arXiv subjects

Helge Øystein Maakestad

Publications and source records attributed to Helge Øystein Maakestad.

At least 19 recordsLinked to original sources

arXiv2025

A universal characteristic class for vector bundles with a connection

In the paper I introduce a new characteristic class $c(E)$ for a finite rank vector bundle $E$ on an affine scheme $S:=Spec(A)$ - the fundamental class of $E$. The class $c(E)$ is not a characteristic class in the classical sense in the sense that it lives in a pointed cohomology torsor $\operatorname{Ext}^1(L, \operatorname{End}_A(E))$. Most characteristic classes lives in a cohomology group. The pointed cohomology torsor $\operatorname{Ext}^1(L, \operatorname{End}_A(E))$ is a torsor on the abelian group $\operatorname{H}^2(L, Z(\operatorname{End}_A(E)))$ where $Z(\operatorname{End}_A(E))$ is the center of the ring of endomorphisms of $E$ and where the cohomology is the Lie-Rinehart cohomology of the center. The class $c(E)$ is trivial if and only if $E$ has a flat algebraic connection. Hence the class $c(T_S)$ where $T_S$ is the tangent bundle, is an an obstruction for $S$ to be algebraically parallelizable. I use a connection $\nabla$ to define $c(E)$ and I also prove the class $c(E)$ is independent of choice of connection, hence $c(E)$ is an invariant of the vector bundle $E$. The class generalize the Chern class, the Pontryagin class, the Euler class and the Teleman characteristic class. I prove using an explicit example that the class $c(E)$ is stronger than the Chern class and the Euler class. I also give a new proof of a formula for the curvature of a connection $\nabla$ in terms of an idempotent endomorphism $ϕ$ defining $E$. This formula was claimed and proved in a paper put out on the arXiv in 2011, and in this paper I give a new proof that is easier to read. The class may be interesting in the study of the "cancellation problem" in affine algebraic geometry and the problem of giving algebraic formulas for the topological Euler characteristic. I also calculate the algebraic deRham cohomology of the complex two sphere and prove it is infinite dimensional.

math.AG↗

arXiv2024

A refined Weyl character formula for comodules on $\operatorname{GL}_{2,A}$

Let $A$ be any commutative unital ring and let $\operatorname{GL}_{2,A}$ be the general linear group scheme on $A$ of rank $2$. We study the representation theory of $\operatorname{GL}_{2,A}$ and the symmetric powers $\operatorname{Sym}^d(V)$, where $(V, Δ)$ is the standard right comodule on $\operatorname{GL}_{2,A}$. We prove a refined Weyl character formula for $\operatorname{Sym}^d(V)$. There is for any integer $d \geq 1$ a (canonical) refined weight space decomposition $\operatorname{Sym}^d(V) \cong \oplus_i \operatorname{Sym}^d(V)^i$ where each direct summand $\operatorname{Sym}^d(V)^i$ is a comodule on $N \subseteq \operatorname{GL}_{2,A}$. Here $N$ is the schematic normalizer of the diagonal torus $T \subseteq \operatorname{GL}_{2,A}$. We prove a character formula for the direct summands of $\operatorname{Sym}^d(V)$ for any integer $d \geq 1$. This refined Weyl character formula implies the classical Weyl character formula. As a Corollary we get a refined Weyl character formula for the pull back $\operatorname{Sym}^d(V \otimes K)$ as a comodule on $\operatorname{GL}_{2,K}$ where $K$ is any field. We also calculate explicit examples involving the symmetric powers, symmetric tensors and their duals. The refined weight space decomposition exists in general for group schemes such as $\operatorname{GL}_{2,A}$ and $\operatorname{SL}_{2,A}$. The study may have applications to the study of groups $G$ such as $\operatorname{SL}(n,k)$ and $\operatorname{GL}(n,k)$ and quotients $G/H$ where $k$ is an arbitrary field (or a Dedekind domain) and $H \subseteq G$ is a closed subgroup. The refined weight space decomposition of $S_λ(V)$ has a relation with irreducible module over a field of positive characteristic. In an example I prove it recovers the irreducible module $V(λ) \subsetneq S_λ(V)$.

