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  In abstract algebra, a Koszul algebra is a graded algebra over which the ground field has a linear minimal graded free resolution, i.e., there exists an exact sequence: It is named after the French mathematician JeanLouis Koszul. Here, is the graded algebra with grading shifted up by , i.e. . The exponents refer to the fold direct sum. We can choose bases for the free modules in the resolution; then the maps can be written as matrices. For a Koszul algebra, the entries in the matrices are zero or linear forms.
 In abstract algebra, a Koszul algebra is a graded algebra over which the ground field has a linear minimal graded free resolution, i.e., there exists an exact sequence: It is named after the French mathematician JeanLouis Koszul. Here, is the graded algebra with grading shifted up by , i.e. . The exponents refer to the fold direct sum. We can choose bases for the free modules in the resolution; then the maps can be written as matrices. The definition of Koszul algebra requires the entries in the matrices to be zero or linear forms.
 In abstract algebra, a Koszul algebra is a graded algebra over which the ground field has a linear minimal graded free resolution, i.e., there exists an exact sequence: Here, is the graded algebra with grading shifted up by , i.e. . The exponents refer to the fold direct sum. Choosing bases for the free modules in the resolution, the chain maps are given by matrices, and the definition requires the matrix entries to be zero or linear forms. The concept is named after the French mathematician JeanLouis Koszul.

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  In abstract algebra, a Koszul algebra is a graded algebra over which the ground field has a linear minimal graded free resolution, i.e., there exists an exact sequence: It is named after the French mathematician JeanLouis Koszul. Here, is the graded algebra with grading shifted up by , i.e. . The exponents refer to the fold direct sum. We can choose bases for the free modules in the resolution; then the maps can be written as matrices. For a Koszul algebra, the entries in the matrices are zero or linear forms. An example of a Koszul algebra is a polynomial ring over a field, for which the Koszul complex is the minimal graded free resolution of the ground field. There are Koszul algebras whose ground fields have infinite minimal graded free resolutions, e.g,
 In abstract algebra, a Koszul algebra is a graded algebra over which the ground field has a linear minimal graded free resolution, i.e., there exists an exact sequence: It is named after the French mathematician JeanLouis Koszul. Here, is the graded algebra with grading shifted up by , i.e. . The exponents refer to the fold direct sum. We can choose bases for the free modules in the resolution; then the maps can be written as matrices. The definition of Koszul algebra requires the entries in the matrices to be zero or linear forms. An example of a Koszul algebra is a polynomial ring over a field, for which the Koszul complex is the minimal graded free resolution of the ground field. There are Koszul algebras whose ground fields have infinite minimal graded free resolutions, e.g,
 In abstract algebra, a Koszul algebra is a graded algebra over which the ground field has a linear minimal graded free resolution, i.e., there exists an exact sequence: Here, is the graded algebra with grading shifted up by , i.e. . The exponents refer to the fold direct sum. Choosing bases for the free modules in the resolution, the chain maps are given by matrices, and the definition requires the matrix entries to be zero or linear forms. An example of a Koszul algebra is a polynomial ring over a field, for which the Koszul complex is the minimal graded free resolution of the ground field. There are Koszul algebras whose ground fields have infinite minimal graded free resolutions, e.g, . The concept is named after the French mathematician JeanLouis Koszul.

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