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We prove that the category of (rigidified) Breuil–Kisin–Fargues modules up to isogeny is Tannakian. We then introduce and classify Breuil–Kisin–Fargues modules with complex multiplication mimicking the classical theory for rational Hodge structures. In particular, we compute an avatar of a ‘
$p$
-adic Serre group’.
In this paper we study consequences of the results of Kang et al. [Monoidal categorification of cluster algebras, J. Amer. Math. Soc. 31 (2018), 349–426] on a monoidal categorification of the unipotent quantum coordinate ring
$A_{q}(\mathfrak{n}(w))$
together with the Laurent phenomenon of cluster algebras. We show that if a simple module
$S$
in the category
${\mathcal{C}}_{w}$
strongly commutes with all the cluster variables in a cluster
$[\mathscr{C}]$
, then
$[S]$
is a cluster monomial in
$[\mathscr{C}]$
. If
$S$
strongly commutes with cluster variables except for exactly one cluster variable
$[M_{k}]$
, then
$[S]$
is either a cluster monomial in
$[\mathscr{C}]$
or a cluster monomial in
$\unicode[STIX]{x1D707}_{k}([\mathscr{C}])$
. We give a new proof of the fact that the upper global basis is a common triangular basis (in the sense of Qin [Triangular bases in quantum cluster algebras and monoidal categorification conjectures, Duke Math. 166 (2017), 2337–2442]) of the localization
$\widetilde{A}_{q}(\mathfrak{n}(w))$
of
$A_{q}(\mathfrak{n}(w))$
at the frozen variables. A characterization on the commutativity of a simple module
$S$
with cluster variables in a cluster
$[\mathscr{C}]$
is given in terms of the denominator vector of
$[S]$
with respect to the cluster
$[\mathscr{C}]$
.
We prove that a generic homogeneous polynomial of degree
$d$
is determined, up to a nonzero constant multiplicative factor, by the vector space spanned by its partial derivatives of order
$k$
for
$k\leqslant \frac{d}{2}-1$
.
Let (R, ) be an analytically unramified local ring of positive prime characteristic p. For an ideal I, let I* denote its tight closure. We introduce the tight Hilbert function $$H_I^*\left( n \right) = \Im \left( {R/\left( {{I^n}} \right)*} \right)$$ and the corresponding tight Hilbert polynomial $$P_I^*\left( n \right)$$, where I is an m-primary ideal. It is proved that F-rationality can be detected by the vanishing of the first coefficient of $$P_I^*\left( n \right)$$. We find the tight Hilbert polynomial of certain parameter ideals in hypersurface rings and Stanley-Reisner rings of simplicial complexes.
We study the realization of acyclic cluster algebras as coordinate rings of Coxeter double Bruhat cells in Kac–Moody groups. We prove that all cluster monomials with
$\mathbf{g}$
-vector lying in the doubled Cambrian fan are restrictions of principal generalized minors. As a corollary, cluster algebras of finite and affine type admit a complete and non-recursive description via (ind-)algebraic group representations, in a way similar in spirit to the Caldero–Chapoton description via quiver representations. In type
$A_{1}^{(1)}$
, we further show that elements of several canonical bases (generic, triangular, and theta) which complete the partial basis of cluster monomials are composed entirely of restrictions of minors. The discrepancy among these bases is accounted for by continuous parameters appearing in the classification of irreducible level-zero representations of affine Lie groups. We discuss how our results illuminate certain parallels between the classification of representations of finite-dimensional algebras and of integrable weight representations of Kac–Moody algebras.
A self-map F of an affine space
${\bf A}_k^n $
over a field k is said to be a Keller map if F is given by polynomials F1, …, Fn ∈ k[X1, …, Xn] whose Jacobian determinant lies in
$k\setminus \{0\}$
. We consider char(k) = 0 and assume, as we may, that the Fis vanish at the origin. In this note, we prove that if F is Keller then its base ideal IF = 〈F1, …, Fn〉 is radical (a finite intersection of maximal ideals in this case). We then conjecture that IF = 〈X1, …, Xn〉, which we show to be equivalent to the classical Jacobian Conjecture. In addition, among other remarks, we observe that every complex Keller map admits a well-defined multidimensional global residue function.
