Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Λ</mi> </math></EquationSource> </InlineEquation> be a <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathbb {k}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">k</mi> </math></EquationSource> </InlineEquation>-algebra where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathbb {k}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">k</mi> </math></EquationSource> </InlineEquation> is a field of arbitrary characteristic, and let <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathscr {A}_\mathbb {k}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">A</mi> <mi mathvariant="double-struck">k</mi> </msub> </math></EquationSource> </InlineEquation> be a full subcategory of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Λ</mi> </math></EquationSource> </InlineEquation>-Mod, the abelian category of left <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\Lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Λ</mi> </math></EquationSource> </InlineEquation>-modules. In particular, <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathscr {A}_\mathbb {k}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">A</mi> <mi mathvariant="double-struck">k</mi> </msub> </math></EquationSource> </InlineEquation> is a <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathbb {k}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">k</mi> </math></EquationSource> </InlineEquation>-category, i.e. the set of morphisms between objects in <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\mathscr {A}_\mathbb {k}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">A</mi> <mi mathvariant="double-struck">k</mi> </msub> </math></EquationSource> </InlineEquation> is a vector space over <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$\mathbb {k}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">k</mi> </math></EquationSource> </InlineEquation> and the composition of morphisms is <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\mathbb {k}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">k</mi> </math></EquationSource> </InlineEquation>-bilinear. Following M. Kleiner and I. Reiten, <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$\mathscr {A}_\mathbb {k}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">A</mi> <mi mathvariant="double-struck">k</mi> </msub> </math></EquationSource> </InlineEquation> is Hom-finite if the hom-space between any two objects in <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$\mathscr {A}_\mathbb {k}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">A</mi> <mi mathvariant="double-struck">k</mi> </msub> </math></EquationSource> </InlineEquation> is finite-dimensional over <InlineEquation ID="IEq14"> <EquationSource Format="TEX">$\mathbb {k}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">k</mi> </math></EquationSource> </InlineEquation>. We further say that <InlineEquation ID="IEq15"> <EquationSource Format="TEX">$\mathscr {A}_\mathbb {k}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">A</mi> <mi mathvariant="double-struck">k</mi> </msub> </math></EquationSource> </InlineEquation> is Ext-finite if <InlineEquation ID="IEq16"> <EquationSource Format="TEX">$\dim _\mathbb {k}\textrm{Ext}^i_\Lambda (X,Y)<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo>dim</mo> <mi mathvariant="double-struck">k</mi> </msub> <msubsup> <mtext>Ext</mtext> <mi mathvariant="normal">Λ</mi> <mi>i</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> for all objects X and Y in <InlineEquation ID="IEq17"> <EquationSource Format="TEX">$\mathscr {A}_\mathbb {k}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">A</mi> <mi mathvariant="double-struck">k</mi> </msub> </math></EquationSource> </InlineEquation>. Let V be an object in <InlineEquation ID="IEq18"> <EquationSource Format="TEX">$\mathscr {A}_\mathbb {k}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">A</mi> <mi mathvariant="double-struck">k</mi> </msub> </math></EquationSource> </InlineEquation>. In this note we prove that if <InlineEquation ID="IEq19"> <EquationSource Format="TEX">$\textrm{End}_\Lambda (V)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>End</mtext> <mi mathvariant="normal">Λ</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is isomorphic to <InlineEquation ID="IEq20"> <EquationSource Format="TEX">$\mathbb {k}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">k</mi> </math></EquationSource> </InlineEquation>, then V has a universal deformation ring <InlineEquation ID="IEq21"> <EquationSource Format="TEX">$R(\Lambda ,V)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Λ</mi> <mo>,</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, which is a local complete Noetherian commutative <InlineEquation ID="IEq22"> <EquationSource Format="TEX">$\mathbb {k}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">k</mi> </math></EquationSource> </InlineEquation>-algebra whose residue field is also isomorphic to <InlineEquation ID="IEq23"> <EquationSource Format="TEX">$\mathbb {k}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">k</mi> </math></EquationSource> </InlineEquation>. We use this result to prove that if <InlineEquation ID="IEq24"> <EquationSource Format="TEX">$\Lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Λ</mi> </math></EquationSource> </InlineEquation> is a two-point infinite-dimensional gentle <InlineEquation ID="IEq25"> <EquationSource Format="TEX">$\mathbb {k}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">k</mi> </math></EquationSource> </InlineEquation>-algebra (in the sense of V. Bekkert et al), then <InlineEquation ID="IEq26"> <EquationSource Format="TEX">$R(\Lambda ,V)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>R</mi> <mo stretchy="false">(</mo> <mi mathvariant="normal">Λ</mi> <mo>,</mo> <mi>V</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is isomorphic either to <InlineEquation ID="IEq27"> <EquationSource Format="TEX">$\mathbb {k}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">k</mi> </math></EquationSource> </InlineEquation>, to <InlineEquation ID="IEq28"> <EquationSource Format="TEX">$\mathbb {k}[\![t]\!]/(t^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">k</mi> <mrow> <mo stretchy="false">[</mo> <mspace width="-0.166667em" /> <mrow> <mo stretchy="false">[</mo> <mi>t</mi> <mo stretchy="false">]</mo> </mrow> <mspace width="-0.166667em" /> <mo stretchy="false">]</mo> </mrow> <mo stretchy="false">/</mo> <mrow> <mo stretchy="false">(</mo> <msup> <mi>t</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> or to <InlineEquation ID="IEq29"> <EquationSource Format="TEX">$\mathbb {k}[\![t]\!]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="double-struck">k</mi> <mo stretchy="false">[</mo> <mspace width="-0.166667em" /> <mo stretchy="false">[</mo> <mi>t</mi> <mo stretchy="false">]</mo> <mspace width="-0.166667em" /> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>.

