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

Let $n\ge 1$ and $A$ be a commutative algebra of the form $\boldsymbolk[x_1,x_2,\dots, x_n]/I$ where $\boldsymbol k$ is a field of characteristic $0$and $I\subseteq \boldsymbol k[x_1,x_2,\dots, x_n]$ is an ideal. Assume thatthere is a Poisson bracket $\{\:,\:\}$ on $S$ such that $\{I,S\}\subseteq I$and let us denote the induced bracket on $A$ by $\{\:,\:\}$ as well. It iswell-known that $[\mathrm d x_i,\mathrm d x_j]:=\mathrm d\{x_i,x_j\}$ defines aLie bracket on the $A$-module $\Omega_{A|\boldsymbol k}$ of K\"ahlerdifferentials making $(A,\Omega_{A|\boldsymbol k})$ a Lie-Rinehart pair. Recallthat $A$ is regular if and only if $\Omega_{A|\boldsymbol k}$ is projective asan $A$-module. If $A$ is not regular, the cotangent complex $\mathbbL_{A|\boldsymbol k}$ may serve as a replacement for the $A$-module$\Omega_{A|\boldsymbol k}$. We prove that there is a structure of an$L_\infty$-algebroid on $\mathbb L_{A|\boldsymbol k}$, compatible with theLie-Rinehart pair $(A,\Omega_{A|\boldsymbol k})$. The $L_\infty$-algebroid on$\mathbb L_{A|\boldsymbol k}$ actually comes from a $P_\infty$-algebrastructure on the resolvent of the morphism $k[x_1,x_2,\dots, x_n]\to A$. Weidentify examples when this $L_\infty$-algebroid simplifies to a dg Liealgebroid. For aesthetic reasons we concentrate on cases when $ \boldsymbolk[x_1,x_2,\dots, x_n]$ carries a (possibly nonstandard) $\mathbb Z_{\ge0}$-grading and both $I$ and $\{\:,\:\}$ are homogeneous.

Higher Koszul brackets on the cotangent complex