math.AG↗

arXiv2023

Explicit formulas for algebraic connections on ellipsoid surfaces

The aim of this paper is to give a new method to construct explicit formulas for algebraic differential operators of any order on a finitely generated projective module $E$ on a commutative unital ring $A$. We moreover give explicit formulas for algebraic connections on a class of finitely generated projective modules on ellipsoid surfaces. The connections we construct are non-flat with trace of curvature equal to zero. We construct these formulas using an idempotent matrix $M$ defining the module $E$. Such an idempotent matrix $M$ is constructed from a "projective basis" $B$ defining the module $E$. Associated to a projective basis $B$ for $E$ we construct a connection $\nabla_B$. The curvature $R_{\nabla_B}$ of the connection $\nabla_B$ is given by a Lie product: $R_{\nabla_B}(x,y):=[\nabla_B(x)(M), \nabla_B(y)(M)]$ involving the matrix $M$, and this Lie product is non-zero in general. Hence the curvature formula indicates that most projective finite rank modules do not have a flat algebraic connection. We also give an explicit formula for a non-flat algebraic connection on the cotangent bundle $Ω$ of the real 2-sphere. The cotangent bundle $Ω$ is topologically non-trivial and it is not clear if it has a flat algebraic connection. All higher Chern classes in deRham cohomology are zero: $c_i(Ω)=0$ for all $i \geq 1$. We relate the construction to non-abelian extensions and a refined characteristic class $c(Ω)$ introduced in another paper on the subject. The class $c(Ω)$ is defined using the connection $\nabla_B$ but it is independent of choice of connection. The class $c(-)$ lives in a torsor. The methods introduced in the paper prove that the underlying complex manifold of any complex affine regular hypersurface is a Calabi-Yau manifold. This is because its canonical bundle is trivial.

math.AG↗

arXiv2023

Higher order differential operators on projective modules

In this paper we give explicit formulas for higher order differential operators on a finitely generated projective module $E$ on an arbitrary commutative unital ring $A$. We use the differential operators constructed to give a simple formula for the curvature of a classical connection and a connection on a Lie-Rinehart algebra in terms of a "projective basis" $B$ for $E$. A "projective basis" is sometimes referred to as a "dual basis". This gives an explicit formula for the curvature $R_{\nabla_B}$ of a connection $\nabla_B$ on $E$ defined in terms of a projective basis $B$ and an idempotent $ϕ$ for $E$. We also consider the notion of a stratification on the module $E$ induced by a projective basis $B$. It turns out few stratifications are induced by a projective basis.

math.AG↗

arXiv2023

Extensions of Lie algebras of differential operators

The aim of this note is to introduce the notion of a $\operatorname{D}$-Lie algebra and to prove some elementary properties of $\operatorname{D}$-Lie algebras, the category of $\operatorname{D}$-Lie algebras, the category of modules on a $\operatorname{D}$-Lie algebra and extensions of $\operatorname{D}$-Lie algebras. A $\operatorname{D}$-Lie algebra is an $A/k$-Lie-Rinehart algebra equipped with an $A\otimes_k A$-module structure and a canonical central element $D$ and a compatibility property between the $k$-Lie algebra structure and the $A\otimes_k A$-module structure. Several authors have studied non-abelian extensions of Lie algebras, super Lie algebras, Lie algebroids and holomorphic Lie algebroids and we give in this note an explicit constructions of all non-abelian extensions a $\operatorname{D}$-Lie algebra $\tilde{L}$ by an $A$-Lie algebra $(W,[,])$ where $\tilde{L}$ is projective as left $A$-module and $W$ is an $A\otimes_k A$-module with $IW=0$ for $I$ the kernel of the multiplication map. As a corollary we get an explicit construction of all non-abelian extensions of an $A/k$-Lie-Rinehart algebra $(L,α)$ by an $A$-Lie algebra $(W,[,])$ where $L$ is projective as left $A$-module.

math.AG↗

arXiv2022

The enveloping algebra of a Lie algebra of differential operators

The aim of this note is to prove various general properties of a generalization of the full module of first order differential operators on a commutative ring - a $\operatorname{D}$-Lie algebra. A $\operatorname{D}$-Lie algebra $\tilde{L}$ is a Lie-Rinehart algebra over $A/k$ equipped with an $A\otimes_k A$-module structure that is compatible with the Lie-structure. It may be viewed as a simultaneous generalization of a Lie-Rinehart algebra and an Atiyah algebra with additional structure. Given a $\operatorname{D}$-Lie algebra $\tilde{L}$ and an arbitrary connection $(ρ, E)$ we construct the universal ring $\tilde{U}^{\otimes}(\tilde{L},ρ)$ of the connection $(ρ, E)$. The associative unital ring $\tilde{U}^{\otimes}(\tilde{L},ρ)$ is in the case when $A$ is Noetherian and $\tilde{L}$ and $E$ finitely generated $A$-modules, an almost commutative Noetherian sub ring of $\operatorname{Diff}(E)$ - the ring of differential operators on $E$. It is constructed using non-abelian extensions of $\operatorname{D}$-Lie algebras. The non-flat connection $(ρ, E)$ is a finitely generated $ \tilde{U}^{\otimes}(\tilde{L},ρ)$-module, hence we may speak of the characteristic variety $\operatorname{Char}(ρ,E)$ of $(ρ, E)$ in the sense of $D$-modules. We may define the notion of holonomicity for non-flat connections using the universal ring $ \tilde{U}^{\otimes}(\tilde{L},ρ)$. This was previously done for flat connections. We also define cohomology and homology of arbitrary non-flat connections. The cohomology and homology of a non-flat connection $(ρ,E)$ is defined using $\operatorname{Ext}$ and $\operatorname{Tor}$-groups of a non-Noetherian ring $\operatorname{U}$. In the case when the $A$-module $E$ is finitely generated we may always calculate cohomology and homology using a Noetherian quotient of $\operatorname{U}$. This was previously done for flat connections.

