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Homological dimension formulas for trivial extension algebras
Rings and Algebras · Submitted 4 Oct 2017
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The main application of the formulas is to give a necessary and sufficient condition for $A$ to be an Iwanaga-Gorenstein algebra.
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The abstract, sentence by sentence
Let $A= \Lambda \oplus C$ be a trivial extension algebra. The aim of this paper is to establish formulas for the projective dimension and the injective dimension for a certain class of $A$-modules which is expressed by using the derived functors $- \otimes^{\mathbb{L}}_{\Lambda}C$ and $\mathbb{R}\text{Hom}_{\Lambda}(C, -)$. Consequently, we obtain formulas for the global dimension of $A$, which gives a modern expression of the classical formula for the global dimension by Palmer-Roos and L\"ofwall that is written in complicated classical derived functors. The main application of the formulas is to give a necessary and sufficient condition for $A$ to be an Iwanaga-Gorenstein algebra. We also give a description of the kernel $\text{Ker} \varpi$ of the canonical functor $\varpi: \mathsf{D}^{\mathrm{b}}(\text{mod} \Lambda) \to \text{Sing}^{\mathbb{Z}} A$ in the case $\text{pd} C < \infty$.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let $A= \Lambda \oplus C$ be a trivial extension algebra. | background 0.99 | 0.15 | 0.02 | asserted 0.94 |
| s1 | The aim of this paper is to establish formulas for the projective dimension and the injective dimension for a certain class of $A$-modules which is expressed by using the derived functors $- \otimes^{\mathbb{L}}_{\Lambda}C$ and $\mathbb{R}\text{Hom}_{\Lambda}(C, -)$. | contribution 0.96 | 0.56 | 0.12 | asserted 0.91 |
| s2 | Consequently, we obtain formulas for the global dimension of $A$, which gives a modern expression of the classical formula for the global dimension by Palmer-Roos and L\"ofwall that is written in complicated classical derived functors. | result 0.88 | 0.90 | 0.20 | asserted 1.00 |
| s3 | The main application of the formulas is to give a necessary and sufficient condition for $A$ to be an Iwanaga-Gorenstein algebra. | result 0.58 | 0.94 | 0.52 | asserted 1.00 |
| s4 | We also give a description of the kernel $\text{Ker} \varpi$ of the canonical functor $\varpi: \mathsf{D}^{\mathrm{b}}(\text{mod} \Lambda) \to \text{Sing}^{\mathbb{Z}} A$ in the case $\text{pd} C < \infty$. | result 0.70 | 0.95 | 0.05 | asserted 1.00 |
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The paper
Original abstract
Let $A= \Lambda \oplus C$ be a trivial extension algebra. The aim of this paper is to establish formulas for the projective dimension and the injective dimension for a certain class of $A$-modules which is expressed by using the derived functors $- \otimes^{\mathbb{L}}_{\Lambda}C$ and $\mathbb{R}\text{Hom}_{\Lambda}(C, -)$. Consequently, we obtain formulas for the global dimension of $A$, which gives a modern expression of the classical formula for the global dimension by Palmer-Roos and L\"ofwall that is written in complicated classical derived functors. The main application of the formulas is to give a necessary and sufficient condition for $A$ to be an Iwanaga-Gorenstein algebra. We also give a description of the kernel $\text{Ker} \varpi$ of the canonical functor $\varpi: \mathsf{D}^{\mathrm{b}}(\text{mod} \Lambda) \to \text{Sing}^{\mathbb{Z}} A$ in the case $\text{pd} C < \infty$.
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How the literature uses it
3 arXiv papers cite this in 38 sentences
Papers citing it · 21 sentences without a confident label
Uses its method or tool 2 papers
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Then, substituting $P' =A(i)$ in the exact sequence (4-5) of [ref] , we obtain an exact triangle in $(Λ^)$ for $i ∈$ [equation] The right most term belongs to $^(Λ^)$.
On finitely graded Iwanaga-Gorenstein algebras and the stable categories of their (graded) Cohen-Macaulay modules · Locally perfect complexes · 0.93
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Our method relies on decompositions of complexes of graded injective or projective modules established in [ref] .
Happel's functor and homologically well-graded Iwanaga-Gorenstein algebras · Results · 1.00
Compares with it 1 paper
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Therefore the proposition above is the same with [ref] .
On finitely graded Iwanaga-Gorenstein algebras and the stable categories of their (graded) Cohen-Macaulay modules · The asid conditions and the asid numbers · 0.99
Cites it as support 2 papers
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It follows from [ref] that $^(Λ) ⊂^(Λ)$.
On finitely graded Iwanaga-Gorenstein algebras and the stable categories of their (graded) Cohen-Macaulay modules · Locally perfect complexes · 0.98
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It follows from [ref] that $_∇A ΔA < ∞$.
Happel's functor and homologically well-graded Iwanaga-Gorenstein algebras · Existence of a generator in $A$ · 1.00
Cites it as background 3 papers
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We recall the quasi-Veronese algebra construction and the relationship with the Beilinson algebra $∇A$ defined in [ref] of a graded algebra from [ref] (see also [ref] ).
Happel's functor and homologically well-graded Iwanaga-Gorenstein algebras · Quasi-Veronese algebra construction · 0.95
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In Section [ref], we recall the notion of asid-bimodules and related results from [ref] .
On finitely graded Iwanaga-Gorenstein algebras and the stable categories of their (graded) Cohen-Macaulay modules · The asid conditions and the asid numbers · 1.00
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We mention that Gorenstein homological properties of trivial extensions are studied in [ref] .
Gorenstein homological properties of tensor rings · Perfect bimodules · 0.99
What it cites · 1 sentence without a confident label
Cites as background 2 papers
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Representation theory of (graded and ungraded) IG-algebra was initiated by Auslander-Reiten [ref] , Happel [ref] and Buchweitz [ref] , has been studied by many researchers and is recently getting interest from other areas [ref] .
Tilting theory of preprojective algebras and $c$-sortable elements · Introduction · 1.00
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Representation theory of (graded and ungraded) IG-algebra was initiated by Auslander-Reiten [ref] , Happel [ref] and Buchweitz [ref] , has been studied by many researchers and is recently getting interest from other areas [ref] .
Tilting and cluster tilting for preprojective algebras and Coxeter groups · Introduction · 1.00
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