‣ TwistedOmegaModule( A, i ) | ( operation ) |
Returns: graded lp
The input is a graded exterior algebra A and an integer i. The output is the graded A-lp \omega_A(i).
‣ TwistedGradedFreeModule( S, i ) | ( operation ) |
Returns: graded lp
The input is a graded polynomial ring S and an integer i. The output is the graded S-lp S(i). The sheafification of S(i) is the structure sheaf \mathcal{O}_{\mathbb{P}^m}(i).
‣ TwistedCotangentModule( S, i ) | ( operation ) |
Returns: graded lp
The input is a graded polynomial ring S and an integer i. The output is the graded S-lp \Omega^i(i). The sheafification of \Omega^i(i) is the twisted cotangent sheaf \Omega^i_{\mathbb{P}^m}(i).
‣ TwistedCotangentModuleAsChain( S, i ) | ( operation ) |
Returns: chain complex
The input is a graded polynomial ring S and an integer i. The output is the chain complex of S-lp's whose objects are direct sums of twists of S and its homology at 0 is \Omega^i(i). I.e., a chain complex that is quasi-isomorphic to \Omega^i(i).
‣ TwistedCotangentModuleAsCochain( S, i ) | ( operation ) |
Returns: cochain complex
The input is a graded polynomial ring S and an integer i. The output is the chain complex of S-lp's whose objects are direct sums of twists of S and its homology at 0 is \Omega^i(i). I.e., a chain complex that is quasi-isomorphic to \Omega^i(i).
‣ BasisBetweenTwistedOmegaModules( A, i, j ) | ( operation ) |
Returns: a list
The input is a graded exterior ring A := KoszulDualRing(S) with S:=k[x_0,\dots,x_m] and two integers i,j with i\geq j. The output is a basis of the external hom: \mathrm{Hom}_A(\omega_A(i), \omega_A(j)). If we denote the indeterminates of A by e_0,\dots, e_m and the \ell'th entry of the output by \sigma_{ij}^{\ell} then we have \sigma_{i+1,i}^{\ell}=e_{\ell-1}, thus \sigma_{i+1,i}^{\ell_1} \sigma_{i,i-1}^{\ell_2}= -\sigma_{i+1,i}^{\ell_2} \sigma_{i,i-1}^{\ell_1}.
‣ BasisBetweenTwistedGradedFreeModules( S, i, j ) | ( operation ) |
Returns: a list
The input is a graded polynomial ring S=k[x_0, \dots, x_m] and two integers i,j with i\leq j. The output is a basis of the external hom: \mathrm{Hom}_S(S(i), S(j)). If we denote the \ell'th entry of the output by \psi_{ij}^{\ell} then we have \psi_{i-1,i}^{\ell}=x_{\ell-1}, thus \psi_{i-1,i}^{\ell_1} \psi_{i,i+1}^{\ell_2}= \psi_{i-1,i}^{\ell_2} \psi_{i,i+1}^{\ell_1}.
‣ BasisBetweenTwistedCotangentModulesAsGLP( S, i, j ) | ( operation ) |
Returns: a list
The input is a graded polynomial ring S=k[x_0, \dots, x_m] and two integers 0\leq j \leq i \leq m. The output is a basis of the external hom: \mathrm{Hom}_S(\Omega^i(i), \Omega^i(j)). If we denote the \ell'th entry of the output by \varphi_{ij}^{\ell} then we have \varphi_{i+1,i}^{\ell} \varphi_{i,i-1}^{\ell}=0 and \varphi_{i+1,i}^{\ell_1} \varphi_{i,i-1}^{\ell_2}= -\varphi_{i+1,i}^{\ell_2} \varphi_{i,i-1}^{\ell_1}.
‣ BeilinsonReplacement( M ) | ( attribute ) |
Returns: a chain or cochain complex
The input is graded S-lp, graded A-lp, chain complex or cochain complex of S-lp's. The output is a Beilinson monad of M.
‣ BeilinsonReplacement( phi ) | ( attribute ) |
Returns: a chain or cochain complex
The input is graded S-lp morphism, graded A-lp morphism, chain morphism or cochain morphism of S-lp's. The output is a Beilinson monad morphism of \phi.
‣ MorphismFromGLPToZerothObjectOfBeilinsonReplacement( M ) | ( attribute ) |
Returns: a morphism
The input is graded S-lp M. The output is a morphism from the sheafification of M to the sheafification of the 0'th object of its Beilinson replacement. This morphism induces a quasi-isomorphism from sheafification of M considered as a chain complex concentrated in degree 0 to the sheafification of the Beilinson replacement of M.
‣ MorphismFromGLPToZerothHomologyOfBeilinsonReplacement( M ) | ( attribute ) |
Returns: a morphism
The input is graded S-lp M. The output is an isomorphism from the Sheafification of M to the sheafification of the 0'th homology of its Beilinson replacement.
‣ MorphismFromZerothObjectOfBeilinsonReplacementToGLP( M ) | ( attribute ) |
Returns: a morphism
The input is graded S-lp M. The output is a morphism from the sheafification of the 0'th object of the Beilinson replacement of M to the sheafification of M. This morphism induces a quasi-isomorphism from the sheafification of the Beilinson replacement of M to sheafification of M considered as a chain complex concentrated in degree 0.
‣ MorphismFromZerothHomologyOfBeilinsonReplacementToGLP( M ) | ( attribute ) |
Returns: a morphism
The input is graded S-lp M. The output is an isomorphism from the the sheafification of the 0'th homology of its Beilinson replacement to Sheafification of M.
‣ ShowMatrix( M ) | ( function ) |
Returns: nothing
The inpute is a homalg matrix or anything that has attribute UnderlyingMatrix. It views the entries using the browse package. To quit: q+y.
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