We know that if $A$ is a commutative ring and $a_1,...,a_k$ are elements on $A$, there is a construction of the Koszul complex of this sequence, and it may have homology or not. If the sequence is regular, then this complex is acyclic is positive dimensions, and in degree $0$, it is $A/I$, where $I$ is the ideal generated by $a_1,...a_k$. If it is not acyclic, there is a construction of Tate by adding variables on the complex to kill cycles and created a bigger complex without homology. The question is, now suppose that $A$ is $\mathbb{Z}$-graded and the elements $a_i$ are homogeneous. What is the correct analogous construction of the Koszul complex of this sequence and does (graded) regularity still implies acyclic complex (in positive dimensions)? And what about Tate construction?
I have one idea of what is the complex. We can take the exterior power of the free-graded module $M$ generated by the symbols $x_1,...,x_k$, $deg(x_i)=deg(a_i)$, and this is a $\mathbb{Z}\times \mathbb{Z}$ commutative graded algebra, and the derivation $d:M \rightarrow A$ is defined by $d(x_i)=a_i$ and we extend by (graded) derivation. So this derivation has degree $(0,-1)$. But I don't know if I need to take care of these two degrees. I hope we can help me.
• If (graded)regularity implies acyclic and my sequence is regular in this sense, then I get what I wanted. Maybe forgetting the grading, this sequence will not be regular (for example, if one the $a_i$ is odd, $a_i^2=0$), and with the ungraded construction I don't get what I want. – ricardopaleari Apr 11 '13 at 21:35
• Trust yourself. Tate's lovely paper works fine in the graded world. It was used, for example, in Claude Schochet's Chicago PhD thesis in 1969 (but not in the short published paper derived from it), and I knew it well before that. Regular sequences are defined in the graded world exactly as in the ungraded world and have long been used regularly in algebraic topology (albeit usually without odd degree elements). A homotopical example is in Rings, modules, and algebras in stable homotopy theory'', page 93, but use for calculating Tor and Ext of graded things goes way back. – Peter May Apr 12 '13 at 0:46