We show that every field that admits a Henselian valuation with non-divisible value group admits a parameter-definable (non-trivial) Henselian valuation. In equicharacteristic 0, we give a complete characterization of Henselian fields admitting a parameter-definable (non-trivial) Henselian valuation. We also obtain partial characterization results of fields admitting -definable (non-trivial) Henselian valuations. We then draw some Galois-theoretic conclusions from our results.

We give a new characterization, in the equicharacteristic case, of Teter rings by using Macaulay inverse systems. We extend the previous characterizations due to Teter, to Huneke and Vraciu and to Ananthnarayan et al., to any characteristic of the ground field and remove the hypothesis on the socle ideal. We construct and describe the variety parametrizing Teter covers and we show how to check if an Artin ring is Teter. If this is the case, we show how to compute a Teter cover.

We transpose the parametric geometry of numbers, recently created by Schmidt and Summerer, to fields of rational functions in one variable and analyze, in that context, the problem of simultaneous approximation to exponential functions.

For every integer $k\geqslant 3$ we construct a $k$ -gonal curve $C$ along with a very ample divisor of degree $2g+k-1$ (where $g$ is the genus of $C$ ) to which the vanishing statement from the Green–Lazarsfeld gonality conjecture does not apply.

We give a descent result for formal smoothness having interesting applications: we deduce that quasiexcellence descends along flat local homomorphisms of finite type, we greatly improve Kunz’s characterization of regular local rings by means of the Frobenius homomorphisms as well as André and Radu relativization of this result, etc. In the second part of the paper, we study a similar question for the complete intersection property instead of formal smoothness, giving also some applications.

The $a$ -invariant, the $F$ -pure threshold, and the diagonal $F$ -threshold are three important invariants of a graded $K$ -algebra. Hirose, Watanabe, and Yoshida have conjectured relations among these invariants for strongly $F$ -regular rings. In this article, we prove that these relations hold only assuming that the algebra is $F$ -pure. In addition, we present an interpretation of the $a$ -invariant for $F$ -pure Gorenstein graded $K$ -algebras in terms of regular sequences that preserve $F$ -purity. This result is in the spirit of Bertini theorems for projective varieties. Moreover, we show connections with projective dimension, Castelnuovo–Mumford regularity, and Serre’s condition $S_{k}$ . We also present analogous results and questions in characteristic zero.

We prove that the generalized Hilbert–Kunz function of a graded module $M$ over a two-dimensional standard graded normal $K$ -domain over an algebraically closed field $K$ of prime characteristic $p$ has the form $gHK(M,q)=e_{gHK}(M)q^{2}+\unicode[STIX]{x1D6FE}(q)$ , with rational generalized Hilbert–Kunz multiplicity $e_{gHK}(M)$ and a bounded function $\unicode[STIX]{x1D6FE}(q)$ . Moreover, we prove that if $R$ is a $\mathbb{Z}$ -algebra, the limit for $p\rightarrow +\infty$ of the generalized Hilbert–Kunz multiplicity $e_{gHK}^{R_{p}}(M_{p})$ over the fibers $R_{p}$ exists, and it is a rational number.