Truncated Polynomial Algebras Research Articles

Derived categories of coherent sheaves on some zero-dimensional schemes

Let R N = k [ y 1 , … , y N ] / ( y 1 , … , y N ) 2 be the truncated polynomial algebra . Geometrically, it is the algebra of functions on the second infinitesimal neighborhood of a closed point in N -dimensional affine space . In this note we study D b ( R N − mod ) , the bounded derived category of finitely generated R N -modules. We show that for N ≥ 2 the lattice of triangulated subcategories in D b ( R N − mod ) has a rich structure (which is probably wild), in contrast to the case of zero-dimensional complete intersections. Our homological methods produce some applications to universal localizations of free associative algebras . These applications are based on a relation between triangulated subcategories in D b ( R N − mod ) and universal localizations of a free graded associative algebra in N variables.

Homological stability and densities of generalized configuration spaces

We prove that the factorization homologies of a scheme with coefficients in truncated polynomial algebras compute the cohomologies of its generalized configuration spaces. Using Koszul duality between commutative algebras and Lie algebras, we obtain new expressions for the cohomologies of the latter. As a consequence, we obtain a uniform and conceptual approach for treating homological stability, homological densities, and arithmetic densities of generalized configuration spaces. Our results categorify, generalize, and in fact provide a conceptual understanding of the coincidences appearing in the work of Farb--Wolfson--Wood. Our computation of the stable homological densities also yields rational homotopy types which answer a question posed by Vakil--Wood. Our approach hinges on the study of homological stability of cohomological Chevalley complexes, which is of independent interest.

REAL K-COHOMOLOGY OF COMPLEX PROJECTIVE SPACES

In this paper, we determine the structure of KO-cohomology of complex projective space P l and its product space P l × P m as algebras over the coefficient ring KO ∗ . We also give a description of the map KO ∗ ( P l+m ) → KO ∗ ( P l × P m ) induced by the map that classifies the tensor product of the canonical line bundles and show that its image is not contained in the image of the cross product KO ∗ ( P l ) ⊗KO∗ KO ∗ ( P m ) → KO ∗ ( P l × P m ) to see that non-existence of the formal group structure on KO ∗ ( P ∞ ). Introduction A commutative ring spectrum E is said to be complex oriented if an element x of the reduced E-cohomology of the infinite dimensional complex projective space CP ∞ is given such that x maps to a generator of the reduced E-cohomology of 1-dimensional complex projective space CP 1 ((2)). We call such an element x a complex orientation of E. On the other hand, if E-homology E∗E of E is a flat over the coefficient ring E∗, E∗E has a structure of a Hopf algebroid and E-homology theory takes values in the category of E∗E-comodule, in other words, the category of representations of the groupoid represented by the affine groupoid scheme represented by E∗E ((1)). If E is a complex oriented ring spectrum, the E-cohomology of the complex projective space is just a truncated polynomial algebra over E∗ and it is shown that E-homology E∗E of E is a flat over E∗. Moreover the product structure of CP ∞ gives a one dimensional formal group law over E ∗ ((5)) which closely relates with the structure of the Hopf algebroid ((2)). The complex K-theory is one of the most basic examples of complex oriented cohomology theories. However, KO-spectrum representing the real K-theory is one of a few well-known examples of spectra E without any complex orientation such that E∗E is flat over E∗ ((2), (7)). In fact, we see that KO-spectrum does not have any complex orientation by showing that the Atiyah-Hirzebruch spectral sequence converging to KO ∗ (CP l ) has a non-trivial

On the K-theory of truncated polynomial algebras, revisited

We revisit the computation, due to Hesselholt and Madsen, of the K-theory of truncated polynomial algebras for perfect fields of positive characteristic. The resulting K-groups are expressed in terms of big Witt vectors of the field. The original proof relied on an understanding of cyclic polytopes in order to determine the genuine equivariant homotopy type of the cyclic bar construction for a suitable monoid. Using the Nikolaus-Scholze framework for topological cyclic homology we achieve the same result using only the homology of said cyclic bar construction, as well as the action of Connes' operator.

Character Formulas for a Class of Simple Restricted Modules over the Simple Lie Superalgebras of Witt Type

Let F be an algebraically closed field of prime characteristic, and W(m, n, 1) be the simple restricted Lie superalgebra of Witt type over F, which is the Lie superalgebra of superderivations of the superalgebra $$\mathfrak{A}(m;1)\otimes\wedge(n)$$, where $$\mathfrak{A}(m;1)$$ is the truncated polynomial algebra with m indeterminants and ∧(n) is the Grassmann algebra with n indeterminants. In this paper, the author determines the character formulas for a class of simple restricted modules of W(m, n, 1) with atypical weights of type I.

