Hey Everyone

Let $A$ be a graded algebra over a field (group rings $k[G]$ for group cohomology, the Steenrod Algebra). We want to compute, say, $Ext_A(k,k)$, so let $F_*\to k$ be an $A$-free resolution. Elements of $Ext^j_A(k,k)$ can be represented as Yoneda extensions, essentially exact sequence $k\to M_1 \to ... \to M_{j} \to k$, which we denote $\mathcal{X}$, and a chain map map $F_*\to \mathcal{X}$ with the cocycle, $F_{j+1}\to k$, on one side and the identity map $k\to k$ on the other. Using pushouts one can, given a cocycle, generate $\mathcal{X}$, but the modules in $\mathcal{X}$ can be rather large.

My question is then this. Is there a way to either generate a "small" extension (measured however you'd like, perhaps the size of an extension should be the largest dimension over $k$ in any bi-digree) representing a cocycle, or given the a Yoneda extension, is there a way to make some sort of local change to it that can make it "smaller" (in which case we can search for a "local optimum").

This question is motivated as a workaround to the problem presented by Bob Bruner's in http://www.math.wayne.edu/~rrb/papers/yoneda.pdf.

Thanks

Hey Everyone

Let $A$ be an algebra over a field (group rings $k[G]$ for group cohomology, the Steenrod Algebra). We want to compute, say, $Ext_A(k,k)$, so let $F_*\to k$ be an $A$-free resolution. Elements of $Ext^j_A(k,k)$ can be represented as Yoneda extensions, essentially exact sequence $k\to M_1 \to ... \to M_{j} \to k$, which we denote $\mathcal{X}$, and a chain map map $F_*\to \mathcal{X}$ with the cocycle, $F_{j+1}\to k$, on one side and the identity map $k\to k$ on the other. Using pushouts one can, given a cocycle, generate $\mathcal{X}$, but the modules in $\mathcal{X}$ can be rather large.

My question is then this. Is there a way to either generate a "small" extension (measured however you'd like, perhaps the size of an extension should be the largest dimension over $k$ in any bi-digree) representing a cocycle, or given the a Yoneda extension, is there a way to make some sort of local change to it that can make it "smaller" (in which case we can search for a "local optimum").

This question is motivated as a workaround to the problem presented by Bob Bruner's in http://www.math.wayne.edu/~rrb/papers/yoneda.pdf.

Thanks

Hey Everyone

Let $A$ be a graded algebra of finite type over a field (group rings $k[G]$ for group cohomology, the Steenrod Algebra). We want to compute, say, $Ext_A(k,k)$, so let $F_*\to k$ be an $A$-free resolution. Elements of $Ext^j_A(k,k)$ can be represented as Yoneda extensions, essentially exact sequence $k\to M_1 \to ... \to M_{j} \to k$, which we denote $\mathcal{X}$, and a chain map map $F_*\to \mathcal{X}$ with the cocycle, $F_{j+1}\to k$, on one side and the identity map $k\to k$ on the other. Using pushouts one can, given a cocycle, generate $\mathcal{X}$, but the modules in $\mathcal{X}$ can be rather large.

My question is then this. Is there a way to either generate a "small" extension (measured however you'd like, perhaps the size of an extension should be the largest dimension over $k$ in any bi-digree) representing a cocycle, or given the a Yoneda extension, is there a way to make some sort of local change to it that can make it "smaller" (in which case we can search for a "local optimum").

This question is motivated as a workaround to the problem presented by Bob Bruner's in http://www.math.wayne.edu/~rrb/papers/yoneda.pdf.

Thanks

Hey Everyone

Let $A$ be a graded algebra over a field (group rings $k[G]$ for group cohomology, the Steenrod Algebra). We want to compute, say, $Ext_A(k,k)$, so let $F_*\to k$ be an $A$-free resolution. Elements of $Ext^j_A(k,k)$ can be represented as Yoneda extensions, essentially exact sequence $k\to M_1 \to ... \to M_{j} \to k$, which we denote $\mathcal{X}$, and a chain map map $F_*\to \mathcal{X}$ with the cocycle, $F_{j+1}\to k$, on one side and the identity map $k\to k$ on the other. Using pushouts one can, given a cocycle, generate $\mathcal{X}$, but the modules in $\mathcal{X}$ can be rather large.

My question is then this. Is there a way to either generate a "small" extension (measured however you'd like, perhaps the size of an extension should be the largest dimension over $k$ in any bi-digree) representing a cocycle, or given the a Yoneda extension, is there a way to make some sort of local change to it that can make it "smaller" (in which case we can search for a "local optimum").

This question is motivated as a workaround to the problem presented by Bob Bruner's in http://www.math.wayne.edu/~rrb/papers/yoneda.pdf.

Thanks