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Combinatorial commutative algebra
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<h2 id="important-notions-of-combinatorial-commutative-algebra">Important notions of combinatorial commutative algebra</h2> <ul><li>Square-free <a href="/facts/Monomial_ideal/UhogyF4z">monomial ideal</a> in a <a href="/facts/Polynomial_ring/ua3vk8Ih">polynomial ring</a> and <a href="/facts/Stanley%25E2%2580%2593Reisner_ring/tzekr8Td">Stanley–Reisner ring</a> of a <a href="/facts/Simplicial_complex/fF2282CF">simplicial complex</a>.</li> <li><a href="/facts/Cohen%25E2%2580%2593Macaulay_ring/4G8wymtC">Cohen–Macaulay rings</a>.</li> <li>Monomial ring, closely related to an affine semigroup ring and to the <a href="/facts/Coordinate_ring/WVynz2Kn">coordinate ring</a> of an <a href="/facts/Affine_variety/WVynz2Kn">affine</a> <a href="/facts/Toric_variety/m1wEbNWw">toric variety</a>.</li> <li>Algebra with a straightening law. There are several versions of those, including Hodge algebras of <a href="/facts/Corrado_de_Concini/VhF4zHmr">Corrado de Concini</a>, <a href="/facts/David_Eisenbud/hw2yEnnx">David Eisenbud</a>, and <a href="/facts/Claudio_Procesi/327Bhv6R">Claudio Procesi</a>.</li></ul> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Algebraic_combinatorics/9b47n3Fh">Algebraic combinatorics</a></li> <li><a href="/facts/Polyhedral_combinatorics/oVFESPjJ">Polyhedral combinatorics</a></li> <li><a href="/facts/Zero-divisor_graph/ATu4hFwp">Zero-divisor graph</a></li></ul> <p>A foundational paper on Stanley–Reisner complexes by one of the pioneers of the theory: </p> <ul><li><a href="/facts/Melvin_Hochster/rbXPNR0X">Hochster, Melvin</a> (1977). <a href="https://archive.org/details/ringtheoryiiproc0026ring">"Cohen–Macaulay rings, combinatorics, and simplicial complexes"</a>. <i>Ring Theory II: Proceedings of the Second Oklahoma Conference</i>. Lecture Notes in Pure and Applied Mathematics. Vol. 26. Dekker. pp. 171–223. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8247-6575-3. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/610144046">610144046</a>. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0351.13009">0351.13009</a>.</li></ul> <p>The first book is a classic (first edition published in 1983): </p> <ul><li><a href="/facts/Richard_P._Stanley/8bQ4QrLM">Stanley, Richard</a> (1996). <a href="https://books.google.com/books?id=tZVCAAAAQBAJ&pg=PR5"><i>Combinatorics and commutative algebra</i></a>. Progress in Mathematics. Vol. 41 (2nd ed.). Birkhäuser. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8176-3836-9. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0838.13008">0838.13008</a>.</li></ul> <p>Very influential, and well written, textbook-monograph: </p> <ul><li>Bruns, Winfried; Herzog, Jürgen (1993). <i>Cohen–Macaulay rings</i>. Vol. 39. Cambridge Studies in Advanced Mathematics: Cambridge University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-41068-1. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/802912314">802912314</a>. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0788.13005">0788.13005</a>.</li></ul> <p>Additional reading: </p> <ul><li>Villarreal, Rafael H. (2001). <a href="https://books.google.com/books?id=rLfKlDJjLewC&pg=PP1"><i>Monomial algebras</i></a>. Monographs and Textbooks in Pure and Applied Mathematics. Vol. 238. Marcel Dekker. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8247-0524-6. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1002.13010">1002.13010</a>.</li> <li>Hibi, Takayuki (1992). <i>Algebraic combinatorics on convex polytopes</i>. Glebe, Australia: Carslaw Publications. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 1875399046. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/29023080">29023080</a>.</li> <li><a href="/facts/Bernd_Sturmfels/BejRHlNI">Sturmfels, Bernd</a> (1996). <a href="https://books.google.com/books?id=K-bxBwAAQBAJ&pg=PR7"><i>Gröbner bases and convex polytopes</i></a>. University Lecture Series. Vol. 8. American Mathematical Society. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-8218-0487-1. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/907364245">907364245</a>. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0856.13020">0856.13020</a>.</li> <li>Bruns, Winfried; Gubeladze, Joseph (2009). <a href="https://books.google.com/books?id=pbgg1pFxW8YC&pg=PP9"><i>Polytopes, Rings, and K-Theory</i></a>. Springer Monographs in Mathematics. Springer. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1007%2Fb105283">10.1007/b105283</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-76355-2. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1168.13001">1168.13001</a>.</li></ul> <p>A recent addition to the growing literature in the field, contains exposition of current research topics: </p> <ul><li>Miller, Ezra; Sturmfels, Bernd (2005). <a href="https://books.google.com/books?id=OYBCAAAAQBAJ&pg=PR11"><i>Combinatorial commutative algebra</i></a>. <a href="/facts/Graduate_Texts_in_Mathematics/VTIfY29j">Graduate Texts in Mathematics</a>. Vol. 227. Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-22356-8. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1066.13001">1066.13001</a>.</li> <li>Herzog, Jürgen; Hibi, Takayuki (2011). <a href="https://books.google.com/books?id=3PxzYIBTibsC&pg=PR11"><i>Monomial Ideals</i></a>. Graduate Texts in Mathematics. Vol. 260. Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-85729-106-6. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1206.13001">1206.13001</a>.</li> <li>Herzog, Jürgen; Hibi, Takayuki; Oshugi, Hidefumi (2018). <a href="https://books.google.com/books?id="><i>Binomial Ideals</i></a>. Graduate Texts in Mathematics. Vol. 279. Springer. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-319-95349-6. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1403.13004">1403.13004</a>.</li></ul>