math.AG↗

arXiv2020

On operations and characteristic classes

In this paper exterior products are used to define operations and characteristic classes with values in the K-theory of an abelian category with tensor and exterior products. We apply the general construction to define Chern and Segre classes with values in algebraic K-theory and the K-theory of connections.

math.KT↗

arXiv2020

On jet bundles and generalized Verma modules

The aim of this paper is to initiate a study of the jet bundles on the grassmannian $X$ over a field of characteristic zero using higher direct images of $G$-linearized sheaves, Lie theoretic methods, enveloping algebra theoretic methods and generalized Verma modules. We calculate the $P$-module of the dual jet bundle $J^l(L)^*$ and prove it equals the $l$'th piece of the canonical filtration for $H^0(X,L)^*$. We use the results obtained to prove the discriminant of any linear system on any grassmannian is irreducible.

math.AG↗

arXiv2020

On jet bundles and generalized Verma modules II

Let G be a semi simple linear algebraic group over a field of characteristic zero and let V be a finite dimensional irreducible G-module with highest weight vector v. Let P in G be the parabolic subgroup fixing v and let g=Lie(G). We get a canonical filtration of V by P-modules U^k(g)v where U^k(g) is the filtration of the universal enveloping algebra U(g). This filtration was in a previous paper studied in the case where P in G=SL(E) is the subgroup fixing an m-dimensional subspace. The aim of this paper is to use higher direct images of G-linearized sheaves, filtrations of generalized Verma modules and annihilator ideals of highest weight vectors to give a basis for U^k(g) and to compute its dimension in the case where P in SL(E) is the parabolic group fixing a flag in E. We also interpret the filtration U^k(g) in terms of SL(E)-linearized jet bundles on SL(E)/P.

math.AG↗

arXiv2020

On jets, extensions and characteristic classes I

In this paper we give general definitions of non-commutative jets in the local and global situation using square zero extensions and derivations. We study the functors Exank(A, I) where A is any k-algebra and I is any left and right A-module and use this to relate affine non-commutative jets to liftings of modules. We also study the Kodaira-Spencer class KS(L) and relate it to the Atiyah class.

math.AG↗

arXiv2020

Discriminants of morphisms of sheaves

The aim of this paper is to give a unified definition of a large class of discriminants arising in algebraic geometry using the discriminant of a morphism of locally free sheaves. The discriminant of a morphism of locally free sheaves has a geometric definition in terms of grassmannian bundles, tautological sequences and projections and is a simultaneous generalization of the discriminant of a morphism of schemes, the discriminant of a linear system on a smooth projective scheme and the classical discriminant of degree $d$ polynomials. We study the discriminant of a morphism in various situations: The discriminant of a finite morphism of schemes, the discriminant of a linear system on the projective line and the discriminant of a linear system on a flag variety. The main result of the paper is that the discrimiant of any linear system on any flag variety is irreducible.

math.AG↗

arXiv2020

Jet bundles on projective space II

Let G=SL(E) be the special linear algebraic group on E where E is a finite dimensional vector space over a field K of characteristic zero. In this paper we study the canonical filtration of the dual G-module of global sections of a G-linearized invertible sheaf L on the grassmannian G/P where P in G is the parabolic subgroup stabilizing a subspace W in E. We classify the canonical filtration as P-module and as a consequence we recover known formulas on the P-module structure of the jet bundle J(L) on projective space. We study the incidence complex for the invertible sheaf O(d) on the projective line and prove it gives a resolution of the incidence scheme I(O(d)) of O(d). The aim of this study is to apply it to the study of resolutions of ideal sheaves of discriminants of invertible sheaves on grassmannians and flag varieties. We also give an elementary proof of the Cauchy formula. Hence the paper introduce the canonical filtration of an arbitrary irreducible SL(E)-module and initiates a study of the canonical filtration as P-module where P in SL(E) is a parabolic subgroup.