Twisted Poisson Homology of Truncated Polynomial Algebras in Four Variables

In this paper, we study the twisted Poisson homology of truncated polynomials algebra A in four variables, and we calculate exactly the dimension of i-th (i = 1, 2, 3, 4) twisted Poisson homology group over A by the induction on the length. The calculation methods provided in this paper can also solve truncated polynomials algebra in a few variables.

Poisson (Co)homology of a Class of Frobenius Poisson Algbras

In this paper, we study the truncated polynomial algebra L in n variables, and discuss the following four problems in detail: 1) Homology complex and homology group of Poisson algebra L; 2) Given a new Poisson bracket by calculation modular derivation of Frobenius Poisson algebra; 3) Calculate the twisted homology group of Poisson algebra L; 4) Verify the theorem of twisted Poincaré duality between twisted Poisson homology and Poisson Cohomology.

On cohomology of a class of nonclassical restricted simple Lie algebras

In this note, we study the cohomology of the nonclassical restricted Lie algebras [Formula: see text], [Formula: see text] over a field [Formula: see text] of characteristic [Formula: see text], which are by definition the Lie algebras of derivations on the truncated polynomial algebras [Formula: see text] (called the Jacobson–Witt algebras), and simple unless [Formula: see text] and [Formula: see text]. We show a vanishing theorem for the cohomology of [Formula: see text] with coefficients in the trivial module [Formula: see text], which says that [Formula: see text] for all [Formula: see text]. Moreover, we compute the cohomology of [Formula: see text] with coefficients in its defining truncated polynomial algebra under a certain assumption on the characteristic of the ground field.

Smooth manifolds with prescribed rational cohomology ring

The Hirzebruch signature formula provides an obstruction to the following realization question: given a rational Poincare duality algebra \({\mathcal {A}}\), does there exist a manifold M such that \(H^*(M;\mathbb {Q})={\mathcal {A}}\)? When \({\mathcal {A}}\) is the truncated polynomial algebra \(\mathbb {Q}[x]/\langle x^3\rangle \), we prove there exists a realizing closed smooth manifold \(M^n\) only if \(n=8(2^a+2^b)\). We also eliminate any existence between dimension 32 and 128. For \(n=32\), we show that such a realizing manifold does not admit a Spin structure, and therefore is not 2-connected. In the case that \({\mathcal {A}}=\mathbb {Q}[x]/\langle x^{m+1}\rangle , |x|=8\), we apply the rational surgery realization theorem to conclude that a rational octonionic projective space exists for m odd. Similar technique is applied to study if the Milnor \(E_8\) manifold has the rational homotopy type of a smooth manifold. The “Appendix” presents a recursive algorithm for efficiently computing the coefficients of the \({\mathcal {L}}\)-polynomials, which arise in the signature formula.

On the algebraic K-theory of truncated polynomial algebras in several variables

AbstractWe consider the algebraic K-theory of a truncated polynomial algebra in several commuting variables, . This naturally leads to a new generalization of the big Witt vectors. If k is a perfect field of positive characteristic we describe the K-theory computation in terms of a cube of these Witt vectors on ℕn. If the characteristic of k does not divide any of the ai we compute the K-groups explicitly. We also compute the K-groups modulo torsion for k = ℤ.To understand this K-theory spectrum we use the cyclotomic trace map to topological cyclic homology, and write as the iterated homotopy cofiber of an n-cube of spectra, each of which is easier to understand.

Comparison morphisms and the Hochschild cohomology ring of truncated quiver algebras

A main contribution of this paper is the explicit construction of comparison morphisms between the standard bar resolution and Bardzell's minimal resolution for truncated quiver algebras over arbitrary fields (TQA's). As a direct application we describe explicitly the Yoneda product and derive several results on the structure of the cohomology ring of TQA's over a field of characteristic zero. For instance, we show that the product of odd degree cohomology classes is always zero. We prove that TQA's associated with quivers with no cycles or with neither sinks nor sources have trivial cohomology rings. On the other side we exhibit a fundamental example of a TQA with nontrivial cohomology ring. Finally, for truncated polynomial algebras in one variable, we construct explicit cohomology classes in the bar resolution and give a full description of their cohomology ring.

