Document Zbl 1532.14049

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Volume 2. Plenary lectures. Berlin: European Mathematical Society (EMS). 712-748 (2023). </div> <div class="abstract">Summary: Fix a prime number $p$. We report on some recent developments in algebraic geometry (broadly construed) over $p$-adically complete commutative rings. These developments include foundational advances within the subject, as well as external applications.<br class="zbmathjax-paragraph">For the entire collection see [<a href="/1532.00036">Zbl 1532.00036</a>].</div> <div class="clear"></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A14G45" title="MSC2020">14G45</a> </td> <td class="space"> Perfectoid spaces and mixed characteristic </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A14F30" title="MSC2020">14F30</a> </td> <td class="space"> $p$-adic cohomology, crystalline cohomology </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A14F17" title="MSC2020">14F17</a> </td> <td class="space"> Vanishing theorems in algebraic geometry </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A11G25" title="MSC2020">11G25</a> </td> <td class="space"> Varieties over finite and local fields </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A11S70" title="MSC2020">11S70</a> </td> <td class="space"> $K$-theory of local fields </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A19F99" title="MSC2020">19F99</a> </td> <td class="space"> $K$-theory in number theory </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A13D99" title="MSC2020">13D99</a> </td> <td class="space"> Homological methods in commutative ring theory </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A14-02" title="MSC2020">14-02</a> </td> <td class="space"> Research exposition (monographs, survey articles) pertaining to algebraic geometry </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3A%5C%28p%5C%29-adic+Hodge+theory">$p$-adic Hodge theory</a>; <a href="/?q=ut%3Aprismatic+cohomology">prismatic cohomology</a>; <a href="/?q=ut%3Atopological+Hochschild+homology">topological Hochschild homology</a>; <a href="/?q=ut%3Aalgebraic+%5C%28K%5C%29-theory">algebraic $K$-theory</a>; <a href="/?q=ut%3Avanishing+theorems">vanishing theorems</a>; <a href="/?q=ut%3ARiemann-Hilbert+correspondence">Riemann-Hilbert correspondence</a>; <a href="/?q=ut%3Aminimal+model+program">minimal model program</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1532.14049" data-ciurl="/ci/07821725" data-biburl="/bibtex/07821725.bib" data-amsurl="/amsrefs/07821725.bib" data-xmlurl="/xml/07821725.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07821725.pdf" title="Zbl 1532.14049 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.4171/ICM2022/113" aria-label="DOI for “Algebraic geometry in mixed characteristic”" title="10.4171/ICM2022/113">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/2112.12010"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> <a href="https://creativecommons.org/licenses/by/4.0/" target="_blank" title="Open Access License" class="cc-license-link no-new-tab-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/cc.svg" alt="Creative Commons CC license icon" class="cc-license-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/by.svg" alt="Creative Commons BY license icon" class="cc-license-icon"> </a> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">A. 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