# Tagged Questions

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We define multiplicity of a module M of dimension $d>0$ as $$mult(M) := lc (P_M) (d-1)!$$ where $P_M$ denotes the Hilbert polynomial of M. Equivalently, we have $mult(M) = Q_M(1)$, where $HP_M (z) ... 2answers 71 views ### Graded rings and modules - which book to refer to? Could you recommend a book with a good treatment of the theory of graded rings and modules? 0answers 57 views ### Extension of multiplication to the tensor algebra. In this wikipedia article http://en.wikipedia.org/wiki/Tensor_algebra#Construction We construct$T(V)$as the direct sum of vector spaces$T^kV$for$k=0,1,2,…$$$T(V)= ... 0answers 117 views ### Computing ext over graded rings This question came up as I was reading Beilinson, Ginzburg, Soergel paper Koszul Duality Patterns in Representation Theory. Suppose that A is a Koszul ring (for the definition of Koszul ring ... 1answer 71 views ### Questions on (subring/ submodule) of a graded (ring/ module) I have a question which seems a bit silly... If we have R a graded ring, does it follow that every subring of R is also graded? Because I have a problem here as such: I have a graded ring R ... 0answers 35 views ### Why do we have \operatorname{Proj}(\oplus_{i=0}^\infty t^i \left< x\right>^i)\cong \operatorname{Proj}(\oplus_{i=0}^\infty t^i \left< x^2\right>^i)? I'm just curious but why is it that$$ \operatorname{Proj}\left(\oplus_{i=0}^\infty t^i \left< x\right>^i\right) $$isomorphic to$$ \operatorname{Proj} \left(\oplus_{i=0}^\infty t^i \left< ... 0answers 30 views ### Understanding$Bl_{\mathcal{I}}(k^4)/S_2$where$\mathcal{I}$is defined by$(x_1-x_2,x_3-x_4)$Let$k_4=Spec(k[x_1, x_2, x_3,x_4])$and$\mathcal{I}$is the ideal sheaf defined by$(x_1-x_2,x_3-x_4)$. Then $$Bl_{\mathcal{I}}(k^4) = Proj (\oplus_{i\geq 0} I^i t^i)$$ where ... 0answers 78 views ### Degree 1 elements in a graded ring from a blow-up perspective This may be an elementary question but I hope this question will benefit others as much as myself. Let$k^4 = Spec \; k[x_1, x_2, x_3, x_4]$. Writing$Bl_{\mathcal{I}}(k^4)$as$R = ...
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In this post I set out to prove an equation holding in a ring $R$ graded over $\mathbb{Z}$ or $\mathbb{N}$. The equation was: $\sqrt{J^\ast}=(\sqrt{J})^\ast$, where $J$ is an ideal of $R$, $J^\ast$ ...
### Isomorphism of ($\mathbb{Z}/{(n)}$-graded) Rings
Let $A=\bigoplus_{d=0}^n A_g$ and $B=\bigoplus_{d=0}^n B_h$ be $\mathbb{Z}/{(n)}$-graded rings. In particular, we assume $A_n\ne 0$ and $B_n\ne 0$. Let $\phi:A\to B$ be an isomorphism of rings. My ...
At the moment I'm mostly interested in commutative graded rings (and in particular those graded over $\mathbb{N}$), but any comments or references about more general graded rings would also be ...