The module of derivations of a Stanley-Reisner ring

Brumatti, Paulo Roberto; Simis, Aron

doi:10.1090/s0002-9939-1995-1243162-7

Cited by 22 publications

(9 citation statements)

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“…The last section describes the Hasse-Schmidt derivations for schemes defined by monomial ideals. This extends results of Brumatti and Simis [3] on the derivations of such algebras. The localization conjecture for Hasse-Schmidt algebras is then verified for monomial rings.…”

Section: Introductionsupporting

confidence: 88%

Localization of the Hasse-Schmidt Algebra

Traves¹

2003

Can. math. bull.

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Section: Introductionsupporting

confidence: 88%

Localization of the Hasse-Schmidt Algebra

Traves¹

2003

Can. math. bull.

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“…In order to prove the second statement, we recall that A = S/I, for some polynomial algebra S = k[x 1 , ..., x n ] in n variables and some ideal I in S, since A is k-algebra of finite type. So, follows from Lemma 2.1.2 in [3] that Der(S/I) ∼ = {∂ ∈ Der(S) | ∂(I) ⊆ I}/IDer(S), which is a submodule of Der(S)/IDer(S) ∼ = (S/I) n = A n . Since A is Noetherian, it is a finitely generated A-module, and therefore also finitely presented.…”

Section: Equivalent Notions and Obstructionmentioning

confidence: 95%

A brief note on the existence of connections and covariant derivatives on modules

Rojas

Mendoza

2017

Proyecciones (Antofagasta)

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In this note we make a review of the concepts of connection and covariant derivative on modules, in a purely algebraic context. Throughout the text, we consider algebras over an algebraically closed field of characteristic 0 and module will always mean left module. First, we concentrate our attention on a k-algebra A which is commutative, and use the Kähler differentials module, Ω 1 A/k , to define connection (see Subsection 2.1). In this context, it is verified that the existence of connections implies the existence of covariant derivatives (cf. Prop. 2.3), and that every projective module admits a connection (cf. Prop. 2.5). Next (in Section 3), we focus our attention in the discussion of some counterexamples comparing these two notions. In fact, it is known that these two notions are equivalent when we consider regular k-algebras of finite type (see [18], Prop. 4.2). As well as, that there exists a connection on M if, and only if, the Atiyah-Kodaira-Spencer class of M , c(M), is zero (see [17], Prop. 4.3). Finally, we take into account the case where A is (not necessarily commutative) and it is used the bimodule, Ω 1 , of noncommutative differentials introduces by Connes ([9], [10]) in place of Kähler differentials to define a connection. In this case, it is proven that a module admits such connection if, and only if, it is a projective module (see [25], Theorem 5.2).

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“…The starting point of our paper is the following theorem due to Brumatti and Simis in [2], Theorem 2.2.1:…”

Section: Preliminariesmentioning

confidence: 99%