English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

In recent work, Dao and Eisenbud define the notion of a Burch index,expanding the notion of Burch rings of Dao, Kobayashi, and Takahashi, and showthat for any module over a ring of Burch index at least 2, its $n$th syzygycontains direct summands of the residue field for $n=4$ or $5$ and all $n\geq7$. We investigate how this behavior is explained by the bar resolution formedfrom appropriate differential graded (dg) resolutions, yielding a new proofthat includes all $n\geq 5$, which is sharp. When the module is Golod, we useinstead the bar resolution formed from $A_\infty$ resolutions to identify such$k$ summands explicitly for all $n\geq 4$ and show that the number of thesegrows exponentially as the homological degree increases.

$k$ Summands of Syzygies over Rings of Positive Burch Index Via Canonical Resolutions