On weak universal deformation rings for objects of Ext-finite categories of modules

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2026年4月24日

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Diego López-García,
Pedro Rizzo,
José A. Vélez-Marulanda

摘要

Let $\Lambda $ $Λ$ be a $\mathbb {k}$ $k$ -algebra where $\mathbb {k}$ $k$ is a field of arbitrary characteristic, and let $\mathscr {A}_\mathbb {k}$ $A_{k}$ be a full subcategory of $\Lambda $ $Λ$ -Mod, the abelian category of left $\Lambda $ $Λ$ -modules. In particular, $\mathscr {A}_\mathbb {k}$ $A_{k}$ is a $\mathbb {k}$ $k$ -category, i.e. the set of morphisms between objects in $\mathscr {A}_\mathbb {k}$ $A_{k}$ is a vector space over $\mathbb {k}$ $k$ and the composition of morphisms is $\mathbb {k}$ $k$ -bilinear. Following M. Kleiner and I. Reiten, $\mathscr {A}_\mathbb {k}$ $A_{k}$ is Hom-finite if the hom-space between any two objects in $\mathscr {A}_\mathbb {k}$ $A_{k}$ is finite-dimensional over $\mathbb {k}$ $k$ . We further say that $\mathscr {A}_\mathbb {k}$ $A_{k}$ is Ext-finite if $\dim _\mathbb {k}\textrm{Ext}^i_\Lambda (X,Y)<\infty $ ${dim}_{k} {Ext}_{Λ}^{i} (X, Y) < \infty$ for all objects X and Y in $\mathscr {A}_\mathbb {k}$ $A_{k}$ . Let V be an object in $\mathscr {A}_\mathbb {k}$ $A_{k}$ . In this note we prove that if $\textrm{End}_\Lambda (V)$ ${End}_{Λ} (V)$ is isomorphic to $\mathbb {k}$ $k$ , then V has a universal deformation ring $R(\Lambda ,V)$ $R (Λ, V)$ , which is a local complete Noetherian commutative $\mathbb {k}$ $k$ -algebra whose residue field is also isomorphic to $\mathbb {k}$ $k$ . We use this result to prove that if $\Lambda $ $Λ$ is a two-point infinite-dimensional gentle $\mathbb {k}$ $k$ -algebra (in the sense of V. Bekkert et al), then $R(\Lambda ,V)$ $R (Λ, V)$ is isomorphic either to $\mathbb {k}$ $k$ , to $\mathbb {k}[\![t]\!]/(t^2)$ $k [[t]] / (t^{2})$ or to $\mathbb {k}[\![t]\!]$ $k [[t]]$ .