@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .
@prefix dc: <http://purl.org/dc/terms/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

psr: a skos:ConceptScheme .
psr:-D655FLJF-X
  skos:prefLabel "algèbre différentielle graduée"@fr, "differential graded algebra"@en ;
  a skos:Concept ;
  skos:narrower psr:-QBPHV79Q-6 .

psr:-QBPHV79Q-6
  skos:broader psr:-D655FLJF-X, psr:-TW7HFXF6-Q ;
  dc:modified "2023-08-23"^^xsd:date ;
  skos:definition """In algebra, the ring of <b>polynomial differential forms</b> on the standard <i>n</i>-simplex  is the differential graded algebra:
<br/>
<br/><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\Omega _{\\	ext{poly}}^{*}()=\\\\mathbb {Q} [t_{0},...,t_{n},dt_{0},...,dt_{n}]/(\\\\sum t_{i}-1,\\\\sum dt_{i}).}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msubsup>
<br/>          <mi mathvariant="normal">Ω<!-- Ω --></mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mtext>poly</mtext>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo>∗<!-- ∗ --></mo>
<br/>          </mrow>
<br/>        </msubsup>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mo stretchy="false">[</mo>
<br/>        <mi>n</mi>
<br/>        <mo stretchy="false">]</mo>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>=</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mi mathvariant="double-struck">Q</mi>
<br/>        </mrow>
<br/>        <mo stretchy="false">[</mo>
<br/>        <msub>
<br/>          <mi>t</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>0</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo>,</mo>
<br/>        <mo>.</mo>
<br/>        <mo>.</mo>
<br/>        <mo>.</mo>
<br/>        <mo>,</mo>
<br/>        <msub>
<br/>          <mi>t</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo>,</mo>
<br/>        <mi>d</mi>
<br/>        <msub>
<br/>          <mi>t</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mn>0</mn>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo>,</mo>
<br/>        <mo>.</mo>
<br/>        <mo>.</mo>
<br/>        <mo>.</mo>
<br/>        <mo>,</mo>
<br/>        <mi>d</mi>
<br/>        <msub>
<br/>          <mi>t</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>n</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo stretchy="false">]</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mo>/</mo>
<br/>        </mrow>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mo>∑<!-- ∑ --></mo>
<br/>        <msub>
<br/>          <mi>t</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>i</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo>−<!-- − --></mo>
<br/>        <mn>1</mn>
<br/>        <mo>,</mo>
<br/>        <mo>∑<!-- ∑ --></mo>
<br/>        <mi>d</mi>
<br/>        <msub>
<br/>          <mi>t</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>i</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\Omega _{\\	ext{poly}}^{*}()=\\\\mathbb {Q} [t_{0},...,t_{n},dt_{0},...,dt_{n}]/(\\\\sum t_{i}-1,\\\\sum dt_{i}).}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8561d29e88a143455f96301cf790bb56fd218cea" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.338ex; width:58.013ex; height:3.843ex;" alt="{\\\\displaystyle \\\\Omega _{\\	ext{poly}}^{*}()=\\\\mathbb {Q} [t_{0},...,t_{n},dt_{0},...,dt_{n}]/(\\\\sum t_{i}-1,\\\\sum dt_{i}).}"></span></dd></dl>
<br/>Varying <i>n</i>, it determines the simplicial commutative dg algebra:
<br/>
<br/><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\Omega _{\\	ext{poly}}^{*}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msubsup>
<br/>          <mi mathvariant="normal">Ω<!-- Ω --></mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mtext>poly</mtext>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo>∗<!-- ∗ --></mo>
<br/>          </mrow>
<br/>        </msubsup>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\Omega _{\\	ext{poly}}^{*}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0c453aa2f16ac70224970c356d75c0d2861d6e2a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -1.338ex; width:4.972ex; height:3.176ex;" alt="{\\\\displaystyle \\\\Omega _{\\	ext{poly}}^{*}}"></span></dd></dl>
<br/>(each <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle u:\\	o }">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>u</mi>
<br/>        <mo>:</mo>
<br/>        <mo stretchy="false">[</mo>
<br/>        <mi>n</mi>
<br/>        <mo stretchy="false">]</mo>
<br/>        <mo stretchy="false">→<!-- → --></mo>
<br/>        <mo stretchy="false">[</mo>
<br/>        <mi>m</mi>
<br/>        <mo stretchy="false">]</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle u:\\	o }</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a39fd8924b9cb09430e0cc1161803300fe55a2b5" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:12.903ex; height:2.843ex;" alt="{\\\\displaystyle u:\\	o }"></span> induces the map <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle \\\\Omega _{\\	ext{poly}}^{*}()\\	o \\\\Omega _{\\	ext{poly}}^{*}(),t_{i}\\\\mapsto \\\\sum _{u(j)=i}t_{j}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <msubsup>
<br/>          <mi mathvariant="normal">Ω<!-- Ω --></mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mtext>poly</mtext>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo>∗<!-- ∗ --></mo>
<br/>          </mrow>
<br/>        </msubsup>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mo stretchy="false">[</mo>
<br/>        <mi>m</mi>
<br/>        <mo stretchy="false">]</mo>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo stretchy="false">→<!-- → --></mo>
<br/>        <msubsup>
<br/>          <mi mathvariant="normal">Ω<!-- Ω --></mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mtext>poly</mtext>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mo>∗<!-- ∗ --></mo>
<br/>          </mrow>
<br/>        </msubsup>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mo stretchy="false">[</mo>
<br/>        <mi>n</mi>
<br/>        <mo stretchy="false">]</mo>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>,</mo>
<br/>        <msub>
<br/>          <mi>t</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>i</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>        <mo stretchy="false">↦<!-- ↦ --></mo>
<br/>        <munder>
<br/>          <mo>∑<!-- ∑ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>u</mi>
<br/>            <mo stretchy="false">(</mo>
<br/>            <mi>j</mi>
<br/>            <mo stretchy="false">)</mo>
<br/>            <mo>=</mo>
<br/>            <mi>i</mi>
<br/>          </mrow>
<br/>        </munder>
<br/>        <msub>
<br/>          <mi>t</mi>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>j</mi>
<br/>          </mrow>
<br/>        </msub>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\Omega _{\\	ext{poly}}^{*}()\\	o \\\\Omega _{\\	ext{poly}}^{*}(),t_{i}\\\\mapsto \\\\sum _{u(j)=i}t_{j}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d60e3ec6f36c3f7bdc8bec9c08cf236e21bcabe" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.505ex; width:36.365ex; height:6.009ex;" alt="{\\\\displaystyle \\\\Omega _{\\	ext{poly}}^{*}()\\	o \\\\Omega _{\\	ext{poly}}^{*}(),t_{i}\\\\mapsto \\\\sum _{u(j)=i}t_{j}}"></span>). 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Polynomial_differential_form">https://en.wikipedia.org/wiki/Polynomial_differential_form</a>)"""@en ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Polynomial_differential_form> ;
  a skos:Concept ;
  skos:prefLabel "forme différentielle polynomiale"@fr, "polynomial differential form"@en ;
  skos:inScheme psr: ;
  dc:created "2023-08-23"^^xsd:date .

psr:-TW7HFXF6-Q
  skos:prefLabel "forme différentielle"@fr, "differential form"@en ;
  a skos:Concept ;
  skos:narrower psr:-QBPHV79Q-6 .