Poisson (co)homology of truncated polynomial algebras in two variables

We study the Poisson (co)homology of the algebra of truncated polynomials in two variables viewed as the semi-classical limit of a quantum complete intersection studied by Bergh and Erdmann. We show in particular that the Poisson cohomology ring of such a Poisson algebra is isomorphic to the Hochschild cohomology ring of the corresponding quantum complete intersection. To cite this article: S. Launois, L. Richard, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

On theK-theory of truncated polynomial algebras over the integers

We show that the K_{2i}(Z[x]/(x^m),(x)) is finite of order (mi)!(i!)^{m-2} and that K_{2i+1}(Z[x]/(x^m),(x)) is free abelian of rank m-1. This is accomplished by showing that the equivariant homotopy groups of the topological Hochschild spectrum THH(Z) are finite, in odd degrees, and free abelian, in even degrees, and by evaluating their orders and ranks, respectively.

A remark on rank varieties for a class of local algebras

We answer a question raised in [9] by showing the equality of two different definitions of rank variety for finitely generated modules over truncated polynomial algebras. We do this by establishing an isomorphism of algebras used in the two definitions of rank variety.

The Bousfield lattice for truncated polynomial algebras

The global structure of the unbounded derived category of a truncated polynomial ring on countably many generators is investigated, via its Bousfield lattice. The Bousfield lattice is shown to have cardinality larger than that of the real numbers, and objects with large tensor-nilpotence height are constructed.

The tower ofK-theory of truncated polynomial algebras

Let A a regular F_p-algebra. The relative K-groups K_q(A[x]/(x^m),(x)) and the Nil-groups Nil_q(A[x]/(x^m)) were evaluated earlier by the author and Madsen in terms of the big de Rham-Witt groups of the ring A. In this paper, we evaluate the maps of relative K-groups and Nil-groups induced by the canonical projection from A[x]/(x^m) to A[x]/(x^n). The result depends strongly on the prime p. It generalizes work of Stienstra on the groups in degrees 2 and 3.

Rank varieties for a class of finite-dimensional local algebras

We develop a rank variety for finite-dimensional modules over a certain class of finite-dimensional local k-algebras, Aq,mn. Included in this class are the truncated polynomial algebras k[X1,…,Xm]/(Xin), with k an algebraically closed field and char(k) arbitrary. We prove that these varieties characterise projectivity of modules (Dade’s lemma) and examine the implications for the tree class of the stable Auslander–Reiten quiver. We also extend our rank varieties to infinitely generated modules and verify Dade’s lemma in this context.

The Avrunin and Scott theorem and a truncated polynomial algebra

We prove an analogue of a theorem of Avrunin and Scott for truncated polynomial algebras Λ m : = k [ X 1 , … , X m ] / ( X i 2 ) over an algebraically closed field of arbitrary characteristic. The Avrunin and Scott theorem relates the support variety for a finite-dimensional kE-module to its rank variety (where char ( k ) = p and E is an elementary abelian p-group). The analogue of the Avrunin and Scott theorem relates the support variety for a finite-dimensional Λ m -module (using Hochschild cohomology) to its rank variety (developed in [K. Erdmann, M. Holloway, Rank varieties and projectivity for a class of local algebras, Math. Z. 247 (2004) 441–460] using Clifford algebras). Along the way to proving our main result we provide a new proof of the Avrunin and Scott theorem for elementary abelian p-group algebras which we are then able to generalise to the setting of Λ m -algebras.

Duan's fixed point theorem: Proof and generalization

Let be an H-space of the homotopy type of a connected, finite CW-complex, any map and the th power map. Duan proved that has a fixed point if . We give a new, short and elementary proof of this. We then use rational homotopy to generalize to spaces whose rational cohomology is the tensor product of an exterior algebra on odd dimensional generators with the tensor product of truncated polynomial algebras on even dimensional generators. The role of the power map is played by a -structure as defined by Hemmi-Morisugi-Ooshima. The conclusion is that and each has a fixed point.

Cyclic polytopes and the $K$ -theory of truncated polynomial algebras

This paper calculates the relative algebraic K -theory K∗(k [x ]/(x n ), (x )) of a truncated polynomial algebra over a perfect field k of positive characteristic p. Since the ideal generated by x is nilpotent, we can apply McCarthy’s theorem: the relative algebraic K -theory is isomorphic to the relative topological cyclic homology, [Mc], and it is the latter groups we actually evaluate. The result is best expressed in terms of big Witt vectors. Let Wm (k ) denote the big Witt vectors in k of length m , i.e. the multiplicative group Wm (k ) = (1 + xk [[x ]])×/(1 + x k [[x ]])×,