Cotangent spaces and separating re-embeddings

Given an affine algebra [Formula: see text], where [Formula: see text] is a polynomial ring over a field [Formula: see text] and [Formula: see text] is an ideal in [Formula: see text], we study re-embeddings of the affine scheme [Formula: see text], i.e. presentations [Formula: see text] such that [Formula: see text] is a polynomial ring in fewer indeterminates. To find such re-embeddings, we use polynomials [Formula: see text] in the ideal [Formula: see text] which are coherently separating in the sense that they are of the form [Formula: see text] with an indeterminate [Formula: see text] which divides neither a term in the support of [Formula: see text] nor in the support of [Formula: see text] for [Formula: see text]. The possible numbers of such sets of polynomials are shown to be governed by the Gröbner fan of [Formula: see text]. The dimension of the cotangent space of [Formula: see text] at a [Formula: see text]-linear maximal ideal is a lower bound for the embedding dimension, and if we find coherently separating polynomials corresponding to this bound, we know that we have determined the embedding dimension of [Formula: see text] and found an optimal re-embedding.

Improved incrementally affine method for viscoelastic-viscoplastic composite by utilizing an adaptive scheme

We propose a micromechanics-based mean-field homogenization scheme for the viscoelastic-viscoplastic particulate-reinforced composite which is applicable to predict its mechanical response under complex loading conditions. We apply a formulation based on an incrementally affine scheme by using algorithmic tangent operators, while adaptively adjusting the strain of each constituent at every step of the loading process to ensure the consistency of the accumulated strain state and the concentration tensor. We name the method adaptive incrementally affine method. Despite mathematically rigorous derivation, the method has some errors in plastic deformation regime. We propose an assumption for better prediction which dropping out the affine strain and affine stress in adaptive scheme. We show that the adaptive incrementally affine scheme is able to predict the viscoelastic response very well. Still, it is inevitable that the plastic deformation of the composite is initiated earlier than our mean-field theoretical prediction because of the local stress concentration near the particulate. Hence, we propose a yield reduction method that enforces the earlier initiation of the plastic deformation in the matrix phase when obtaining an effective mechanical response. We show that the predictions from the adaptive incrementally affine scheme adjusted with the yield reduction match well with various numerical simulations on particulate-reinforced composites considering viscoelastic, elastic-viscoplastic, and viscoelastic-viscoplastic matrices under uniaxial, cyclic, and bi-axial loadings.

Symmetric products, linear representations and trace identities

We give the equations of the n-th symmetric product $$X^n/S_n$$ X n / S n of a flat affine scheme $$X=\mathrm {Spec}\,A$$ X = Spec A over a commutative ring F. As a consequence, we find a closed immersion into the coarse moduli space parameterizing n-dimensional linear representations of A. This is done by exhibiting an isomorphism between the ring of symmetric tensors over A and the ring generated by the coefficients of the characteristic polynomial of polynomials in commuting generic matrices giving representations of A. Using this we derive an isomorphism of the associated reduced schemes over an infinite field. When the characteristic is zero we show that this isomorphism is an isomorphism of schemes and we express it in term of traces.

Lower-Estimates on the Hochschild (Co)Homological Dimension of Commutative Algebras and Applications to Smooth Affine Schemes and Quasi-Free Algebras

The Hochschild cohomological dimension of any commutative k-algebra is lower-bounded by the least-upper bound of the flat-dimension difference and its global dimension. Our result is used to show that for a smooth affine scheme X satisfying Pointcaré duality, there must exist a vector bundle with section M and suitable n which the module of algebraic differential n-forms Ωn(X,M). Further restricting the notion of smoothness, we use our result to show that most k-algebras fail to be smooth in the quasi-free sense. This consequence, extends the currently known results, which are restricted to the case where k=C.

Quasi-Affineness and the 1-Resolution Property

We prove that, under mild hypothesis, every normal algebraic space that satisfies the $1$-resolution property is quasi-affine. More generally, we show that for algebraic stacks satisfying similar hypotheses, the 1-resolution property guarantees the existence of a finite flat cover by a quasi-affine scheme.

The test function conjecture for local models of Weil-restricted groups

We prove the test function conjecture of Kottwitz and the first named author for local models of Shimura varieties with parahoric level structure attached to Weil-restricted groups, as defined by B. Levin. Our result covers the (modified) local models attached to all connected reductive groups over $p$-adic local fields with $p\geqslant 5$. In addition, we give a self-contained study of relative affine Grassmannians and loop groups formed using general relative effective Cartier divisors in a relative curve over an arbitrary Noetherian affine scheme.

On middle cohomology of special Artin–Schreier varieties and finite Heisenberg groups

We introduce a certain Artin–Schreier scheme over a finite field associated to a pair of coprime integers {(m,n)} with {n\geq 3} divisible by the characteristic of the base field, and study the middle étale cohomology group of it. If m is even, the variety admits actions of some finite Heisenberg groups. We study the middle cohomology as representations of the Heisenberg groups. If m is odd, we compute the Frobenius eigenvalues of it concretely. This affine scheme comes from the reduction of a certain affinoid in a Lubin–Tate space.

Representations of the fundamental group and the torsor of deformations. Local study

This chapter focuses on representations of the fundamental group and the torsor of deformations. It considers the case of an affine scheme of a particular type, qualified also as small by Faltings. It introduces the notion of Dolbeault generalized representation and the companion notion of solvable Higgs module, and then constructs a natural equivalence between these two categories. It proves that this approach generalizes simultaneously Faltings' construction for small generalized representations and Hyodo's theory of p-adic variations of Hodge–Tate structures. The discussion covers the relevant notation and conventions, results on continuous cohomology of profinite groups, objects with group actions, logarithmic geometry lexicon, Faltings' almost purity theorem, Faltings extension, Galois cohomology, Fontaine p-adic infinitesimal thickenings, Higgs–Tate torsors and algebras, Dolbeault representations, and small representations. The chapter also describes the descent of small representations and applications and concludes with an analysis of Hodge–Tate representations.

Representations of the fundamental group and the torsor of deformations. An overview

This chapter provides an overview of a new approach to the p-adic Simpson correspondence, focusing on representations of the fundamental group and the torsor of deformations. The discussion covers the notation and conventions, small generalized representations, the torsor of deformations, Faltings ringed topos, and Dolbeault modules. The chapter begins with a short aside on small generalized representations in the affine case, which will be used as intermediary for the study of Dolbeault representations. It then introduces the notion of generalized Dolbeault representation for a small affine scheme and the companion notion of solvable Higgs module, and constructs a natural equivalence between these two categories. It establishes links between these notions and Faltings smallness conditions and relates this to Hyodo's theory. It also describes the Higgs–Tate algebras and concludes with an analysis of the logical links for a Higgs bundle, between smallness and solvability.

Macaulay-like marked bases

We define marked sets and bases over a quasi-stable ideal [Formula: see text] in a polynomial ring on a Noetherian [Formula: see text]-algebra, with [Formula: see text] a field of any characteristic. The involved polynomials may be non-homogeneous, but their degree is bounded from above by the maximum among the degrees of the terms in the Pommaret basis of [Formula: see text] and a given integer [Formula: see text]. Due to the combinatorial properties of quasi-stable ideals, these bases behave well with respect to homogenization, similarly to Macaulay bases. We prove that the family of marked bases over a given quasi-stable ideal has an affine scheme structure, is flat and, for large enough [Formula: see text], is an open subset of a Hilbert scheme. Our main results lead to algorithms that explicitly construct such a family. We compare our method with similar ones and give some complexity results.