math.AG↗

arXiv2020

On the canonical filtration of an irreducible representation

The aim of this paper is to study the canonical filtration $L(λ)_l$ of an irreducible finite dimensional $\operatorname{SL}(V)$-module $L(λ)$ using the universal enveloping algebra $U(\mathfrak{sl}(V))$ and the annihilator ideal $ann(v)$ of a highest weight vector $v$ in $L(λ)$. We give a basis for $L(λ)_l$ and calculate the dimension of $L(λ)_l$ as a function of $l$. This is done in terms of the universal enveloping algebra of the nilpotent radical of an opposite parabolic sub algebra of the stabilizer Lie algebra of a flag $V_*$ in $V$ with respect to a choice of roots for $\mathfrak{sl}(V)$.

math.RT↗

arXiv2020

On discriminants and incidence resolutions

In this paper we study the incidence complex of an arbitrary morphism of locally free sheaves relative to an arbitrary quasi compact morphism of schemes. We prove it is a local complete intersection in the case when the sheaf morphism is surjective. We construct a complex - the incidence complex - which is a candidate for a resolution of the ideal sheaf of the incidence scheme. When the initial scheme is Cohen-Macaulay we prove the incidence complex is a resolution. We also study the rational points of the incidence scheme and discriminant scheme of any linear system on the projective line over any field of characteristic zero. We use this study to relate the discriminant to the classical discriminant of degree d polynomials.

math.AG↗

arXiv2020

On jets, extensions and characteristic classes II

In this paper we define and study generalized Atiyah classes for quasi coherent sheaves relative to arbitrary morphisms of schemes. We use derivations and quasi coherent sheaves of left and right O-modules to define a generalized first order jet bundle J(E) and a generalized Atiyah sequence for E. The generalized jet bundle J(E) is a left and right module over a sheaf J of associative rings on X. The sheaf J is an extension of O with a sheaf I of two sided ideals of square zero. The Atiyah sequence gives rise to a generalized Atiyah class c(E) with the property that c(E)=0 if and only if the left structure on J(E) is O-isomorphic to the right structure on J(E). We give examples where c(E)=0 and c(E)\neq 0 hence the class c(E) is a non trivial characteristic class.

math.AG↗

arXiv2020

Scalar generalized Verma modules

In this paper we study the scalar generalized Verma module $M$ associated to a character of a parabolic subgroup of $\operatorname{SL}(E)$. Here $E$ is a finite dimensional vector space over an algebraically closed field $K$ of characteristic zero. The Verma module $M$ has a canonical simple quotient $L$ with a canonical filtration $F$. In the case when the quotient $L$ is finite dimensional we use left annihilator ideals in $U(\mathfrak{sl}(E))$ and geometric results on jet bundles to generalize to an algebraically closed field of characteristic zero a classical formula of W. Smoke on the structure of the jet bundle of a line bundle on an arbitrary quotient $\operatorname{SL}(E)/P$ where $P$ is a parabolic subgroup of $\operatorname{SL}(E)$. This formula was originally proved by Smoke in 1967 using analytic techniques.

math.RT↗

arXiv2020

Jet bundles on projective space: New examples

In previous papers it was shown that the left and right O-module structure of the jet bundles on the projective line differed. In this paper we show that similar statements hold for jet bundles on projective space in any dimension. We also consider some classes associated to jet bundles in the algebraic K-theory of projective space.

math.AG↗

arXiv2020

Algebraic connections on projective modules with prescribed curvature

In this paper we generalize classical results on Lie algebras and universal enveloping algebras of Lie algebras to Lie-Rinehart algebras. We define for any Lie-Rinehart algebra $L$ and any cocycle $f$ in $Z^2(L,B)$, a universal enveloping algebra $U(B,L,f)$ with the property that the category of left modules on $U(B,L,f)$ is equivalent to the category of modules with an $L$-connection where the curvature has "type $f$". A connection of curvature "type $f$" is a special case of a non-flat connection. We also study deformations of filtered algebras and prove a relationship between the deformation groupoid $A(\operatorname{Sym}_B^*(L))$ of the Lie-Rinehart algebra $L$ and $H^2(L,B)$. We also give an explicit realization of the category of $L$-connections as a category of left modules on an associative ring $U(L)$. We use the associative ring $U(L)$ to give a definition of cohomology and homology of arbitrary $L$-Connections. In the construction of the ring $U(L)$ we implicitly introduce the notion "D-Lie algebra" for the first time.

math.AG↗