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Harry Gindi
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Let $k$ be a field of characteristic $0$. Consider the following $k$-algebra $R$, which is the quotient of a tensor algebra generated by elements $x_i$ in degree $1$ with the relation $x_ix_j=-x_jx_i$ when $i\neq j$. 

In other words, $x_i^2$ does not vanish. Has anyone studied this graded ring? Does this graded ring have finite homological dimension? Is there a proof of Hilbert's syzygy theorem that can be doctored to apply to it?

Let $k$ be a field of characteristic $0$. Consider the following $k$-algebra $R$, which is the quotient of a tensor algebra generated by elements $x_i$ in degree $1$ with the relation $x_ix_j=-x_jx_i$ when $i\neq j$. In other words, $x_i^2$ does not vanish. Has anyone studied this graded ring? Does this graded ring have finite homological dimension? Is there a proof of Hilbert's syzygy theorem that can be doctored to apply to it?

Let $k$ be a field of characteristic $0$. Consider the following $k$-algebra $R$, which is the quotient of a tensor algebra generated by elements $x_i$ in degree $1$ with the relation $x_ix_j=-x_jx_i$ when $i\neq j$. 

In other words, $x_i^2$ does not vanish. Has anyone studied this graded ring? Does this graded ring have finite homological dimension? Is there a proof of Hilbert's syzygy theorem that can be doctored to apply to it?

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Harry Gindi
  • 19.6k
  • 16
  • 123
  • 215

ConsiderLet $k$ be a field of characteristic $0$. Consider the following ring R$k$-algebra $R$, which is the quotient of a tensor algebra generated by elements x_i$x_i$ in degree 1$1$ with the relation x_ix_j=-x_jx_i$x_ix_j=-x_jx_i$ when i doesn't equal j$i\neq j$. In other words, (x_i)^2$x_i^2$ does not equal to zerovanish. Has anyone studied this graded ring? Does this graded ring have finite homological dimension? Is there a proof of Hilbert's szy.syzygy theorem that can be doctored to apply to it?

Consider the following ring R which is the quotient of a tensor algebra generated by elements x_i in degree 1 with the relation x_ix_j=-x_jx_i when i doesn't equal j. In other words, (x_i)^2 does not equal to zero. Has anyone studied this graded ring? Does this graded ring have finite homological dimension? Is there a proof of Hilbert's szy. theorem that can be doctored to apply to it?

Let $k$ be a field of characteristic $0$. Consider the following $k$-algebra $R$, which is the quotient of a tensor algebra generated by elements $x_i$ in degree $1$ with the relation $x_ix_j=-x_jx_i$ when $i\neq j$. In other words, $x_i^2$ does not vanish. Has anyone studied this graded ring? Does this graded ring have finite homological dimension? Is there a proof of Hilbert's syzygy theorem that can be doctored to apply to it?

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Consider the following ring R which is the quotient of a tensor algebra generated by elements x_i in degree 1 with the relation x_ix_j=-x_ix_jx_jx_i when i doesn't equal j. In other words, (x_i)^2 does not equal to zero. Has anyone studied this graded ring? Does this graded ring have finite homological dimension? Is there a proof of Hilbert's szy. theorem that can be doctored to apply to it?

Consider the following ring R which is the quotient of a tensor algebra generated by elements x_i in degree 1 with the relation x_ix_j=-x_ix_j when i doesn't equal j. In other words, (x_i)^2 does not equal to zero. Has anyone studied this graded ring? Does this graded ring have finite homological dimension? Is there a proof of Hilbert's szy. theorem that can be doctored to apply to it?

Consider the following ring R which is the quotient of a tensor algebra generated by elements x_i in degree 1 with the relation x_ix_j=-x_jx_i when i doesn't equal j. In other words, (x_i)^2 does not equal to zero. Has anyone studied this graded ring? Does this graded ring have finite homological dimension? Is there a proof of Hilbert's szy. theorem that can be doctored to apply to it?

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