We show that the virtual cohomological dimension of a Coxeter group is essentially the regularity of the Stanley–Reisner ring of its nerve. Using this connection between geometric group theory and commutative algebra, as well as techniques from the theory of hyperbolic Coxeter groups, we study the behavior of the Castelnuovo–Mumford regularity of square-free quadratic monomial ideals. We construct examples of such ideals which exhibit arbitrarily high regularity after linear syzygies for arbitrarily many steps. We give a doubly logarithmic bound on the regularity as a function of the number of variables if these ideals are Cohen–Macaulay.
Let R be a Mori domain with complete integral closure
$\widehat R$
, nonzero conductor
$\mathfrak f = (R: \widehat R)$
, and suppose that both v-class groups
${{\cal C}_v}(R)$
and
${{\cal C}_v}(3\widehat R)$
are finite. If
$R \mathfrak f$
is finite, then the elasticity of R is either rational or infinite. If
$R \mathfrak f$
is artinian, then unions of sets of lengths of R are almost arithmetical progressions with the same difference and global bound. We derive our results in the setting of v-noetherian monoids.
We investigate whether the property of having linear quotients is inherited by ideals generated by multigraded shifts of a Borel ideal and a squarefree Borel ideal. We show that the ideal generated by the first multigraded shifts of a Borel ideal has linear quotients, as do the ideal generated by the
$k$
th multigraded shifts of a principal Borel ideal and an equigenerated squarefree Borel ideal for each
$k$
. Furthermore, we show that equigenerated squarefree Borel ideals share the property of being squarefree Borel with the ideals generated by multigraded shifts.
Let
$S$
be a surface,
$G$
a simply connected classical group, and
$G^{\prime }$
the associated adjoint form of the group. We show that the moduli spaces of framed local systems
${\mathcal{X}}_{G^{\prime },S}$
and
${\mathcal{A}}_{G,S}$
, which were constructed by Fock and Goncharov [‘Moduli spaces of local systems and higher Teichmuller theory’, Publ. Math. Inst. Hautes Études Sci.103 (2006), 1–212], have the structure of cluster varieties, and thus together form a cluster ensemble. This simplifies some of the proofs in that paper, and also allows one to quantize higher Teichmuller space, which was previously only possible when
$G$
was of type
$A$
.
A number field K with a ring of integers 𝒪K is called a Pólya field, if the 𝒪K-module of integer-valued polynomials on 𝒪K has a regular basis, or equivalently all its Bhargava factorial ideals are principal [1]. We generalize Leriche's criterion [8] for Pólya-ness of Galois closures of pure cubic fields, to general S3-extensions of ℚ. Also, we prove for a real (resp. imaginary) Pólya S3-extension L of ℚ, at most four (resp. three) primes can be ramified. Moreover, depending on the solvability of unit norm equation over the quadratic subfield of L, we determine when these sharp upper bounds can occur.
A square-free monomial ideal
$I$
of
$k[x_{1},\ldots ,x_{n}]$
is said to be an
$f$
-ideal if the facet complex and non-face complex associated with
$I$
have the same
$f$
-vector. We show that
$I$
is an
$f$
-ideal if and only if its Newton complementary dual
$\widehat{I}$
is also an
$f$
-ideal. Because of this duality, previous results about some classes of
$f$
-ideals can be extended to a much larger class of
$f$
-ideals. An interesting by-product of our work is an alternative formulation of the Kruskal–Katona theorem for
$f$
-vectors of simplicial complexes.
We consider a family of nonlinear rational recurrences of odd order which was introduced by Heideman and Hogan, and recently rediscovered in the theory of Laurent phenomenon algebras (a generalization of cluster algebras). All of these recurrences have the Laurent property, implying that for a particular choice of initial data (all initial values set to 1) they generate an integer sequence. For these particular sequences, Heideman and Hogan gave a direct proof of integrality by showing that the terms of the sequence also satisfy a linear recurrence relation with constant coefficients. Here we present an analogous result for the general solution of each of these recurrences.
