论文标题
什么时候两个HKR同构相等？
When are two HKR isomorphisms equal?
论文作者
论文摘要
令$ x \ hookrightarrow s $为平滑方案的封闭嵌入，将其拆分为第一阶。 HKR同构是移位的正常束$ \ MATHBB {N} _ {X/S} [ - 1] $与派生的自我交叉$ x \ times^r_sx $之间的同构。鉴于封闭嵌入的两个不同的一阶分裂，可以使用Arinkin和Căldăraru的结构获得两个HKR同构。先验，尚不清楚这两个同构是否相等。我们定义了与封闭嵌入相关的$ x $的矢量捆绑包的广义atiyah类，并定义了两个一阶分组。我们使用广义的Atiyah类来提供足够和必要的条件，以分别超过$ x $和$ x \ times x $等于$ x $。当$ i $是对角线嵌入时，有两个自然预测从$ x \ times x $到$ x $。我们表明，这两个预测定义的HKR同构相当于$ x $，但总体上不等于$ x \ times x $。
Let $X\hookrightarrow S$ be a closed embedding of smooth schemes which splits to first order. An HKR isomorphism is an isomorphism between the shifted normal bundle $\mathbb{N}_{X/S}[-1]$ and the derived self-intersection $X\times^R_SX$. Given two different first order splittings of a closed embedding, one can obtain two HKR isomorphisms using a construction of Arinkin and Căldăraru. A priori, it is not known if the two isomorphisms are equal or not. We define the generalized Atiyah class of a vector bundle on $X$ associated to a closed embedding and two first order splittings. We use the generalized Atiyah class to give sufficient and necessary conditions for when the two HKR isomorphisms are equal over $X$ and over $X\times X$ respectively. When $i$ is the diagonal embedding, there are two natural projections from $X\times X$ to $X$. We show that the HKR isomorphisms defined by the two projections are equal over $X$, but not equal over $X\times X$ in general.
