Let G be a finite group. We investigate the cohomology ring <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$H^(G,b;A)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mo>∗</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo>,</mo> <mi>b</mi> <mo>;</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of a block ideal b of the finite group algebra kG, where k is an algebraically close field of characteristic p and A is a source algebra of b. The author expected in Sasaki (Algebr. Represent. Theory 16, 1039–1049, <CitationRef CitationID="CR15">2013</CitationRef>) that, P being its defect group, the image of the transfer map <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$t_A$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>t</mi> <mi>A</mi> </msub> </math></EquationSource> </InlineEquation> of the cohomology ring <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$H^(P,k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mo>∗</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>P</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> induced by A coincides with <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$H^(G,b;A)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>H</mi> <mo>∗</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo>,</mo> <mi>b</mi> <mo>;</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>: <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\text {Im} t_A=H^(G,b;A)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>Im</mtext> <msub> <mi>t</mi> <mi>A</mi> </msub> <mo>=</mo> <msup> <mi>H</mi> <mo>∗</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo>,</mo> <mi>b</mi> <mo>;</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. This expectation has previously been settled in Sasaki (Hokkaido Math. J. 53, 443–462, <CitationRef CitationID="CR17">2024</CitationRef>) for blocks of tame representation type and blocks with extraspecial p-groups as P. In this note, we prove that this expectation holds for b whose defect group P is isomorphic to a wreathed 2-group. The proof relies on prior works Okuyama and Sasaki (Algebr. Represent. Theory 4, 405–444, <CitationRef CitationID="CR11">2001</CitationRef>; J. Algebra 497, 92–101, <CitationRef CitationID="CR12">2018</CitationRef>), Kawai and Sasaki (J. Algebra 306(2), 301–321, <CitationRef CitationID="CR5">2006</CitationRef>), and Sasaki (J. Algebra 666, 777–793, <CitationRef CitationID="CR18">2025</CitationRef>).

The Transfer Maps for the Cohomology Rings of Block Ideals with Wreathed Defect Groups

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发表日期：

2026年2月13日

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Hiroki Sasaki

摘要

Let G be a finite group. We investigate the cohomology ring $H^*(G,b;A)$ $H^{*} (G, b; A)$ of a block ideal b of the finite group algebra kG, where k is an algebraically close field of characteristic p and A is a source algebra of b. The author expected in Sasaki (Algebr. Represent. Theory 16, 1039–1049, 2013) that, P being its defect group, the image of the transfer map $t_A$ $t_{A}$ of the cohomology ring $H^*(P,k)$ $H^{*} (P, k)$ induced by A coincides with $H^*(G,b;A)$ $H^{*} (G, b; A)$ : $\text {Im} t_A=H^*(G,b;A)$ $Im t_{A} = H^{*} (G, b; A)$ . This expectation has previously been settled in Sasaki (Hokkaido Math. J. 53, 443–462, 2024) for blocks of tame representation type and blocks with extraspecial p-groups as P. In this note, we prove that this expectation holds for b whose defect group P is isomorphic to a wreathed 2-group. The proof relies on prior works Okuyama and Sasaki (Algebr. Represent. Theory 4, 405–444, 2001; J. Algebra 497, 92–101, 2018), Kawai and Sasaki (J. Algebra 306(2), 301–321, 2006), and Sasaki (J. Algebra 666, 777–793, 2025).