For a skew-symmetrizable cluster algebra
${\mathcal{A}}_{t_{0}}$
with principal coefficients at
$t_{0}$
, we prove that each seed
$\unicode[STIX]{x1D6F4}_{t}$
of
${\mathcal{A}}_{t_{0}}$
is uniquely determined by its
$C$
-matrix, which was proposed by Fomin and Zelevinsky (Compos. Math. 143 (2007), 112–164) as a conjecture. Our proof is based on the fact that the positivity of cluster variables and sign coherence of
$c$
-vectors hold for
${\mathcal{A}}_{t_{0}}$
, which was actually verified in Gross et al. (Canonical bases for cluster algebras, J. Amer. Math. Soc. 31(2) (2018), 497–608). Further discussion is provided in the sign-skew-symmetric case so as to obtain a weak version of the conjecture in this general case.
We establish a combinatorial realization of continued fractions as quotients of cardinalities of sets. These sets are sets of perfect matchings of certain graphs, the snake graphs, that appear naturally in the theory of cluster algebras. To a continued fraction
$[a_{1},a_{2},\ldots ,a_{n}]$
we associate a snake graph
${\mathcal{G}}[a_{1},a_{2},\ldots ,a_{n}]$
such that the continued fraction is the quotient of the number of perfect matchings of
${\mathcal{G}}[a_{1},a_{2},\ldots ,a_{n}]$
and
${\mathcal{G}}[a_{2},\ldots ,a_{n}]$
. We also show that snake graphs are in bijection with continued fractions. We then apply this connection between cluster algebras and continued fractions in two directions. First we use results from snake graph calculus to obtain new identities for the continuants of continued fractions. Then we apply the machinery of continued fractions to cluster algebras and obtain explicit direct formulas for quotients of elements of the cluster algebra as continued fractions of Laurent polynomials in the initial variables. Building on this formula, and using classical methods for infinite periodic continued fractions, we also study the asymptotic behavior of quotients of elements of the cluster algebra.
We show that the canonical lift construction for ordinary elliptic curves over perfect fields of characteristic
$p>0$
extends uniquely to arbitrary families of ordinary elliptic curves, even over
$p$
-adic formal schemes. In particular, the universal ordinary elliptic curve has a canonical lift. The existence statement is largely a formal consequence of the universal property of Witt vectors applied to the moduli space of ordinary elliptic curves, at least with enough level structure. As an application, we show how this point of view allows for more formal proofs of recent results of Finotti and Erdoğan.
We construct a norm on the nonzero elements of a Prüfer domain and extend this concept to the set of ideals of a Prüfer domain. These norms are used to study factorization properties Prüfer of domains.
Given a smooth variety
$X$
and an effective Cartier divisor
$D\subset X$
, we show that the cohomological Chow group of 0-cycles on the double of
$X$
along
$D$
has a canonical decomposition in terms of the Chow group of 0-cycles
$\text{CH}_{0}(X)$
and the Chow group of 0-cycles with modulus
$\text{CH}_{0}(X|D)$
on
$X$
. When
$X$
is projective, we construct an Albanese variety with modulus and show that this is the universal regular quotient of
$\text{CH}_{0}(X|D)$
. As a consequence of the above decomposition, we prove the Roitman torsion theorem for the 0-cycles with modulus. We show that
$\text{CH}_{0}(X|D)$
is torsion-free and there is an injective cycle class map
$\text{CH}_{0}(X|D){\hookrightarrow}K_{0}(X,D)$
if
$X$
is affine. For a smooth affine surface
$X$
, this is strengthened to show that
$K_{0}(X,D)$
is an extension of
$\text{CH}_{1}(X|D)$
by
$\text{CH}_{0}(X|D)$
.
We advocate the use of cluster algebras and their
$y$
-variables in the study of hyperbolic 3-manifolds. We study hyperbolic structures on the mapping tori of pseudo-Anosov mapping classes of punctured surfaces, and show that cluster
$y$
-variables naturally give the solutions of the edge-gluing conditions of ideal tetrahedra. We also comment on the completeness of hyperbolic structures.
The article concerns the existence and uniqueness of quantisations of cluster algebras. We prove that cluster algebras with an initial exchange matrix of full rank admit a quantisation in the sense of Berenstein-Zelevinsky and give an explicit generating set to construct all quantisations.